Eurasian Snow Cover and the Role of Linear Interference in Stratosphere-Troposphere Interactions

by

Karen L. Smith

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy Department of Physics University of Toronto

© Copyright by Karen L. Smith 2012

ii

Eurasian Snow Cover and the Role of Linear Interference in Stratosphere-Troposphere Interactions

Karen L. Smith

Doctor of Philosophy

Department of Physics University of Toronto

2012

Abstract

The classical problem of predicting the atmospheric circulation response to extratropical surface

forcing is revisited in the context of the observed connection between autumn snow cover

anomalies over Eurasia and the wintertime Northern Annular Mode (NAM). In general

circulation model (GCM) simulations with prescribed autumn Siberian snow forcing, a vertically

propagating Rossby wave train is generated, driving dynamical stratospheric warming and a

negative NAM response that couples to the troposphere. It is shown that unexplained aspects of

the evolution of this response can be clarified by examining the time evolution of the phasing,

and hence the linear interference, between the wave response and the background climatological

wave. When the wave response and background wave are in phase (out of phase), wave activity

into the stratosphere is amplified (attenuated) and the zonal mean stratosphere-troposphere NAM

response displays a negative (positive) tendency. This effect is probed further in a simplified

GCM with imposed lower tropospheric cooling. As in the comprehensive GCM, linear

interference strongly influences the NAM response. The transition from linear to nonlinear

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behaviour is shown to depend on forcing strength. Linear interference also plays a key role in the

observed October Eurasian snow cover-NAM connection. It is shown that the time lag between

October Eurasian snow anomalies and the peak wave activity flux arises because the Rossby

wave train associated with the snow is out of phase with the climatological stationary wave from

October to mid-November. Beginning in mid-November, the associated wave anomaly migrates

into phase with the climatological wave, leading to constructive interference and anomalously

positive upward wave activity fluxes. Current generation climate models do not capture this

behaviour.

Linear interference is not only associated with stratospheric warming due to Eurasian

snow cover anomalies but is a general feature of both Northern and Southern Hemisphere

stratosphere-troposphere interactions, and in particular dominated the negative NAM events of

the fall-winter of 2009-2010. The interannual variability in upward wave activity flux during the

season of strongest stratosphere-troposphere interactions is primarily determined by linear

interference of quasi-stationary waves. The persistence of the linear interference component of

this flux may help improve wintertime extratropical predictability.

iv

Acknowledgments

After a three year absence from the field of atmospheric dynamics, it was challenging for me to

start over at the University of Toronto. With the guidance and support of the many incredible

scientists in the Atmospheric Physics Group at U of T, my transition back into the field has been

very rewarding.

I am grateful to my supervisor, Dr. Paul Kushner, for his guidance and encouragement.

Within his diverse group, he has challenged me and encouraged me to pursue my own scientific

interests. He has generously sent me to numerous conferences, summer schools and invited me to

spend two months at the National Center for Atmospheric Research (NCAR) during his

sabbatical, introducing me to world-renowned scientists and exposing me to exciting new

research. Both his scientific advice and professional advice have been invaluable. I have learned

a great deal from him and feel that his influence has truly made me a better scientist.

I would also like to thank my committee members, Dr. W. Richard Peltier and Dr.

Kimberly Strong. They have been very supportive of my work and provided me with excellent

feedback and advice at my annual committee meetings and throughout our interactions in the

department.

I would also like to thank my collaborators, Chris Fletcher and Judah Cohen. Chris has

been an excellent mentor over the past few years, advising me on everything from my

fellowships to shell scripts. Our bi-weekly meetings provided a non-judgmental setting for the

exchange of ideas and greatly influenced my research. I will always be grateful for his patience

and support. I would also like to acknowledge Judah for hosting me at Atmospheric and

Environmental Research (AER) and for providing me with insightful feedback on my work. He

has been generous with his time by meeting with me at conferences to discuss new ideas.

The Atmospheric Physics Group at U of T is a stimulating environment for research and I

am grateful to my fellow students and the many post-docs and faculty for creating such an

environment. I am particularly grateful to Isla Simpson, Peter Hitchcock, Heather Andres, Lei

v

Wang and Lawrence Mudryk for interesting conversations, technical help and most of all, moral

support.

These acknowledgments would not be complete without thanking my dear family; my

mother and father for their steadfast support over the years and my sister for challenging me to

be myself. And thank you to my extended family of aunts, uncles, cousins, in-laws and friends.

Finally, I owe immeasurable thanks to my husband, Jay Cleary. He has supported me

unconditionally over the past four years. His patience, positivity and love have helped me

through the ups and downs of graduate life.

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Table of Contents

Acknowledgements iv

List of Tables ix

List of Figures x

Chapter 1 Introduction 1

1.1 Preface 1

1.2 The Northern Annular Mode 3

1.3 Annular Mode responses to External Forcings

and Surface Boundary Conditions 5

1.4 Stratosphere-Troposphere Interactions 12

1.5 Eurasian Snow and the NAM 17

1.6 Conclusion 24

Chapter 2 The Role of Linear Interference in the Annular Mode

Response to Extratropical Surface Forcing 26

2.1 Introduction 26

2.2 Methods 27

2.2.1 Model Descriptions 27

2.2.2 Snow/Surface Cooling Method 29

2.3 Results 31

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2.3.1 Revisiting the Transient Response to

Siberian Snow Forcing in F09 31

2.3.2 Comparison between AM2 and the SGCM 36

2.3.3 Sensitivity to Position and Sign of the Forcing in the SGCM 41

2.3.4 Sensitivity to Forcing Strength in the SGCM 45

2.4 Sensitivity to Polar Vortex Strength 47

2.5 Conclusions 52

Chapter 3 The Role of Linear Interference in Northern Annular Mode

Variability associated with Eurasian Snow Cover Extent 57

3.1 Introduction 57

3.3 Methods 59

3.3 Results 61

3.3.1 Linear Inteference Effects in Interannual

Variability of Wave Activity 61

3.3.2 Linear Interference in the Snow-NAM Link 64

3.3.3 Case Study: Winter 2009 – 2010 74

3.3.4 Linear Interference and the Snow-NAM link in IPCC models 77

3.4 Conclusions 82

Chapter 4 Extratropical Linear Interference in

Stratosphere-Troposphere Interactions 86

4.1 Introduction 86

4.2 Methods 89

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4.3 Results 92

4.3.1 Northern Hemisphere Seasonal Heat Flux Characteristics 92

4.3.2 Northern Hemisphere Anomalous Heat Flux Composites. 97

4.3.3 Stratospheric Sudden Warming Events and Linear Interference 110

4.3.4 Comparison between Northern and Southern Hemisphere 113

4.3.5 Stratospheric Final Warmings and Linear Interference 119

4.4 Conclusions 123

Chapter 5 Conclusions and Discussion 126

5.1 Summary 126

5.2 Future Work 131

References 142

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List of Tables

TABLE 2.1: List of SGCM simulations 30

TABLE 2.2: Ratio of covariance between EMLIN and wave components of EMLIN

to the variance of EMLIN across Simulations A-L for each SGCM Suite. 51

TABLE 3.1: Variance decomposition for December mean {v*T*} at 100 hPa

calculated using daily-averaged NCEP-NCAR data from 1972-2007 63

TABLE 3.2: Variance decomposition for December mean {v*T*} at 100 hPa

calculated using monthly-averaged IPCC model archive data

for 20th century runs 79

TABLE 3.3: Amplitude of wave-1 component of December-January-February

Z*c at 60°N and 50 hPa for NCEP-NCAR (1972-2007) and the

IPCC model archive data for 20th century runs 80

x

List of Figures

FIG. 1.1. (top) Zonal-mean geostrophic wind and (bottom) lower-tropospheric geopotential

height regressed on the standardized indices of the annular modes (the NAM and the SAM)

based upon monthly data, Jan 1958–Dec 1997. Left panels are for the SH, right panels are for the

NH. Units are m s-1 (top) and m per std dev of the respective index time series (bottom). Contour

intervals are 10 m (-15, -5, 5, . . . ) for geopotential height and 0.5 m s-1 (-0.75, -0.25, 0.25, . . .)

for zonal wind (from Thompson and Wallace 2000).

……………………………………………… 5

FIG. 1.2. (a) The time- and zonal-mean zonal winds for γ = 2 K km-1. The contour interval is 10

m s-1 and the zero contour is not plotted. The latitude 30°S is marked with a heavy vertical line.

(b) As in (a) but for γ = 4 K km-1. (c) The difference in the time- and zonal-mean zonal wind

between γ = 4 K km-1 and γ = 2 K km-1, that is, (b) minus (a). The contour interval is 5 m s-1 and

the zero contour is not plotted (from Kushner and Polvani 2004).

……………………………………………… 11

FIG. 1.3. Composites of time-height development of the northern annular mode for (A) 18 weak

vortex events and (B) 30 strong vortex events. The events are determined by the dates on which

the 10-hPa annular mode values cross -3.0 and +1.5, respectively. The indices are

nondimensional; the contour interval for the color shading is 0.25, and 0.5 for the white contours.

Values between -0.25 and 0.25 are unshaded. The thin horizontal lines indicate the approximate

boundary between the troposphere and the stratosphere (from Baldwin and Dunkerton 2001).

……………………………………………… 13

FIG. 1.4. January 500 hPa GPH for 1972-2009 (NCEP-NCAR) regressed on the (a) October

Eurasian snow index (OCTSNW) and the (b) January NAM Index. Contour intervals are 5 m and

20 m for (a) and (b), respectively. (c) and (d) same as (a) and (b) but for zonal mean GPH.

Contour intervals 10 m and 20 m for (c) and (d), respectively. Blue dashed and red solid contours

are for negative and positive values. Gray shading indicates 95% significance.

xi

……………………………………………… 19

FIG. 1.5. Conceptual model for NAM response to Eurasian snow cover anomalies. Eurasian snow

cover exhibits its largest variability in autumn. In years with anomalously high snow cover, the

increase in surface albedo diabatically cools the Eurasian region, exciting vertically propagating

Rossy Waves. The waves dissipate in the stratosphere and weaken the polar vortex generating a

negative NAM response in the stratosphere. The NAM response propagates downward, reaching

the troposphere by mid-winter (adapted from Cohen et al. 2007).

……………………………………………… 21

FIG. 2.1. (a) Time series of ensemble mean polar cap-averaged 50 hPa geopotential height

response (∆‹Zpcap›) to a switch-on snow forcing in the AM2 GCM. Thick, solid portions of the

line indicate 95% significance (the statistical significance of the response is assessed for each

simulation day using the one-sample Student’s t-test assuming independence of the realizations

that start 1 year apart). The solid horizontal line indicates the zero line. (b) Day 1-65 averaged

ensemble mean wave GPH response (∆‹Z*›) at 60°N. (c) as in (b) but for days 66-92. The solid

contours correspond to positive values and the dashed contours correspond to negative values.

The contour interval is 5 m. The gray shading shows 95% significance.

……………………………………………… 32

FIG. 2.2. (a) Day 1-65 averaged ensemble and zonal mean total wave heat flux response,

∆{‹v*T*›}. (b) the linear contribution, EMLIN, to ∆{‹v*›‹T*›}. (c) the nonlinear contribution,

EMNL, to ∆{‹v*›‹T*›}. (d), (e) and (f) are as in (a), (b) and (c) but for days 66-92. The contour

interval is 0.5 m K s-1. For panel (a), the Student’s t-test is computed using the time-averaged

fields; the gray shading shows 95% significance. Deriving straightforward significance tests for

the EMLIN and EMNL terms that are consistent with the t-test on ∆{‹v*T*›} has been difficult.

Thus, significance shading is not included in panels (b), (c), (e) and (f), but that the main features

are robust by subsampling the ensemble has been verified.

……………………………………………… 35

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FIG. 2.3. Time series of 60°N wave-1 phase in degrees for the control state wave, ‹Z*c›, (solid

line) and for the wave response, ∆‹Z*›, with (dotted line) and without (dashed line) a 10-day

running mean applied at (a) 50 hPa and (b) 500 hPa. The gray shading indicates regions where

‹Z*c› and ∆‹Z*› are out of phase.

……………………………………………… 37

FIG. 2.4. Day 1-22 averaged ensemble mean response to Siberian lower tropospheric cooling in

the SGCM. (a) wave response (∆‹Z*›)at 60°N and (b) zonal mean GPH response ({∆‹Z ›}) (c)

and (d) are as in (a) and (b) but for the Pacific lower tropospheric cooling case. The solid

contours correspond to positive values, the dashed contours correspond to negative values and

the gray shading shows 95% significance. The contour interval is 5 m.

……………………………………………… 39

FIG. 2.5. Day 1-22 averaged ensemble mean wave response (∆‹Z*›; black contours) at 60°N to

Siberian lower tropospheric cooling in the SGCM superimposed on the control state wave at

60°N (‹Z*c›; gray shading) for (a) all-waves, (b) wave-1, and (c) wave-2. The contour interval is

5 m.

……………………………………………… 40

FIG. 2.6. Difference between ∆‹Zpcap› as a function of time for the “Pacific Case” and “Siberian

Case”. Contour interval is 5 m. Solid contours indicate positive and negative values. Gray

shading indicates regions where the difference between the two cases are significant at the 95%

level.

……………………………………………… 43

FIG. 2.7. Dependence of the SGCM response on forcing location (Simulations A-L in Table 2.1).

(a) the TOTAL E-P flux divergence response averaged over 40-80°N, 10-1 hPa, and days 1-22

versus the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days 10-40.

(b) ∆{‹v*›‹T*›} (EM) at 10 hPa, averaged over 40-80°N, and cumulative to day 22 versus the E-P

flux divergence response. (c) the linear contribution, EMLIN, to EM, versus EM. (d) the all-wave

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(solid circles) and wave-1 (open circles) spatial correlation between ∆‹Z*› and ‹Z*c› versus

EMLIN.

……………………………………………… 44

FIG. 2.8. (a) and (c) as in Fig. 2.5b but for Simulation A and Simulation S, respectively. (b) and

(d) as in Fig. 4b but for Simulation A and Simulation S, respectively; gray shading shows 95%

significance.

……………………………………………… 46

FIG. 2.9. Dependence of the SGCM response on forcing strength (Simulations M, B, N-R in

Table 1). (a) the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days

10-40. (b) day 1-22 ∆{‹v*›‹T*›}(EM) at 10 hPa, averaged over 40-80°N. (c) linear contribution,

EMLIN, to EM. (d) the nonlinear contribution, EMNL, to EM, as a function of forcing strength.

The forcing strength has been normalized such that a forcing strength of 1 corresponds to the

forcing discussed in Section 2. The solid lines in (c) and (d) show the linear and quadratic fits

passing through the origin, respectively.

……………………………………………… 48

FIG. 2.10. Control state zonal mean zonal wind, uc, for the (a) γ = 1, (b) γ = 2, and (c) γ = 3 K km-

1 SGCM configurations. Positive and negative contours are red and blue, respectively. Contour

interval is 5 m s-1. Control state stationary wave field, Z*c, at 60°N for the (d) γ = 1, (e) γ = 2, and

(f) γ = 3 K km-1 SGCM configurations. Wave response, ∆Z*, at 60°N for the “Siberian Case” for

the (d) γ = 1, (e) γ = 2, and (f) γ = 3 K km-1 SGCM configurations. Note the difference in colour

bar scale for panels (d)-(f) and (g)-(i).

……………………………………………… 49

FIG. 2.11. Dependence of the SGCM response on forcing location (Simulations A-L in Table 2.1)

for three polar vortex configurations, γ = 1 (green), γ = 2 (red) and γ = 3 K km-1 (blue). (a) the

TOTAL E-P flux divergence response averaged over 40-80°N, 10-1 hPa, and days 1-22 versus

the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days 5-22 (γ = 1

and 3) or days 10-40 (γ = 2). (b) ∆{‹v*›‹T*›} (EM) at 10 hPa, averaged over 40-80°N, and

xiv

cumulative to day 22 versus the E-P flux divergence response. (c) the linear contribution, EMLIN,

to EM, versus EM. (d) sum of wave-1 and wave-2 components of EMLIN versus EMLIN.

……………………………………………… 53

FIG. 3.1. (first row) Composite mean of time evolution of the 40-day cumulative mean total

meridional wave heat flux (v*T*; black curve) anomalies at 100 hPa and the corresponding LIN

(red curve) and NONLIN (blue curve) components for 22 high (left) and 15 low (right)

anomalous v*T* events in November-December-January. Solid sections of the heat flux curves

indicate times when anomalies are different from zero at the level of 95% significance. (second

row) Composites of the time evolution of the standardized anomaly polar cap GPH

corresponding to these anomalous v*T* events as a function of pressure. The GPH contour

interval is [0.25, 0.5, 1.0, 1.5], warm and cold shading are positive and negative contours, and

the black contour indicates pressures and times for which anomalies are different from zero at

the level of 95% significance.

……………………………………………… 65

FIG. 3.2. Correlations of OCTSNW with daily (a) polar cap GPH, (b) the 40-day cumulative

mean total meridional wave heat flux averaged over 40-80°N, (c) the LIN component of (b), (d)

the NONLIN component of (b), (e) the wave-1 component of (c), and (f) the wave-2 component

of (c). Time-axis begins on October 10. Contour interval is 0.1, warm and cold shading are

positive and negative contours, and the black contour indicates pressures and times for which

correlations are different from zero at the level of 95% significance.

……………………………………………… 67

FIG. 3.3. Covariance of Z* with OCTSNW (black contours) superimposed on Z*c (shading) at

60°N for (a)-(c) October 16th – November 30th (ON) and (d)-(e) December 1st – January 15th. (b)

and (e) show the wave-1 component and (c) and (d) show the wave-2 of Z*snow and Z*

c. Black

solid and dashed contours show positive and negative values, respectively. Contour interval is 5

m.

……………………………………………… 69

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FIG. 3.4. Daily time series of (a)wave-1 40-80°N averaged zonal mean wave meridional heat flux

components at 100hPa regressed on the snow index (total – black line; linear – red line;

nonlinear – blue line) and (b) the phase of wave-1 component of Z*c for 1972-2009 mean (solid

line) the phase of Z*snow (dashed line) at 60°N and 100hPa. (c) and (d) as (a) and (b) but for

wave-2 at 500hPa. (e) as (d) but for all wave numbers greater than wave-2. Gray shading in (a)

and (c) indicates regions where Z*c and Z*

snow are out-of-phase.

……………………………………………… 70

FIG. 3.5. As described in text, distribution of potential temperature (black contours) and wave

geopotential (red contours for positive, blue contours for negative) at 60°N associated with

climatology (solid contours) and the climatology plus two times the regression on OCTSNW

(dashed contours) for (a) October 16th –November 30th (ON) and (b) December 1st – January 15th

(DJ).

……………………………………………… 73

FIG. 3.6. October 16th – November 30th (ON) temperature advection. (a) ZON_ADVsnow, (b)

MER_ADVsnow, (c) VERT_ADVsnow, and (d) TOT_ADVsnow vertically integrated from 925-700

hPa and filtered to retain wavenumbers 1-3. Contour interval of 0.03 K day-1, warm and cold

shading are positive and negative contours, and the black contour indicates regions for which

correlations are different from zero at the level of 95% significance.

……………………………………………… 74

FIG. 3.7. December 1st – January 15th (DJ) temperature advection. (a) ZON_ADVsnow, (b)

MER_ADVsnow, (c) VERT_ADVsnow, and (d) TOT_ADVsnow vertically integrated from 925-700

hPa and filtered to retain wavenumbers 1-3. Contour interval of 0.03 K day-1, warm and cold

shading are positive and negative contours, and the black contour indicates regions for which

correlations are different from zero at the level of 95% significance.

……………………………………………… 75

FIG. 3.8. Daily standardized (a) Zpcap′, and 40-day averaged (b) 40-80°N averaged v*T*′, (c) the

LIN component of (b) and (d) the NONLIN component of (b). X-axis begins on October 10,

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2009 and ends on February 29, 2010. Contour interval is 0.2 standard deviation units and warm

and cold shading are positive and negative contours.

……………………………………………… 76

FIG. 3.9. Z*′ (black contours) superimposed on Z*c (shading) at 60°N for (a)-(b) November 2009

and (c)-(d) December 2009 (b) and (d) show the wave-1 component Z*′ and Z*c. Black solid and

dashed contours show positive and negative values, respectively. Contour interval is 40 m.

……………………………………………… 78

FIG. 3.10. Scatter plot of the correlation between December v*T* and OCTSNW-M and the

correlation between December LIN and OCTSNW-M for each model.

……………………………………………… 82

FIG. 3.11. October-November mean Z*snow (black contours) superimposed on the Z*

c (shading) at

60°N for (a) the GISS model and (b) the GFDL CM2.1 model. (c) and (d) as (a) and (b) for

December-January. Contour interval is 3 m.

……………………………………………… 83

FIG. 4.1. NH meridional wave heat flux decomposition at 100 hPa averaged over 40-80°N. (a)

Climatological monthly mean (see Eqn. (4.1) and (4.4)). Total and selected high- and low-

frequency components are plotted (see legend). (b) Monthly variance decomposition (see Eqn.

(4.3)). The asterisks denote months when the correlation between LIN and NONLIN is

statistically significant at the 95% level.

……………………………………………… 93

FIG. 4.2. Contributions of terms in Eqns. (4.2) and (4.5) to interannual variability of NH {v* T*}

at 100 hPa and averaged over 40-80°N for each climatological month (in units of m2 K2 s-2).

Colour scheme corresponds to different terms in Eqn. (4.5): blue – var({v*low T*

low}); red -

var({v*high T*

high}); green - var({v*low T*

high}) + var({v*high T*

low}); yellow – R. Note the different

scales on the ordinate axes.

……………………………………………… 96

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FIG. 4.3. Heat flux anomaly autocorrelations for {v*T*}′ (black curve), LIN (red curve) and

NONLIN (blue curve) and the cross-correlation of LIN and NONLIN (green curve) as a function

of lag.

……………………………………………… 97

FIG. 4.4. Weak vortex composite mean 40-day averaged heat flux anomaly decomposition for (a)

{v*T*}′, (c) LIN and (e) NONLIN as a function of lag and pressure. (b), (d) and (f) same as (a),

(c) and (e) but for the strong vortex composite. Black contour indicates 95% significance.

……………………………………………… 98

FIG. 4.5. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′

(black curve), LIN (red curve) and NONLIN (blue curve) for the (a) weak and (b) strong vortex

events. Solid sections of the curves indicate 95% significance. Composite mean S(Zpcap′ ) for the

(c) weak and (d) strong vortex events. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5 1 1.5].

Black contour indicates 95% significance.

……………………………………………… 100

FIG. 4.6. Fraction of {v*T*}′ from LIN and NONLIN for November-December-January (NDJ),

December-January-February (DJF) and January-February-March (JFM) for the (a) weak and (c)

strong vortex composites. {v*T*}′, LIN and NONLIN for NDJ, DJF and JFM for the (b) weak

and (d) strong vortex composites.

……………………………………………… 101

FIG. 4.7. NH 40-day averaged heat flux anomaly histogram for (a) {v*T*}′, (b) LIN and (c)

NONLIN.

……………………………………………… 102

FIG. 4.8. Sensitivity of composite mean 40-day averaged {v*T*}′ (black curve), LIN (red curve)

and NONLIN (blue curve) at lag zero and of the number of events in each composite (green

xviii

curve) to a standardized {v*T*}′ threshold value (in units standard deviation) for NH (a) weak

and (b) strong vortex events.

……………………………………………… 103

FIG. 4.9. Composite mean 40-day averaged heat flux anomaly decompostion at 100 hPa; {v*T*}′

(black curve), LIN (red curve) and NONLIN (blue curve) for (a) LIN and (b) NONLIN weak

vortex events. Solid sections of the curves indicate 95% significance. Composite mean S(Zpcap′ )

for (c) LIN and (d) NONLIN weak vortex events. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5

1 1.5]. Black contour indicates 95% significance.

……………………………………………… 105

FIG. 4.10. (a) Phase difference between the composite mean Z*′ and Z*c at 60°N averaged over

days [-30,-1] for the weak (red curve) and strong vortex composites (blue curve). (b) and (d)

stratospheric anomaly correlation between the composite mean Z*′ and Z*c at 60°N at 100 hPa for

the full wave field (solid curve) and the wave-1 component (dashed curve) for the weak and

strong vortex composites, respectively. (c) and (e) same as (b) and (d) but for the tropospheric

anomaly correlation.

……………………………………………… 108

FIG. 4.11. Composite mean Z*′ (contours) and Z*c (shading) at 60°N averaged over days [-15,-1]

for the (a) weak and (b) strong vortex composites. Contour interval is 5 m.

……………………………………………… 110

FIG. 4.12. SSW composite mean daily heat flux anomaly decomposition for (a) {v*T*}′, (d) LIN

and (g) NONLIN. (b), (e) and (h) and (c), (f) and (i) same as (a), (d) and (g) but for D SSWs

(LIN fluxes are wave-1 only) and S SSWs (NONLIN fluxes are wave-2 only). (j)-(l) shows the

composite mean S(Zpcap′ ) for SSWs, displacement (D) SSWs and split (S) SSWs, respectively.

Black contour indicates 95% significance.

……………………………………………… 112

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FIG. 4.13. SH meridional wave heat flux decomposition at 100 hPa averaged over 40-80°S. (a)

Climatological monthly mean (see Eqns. (4.1) and (4.4)). Total and high- and low-frequency

components are plotted (see legend). (b) Monthly variance decomposition (see Eqn. (4.3)). No

points on green curve in (b) are statistically significant at the 95% level.

……………………………………………… 115

FIG. 4.14. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa;

{v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) weak and (b) strong

vortex composites. Solid sections of the curves indicate 95% significance. Composite mean

S(Zpcap′) for (c) NH_HIGH and (d) SH_HIGH. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5 1

1.5]. Black contour indicates 95% significance.

……………………………………………… 117

FIG. 4.15. Sensitivity of composite mean 40-day averaged {v*T*}′ (black curve), LIN (red curve)

and NONLIN (blue curve) and of the number of events per composite (green curve) to a

standardized {v*T*}′ threshold value (in units standard deviation) for SH (a) weak and (b) strong

vortex events.

……………………………………………… 119

FIG. 4.16. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa;

{v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) “early” and (b) “late”

NH SFWs. Solid sections of the curves indicate 95% significance. Composite mean Zpcap′ for (c)

NH and (d) SH final warmings. Contour interval is […, -40, -20, -10, -5, 5, 10, 20, 40,...]. Black

contour indicates 95% significance.

……………………………………………… 120

FIG. 4.17. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa;

{v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) “early” and (b) “late”

SH SFWs. Solid sections of the curves indicate 95% significance. Composite mean Zpcap′ for (c)

xx

SH and (d) SH final warmings. Contour interval is […, -40, -20, -10, -5, 5, 10, 20, 40,...]. Black

contour indicates 95% significance.

……………………………………………… 122

FIG. 5.1. Thermodynamic response to prescribed Siberian snow forcing in AM2 at 800hPa and

averaged over the Siberian region for (a) the linearized thermodynamic equation, (b) the full

thermodynamic equation, and (c) the EMLIN component of the full thermodynamic equation. The

black solid and dashed lines are the same in (b) and (c).

……………………………………………… 133

FIG. 5.2. Correlation between OCTSNW and October NCEP net incoming surface short wave

radiation flux for 1972-2008 over the Eurasian region. Positive and negative contours are red and

blue, respectively, and gray shading indicates regions where the correlation is significant at the

95% level.

……………………………………………… 136

FIG. 5.3. Variance decomposition of NH January wave momentum fluxes (a) var({u*v*}), (b)

var(LIN), (c) var(NONLIN), and (d) 2*cov(LIN,NONLIN) . (e) shows the difference between

panels (b) and (c). Positive and negative contours are red and blue. Contour interval is 21, 22, 23,

etc. Gray shading shows regions where the correlation of LIN and NONLIN is statistically

significant at the 95% level.

……………………………………………… 140

1

Chapter 1

Introduction

1.1 Preface

Terrestrial snow cover constitutes the largest component of the cryosphere by area and

experiences the greatest spatial and temporal fluctuations of Earth’s surface conditions (Cohen

and Rind, 1991; Vavrus, 2007). Consequently, it exerts a strong influence over the global surface

energy and moisture budgets (Groisman et al. 1994; Brown and Mote, 2009). While the

overlying atmosphere and the underlying soil and vegetation are strongly coupled to snow cover

on regional scales, studies have shown that Eurasian snow cover anomalies are also associated

with remote, large-scale atmospheric circulation anomalies coherent with the leading mode of

extratropical variability, the Northern Annular Mode (NAM; Thompson and Wallace, 1998;

Cohen and Entekhabi, 1999; Gong et al, 2003; Cohen et al., 2007; Fletcher et al., 2007; Fletcher

et al., 2009a; Henderson and Leathers 2009; Allen and Zender, 2010). It is observed that years in

which snow cover is anomalously extensive over Eurasia in October tend to be years in which

the NAM in the following winter is in its negative phase (see Fig. 1.4). This thesis explores the

underlying dynamics of this relationship.

In Chapter 2 the dynamical forcing of large-scale circulation anomalies by terrestrial

snow cover in general circulation model (GCM) simulations is examined. Predicting the

response of large-scale modes like the North Atlantic Oscillation and the NAM to extratropical

surface anomalies represents a classical challenge in climate science (e.g. Robinson et al. 2000,

Kushnir et al. 2006). These modes are intrinsically difficult to predict because they are internally

generated by tropospheric wave-mean flow interactions that are stochastic in character and

because they are modulated by multiple influences, including interactions with the ocean surface,

land surface, and the stratosphere (DeWeaver and Nigam 2004, Limpasuvan and Hartmann

2000, Czaja and Frankignoul 2002, Gong et al. 2002, Baldwin et al. 2003). In simulations, the

response of the modes to a prescribed forcing is model dependent because many details,

including the characteristics of the modes, the temporal and spatial structure of the forcing, the

2

background flow, and model configuration, all appear to matter for the extratropical response to

surface forcing. The approach taken here is to use simple and comprehensive GCMs to diagnose

the dynamical processes involved in the circulation response to Eurasian snow cover. It is found

that a particular interaction between anomalous and climatological waves, referred to as linear

interference, is key to determining the stratospheric NAM response to prescribed Eurasian snow

cover forcings in a GCM. In Chapter 3 the observed Eurasian snow-NAM connection is

revisited. Linear interference effects are also shown to be the dominant process associated with

the snow-related stratosphere-troposphere interaction, illustrating a robust dynamical process in

both models and observations.

Although the Eurasian snow cover-NAM relationship provided much of the motivation

for this thesis, the dynamical diagnostics developed in Chapters 2 and 3 opened up a novel line

of inquiry applicable to the broader question of stratosphere-troposphere interactions (Baldwin

and Dunkerton 2001; Polvani and Waugh 2004; Perlwitz and Harnik 2004; Shaw et al. 2010). In

Chapter 4, stratosphere-troposphere interactions are examined through the lens of linear

interference. It is shown that many of the dynamical features that are important for the snow-

NAM relationship are universally important for extratropical stratosphere-troposphere coupled

variability in both the Northern and Southern hemispheres.

The autumn Eurasian snow-NAM relationship is a complex problem involving the

radiative effects of snow cover at the local scale, the communication of changes in the surface

energy balance to the free troposphere as well as a large-scale circulation response involving

coupled stratosphere-troposphere NAM dynamics. As a means of introduction to the topic of

extratropical variability, Section 1.2 describes the characteristics and significance of the NAM in

the extratropical atmosphere. The NAM arises primarily from internal atmospheric instability;

however, it has the potential to be excited by external forcings and/or anomalous boundary

conditions. In Section 1.3, observational and modeling evidence is presented in support of

Annular Mode circulation responses to various forcings. Within the scientific community that

studies the Eurasian snow-NAM relationship, some describe this relationship as an atmospheric

response to an anomalous boundary condition. However, it is important to recognize that snow

cover is intrinsically related to the atmospheric circulation. While the majority of the Eurasian

snow-NAM discussion is reserved for Section 1.5, some of this literature is introduced in Section

3

1.3. Section 1.4 outlines the fundamentals of NAM dynamics with an emphasis on stratosphere-

troposphere interactions. The emphasis on stratosphere-troposphere interactions reflects the

importance of these dynamical processes in the Eurasian snow-NAM relationship. This section

also describes the recent literature on the role of linear interference in stratosphere-troposphere

dynamics, which turns out to be critically important to understanding the circulation anomalies

associated with autumn Eurasian snow cover. A review of the observational and modeling

literature on Eurasian snow and the NAM is presented in Section 1.5. Finally, Section 1.6

outlines some of the outstanding questions in this area of study and identifies those that will be

addressed in the following chapters.

1.2 The Northern Annular Mode

The study of extratropical atmospheric variability dates back to the diary of Hans Egede Saabye,

a missionary in Greenland from 1770-1778 (van Loon and Rogers 1978). Saabye observed a

seesaw in winter temperatures between Greenland and Northern Europe. Van Loon and Rogers

(1978) list several other historical references to this seesaw illustrating that it has been a robust

feature of the North Atlantic climate over the past two centuries. Walker and Bliss (1932)

describe the seesaw in terms of a sea level pressure (SLP) difference between Iceland and the

Azores and subsequently named it the North Atlantic Oscillation (NAO). An empirical

orthogonal function (EOF) analysis of extratropical SLP in the North Atlantic domain reveals the

NAO as the primary mode of variability in this region (Wallace and Gutzler 1981). Walker and

Bliss (1932) also describe an analogous North Pacific Oscillation (NPO). Together the NAO and

NPO teleconnection patterns form part of a larger zonally symmetric seesaw in SLP between the

polar and mid-latitudes. An EOF analysis of the entire extratropical Northern Hemisphere SLP

reveals this “annular” zonally symmetric seesaw (Wallace and Gutzler 1981) known as the

Arctic Oscillation (AO) or the Northern Annular Mode (NAM; Thompson and Wallace 1998).

Although the NAM has several definitions, a commonly used definition is the first EOF of the

extratropical (poleward of 20ºN) SLP (Baldwin and Thompson 2009). When regressed onto

geopotential height (GPH), the spatial pattern reveals a coherent vertical structure within the

troposphere and stratosphere. The NAM explains 20-30% of the observed variance in GPH and

4

zonal wind, depending on the pressure level and/or timescale considered (Thompson and

Wallace 2000). Figure 1.1a-d (Thompson and Wallace 2000) shows the positive phase regression

patterns of zonal wind and GPH onto the Southern Annular Mode (SAM; Figs. 1.1a and c) and

NAM indices (the principal component time series of the SAM and NAM; Figs. 1.1b and d). The

SAM is the first EOF of extratropical SLP in the Southern Hemisphere. Notice the annular

pattern of mass redistribution from the poles to the mid-latitudes and the poleward intensification

(in the positive phase) of the tropospheric jet. The positive phase of the NAM is characterized by

anomalously lower pressures over the polar region, anomalously higher pressures in mid-

latitudes, a stronger polar stratospheric jet and a poleward-shifted tropospheric jet. Given the

zonally symmetric nature of the NAM (and to an even greater extent, the SAM), the Annular

Modes may be viewed as the primary mode of variability of the zonal mean circulation. Baldwin

and Thompson (2009) demonstrate that the first principal component time series based on

Northern Hemisphere zonal mean GPH agrees very well with the NAM index based on the full,

zonally varying GPH field throughout the depth of the troposphere and stratosphere.

With timescales ranging from 10 days to a season, the NAM is considered a feature of

low-frequency atmospheric variability. Given this range of time scales, changes in boundary

conditions, such as fluctuations in sea-surface temperatures (SST), which typically occur on

longer time scales, are likely unimportant in establishing the NAM. The balance of literature on

the NAM argues that it arises primarily from feedbacks between the mean flow and transient,

synoptic scale waves (Lee and Feldstein 1996; Robinson 2000; Polvani and Kushner 2002;

Lorenz and Hartmann 2003; Vallis et al. 2004; Ring and Plumb 2007). However, DeWeaver and

Nigam (2000a) argue that the interaction between anomalous waves and the large-scale

climatological stationary waves is the dominant process maintaining tropospheric NAM

anomalies. Thus, although the NAM represents variability in the zonal mean extratropical

circulation, the large-scale zonally asymmetric circulation may play an important role in NAM

dynamics.

5

FIG. 1.1. (top) Zonal-mean geostrophic wind and (bottom) lower-tropospheric geopotential height regressed on the standardized indices of the annular modes (the NAM and SAM) based upon monthly data, Jan 1958–Dec 1997. Left panels are for the SH, right panels are for the NH. Units are m s-1 (top) and m per std dev of the respective index time series (bottom). Contour intervals are 10 m (-15, -5, 5, . . . ) for geopotential height and 0.5 m s-1 (-0.75, -0.25, 0.25, . . .) for zonal wind (from Thompson and Wallace 2000).

1.3 Annular Mode Responses to External Forcings and Surface

Boundary Conditions

In addition to being the dominant mode of internal atmospheric variability, the Annular Modes

(hereafter, AMs) are also the preferred pattern of atmospheric response to external forcings

(Thompson and Wallace 1998; Cohen and Entekhabi 1999; Thompson et al. 2000; Kushner et al.

6

2001; Deser et al. 2004; Ring and Plumb 2007). Formally, this finding is consistent with the

fluctuation-dissipation theorem (Leith 1975). For the climate system, this theorem implies that

the system’s response to external forcings will project strongly onto its leading modes of internal

variability and the magnitude of the response will be proportional to the decorrelation timescale

of the internal modes. Although there are only a few studies that look at the validity of

fluctuation-dissipation theory in the climate system in a quantitative manner (Gritsun and

Branstator 2007; Ring and Plumb 2008), there is a great deal of qualitative evidence supporting

this theory. This section explores the evidence specific to AM-like responses to external

forcings.

Thompson and Wallace (1998) document that both the NAM and SAM index time series

exhibited positive trends over the last several decades of the 20th century and suggested that these

trends may be related in part to externally forced, anthropogenic climate change. In following

papers, Thompson et al. (2000) show that 30% of the observed January-February-March surface

temperature trend over the NH was linearly related to the NAM index and Thompson and

Solomon (2002) propose that the positive SAM trend resulted from the effect of ozone depletion

in the Southern Hemisphere (SH) stratosphere on the tropospheric circulation. Although the

trend in the NAM has reversed since these papers were published (Cohen and Barlow 2005;

Overland and Wang 2005), the idea that the AMs could be forced, in part, by external factors has

inspired a significant number of publications over the past decade.

One of the first publications to document this phenomenon in model simulations was

Kushner et al. (2001; see also Fyfe et al. 1999 and Shindell et al. 1999). This study investigates

the transient response to greenhouse gas (GHG) forcing in the Southern Hemisphere using a

coupled atmosphere-ocean-land-ice GCM. The authors find that the tropospheric zonal wind

response in the SH (a poleward shift of the tropospheric jet) projects strongly onto the positive

phase of the model’s SAM. The residual response is described as the direct response to the GHG

forcing. Miller et al. (2006) examine the suite of Coupled Model Intercomparison Project 3

(CMIP3) simulations and found a similar relationship in the SH for the SLP response to climate

change across all of the models. It is now well established that the positive trend in the SAM

index is due, in large part, to ozone depletion (Arblaster and Meehl 2006; Son et al. 2008, 2010;

Polvani et al. 2011; McLandress et al. 2011). Many of the CMIP3 models did not account for

7

ozone recovery and consequently exhibit an exaggerated positive SAM trend (Son et al. 2008).

Several recent studies investigate the relative effects of ozone depletion and GHG warming on

the SAM. Son et al. (2008, 2010) use the CMIP3 and Chemistry-Climate Model Validation

(CCMVal) model archives to compare the SAM response to climate change in model simulations

that included ozone recovery and those that did not. The authors show that models with (without)

ozone recovery exhibited a trend in the zonal winds consistent with a negative (positive) SAM

response. Polvani et al. (2011) and McLandress et al. (2011) show similar results in GCM

simulations with independently prescribed ozone or ozone depleting substances and GHG

forcings.

The projection of the NH circulation response to climate change onto the NAM, however,

is non-robust across the CMIP3 models (Miller et al. 2006). In addition to the effect of inter-

model differences, Deser et al. (2010) show that internal climate variability renders the

prediction of the NAM response to climate change on multi-decadal timescales difficult. The

authors construct a 40-member ensemble of 60-year coupled climate change simulations with the

same GCM by perturbing only the atmospheric initial conditions for each member. The

atmospheric initial conditions are taken from 40 different days in the December 1999-January

2000 time period of a twentieth century control run. Using this 40-member ensemble, they find

that between 20 and 30 ensemble members are required to detect a significant change in the

NAM between the years 2010 and 2050. They estimate that approximately half of the inter-

model spread in the CMIP3 projected 2005-2060 climate change trends can be attributed to

natural climate variability.

Given that anthropogenic climate change involves not only changes in radiative forcings,

including carbon dioxide and ozone concentrations, but also changes in surface boundary

conditions, including sea-ice, SST’s and snow cover, studies have examined the AM response to

several of these forcings independently as a means of quantifying their relative importance.

Deser et al. (2004) apply Northern Hemisphere SST and sea-ice boundary conditions derived

from 20th century trends to an atmosphere-land GCM and produce circulation responses that can

be partitioned into a response that projects onto the NAM of the control run (the indirect

response) and a response that is the residual from that projection (the direct response). The

indirect response is a remote teleconnection pattern maintained by the wave-driven circulation

8

(i.e., circulation driven by convergence of momentum and heat by baroclinic waves) while the

direct response is more localized to the forcing region and maintained by diabatic heating (Deser

et al. 2007). In a more recent study, Deser et al. (2010) examine the effect of NH sea-ice loss on

the NAM response to climate change by analyzing the difference between two ensembles of

atmosphere-land GCM simulations with fixed seasonally varying 2080-2099 (21C) and 1980-

1999 (20C) sea-ice, respectively; SSTs and atmospheric composition were set to 20C values in

both ensembles. The wintertime response to sea-ice loss resembles the negative phase of the

NAM and the authors attribute the NAM response to boundary layer heating due to sea-ice loss.

A complementary study examining the effect of reductions in snow cover on the NAM response

to climate change shows a weak stratospheric positive NAM response in winter and spring

(Alexander et al. 2011).

In addition to the considerable research that has been devoted to understanding the AM

response to anthropogenic climate change, a large body of work has also been aimed at

understanding the influence of the El Niño-Southern Oscillation (ENSO) on the AMs. Evidence

for ENSO-related SST anomalies influencing the NAM is somewhat non-robust (Free and Seidel

2009; Butler et al. 2011). The balance of evidence from both observations and modeling studies

suggests that warm ENSO SST anomalies are associated with the negative phase of the NAM via

an amplification of the positive phase Pacific-North-American (PNA) pattern and subsequent

stratospheric wave-mean-flow interaction (Garfinkel and Hartmann 2008; Ineson and Scaife

2009; Manzini 2009; Bell et al. 2009; Fletcher and Kushner 2011). Some observational evidence

suggests that anomalously warm SSTs in the Pacific warm pool region (i.e. the Niño 4 region)

may also be negatively correlated with the stratospheric SAM (Hurwitz et al. 2011).

Seasonal forecasters are particularly interested in the relationship between ENSO and the

NAM. During the Tropical Ocean-Global Atmosphere (TOGA) program, marked progress was

made in seasonal prediction stemming from enhanced observations of tropical SST’s and

improved understanding of tropical-extratropical interactions (Trenberth et al. 1998). Many long-

range statistical forecasts rely on ENSO as the primary predictor of seasonal climate (e.g. the

Climate Prediction Center uses three different statistical forecast models); however, despite

improvements in the tropical observing system during TOGA, ENSO lacks skill in predicting

NH surface temperatures in the extratropics in winter (particularly over Europe), much of which

9

is NAM-related (Palmer and Anderson 1994; Trenberth et al. 1998; Barnston et al. 1999;

McPhaden 1999). Improving prediction of the NAM would constitute a significant advance in

seasonal forecasting, and identifying slowly evolving boundary conditions associated with

patterns of extratropical variability such as the NAM is an important field of research. However,

a complete review of this literature is beyond the scope of this thesis.

With respect to the observed correlation between October Eurasian snow cover and the

NAM, the motivation for this thesis work partially stems from the potential utility of October

Eurasian snow cover as a predictor of winter climate in the NH extratropics. It has been shown

that when October Eurasian snow cover extent is included as a predictor in a statistical forecast

model, the forecast skill for the NH extratropics improves (Cohen and Saito 2003; Cohen and

Fletcher 2007; Cohen et al. 2010; Orsolini and Kvamsto 2009). Cohen and Saito (2003) and

Cohen and Fletcher (2007) demonstrate that a simple statistical forecast model including October

Eurasian snow cover and SLP anomalies had greater skill in predicting NH winter surface

temperature over the Eastern United States, Europe and Asia compared to several dynamical

forecast models. Using the Arpege Climat forecast model, Orsolini and Kvamsto (2009) generate

two sets of five-member ensemble hindcasts to investigate the predictive utility of autumn

Eurasian snow cover, one with prognostic snow cover and one with imposed observational snow

cover. Although they do not find a clear difference between the two sets of hindcasts with

respect to the association between observed snow cover and the NAM, they do find that forecast

skill improves over the Aleutian and Icelandic Low regions when observed snow cover was

prescribed. This work implies that the knowledge of autumn snow cover improves the accuracy

of winter forecasts and suggests that snow cover exerts an influence on the atmospheric

circulation on seasonal timescales. A broader discussion of the NAM response to snow cover

anomalies will be presented in depth in Section 1.5.

To conclude this section, AM responses to external forcings in simplified atmospheric

models are discussed. This class of models provides a framework for studying complex,

nonlinear atmospheric variability in the absence of other climate variations. A model that is

commonly used is a primitive equation, atmospheric GCM with idealized radiation, boundary

layer and gravity wave drag schemes (Held and Suarez 1994). This model has both a

tropospheric configuration and a configuration including a winter hemisphere stratospheric polar

10

vortex (Held and Suarez 1994; Polvani and Kushner 2002). Polvani and Kushner (2002) use this

model to simulate the observed effect of ozone depletion in the SH. They demonstrate that a

polar stratospheric cooling generates a positive AM response. Stratospheric cooling is imposed

by changing the stratospheric lapse rate, γ (in K km-1), in the equilibrium temperature profile in

the Newtonian cooling scheme. Figure 1.2a-c shows the zonal mean zonal wind for a relatively

warm (γ = 2) and cold (γ = 4) stratosphere and the difference between the two. The difference

represents a positive AM response, i.e. a poleward shift of the tropospheric jet. In a follow-on

study Kushner and Polvani (2004) use the zonally symmetric version of the GCM to investigate

the relative roles of the wave feedbacks and the stratospheric forcing in generating the response.

They find that the non-local zonal tropospheric response to stratospheric cooling is mainly

captured by the wave-zonal flow feedbacks alone while the response in the stratosphere was only

reproducible with both the wave feedbacks and thermal forcings. Using a similar GCM, Ring and

Plumb (2007) demonstrate that zonally symmetric angular momentum forcings produce AM-like

responses. Specifically, such responses were found only when the imposed forcing projects

strongly onto the model’s AMs. As in Kushner and Polvani (2004), the zonally symmetric

version of the model without wave feedbacks fails to capture both the strength and structure of

the AM responses to the applied forcings. Consistent with the above, Simpson et al. (2009)

generate a positive AM response to imposed tropical stratospheric heating in a simple GCM; the

forcing in this case is constructed to mimic the tropical stratospheric temperature response to the

11-year solar cycle. But Simpson et al. (2009) are unable to reproduce a similar response in the

zonally symmetric version of the model, confirming that eddy feedbacks are essential.

It is important to note that the simple GCMs used in these studies typically have

unrealistically long AM timescales, resulting in exaggerated AM responses to external forcings

(Chan and Plumb 2009; Gerber and Polvani 2009; Simpson et al. 2010). This is consistent with

the fluctuation-dissipation theorem which relates the magnitude of a system’s response to a

perturbation to the length of the timescale of its leading mode of internal variability. The length

of the AM timescale and also the magnitude of the response to a particular forcing are correlated

with tropospheric jet structure, particularly the jet latitude, with low-latitude jets exhibiting

greater AM persistence and larger responses (Chan and Plumb 2008; Simpson et al. 2010;

Kidston and Gerber 2010; Barnes et al. 2010). Bearing in mind their limitations, simple GCMs

are useful tools for qualitative and quantitative investigation of fluctuation-dissipation theory and

11

for demonstrating the importance of wave-driven dynamics in generating and maintaining AM

responses.

FIG. 1.2. (a) The time- and zonal-mean zonal winds for γ = 2 K km-1. The contour interval is 10 m s-1 and the zero contour is not plotted. The latitude 30°S is marked with a heavy vertical line. (b) As in (a) but for γ = 4 K km-1. (c) The difference in the time- and zonal-mean zonal wind between γ = 4 K km-1 and γ = 2 K km-1, that is, (b) minus (a). The contour interval is 5 m s-1 and the zero contour is not plotted (from Kushner and Polvani 2004).

In summary, there is extensive evidence that external forcings, such as GHG warming,

ozone depletion, and anomalous boundary conditions such as SSTs, snow cover and sea-ice

anomalies, generate climate responses that project onto the AMs. Within the context of

fluctuation-dissipation theory, it is perhaps not surprising that such responses exist; however, the

dynamical processes involved in generating these responses are complex and unique to the

forcing involved. In the following section, specific aspects of AM dynamics involving

stratosphere-troposphere interactions are discussed.

12

1.4 Stratosphere-Troposphere Interactions

The AMs describe the variability of the extratropical zonal mean circulation and, consequently,

involve wave-mean flow interactions (Limpasuvan and Hartmann 1999, 2000; Polvani and

Waugh 2004; Kushner and Polvani 2004; Kushner 2010). Although the AMs are often described

as deep equivalent barotropic patterns, Kushner (2010) argues that the AM dynamics of the

troposphere and stratosphere are distinct. During the active season of the stratospheric polar

vortex, the vertical coherence in the AM patterns suggests a degree of coupling between the

stratosphere and troposphere; however, outside of this season the tropospheric AMs are still

present while those in the stratosphere are not. In this section, stratospheric AM dynamics will be

emphasized. The reason for this emphasis reflects the fact that the observed and simulated

relationship between October Eurasian snow cover and the NAM involves a coupled

stratosphere-troposphere NAM circulation anomaly.

Stratospheric NAM variability is characterized by a strengthening (positive phase) or

weakening (negative phase) of the stratospheric polar vortex. Baldwin and Dunkerton (1999,

2001) identify extreme stratospheric NAM events as downward-propagating stratosphere-

troposphere coupling events. Figure 1.3 shows the positive and negative stratospheric NAM

composites as a function of pressure and lag from Baldwin and Dunkerton (2001). These types of

plots have been termed “dripping paint plots”, a name which fittingly describes the downward

propagation of the tropospheric AM anomalies. These coupling events lead to anomalies that

persist in the troposphere for up to 60 days. Downward propagation of NH stratospheric wind

anomalies subsequently affecting the tropospheric winds was previously documented by Kodera

et al. (1990) and Kodera and Koide (1997).

13

FIG 1.3. Composites of time-height development of the NAM for (A) 18 weak vortex events and (B) 30 strong vortex events. The events are determined by the dates on which the 10-hPa annular mode values cross -3.0 and +1.5, respectively. The indices are nondimensional; the contour interval for the color shading is 0.25, and 0.5 for the white contours. Values between -0.25 and 0.25 are unshaded. The thin horizontal lines indicate the approximate boundary between the troposphere and the stratosphere (from Baldwin and Dunkerton 2001).

Viewing the NAM as the dominant mode of variability of the zonal mean circulation, the

essential features of the NAM in the stratosphere can be described using the polar cap-averaged

GPH anomaly field (Cohen et al. 2002; Baldwin and Thompson 2009). The zonal mean GPH is

directly related to the zonal mean potential vorticity (PV) via the principle of inversion of PV

(Hoskins et al. 1985). Thus, anomalies in the zonal mean circulation - anomalies that the AMs

are constructed to represent - can be entirely described by PV anomalies. Following Kushner

(2010), the density-weighted polar cap-averaged quasi-geostrophic (QG) PV anomaly tendency

may be written as

( )∫ +′′==

T

B

z

zp

aysy

ao

p

at Sqvdzq ρ , (1.1)

14

where q is the QG PV, v is the meridional velocity, S represents the forcing and dissipation, p

denotes the density-weighted average over the polar cap, superscript a indicates the anomaly,

prime indicates the deviation from the zonal mean, ys is the southern edge of the polar cap, and zT

and zB are the top and bottom of the atmospheric column. Using the QG Eliassen-Palm (E-P)

flux, the meridional wave flux of PV may be written as,

( ) Fvfvuqv ozoz

ooyoo

⋅∇=

′′+′′−=′′ −− 11 ρθθρρρ ,

where,

[ ]

′′′′−==oz

ooo

vfvuzFyFF θθρρ ,)(),(

.

Equation (1.1) may be expressed as follows,

( )∫ ++= ==

===

T

B

sT

sBs

z

zp

ayyzz

yyzza

yyya

p

at SzFyFdzq ,

,)()( . (1.2)

In the stratosphere, the first term on the RHS of Eqn. (1.2) is small (Newman et al. 2001). Taking

the meridional wave heat flux anomaly as a proxy for the vertical component of the E-P flux

anomaly, the anomalous polar cap-averaged stratospheric zonal mean circulation is, therefore,

primarily controlled by the meridional wave heat flux anomaly in the stratosphere at the southern

edge of the polar cap (and by anomalous polar cap-averaged forcing and dissipation, Sa). Polar

cap-averaged GPH anomalies, which have been shown to agree very well with the NAM index,

will be used as a measure of the NAM index throughout this thesis (Baldwin and Thompson

2009). Polvani and Waugh (2004) demonstrate that composites of the NAM index based on

anomalous extratropical meridional wave heat fluxes at 100 hPa show downward-propagating

positive and negative NAM events following anomalously low and high heat flux events.

Stratospheric sudden warmings (SSWs), characterized by stratospheric zonal wind reversals and

downward propagating positive polar cap-averaged GPH anomalies, are also preceded by large

heat flux anomalies (Matsuno 1971; Dunkerton et al. 1981; Limpasuvan et al. 2004; Charlton-

Perez and Polvani 2007).

15

Although the dynamics of the initiation of stratosphere-troposphere coupling events is

well characterized by E-P flux convergence associated with meridional wave heat flux

anomalies, the dynamics of the subsequent downward-propagation of AM anomalies is less

clear. Several diagnostic approaches have been applied to elucidate the relevant mechanisms.

One class of diagnostics consists of balanced responses to E-P flux convergence in the

stratosphere. For example, E-P flux convergence in the stratosphere induces a meridional

circulation below, which, in the steady-state limit, is exactly that predicted by downward control

theory (Haynes et al. 1991). In the instantaneous balanced response, upper level poleward

meridional flow causes an increase in mass in the polar stratosphere (Haynes and Shepherd 1989;

Baldwin and Dunkerton 1999; Sigmond et al. 2003). This mass redistribution alters the surface

pressure over the pole and extratropics, the meridional pressure gradient and, thus, the

tropospheric circulation. PV inversions provide an alternative balanced response diagnostic.

Specifically, PV is related to the GPH through a second order differential operator. Thus, by

definition, PV anomalies can induce non-local anomalies in GPH (Hartley et al. 1998; Ambaum

and Hoskins 2002; Black 2002). In addition, Thompson et al. (2006) demonstrate that, on weekly

time-scales, the persistence of tropospheric wind anomalies associated with stratosphere-

troposphere coupling is consistent with a balanced response to stratospheric heating. Although

these balanced response diagnostics can explain some of the observed tropospheric response to

stratospheric anomalies, they are not able to explain the entire response because they neglect the

critical role of wave feedbacks (Thompson et al. 2006). The second class of diagnostics

describing the downward propagation of AM anomalies involves wave-mean flow feedbacks.

Kushner and Polvani (2004) and Simpson et al. (2009) demonstrate that synoptic wave feedbacks

were necessary to reproduce the tropospheric response to stratospheric forcing. Other work

shows that stratospheric anomalies alter the propagation of planetary waves into the stratosphere,

resulting in a downward propagation of the region of momentum deposition by these waves

(Scott and Polvani 2004). Finally, planetary wave reflection has also been identified as a

mechanism by which stratospheric anomalies (in this case reflected zonal wave number 1)

propagate down to the troposphere (Perlwitz and Harnik 2003, 2004; Harnik 2009; Shaw et al.

2010).

The identification of numerous downward-propagating NAM events in the climate record

suggests that knowledge of stratospheric conditions could provide useful information for

16

predicting tropospheric weather on seasonal timescales (Baldwin and Dunkerton 2001).

Recently, several studies have identified tropospheric circulation patterns prior to stratospheric

NAM events, sometimes referred to as “tropospheric precursors”, further fueling the idea that

tropospheric NAM variability may be, in part, predictable (Garfinkel et al. 2010; Ineson and

Scaife 2009; Kolstad and Charlton-Perez 2010; Charlton-Perez et al. 2010; Nishii et al. 2010;

Fletcher and Kushner 2011). These tropospheric precursors are themselves circulation anomalies

and, thus, cannot necessarily be considered “external forcings” or “anomalous boundary

conditions” of the kind discussed in Section 1.3. However, these precursors may sometimes

represent atmospheric teleconnections linked to anomalous boundary conditions, such as SST

anomalies (Fletcher and Kushner 2011). For example, Garfinkel et al. (2010) show that

stratospheric NAM variability is negatively correlated with the amplitude of the wave pattern

that corresponds to the mid-tropospheric climatological stationary wave field, particularly its

wave-1 and wave-2 components. They find that when the climatological stationary wave field is

amplified or attenuated, vertical wave activity flux anomalies are enhanced or suppressed and the

stratospheric jet correspondingly weakens or strengthens. For the purposes of this thesis, the

Garfinkel et al. (2010) results are described as linear interference between wave anomalies and

the climatological stationary wave. DeWeaver and Nigam (2000a) show that linear interference

is also an important component of tropospheric NAM variability. They show that the wave

momentum flux anomalies associated with the tropospheric NAM are dominated by linear

interference between the anomalous wave and the climatological stationary wave. Linear

interference effects underlie the primary findings of this thesis and will be described in greater

detail in the subsequent chapters.

Stratosphere-troposphere interactions are an integral component of wintertime AM

variability. Although promising research has been done in this field, the predictive power of

stratospheric circulation anomalies for tropospheric weather is mixed (Scaife et al. 2005;

Douville 2009; Cohen et al. 2010; Jung et al. 2011). Douville (2009) demonstrate that simulation

of the NAM and related winter surface temperature is significantly improved when the model’s

stratosphere is nudged towards the observations using reanalysis data. However, Jung et al.

(2011) show little improvement in the forecast skill for the early winter 2009-2010 downward-

propagating NAM event (Wang and Chen 2010; Seager et al. 2010) when stratospheric nudging

is included in their simulations. Contrary to Cohen et al.’s (2010) argument that Eurasian snow

17

cover initiated a tropospheric precursor to the early winter 2009-2010 NAM event, Jung et al.

(2011) also show no improvement in forecast skill when autumn Eurasian surface temperatures

(a proxy for Eurasian snow cover) are nudged to the observations. In Section 1.5, the specific

case of stratosphere-troposphere interactions associated with autumn Eurasian snow cover

anomalies is presented in greater detail.

1.5 Eurasian Snow and the NAM

In this Section, observational and modeling support for the snow-NAM connection is reviewed.

In line with much of the literature discussed in Section 1.3, there is a body of work that argues

that regional snow cover variability can “force” the atmosphere and lead to large-scale, NAM

climate variability. However, unlike some of the external forcings discussed above, such as

increasing GHG forcing or ozone depletion, snow fall and, thus, snow cover is internal to the

climate system and coupled with the atmospheric circulation itself. The literature reviewed

below shows that many aspects of the observed relationship between Eurasian snow cover

anomalies in the autumn and the subsequent winter NAM can be simulated in GCMs with

prescribed snow forcings. Nevertheless, the question of the role of snow as a climate forcing

remains open.

The observed relationship between autumn Eurasian snow and winter climate was first

identified at the regional scale. Foster et al. (1983) demonstrate a negative relationship between

autumn Eurasian snow cover anomalies and winter Eurasian surface temperatures. When snow

cover over Eurasia is anomalously high in the autumn, winter temperatures are anomalously

cold. The authors attribute the relationship to the persistence of snow cover anomalies and an

enhancement of the cold Siberian High circulation. The first studies to relate Eurasian snow

cover anomalies to large-scale atmospheric circulation anomalies (Watanabe and Nitta 1998,

1999) suggest that the dramatic negative Eurasian snow cover anomalies of 1988 contributed to a

hemispheric change in the atmospheric circulation. Indeed, the circulation anomalies for the

1988/89 winter resemble the positive phase of what is now identified as the NAM, which is

consistent with the general snow-NAM relationship discussed below. GCM simulations with

18

prescribed Eurasian snow and NH SST anomalies from 1988 confirm that the circulation

anomaly is due, in part, to the large snow cover anomaly (Watanabe and Nitta 1998).

Cohen and Entekhabi (1999) demonstrate that observed September-October-November

(SON) Eurasian snow cover is significantly correlated with extratropical DJF 500 hPa GPH

throughout the second half of the twentieth century. The pattern of the spatial correlation map

resembles the negative NAM and is particularly strong over the North Atlantic. Figures 1.4a and

b show the January 500 hPa GPH regressed on the October Eurasian snow index (OCTSNW is a

standardized anomaly index of snow cover extent over Eurasia in October; see Section 3.2 for

more information about OCTSNW) and on the January NAM index for the years 1972-2009,

respectively. Figure 1.4b shows the negative phase of the NAM in this field. The similarity

between Figs. 1.4a and b clearly illustrates the snow-NAM relationship. Thus, years in which

snow cover is anomalously extensive over Eurasia in October tend to be years in which the NAM

in the following winter is in its negative phase. Cohen and Entekhabi (1999) and Cohen et al.

(2001) explain the relationship as follows: early autumn snow cover over Eurasia leads to

anomalous regional cooling due to the high albedo of snow. The anomalous cooling enhances the

dense and shallow Siberian High, which expands westward and northward due to topographic

restrictions to the south and east. The Icelandic Low is then forced southward and the resulting

pattern of highs and lows projects onto the negative phase of the NAM. The explanation of

Cohen and Entekhabi (1999) and Cohen et al. (2001), however, provides only a qualitative

description of the circulation anomaly from a surface/lower tropospheric perspective. For

example, the authors do not clearly explain the cause of the observed migration of the Siberian

high.

Closer examination of the circulation anomaly associated with autumn Eurasian snow

cover reveals not only a NAM anomaly in the lower troposphere, but also a NAM anomaly

extending well into the stratosphere. Figures 1.4c and 1.4d show the January zonal mean GPH

regressed on OCTSNW and January NAM index, respectively. Figure 1.4c reveals a deep

vertical coherence between the NAM and the circulation pattern associated with October

Eurasian snow suggestive of coupling between the troposphere and stratosphere (Baldwin and

Dunkerton 2001). This key observation implies that stratosphere-troposphere coupling is crucial

to understanding the snow-NAM connection.

19

FIG. 1.4. January 500 hPa GPH for 1972-2009 (NCEP-NCAR) regressed on the (a) October Eurasian snow index (OCTSNW) and the (b) January NAM Index. Contour intervals are 5 m and 20 m for (a) and (b), respectively. (c) and (d) same as (a) and (b) but for zonal mean GPH. Contour intervals 10 m and 20 m for (c) and (d), respectively. Blue dashed and red solid contours are for negative and positive values. Gray shading indicates 95% significance.

Several subsequent studies demonstrate that years when autumn Eurasian snow cover is

anomalously extensive are also years with anomalously upward vertical wave activity flux

emanating from the Eurasian region (Saito et al. 2001; Cohen et al. 2002; Cohen and Saito

2003). The authors suggest that anomalously extensive snow cover anomalies are associated with

wave-driven stratosphere-troposphere coupling. For example, Cohen et al. (2007) calculate a

multivariate EOF (MVEOF) of January SLP and December 100 hPa vertical wave activity flux

to define a stratosphere-troposphere coupling index (STCI) and show that it is significantly

20

correlated with OCTSNW. Regressions of daily wave activity flux and polar cap-averaged GPH

onto both the STCI and OCTSNW reveal positive wave activity flux anomalies preceding

positive downward propagating polar cap-averaged GPH anomalies from the stratosphere to the

troposphere (for example, see Fig. 3.2). Based on this analysis, the authors present a

stratosphere-troposphere coupling mechanism linking snow and the NAM. Figure 1.5 shows a

conceptual model for this mechanism (adapted from Cohen et al. 2007). Positive snow cover

extent anomalies generate strong regional surface cooling via increasing surface albedo, which

excites a vertically propagating Rossby wave train in late fall and early winter. The waves are

dissipated in the stratosphere leading to a negative NAM response consisting of a stratospheric

warming and downward propagation of a negative NAM signal to the troposphere by January.

Although the relationship between Eurasian snow cover anomalies and the NAM is

robust in observations, several authors have questioned the role of snow cover in influencing the

wintertime NAM. Limpasuvan et al. (2005) question the ability of snow cover anomalies to

excite a wave of large enough zonal scale to propagate from the troposphere to the stratosphere

and Kushnir et al. (2006) argue that snow cover likely has little influence given the lack of

persistence of snow cover anomalies. In addition, the observation that GCMs are unable to

simulate this relationship within their natural variability raises more questions about how autumn

Eurasian snow can influence the NAM (Gong et al. 2002; Hardiman et al. 2008).

Despite these arguments, an atmosphere-land GCM (ECHAM3) simulation in which

snow is allowed to vary freely from year to year exhibits somewhat stronger NAM variability

over the North Atlantic and Siberia relative to a simulation in which the seasonal cycle of snow

cover is fixed (Gong et al. 2002). Interestingly, the correlations between the extratropical DJF

SLP and height-dependent GPH EOFs are significant throughout most of the atmosphere for the

freely varying snow runs but only up to the upper troposphere for the fixed snow runs. The

authors suggest, “that without interannual snow variations, the dominant modes of variability in

the troposphere and stratosphere may be essentially uncoupled.” The statistical robustness and

dynamics of this result are not well understood yet it suggests a fundamental role for snow cover

anomalies in modulating NAM variability.

21

FIG. 1.5. Conceptual model for NAM response to Eurasian snow cover anomalies. Eurasian snow cover exhibits its largest variability in autumn. In years with anomalously high snow cover, the increase in surface albedo diabatically cools the Eurasian region, exciting vertically propagating Rossy Waves. The waves dissipate in the stratosphere and weaken the polar vortex generating a negative NAM response in the stratosphere. The NAM response propagates downward, reaching the troposphere by mid-winter (adapted from Cohen et al. 2007).

Following this study, Gong et al. (2003a) conduct two 20-member ensembles of

ECHAM3 simulations in which satellite-derived Siberian snow forcings are prescribed, one with

low snow cover (observed 1988 snow cover) and one with high snow cover (observed 1976

snow cover). This study and many subsequent GCM simulations with prescribed snow forcings

restrict the forcing to the Siberian region rather than the entire Eurasian region, given the large

variations in snow cover in this region in autumn (Robinson et al. 1993; Cohen et al. 2001; Gong

et al. 2002). The ensemble-mean difference between the high and low ensembles reveals a

negative NAM response involving enhanced upward and poleward wave activity flux from the

stationary waves and a downward propagating positive polar cap averaged GPH response. The

authors argue for the enhancement of stationary wave activity via snow-induced thermal cooling

22

(Gong et al. 2003b). To test this theory, the authors perform a complementary set of simulations

to those in Gong et al. (2003a) in which they remove the Siberian topography (Gong et al.

2004a). The simulations fail to produce a negative NAM response. The wave activity flux

response shows greatly diminished upward flux and enhanced equatorward flux. The authors

explain that because the stationary wave field is diminished the wave response is unable to

propagate vertically. The details of this explanation are vague and, although intriguing, the study

leaves many unanswered questions. The question of how the background climatological wave

field influences the response to snow cover is the central question of Chapters 2 and 3 of this

thesis.

Many of the early prescribed-snow-forcing simulations involved the same GCM, very

similar snow forcings and a relatively small number of ensemble members (Gong et al. 2002,

2003a, 2003b, 2004a, 2004b). Additional simulations with different GCMs, alternative snow

forcings and larger ensembles confirmed the robustness of the GCM response to October

Siberian snow forcings. Fletcher et al. (2009a) conduct a pair of 100-member ensembles of high

and low Siberian snow forcing simulations using both the standard and stratosphere-resolving

Geophysical Fluid Dynamics Laboratory (GFDL) GCMs. The ensemble mean response to a high

snow forcing in the standard GCM exhibits the development of an upstream high/downstream

low vertically propagating Rossby wave pattern around the forcing region. Fletcher et al. (2009a)

argue that the regional surface cooling associated with the snow forcing causes the isentropes to

dome up, forming effective topography for approximately adiabatic wave motions. The wave

pattern that develops is analogous to the wave pattern that develops for westerly flow over

topography (Holton 1994). The form stress resulting from the regional wave and potential

temperature responses can be interpreted as a zonal mean heat flux response, a proxy for the

zonal mean vertical wave activity flux response (Vallis 2006). After roughly 20 days, a

downward-propagating negative NAM response develops, consistent with Gong et al. (2002),

with the negative NAM signal evident at the surface in December. The coupling is more robust

when the stratosphere is initially warm (Fletcher et al. 2007). Complementary simulations with a

stratosphere-resolving GCM show a similar yet markedly weaker response. The authors attribute

this to the shorter decorrelation timescales in the stratosphere-resolving GCM, which is likely

related to its weaker lower stratospheric winds.

23

The relative importance of snow cover versus snow depth in generating a negative NAM

response to prescribed snow forcings is somewhat unclear and likely depends in part on the

manner in which snow processes are handled by a particular GCM. Snow cover and snow depth

have different effects on the surface energy balance. Snow cover is the component of the snow

forcing that is primarily associated with a reduction in short wave absorption at the surface,

while the snow depth component acts to reduce upward heat flux from the soil to the surface and

increase latent heat flux associated with snowmelt. For example, Gong et al. (2004b) show that

both snow cover and snow depth anomalies are required to simulate the negative NAM response

to autumn Eurasian snow cover anomalies in the ECHAM3 GCM. However, Fletcher et al.

(2009) find that their response to a prescribed autumn Siberian snow forcing is insensitive to the

depth of the snow forcing in the AM2 GCM. In addition, Allen and Zender (2010) find a

negative NAM response to prescribed autumn Eurasian albedo anomalies, which the authors use

as a proxy for snow cover, in simulations with the National Center for Atmospheric Research

(NCAR) Community Climate System Model 3 (CCSM3) GCM.

The response to prescribed snow cover anomalies is clearly robust in atmosphere-land

GCM simulations and several studies have confirmed a similar stratosphere-troposphere

coupling pathway by which a negative tropospheric NAM response is generated. However, many

questions remain regarding the true character of the snow forcing and the snow-NAM

relationship in nature. Is it reasonable to consider snow cover as simply an anomalous boundary

condition on the atmosphere or is the atmospheric circulation anomaly that brings about snow

fall important? A recent study by Allen and Zender (in press) shows that although a transient

atmosphere-land GCM control run does not display the October Eurasian snow-NAM

relationship, when the GCM is run with prescribed satellite-derived snow cover fraction, a snow-

NAM relationship similar to observations is reproduced. Although this experimental design does

not provide a full answer to the question of whether snow cover “forces” the atmosphere in

nature, it shows that GCM simulations with realistic interannual snow cover anomalies give

consistent results to previous snow forcing simulations.

The following section outlines several of the outstanding questions that this thesis aims to

address. A broader discussion of ongoing areas of research related to snow and circulation is

presented in Chapter 5.

24

1.6 Conclusion

The connection between snow cover and the NAM raises several immediate questions that have

yet to be addressed in the literature. Firstly, what determines the lag between autumn Eurasian

snow anomalies and wintertime anomalies in the vertical flux of wave activity from the

troposphere to the stratosphere? Negative surface temperature anomalies over Eurasia associated

with OCTSNW appear in October yet the wave activity flux anomaly is delayed until December.

In GCM simulations with prescribed snow forcings, there is very little delay between the

initiation of the snow forcing and the wave activity flux response. In Chapter 2, the simulations

of Fletcher et al. (2009a) are revisited; the timing of the circulation response in these simulations

is explored in detail and the robustness of the dynamical features is investigated using a suite of

simplified GCM simulations. It is shown that the interaction between the wave response to the

prescribed snow forcing and the background climatological wave, i.e. linear interference,

determines the timing of the transient wave activity flux response and, consequently, the

stratospheric NAM response. The findings of Chapter 2 are applied to the timing of the observed

Eurasian snow-NAM relationship in Chapter 3. It is shown that linear interference also plays a

role in determining the timing of the observed relationship between October Eurasian snow

cover, wave activity flux and the NAM.

Secondly, how does the background climatological wave influence the wave activity flux

and NAM responses to prescribed snow forcings? The simulated wave response to Siberian snow

forcings is typically a large, wave-1 Rossby wave, which should be able to propagate vertically

into the stratosphere (Charney and Drazin 1961; Hardiman et al. 2008; Fletcher et al. 2009a). Yet

when the Eurasian topography is removed, Gong et al. (2003b) find that they are unable to

reproduce the wave activity flux and NAM response from their previous study. Although this

thesis does not revisit these simulations, Chapter 2 presents evidence that the magnitude of the

background climatological wave is an important aspect of generating a NAM response to

prescribed Eurasian snow cover forcings due to the dominance of the linear interference effect.

Thirdly, can the observation that GCMs do not capture the correlation between Eurasian

snow and the NAM within their natural variability be understood within the context of linear

interference (Hardiman et al. 2008)? Chapter 3 presents new evidence that the transient evolution

25

of the wave associated with October Eurasian snow cover in GCMs differs from observations;

for example, autumn Eurasian snow cover related wave anomalies in GCMs typically display

destructive interference with the background climatological wave, opposite to observations.

Finally, motivated by the findings of Chapters 2 and 3, Chapter 4 addresses broader

questions about the nature of the vertical wave activity flux anomalies and the importance of

linear interference in climatological stratosphere-troposphere interactions, in general. Using

zonal mean extratropical wave heat fluxes as a proxy for the vertical component of the zonal

mean wave activity flux, Chapter 4 demonstrates that interannual heat flux anomalies

representing linear interference are a key component of heat flux anomalies from the troposphere

to the stratosphere in both the Northern and Southern hemispheres. In particular, it is shown that

linear interference enhances the persistence of heat flux anomalies. In addition, Chapter 4

demonstrates that linear interference diagnostics highlight important dynamical features of

stratospheric variability including different classes of stratospheric sudden warmings and the

timing of stratospheric final warmings1.

This thesis includes research that has been published in, accepted by, or is to be

submitted to peer-reviewed journals. Chapter 2, Sections 2.1-2.3.4 and most of 2.4 have been

published in the Journal of Climate (Smith et al. 2010). Chapter 3 has been accepted by the

Journal of Climate and is currently in press (Smith et al. in press). In addition, paragraphs in

Chapter 1, Sections 1.4 and 1.5, include introductory material extracted from the two above

mentioned publications. Chapter 4 is in preparation for submission.

1 Stratospheric sudden warmings are defined as reversals of the westerly zonal mean zonal wind in the stratospheric polar vortex at 60°N and 10 hPa and represent the most extreme events in the wintertime stratospheric circulation. Stratospheric final warmings characterize the seasonal breakdown of the stratospheric polar vortex in the spring as the zonal mean zonal wind transitions from westerly to easterly.

26

Chapter 2

The Role of Linear Interference in the Annular Mode Response to

Extratropical Surface Forcing

2.1 Introduction

As introduced in Chapter 1, several GCM studies with prescribed October Siberian snow forcing

have been conducted to gain a better mechanistic understanding of the snow-NAM relationship.

Although these studies capture important features of the physical mechanism (Gong et al. 2003;

Fletcher et al. 2009a), questions remain regarding the transient evolution and dynamics of the

response. For example, Fletcher et al. (2009a) focus on a downward-propagating negative NAM

(weak vortex) response to prescribed snow forcing that corresponds to the observed behavior, but

their simulation also includes an initial weak positive NAM (strong vortex) response that

remains unexplained and that is not detected in observations. In addition, only partially

understood is the mechanism whereby the negative NAM response peaks and decays, despite the

fact that the snow forcing remains switched on and a robust upward-propagating Rossby wave

train response persists. The unexplained aspects of the Fletcher et al. (2009a) simulations add to

outstanding questions regarding the inability of GCMs with predicted (as opposed to prescribed)

snow cover to capture the snow-circulation connection (Hardiman et al. 2008).

In an attempt to better constrain some aspects of the problem, Chapter 2 presents a

straightforward diagnostic approach based on linear interference effects found in the simulated

response to a particular extratropical forcing. The approach is only partially predictive and in this

study applies mainly to the high-latitude stratospheric circulation response. But it accounts for

some previously unexplained and non-robust aspects of the response, and it is found to apply

more broadly to other problems of this class (Fletcher and Kushner 2011). The emphasis on

linear interference effects is motivated by recent observational and modeling work on the

coupled stratosphere-troposphere response to tropospheric forcing. For example, Garfinkel and

Hartmann (2008) show that ENSO primarily affects the stratospheric polar vortex through a

Pacific North American-like teleconnection pattern that constructively or destructively interferes

27

with the wave-1 quasi-stationary wave field in the troposphere, and, thus, strengthens or weakens

the E-P flux into the stratosphere. This, in turn, weakens or strengthens the polar vortex via wave

induced stresses on the zonal mean. Besides ENSO-related forcing of climate, Garfinkel et al.

(2010) show that Eurasian snow cover anomalies similarly influence the wave-1 quasi-stationary

wave field and hence the E-P flux into the stratosphere. Other studies indicate that linear

interference effects are at work in stratosphere-troposphere coupling. For example, Martius et al.

(2009) show that tropospheric blocking events that are co-located with the climatological wave-1

and wave-2 quasi-stationary wave fields are associated with wave-1 and wave-2 stratospheric

sudden warmings, respectively. In addition, Ineson and Scaife (2009) show that the extratropical

stratosphere-troposphere response to ENSO in simulations is controlled by the coherence

between the ENSO wave anomaly and the background waves.

In this study, these insights are applied to study linear interference effects in the snow-

forced teleconnection to the NAM. The main point is that the phase of the wave response to the

surface forcing relative to the phase of the background stationary wave plays a key role in

determining the zonal mean response to surface cooling. Section 2.2 outlines the models

employed in this study and the experimental design. In Section 2.3.1, details of the transient

response to a Siberian snow forcing in a comprehensive GCM simulation are explored. A

simplified GCM is used to investigate and diagnose the dynamics in greater detail in Sections

2.3.2-2.3.4 and Section 2.4. Finally, Section 2.5 summarizes the conclusions.

2.2 Methods

2.2.1 Model Descriptions

The simulations performed by Fletcher et al. (2009a) (henceforth F09) with the low-top

Geophysical Fluid Dynamics Laboratory (GFDL) atmospheric/land GCM AM2/LM2 (Anderson

et al. 2004; denoted AM2-LO in F09) are revisited. Although this model does not have a well-

resolved stratospheric circulation, it has been argued that this is not critical in determining the

response (F09). The specifics of the model configurations are discussed in detail in F09. Owing

to an irrecoverable data loss, the number of ensemble members used in this study is 52 versus the

28

100 ensemble members used in F09. The ensemble mean results considered are unaffected by

this change.

A simplified GCM (henceforth SGCM) that solves the dry, hydrostatic, primitive

equations in hybrid coordinates is also used (Polvani and Kushner 2002; Held and Suarez 1994).

It is forced with a Newtonian relaxation of the temperature to a prescribed, zonally symmetric

and time-independent equilibrium temperature profile, Teq. The SGCM is not constructed to

closely correspond to the comprehensive GCM, AM2, but will serve to test dynamical ideas (for

example, while AM2 has a seasonal cycle, the SGCM equilibrium temperature profile is

independent of time such that the climatology is representative of Northern Hemisphere winter

solstice conditions). In this model, the strength of the NAM response to tropospheric forcings is

sensitive to the strength of the polar vortex (Reichler et al. 2005; Gerber and Polvani 2009),

which can be adjusted by specifying γ, the temperature lapse rate over the winter pole in the

equilibrium temperature profile. A vortex strength of γ = 2 K km-1 is chosen in this study for the

majority of the analysis. In Section 2.3.5, the sensitivity of the results to polar vortex strength is

investigated by running perturbation simulations with the SGCM with γ = 1 K km-1 and γ = 3 K

km-1. The model has 40 vertical levels (model lid height of 0.02 hPa), has a horizontal resolution

of T42, and is run with a time-step of 800 s. Additional details are given in Polvani and Kushner

(2002) and Kushner and Polvani (2004). The most important difference from previous studies is

that in this study the SGCM uses realistic topography (i.e. the T42 spectral representation of the

observed topographic distribution), instead of idealized topography or no topography, which

allows for the generation of a planetary stationary wave field (Fig. 2.5, shading) with a fairly

realistic phase structure. But the amplitude of the resulting stationary wave in the SGCM is too

weak compared to observations, and also to the comprehensive GCM, AM2. For example, the

amplitude of the climatological stationary wavenumber 1 at 60°N and 50 hPa is 45 m in the

SGCM, 141 m in AM2 (December-January-February (DJF)) and 229 m in NCEP for 1967-2007

(DJF). The consequences the weak stationary wave field has on the SGCM results are discussed

in Section 2.5.

29

2.2.2 Snow/Surface Cooling Method

F09 apply a “switch-on” positive snow forcing in the comprehensive GCM, AM2, over the

Siberian region for October-December; this increases surface albedo and generates a surface

shortwave cooling. F09 branch “high-snow” and “low-snow” integrations from independent

initial states taken from a long “climatological SST” control integration. In these integrations, the

difference between the snow depth in the high-snow and low-snow cases is time independent.

Further details of the snow forcing perturbation method for the comprehensive GCM are

provided in Fletcher et al. (2007) and F09. Regarding the low-snow state as the background state

or control state, c, and the high-snow state as the perturbed state, p, the response in X is defined

as

∆X = Xp - Xc , (2.1)

for a given realization. Then, denoting an ensemble mean by angled brackets, ‹∙›, the ensemble

mean response is

∆‹X› = ‹Xp› - ‹Xc› . (2.2)

The SGCM is used to explore a broad range of surface thermal forcings, including

variations in the strength, position and sign of the forcings. These simulations are listed in Table

2.1 and the motivation for using them will be presented in Sections 2.3.2-3. A lower tropospheric

cooling is prescribed over a region bounded to the west at longitude λ1 and to the east at

longitude λ2 and to the south and north by latitudes 40°N and 80°N. A term, Q(λ,φ,σ), is added to

the temperature tendency equation, where φ is latitude and σ is the vertical sigma coordinate. Q

is defined by

Q = Qo (φo/φ)3max[0,( σ- σb/1- σb)], λ1 ≤ λ ≤ λ2, 40°N ≤ φ ≤ 80°N (2.3)

0, everywhere else,

where φo is 40°N and Qo is chosen to achieve a cooling response that resembles the cooling

response to the snow forcing in AM2 (Qo = -3.125 K day-1 gives an area-averaged forcing of -1.4

K day-1 at σ = 1). The forcing is strongest at the surface and decreases linearly to zero in the

vertical up to σb = 0.7 and decreases meridionally as φ-3 from 40°N-80°N to mimic the effect of

30

the decreasing meridional insolation gradient on snow-related shortwave diabatic cooling in fall

and winter. For a Siberian forcing, corresponding to F09, λ1 = 60°E and λ2 = 140°E. Similar

simulations using other forcing shapes have been run, e.g. two-dimensional sine-squared forcings

centered at 60°N and 100°E, and the results are found to be qualitatively similar. In addition to

the simulations run with the forcing centered over Siberia (Simulation B in Table 2.1; see Section

2.3.2), eleven additional simulations have also been run (Simulations A and C-L) in which the

forcing is shifted zonally at intervals of 30° longitude (that is, λ1 and λ2 in Eqn. (2.3) are

increased in 30° increments), another series of simulations in which the strength of the forcing

over Siberia is varied (Simulations M-R), and one additional simulation in which the applied

forcing was a heating rather than a cooling (Simulation S).

TABLE 2.1: List of SGCM simulations. Each simulation consists of 90, 100-day ensemble members.

Simulation Forcing Location (λ1 - λ2) Forcing Strength* A 30° - 110° 1 B 60° - 140° 1 C 90° - 170° 1 D 120° - 200° 1 E 150° - 230° 1 F 180° - 260° 1 G 210° - 290° 1 H 240° - 320° 1 I 270° - 350° 1 J 300° - 20° 1 K 330° - 50° 1 L 0° - 80° 1 M 60° - 140° 0.5 N 60° - 140° 1.5 O 60° - 140° 2 P 60° - 140° 2.5 Q 60° - 140° 3 R 60° - 140° 3.5 S 30° - 110° -1

*A forcing strength of 1 corresponds to the standard forcing strength, Qo = -3.125 K day-1, in Eqn. (2.3).

For all SGCM perturbation simulations, a 9,000-day control run with time-independent

forcing is used to provide initial conditions for 90, 100-day realizations. The forcing is switched

31

on and held constant for 100 days. For these simulations, the response is given by Eqns. (2.1)

and (2.2) with p referring to the forced state and c referring to the corresponding control state.

2.3 Results

2.3.1 Revisiting the Transient Response to Siberian Snow Forcing in F09

Figure 2.1a shows the ensemble-mean time series of the 60°N-90°N polar cap averaged 50 hPa

geopotential height (GPH) response (∆‹Zpcap›) to snow forcing in AM2. ∆‹Zpcap› is a proxy for

the NAM index with positive values of ∆‹Zpcap› corresponding to a negative NAM index and vice

versa (Baldwin and Thompson 2009). Fletcher et al. (2007) and F09 discuss the vertical structure

of this response, its downward propagation in the stratosphere, and its coupling to the

troposphere; the 50 hPa response is a good proxy for the lower stratospheric response as a whole.

Over the first 15 days of the simulation, there is a slight decrease in ∆‹Zpcap›, followed by a linear

increase until day 65, and a sharp drop off afterwards. Two distinct periods of evolution are

identified: the time interval before the peak in ∆‹Zpcap› in Fig. 2.1a (days 1-65), which is

characterized by an overall negative NAM tendency, and the time interval after the peak in

∆‹Zpcap› (days 66-92), which is characterized by a positive NAM tendency (the issue of the weak

positive NAM feature during the first 15 days of the transient simulation is discussed below).

Figures 2.1b-c show the ensemble-mean longitude-level cross-section of the wave

response (∆‹Z*›) at 60°N averaged over days 1-65 and days 66-92, respectively. Here, the

asterisk superscript, (∙)*, indicates the deviation from the zonal mean. The wave response at 60°N

is representative of the high extratropical wave response in the latitude band 50°N to 70°N.

During both periods, a characteristic westward-tilting wave structure is observed that is

associated with upward-propagating Rossby waves originating from the forcing region. At first,

this seems to imply that snow forcing is consistently associated with a positive net upward wave

activity response. But the GPH tendency in Fig. 2.1a is positive in the first period, and negative

in the second period. Consistently, the response of the ensemble, time, and zonal mean

meridional wave heat flux, which is a proxy for the vertical component of the

32

FIG. 2.1. (a) Time series of ensemble mean polar cap-averaged 50 hPa geopotential height response (∆‹Zpcap›) to a switch-on snow forcing in the AM2 GCM. Thick, solid portions of the line indicate 95% significance (the statistical significance of the response is assessed for each simulation day using the one-sample Student’s t-test assuming independence of the realizations that start 1 year apart). The solid horizontal line indicates the zero line. (b) Day 1-65 averaged ensemble mean wave GPH response (∆‹Z*›) at 60°N. (c) as in (b) but for days 66-92. The solid contours correspond to positive values and the dashed contours correspond to negative values. The contour interval is 5 m. The gray shading shows 95% significance.

Eliassen-Palm (E-P) flux (Newman et al. 2001), shows an increase in the wave heat flux during

the day 1-65 period (Fig. 2.2a) and a decrease of the wave heat flux during the day 66-92 period

(Fig. 2.2d). The change in the sign of the ∆‹Zpcap› tendency during the simulation corresponds to

33

the change in sign of the wave heat flux response in an analogous manner to that observed during

natural negative and positive NAM events (McDaniel and Black 2005). Thus, the original

inference about the change in wave activity, based on examining the wave GPH response alone,

is incorrect.

To explain the change in sign of the wave activity response between the two periods, it is

necessary to consider the nonlinear nature of the wave heat flux response. The ensemble mean

wave heat flux response is denoted as ∆{‹v*T*›}, where the braces, {∙}, indicate zonal and time

averaging. For each ensemble member, v and T may be defined as

v = ‹v› + v′, T = ‹T› + T′,

where the prime superscript, (∙)′, denotes the departure from the ensemble mean. The ensemble

mean wave heat flux response can then be decomposed as

∆{‹v*T*›} = ∆{‹v*›‹T*›} + ∆{‹v*′T*′›}. (2.4)

The first term on the right-hand-side of Eqn. (2.4) is denoted “EM” as it characterizes the

contributions from the ensemble mean response and the second term is denoted “FL” as it

characterizes the contributions from the fluctuations about the ensemble mean. Thus, Eqn. (2.4)

may be written as

∆{‹v*T*›} = TOTAL = EM + FL, (2.5)

where

EM ≡ ∆{‹v*›‹T*›} and FL ≡ ∆{‹v*′T*′›}.

It is found, and will be shortly confirmed, that TOTAL is dominated by EM, i.e. ∆{‹v*T*›}≈

∆{‹v*›‹T*›}. The ensemble averaging effectively low-pass filters the dynamics and so the EM

term represents contributions to the response from relatively low-frequency waves. The FL term

is relatively small here and is dominated by contributions from relatively high-frequency waves,

e.g. synoptic waves in the troposphere, which are independent in each realization but contribute

systematically to the wave fluxes from one realization to the next.

34

Using Eqn. (2.1), the response of the EM wave heat flux can be decomposed

straightforwardly into terms that are linear and quadratic in the ensemble mean response,

EM = EMLIN + EMNL, (2.6)

where

EMLIN = {‹v*c›∆‹T*› + ∆‹v*›‹T*

c›} and EMNL = {∆‹v*›∆‹T*›}.

The term EMLIN represents the wave heat flux response associated with the covariance (under a

time and zonal mean, {∙}) between the ensemble mean wave response (∆‹v*› and ∆‹T*›) and the

control state (‹v*c› and ‹T*

c›). The term EMLIN is linear in the ensemble mean wave response –

for example, EMLIN would double if the amplitude of the wave response doubled. The term

EMNL involves only the ensemble mean wave response and, thus, represents the wave heat flux

response intrinsic to the wave response itself. The term EMNL is quadratic in the ensemble mean

wave response – for example, EMNL would quadruple if the amplitude of the wave response

doubled.

TOTAL, EMLIN and EMNL are plotted in Figs. 2.2a-c for the day 1-65 period and in Figs.

2.2d-f for the day 66-92 period. In both periods, TOTAL, i.e., ∆{‹v*T*›}, is seen to be dominated

by EMLIN; EMNL is relatively small. Figure 2.2 also confirms that ∆{‹v*T*›}≈ ∆{‹v*›‹T*›}, which

itself represents a considerable simplification (it has been separately verified that ∆{‹v*T*›}-

∆{‹v*›‹T*›} is generally small throughout the integration).

The key point arising from Fig. 2.2 is that the EMLIN term changes sign from positive to

negative in the stratosphere from the first to the second period and thereby accounts for the

switch in sign of ∆{‹v*T*›}. It will be shown that the switch in sign comes about because the

wave response reinforces the control state wave in the first period and attenuates the control state

wave in the second period. In other words, the wave response at first constructively interferes

with the control state wave and then destructively interferes with it. The EMNL term, on the other

hand, is smaller but remains positive for both periods, consistent with the upward-propagating

wave activity inferred from Figs. 2.1b-c. EMNL is small primarily because the wave response

amplitude is small compared to the control state wave (see Section 2.3.3). Thus, the change in

35

sign of the wave activity between the two periods, and hence the change in the sign of the

tendency of the zonal mean response, is controlled by linear interference effects.

FIG. 2.2. (a) Day 1-65 averaged ensemble and zonal mean total wave heat flux response, ∆{‹v*T*›}. (b) the linear contribution, EMLIN, to ∆{‹v*›‹T*›}. (c) the nonlinear contribution, EMNL, to ∆{‹v*›‹T*›}. (d), (e) and (f) are as in (a), (b) and (c) but for days 66-92. The contour interval is 0.5 m K s-1. For panel (a), the Student’s t-test is computed using the time-averaged fields; the gray shading shows 95% significance. Deriving straightforward significance tests for the EMLIN and EMNL terms that are consistent with the t-test on ∆{‹v*T*›} has been difficult. Thus, significance shading is not included in panels (b), (c), (e) and (f), but that the main features are robust by subsampling the ensemble has been verified.

To more clearly demonstrate the interference effect, it is useful to compare the relative

phase of the wave response and the control state wave over time. The wave-1 component is the

36

focus, since this controls the overall character of the response. Figure 2.3 shows the time series

of the phase of the wave-1 wave GPH (i.e. the longitude of the positive wave GPH anomaly) for

the control state wave, ‹Z*c›, and for the ensemble mean response, ∆‹Z*›, at 60°N and 50 hPa

(Fig. 2.3a) and at 60°N and 500 hPa (Fig. 2.3b). The time series with a 10-day running mean

applied is also shown. In the stratosphere, ∆‹Z*› starts roughly 105° out-of-phase with the control

state, ‹Z*c›, at the outset of the simulation and rapidly shifts 65° further out-of-phase (westward)

over the first few days. This results in destructive interference and a suppression of the wave

activity flux into the stratosphere up to approximately day 19 (not shown) and accounts for the

weak positive NAM feature observed during the first 20 days of the simulation (Fig. 2.1a). As

the simulation progresses through the first 65 days, ∆‹Z*› shifts eastward until roughly day 26 in

both the troposphere and the stratosphere and becomes relatively in-phase with ‹Z*c›. After day

26, the phase of ∆‹Z*› is more variable in the troposphere, reflecting greater synoptic variability

in this region; however, ∆‹Z*› clearly shifts westward in the stratosphere and becomes relatively

out-of-phase with ‹Z*c› during days 66-92.

It is noteworthy that the east-west shifting of the wave response was identified in F09 but

its significance was not appreciated in that study. The character of these linear interference

effects in the SGCM will be examined in the next subsection.

2.3.2 Comparison between AM2 and the SGCM

Dynamical ideas about the snow-NAM teleconnection are now tested by conducting idealized

perturbation simulations in the SGCM. The initial regional response to the imposed Siberian

lower tropospheric cooling in the SGCM (henceforth the “Siberian case”, which corresponds to

Simulation B in Table 2.1) bears some similarity to that in AM2. As in AM2, the direct response

to the forcing is a surface cooling localized over the forcing region. The ensemble mean surface

cooling over the forcing region stabilizes at approximately -5.5 K by day 30. This is

approximately two degrees cooler than the surface cooling observed in the F09 simulations but

this discrepancy is not qualitatively important for this analysis (see Section 2.3.4). Over the first

several weeks, a local surface high in sea-level pressure (SLP) and a deep upper-level low in

GPH extending to the upper troposphere develop over the forcing region in the SGCM

37

simulation (not shown), which are broadly consistent with F09. Despite these similarities with

AM2, the hemispheric zonal mean response is significantly different.

FIG. 2.3. Time series of 60°N wave-1 phase in degrees for the control state wave, ‹Z*c›, (solid

line) and for the wave response, ∆‹Z*›, with (dotted line) and without (dashed line) a 10-day running mean applied at (a) 50 hPa and (b) 500 hPa. The gray shading indicates regions where ‹Z*

c› and ∆‹Z*› are out of phase.

Figure 2.4a shows the ensemble-mean longitude-level cross-section of ∆‹Z*› at 60°N

averaged over days 1-22. The SGCM Rossby wave response is quite coherent and has a more

38

pronounced westward tilt with height than the AM2 wave response (Fig. 2.1b). Seeing this, it

was at first anticipated that a similar or stronger negative NAM response to the forcing in the

SGCM would be observed. But for this case, a negative-signed zonal mean GPH response is

obtained, which corresponds to a positive NAM response. The zonal mean GPH, {∆‹Z›}, for

days 1-22 is shown in Fig. 2.4b. This positive NAM response develops early and retains the

same sign throughout the 100-day simulation, but is not significant beyond day 22 (not shown).

Unlike the AM2 case, there are no reversals of sign of the GPH tendency. The positive NAM

response is inconsistent with the dominant negative NAM response in AM2, with observational

results (Cohen and Entekahbi 1999; Cohen et al. 2007; Hardiman et al. 2008) and with other

modeling studies (Gong et al. 2003).

Besides the difference in sign, there are other quantitative differences between the AM2

and SGCM response. First, the SGCM stratospheric NAM response is relatively confined to the

mid-to-upper stratosphere whereas the AM2 stratospheric NAM response extends into the lower

stratosphere (F09). The reduced SGCM response in the lower stratosphere is consistent with

generally weak stratosphere-troposphere coupling in the SGCM. Stratosphere-troposphere

coupling in this model is particularly sensitive to the choice of equilibrium temperature profile

and topographic configuration (Gerber and Polvani 2009; Chan and Plumb, 2009). Second, the

AM2 response, which remains significant up to days 60-90, is more persistent than the SGCM

response, which is only significant up to day 22. The cause of this behaviour is unclear, given

that the annular mode signals tend to be more persistent than observed in this model (Gerber et

al. 2008a). Related to the current study, in F09, a version of AM2 with enhanced stratospheric

resolution also showed a much less persistent response to snow forcing than the standard AM2

used here. It is beyond the scope of this study to further address the important differences

between the simplified and comprehensive GCMs, but it is important to note that there are

several potential dynamical controls, many of which were discussed in F09 and Hardiman et al.

(2008), that need to be considered.

39

FIG. 2.4. Day 1-22 averaged ensemble mean response to Siberian lower tropospheric cooling in the SGCM. (a) wave response (∆‹Z*›) at 60°N and (b) zonal mean GPH response ({∆‹Z ›}) (c) and (d) are as in (a) and (b) but for the Pacific lower tropospheric cooling case. The solid contours correspond to positive values, the dashed contours correspond to negative values and the gray shading shows 95% significance. The contour interval is 5 m.

The main aim in the remainder of this analysis will be to answer the more focused

question of what controls the sign of the NAM response; this can be understood more clearly

when the role of linear interference is investigated in the SGCM simulations. It is instructive to

compare the wave structure of the response and the control state in detail to assess linear

interference effects. Figure 2.5a again plots the day 1-22 ∆‹Z*› at 60°N (as in Fig. 2.4a) but this

40

FIG. 2.5. Day 1-22 averaged ensemble mean wave response (∆‹Z*›; black contours) at 60°N to Siberian lower tropospheric cooling in the SGCM superimposed on the control state wave at 60°N (‹Z*

c›; shading) for (a) all-waves, (b) wave-1, and (c) wave-2. The contour interval is 5 m.

time superimposed on the control state wave, ‹Z*c›. Figures 2.5b-c show the wave-1 and wave-2

components of both fields. It is found that the log-pressure weighted spatial correlation of ∆‹Z*›

and ‹Z*c› is -0.36 for the all-wave response (indicating destructive interference), -0.55 for the

wave-1 component (destructive interference) and 0.84 for the wave-2 component (constructive

41

interference). Although the magnitude of the correlation for the wave-1 response is smaller than

for the wave-2 response, the sign of the all-wave correlation is determined by the larger-

amplitude wave-1 component, particularly in the stratosphere.

As will be demonstrated in the discussion of Fig. 2.7 below, in the SGCM simulations, as

for the AM2 case, the term ∆{‹v*›‹T*›} (EM) dominates the wave driving response and the

EMLIN contribution is larger than the EMNL contribution. Following from the linear interference

effects illustrated in Fig. 2.5, the EMLIN term is negative throughout the entire stratosphere and

most of the troposphere during days 1-22 for the “Siberian case”, while the westward-tilting

structure of the waves indicates that the EMNL term is again positive (see also Fig. 2.9d).

Note that the structure of ∆‹Z*› in the “Siberian case” looks strikingly similar to that in

the second period of the AM2 simulation, that is, the day 1-22 SGCM response is westward-

shifted, especially in the stratosphere, relative to the day 1-65 AM2 response. The phase of ∆‹Z*›

in the “Siberian case” in the SGCM shifts westward and out-of-phase with ‹Z*c› over the first

few days of the run. This also occurs in AM2, albeit more quickly; but unlike the AM2 case (Fig.

2.3) the SGCM wave response does not then shift eastward and in-phase with ‹Z*c› (not shown).

As is the case for other differences between the SGCM and AM2 simulations, it is not simple to

explain why the transient ensemble mean wave response differs so significantly between the two

simulations (see Section 2.4). But given the different wave responses, it is now understood how

differences in phasing between ∆‹Z*› and ‹Z*c› exert correspondingly different linear interference

effects on wave driving, and, hence, why opposite sign NAM responses are obtained in the two

simulations.

2.3.3 Sensitivity to Position and Sign of the Forcing in the SGCM

In order to further probe the linear interference effect, eleven additional forcing simulations with

the SGCM are conducted in which the forcing is shifted zonally at intervals of 30° longitude

(that is, λ1 and λ2 in Eqn. (2.3) are increased in 30° increments; Table 2.1 Simulations A-L). For

these experiments, this forcing should no longer be interpreted as an idealized snow forcing, but

rather more generally as a low-level cooling.

42

One of these additional simulations, a simulation with the forcing location given by λ1 =

150°E and λ2 = 230°E (henceforth the “Pacific case”, Simulation E in Table 2.1), illustrates the

response to the forcing when the wave response constructively interferes with the control state

wave. The wave response, ∆‹Z*›, for this case is shown in Fig. 2.4c and the {∆‹Z›} response is

shown in Fig. 2.4d. In this case, the all-wave spatial correlation between ∆‹Z*› and ‹Z*c› is 0.58

and is determined by a very strong positive correlation in wave-1 of 0.96. This positive phasing

of the wave fields translates into increased vertical wave activity propagation into the

stratosphere. As in the cases discussed so far, the linear wave heat flux term, EMLIN, dominates

but unlike the “Siberian case” (Simulation B) it is positive in the stratosphere. Correspondingly,

wave activity is absorbed in the stratosphere, leading to a negative NAM response in the polar

stratosphere, illustrated by the positive zonal-mean GPH response poleward of 60°N and above

10 hPa (Fig. 2.4d). While the coupling of the response into the troposphere for both cases is

weak, the negative GPH response in the troposphere is weaker in the “Pacific case” (Simulation

E, Fig. 2.4d) than in the “Siberian case” (Simulation B, Fig. 2.4b). Thus, the NAM signature of

the “Pacific case” minus that of the “Siberian case” is consistently positive in the troposphere

and stratosphere as shown in Fig. 2.6.

Figure 2.7 summarizes the results of the sensitivity study on the location of the forcing.

Figure 2.7a shows that for the twelve sensitivity simulations, the polar cap-averaged thickness

response, ∆‹Zt›, from 10-1 hPa for days 10-40 is negatively correlated with the zonal mean

TOTAL E-P flux divergence response for a 40-80°N and 10-1 hPa box for days 1-22 (variance

explained: 88%)2. The 10-1 hPa thickness response is used to highlight the changes in this layer,

because it is in the mid-to-upper stratosphere where the response is most sensitive to the change

in the forcing in the SGCM (as seen in Fig. 2.4). Figure 2.7b shows that the E-P flux divergence

response is itself negatively correlated with the EM wave heat flux response at 10 hPa,

∆{‹v*›‹T*›} (variance explained: 81%). Thus, as expected, the total wave driving in the

2 Although the decomposition TOTAL = EM + FL = EMLIN + EMNL + FL was derived for the meridional wave heat flux, the decomposition generalizes simply to the total E-P Flux.

43

FIG. 2.6. Difference between ∆‹Zpcap› as a function of time for the “Pacific Case” and the “Siberian Case”. Contour interval is 5 m. Solid contours indicate positive and negative values. Gray shading indicates regions where the difference between the two cases is significant at the 95% level.

stratosphere is dominated by the vertical wave activity flux (Newman et al., 2001) and this is in

turn dominated by the EM term, as was found for the AM2 snow simulations. Figure 2.7c shows

that the EM term (as in Fig. 2.7b) is in turn positively correlated with the linear term, EMLIN,

(variance explained: 98%). This correlation is close to perfect and points to the importance of

linear regime dynamics in the interaction of surface forcings and the atmospheric circulation.

Finally, Fig. 2.7d shows that EMLIN (as in Fig. 2.7c) is positively correlated with the anomaly

correlation of the response, ∆‹Z*›, and the control state, ‹Z*c›, at 60°N for days 1-22 (as was

calculated in Fig. 2.5; variance explained: 89%). This correlation reflects the wave-1 anomaly

correlation which explains roughly 86% of the variance in EMLIN. Figure 2.7d also illustrates the

relationship between forcing location (indicated by the labeled data points) and the wave-1

anomaly correlation. In general, forcing location is an excellent predictor of linear interference

effects in the SGCM simulations.

44

In conclusion, the phase of the wave-1 ensemble mean response relative to the phase of

the control state wave explains most of the wave driving response, and, hence, the NAM

response for this amplitude of forcing.

FIG. 2.7. Dependence of the SGCM response on forcing location (Simulations A-L in Table 2.1). (a) the TOTAL E-P flux divergence response averaged over 40-80°N, 10-1 hPa, and days 1-22 versus the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days 10-40. (b) ∆{‹v*›‹T*›} (EM) at 10 hPa, averaged over 40-80°N, and cumulative to day 22 versus the E-P flux divergence response. (c) the linear contribution, EMLIN, to EM, versus EM. (d) the all-wave (solid circles) and wave-1 (open circles) spatial correlation between ∆‹Z*› and ‹Z*

c› versus EMLIN.

45

The role of phasing may also be tested by switching the sign of the forcing, i.e. by

imposing a lower tropospheric heating instead of a lower tropospheric cooling in the SGCM. The

forcing centered at 70°E (λ1 = 30°E and λ2 = 110°E; Simulation A in Table 2.1 and Fig. 2.7d) is

chosen as the one for which the sign of the forcing is switched from a cooling to a heating; this

simulation is Simulation S in Table 2.1. In the cooling case, Simulation A, the forcing generates a

wave train that strongly destructively interferes with the control state wave: the all-wave and

wave-1 ∆‹Z*›-‹Z*c› anomaly correlations in this case are the most strongly negative in response to

cooling among the twelve simulations, and a strong positive NAM response results (Figs. 2.8a

and b). When the sign of the forcing is switched, an opposite-signed wave response that is in-

phase with ‹Z*c› is achieved, generating strong constructive interference, and a strong negative

NAM response (Figs. 2.8c and d). The corresponding anomaly correlations for the all-wave,

wave-1 and wave-2 wave GPH fields are -0.71, -0.88 and 0.59 for the cooling and 0.59, 0.83 and

-0.75 for the heating. While linear effects explain most of the responses, it is noted that there is

some nonlinearity acting in the wave response: the anomaly correlations are not equal and

opposite, and there is a slight but statistically significant eastward shift of the response wave in

the heating case relative to the cooling case.

Collectively, these sensitivity simulations demonstrate the importance of the phasing of

the wave response with the control state wave in generating a teleconnected response to a

localized lower tropospheric forcing. The high correlations between dynamical fields illustrated

in Fig. 2.7 provide quantitative support for this feature of stratosphere-troposphere interaction.

2.3.4 Sensitivity to Forcing Strength in the SGCM

The dominance of the linear term, EMLIN, in EM, ∆{‹v*›‹T*›}, is contingent on the fact

that the wave response generated by the forcing is relatively small. If the magnitude of the

forcing is increased, a regime is entered where the EMNL term dominates. Starting with the

standard “Siberian case” discussed in Sections 2.2, 2.3.2 and 2.3.3, in which destructive

interference is acting, the sensitivity to the strength of the forcing is tested by decreasing and

increasing Qo in Eqn. (2.3) in the SGCM simulations.

46

FIG. 2.8. (a) and (c) as in Fig. 2.5b but for Simulation A and Simulation S, respectively. (b) and (d) as in Fig. 4b but for Simulation A and Simulation S, respectively; gray shading shows 95% significance.

Figure 2.9 shows the response in polar cap-averaged stratospheric thickness (∆‹Zt› from

10-1 hPa for days 10-40), the EM wave heat flux response at 10 hPa averaged over 40-80°N, the

linear contribution EMLIN, and the nonlinear contribution, EMNL as a function of forcing strength

for seven simulations. The forcing strength has been normalized such that a forcing strength of 1

47

corresponds to the standard case previously discussed. Figure 2.9a shows, interestingly, that the

circulation response is not monotonic in the strength of the forcing in this case, but in fact is a

minimum at the standard forcing. This also holds true for the EM wave heat flux response

although the minimum is weaker (Fig. 2.9b). The reason is that the linear response destructively

interferes with the control state wave, so that the EMLIN term is negative and decreases more or

less linearly with the forcing (Fig. 2.9c), while the EMNL term is consistently positive and

increases more rapidly than linearly with the forcing (Fig. 2.9d). If the wave response is linear in

the forcing, then it is expected that that the EMLIN and EMNL terms should be, respectively, linear

and quadratic in the forcing. Indeed, an excellent fit is found when a linear dependence on the

forcing strength is fit to the EMLIN term by linear regression (94% variance explained for a linear

fit passing through the origin; solid curve in Fig. 2.9c) and a quadratic dependence on the forcing

strength is fit to the EMNL term, again by linear regression (99% variance explained for a

quadratic fit passing through the origin; solid curve in Fig. 2.9d). EMNL dominates EM once the

forcing strength is sufficiently large, which occurs at roughly a doubling of the standard forcing

strength.

2.4 Sensitivity to Polar Vortex Strength

Section 2.3 was published in Smith et al. (2010). The following section presents additional

results related to the SGCM simulations that are not published to date. These results complement

the findings of Smith et al. (2010) and attempt to address some of the outstanding questions

related to the atmospheric response to a prescribed surface cooling.

The climatological zonal mean wind field can influence both the climatological stationary

wave field and the wave response to an external forcing (Charney and Drazin 1961; Simmons

1974; Wang and Kushner, in press). For example, F09a demonstrate that although the response

to Siberian snow forcing in a high-top, stratosphere-resolving version of AM2 is qualitatively

similar to the response in the standard low-top AM2, there are clear quantitative differences

between the responses, which appear to be related to differences in the strength of the lower

stratospheric winds in the two GCMs. With this in mind, Section 2.4 investigates to what extent

48

the findings of Section 2.3 generalize to SGCM configurations with weaker or stronger polar

vortices. As in Section 2.3.3, the focus is on the sensitivity of the NAM response to the

longitudinal position of the cooling.

FIG. 2.9. Dependence of the SGCM response on forcing strength (Simulations M, B, N-R in Table 2.1). (a) the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days 10-40. (b) day 1-22 ∆{‹v*›‹T*›} (EM) at 10 hPa, averaged over 40-80°N. (c) linear contribution, EMLIN, to EM. (d) the nonlinear contribution, EMNL, to EM, as a function of forcing strength. The forcing strength has been normalized such that a forcing strength of 1 corresponds to the forcing discussed in Section 2.2. The solid lines in (c) and (d) show the linear and quadratic fits passing through the origin, respectively.

49

FIG. 2.10. Control state zonal mean zonal wind, uc, for the (a) γ = 1, (b) γ = 2, and (c) γ = 3 K km-

1 SGCM configurations. Positive and negative contours are red and blue, respectively. Contour interval is 5 m s-1. Control state stationary wave field, Z*

c, at 60°N for the (d) γ = 1, (e) γ = 2, and (f) γ = 3 K km-1 SGCM configurations. Wave response, ∆Z*, at 60°N for the “Siberian Case” for the (d) γ = 1, (e) γ = 2, and (f) γ = 3 K km-1 SGCM configurations. Note the difference in colour bar scale for panels (d)-(f) and (g)-(i).

In order to examine the effect of stratospheric conditions on the sensitivity of the

response to surface cooling on forcing location, two additional suites of SGCM simulations are

conducted in which the position of the forcing is shifted zonally (forcings A-L in Table 2.1). The

two additional suites differ from the one presented in Section 2.3 in that the SGCM is configured

with γ = 1 K km-1 and γ = 3 K km-1 (hereafter γ = 1 and γ = 3) which lead to weaker and stronger

polar vortices than the γ = 2 K km-1 (hereafter γ = 2) case. Figures 2.10a-c show the control state

50

zonal mean zonal wind for the γ = 1, γ = 2 and γ = 3 configurations, respectively. As γ increases,

the tropospheric zonal winds are roughly constant, yet the polar stratospheric winds intensify.

The rest of Fig. 2.10 will be discussed further below.

With respect to the sensitivity of the response to forcing location, Figures 2.11a-c show

similar diagnostics to those shown in Figure 2.7a-c but extended to include the γ = 1 and γ = 3

suite and demonstrate that similar relationships to those shown in Figs. 2.7a-c hold for the

weaker and stronger polar vortex configurations. In particular, the relationship between EMLIN

and EM lies close to the one-to-one line for each suite (Fig. 2.7c) and the variance in EM

explained by EMLIN across the three suites is 98%. In other words, the ratio of the covariance

between EMLIN to EM to the variance of EM across Simulations A-L for all three suites is 0.91.

But an important difference does emerge for the new simulations: recall that for the γ = 2 suite

the wave-1 anomaly correlations between the wave response and the control state wave at 60ºN

explain 86% of the variance in EMLIN (see Fig. 2.7d). This simple relationship between the

phasing of the wave-1 response and the control state wave-1 and EMLIN found for the γ = 2 suite,

however, does not exist for either the weaker or stronger polar vortex suites. For the γ = 1 and γ =

3 suites, the wave-1 anomaly correlations only explain 1% and 11% of the variance in EMLIN,

respectively.

The lack of a simple wave-1 phasing relationship to account for differences in EMLIN due

to forcing location suggests that the contributions to EMLIN from waves of smaller zonal scale

but larger amplitude become important for both the weaker and stronger vortex configurations.

Examination of the control state wave field, Z*c, for the three polar vortex configurations reveals

that wave-2 is of relatively larger amplitude for the γ = 1 and γ = 3 configurations. Figures 2.10d-

f show the control state wave field, Z*c, at 60°N for the γ = 1, γ = 2 and γ = 3 configurations,

respectively. The ratio of wave-1 to wave-2 amplitude at 60ºN and 10 hPa is 0.67 for γ = 1, 1.4

for γ = 2 and 0.64 for γ = 3. In both the weaker and stronger vortex configurations, the amplitude

of the wave-2 component of the control state wave in the stratosphere is larger than the wave-1

component. Wave-2 also appears to be of relatively larger amplitude in the response to surface

cooling, ∆Z*, in the weaker and stronger vortex configurations (Figs. 2.10g-i): the ratio of the

mean wave-1 to wave-2 amplitude response at 60ºN and 10 hPa is 1.61, 1.73 and 0.96 for γ = 1, γ

= 2 and γ = 3, respectively.

51

To show what these differences imply for the wave activity flux response, Table 2.2

shows the ratio of the covariance between EMLIN and either the wave-1 or wave-2 component of

EMLIN to the variance of EMLIN across Simulations A-L for each suite. For the γ = 2 suite, both

the wave-1 amplitude of ∆Z* and Z*c are much larger than the wave-2 counterparts resulting in

the wave-1 component determining the overall sign and strength of EMLIN. However, for the γ =

1 and γ = 3 suites, the wave-2 contribution to EMLIN is larger than the wave-1 contribution in at

least half the simulations for both the γ = 1 and γ = 3 suites, suggesting that the response to

surface cooling in these cases is more localized to the forcing region. In the γ = 1 and γ = 3

suites, EMLIN depends in a more complex way upon to the sum of its wave-1 and wave-2

components: the amplitude and the phasing of both components are important in these suites. For

example, the wave-1 and wave-2 phasing is often of opposite sign, which complicates the

interpretation of EMLIN in these two additional suites of simulations. The variance of EMLIN

explained by the sum of its wave-1 and wave-2 components is shown in Table 2.2 for each

SGCM suite individually and in Fig. 2.10d for all three suites. Despite the complexity, wave-1

and wave-2 together account for most of the variance in EMLIN, i.e. the bottom row of Table 2.2

is near to one for all three suites (the higher wavenumbers are unimportant for the polar lower

stratosphere as expected by Charney and Drazin (1961)).

TABLE 2.2: Ratio of covariance between EMLIN and wave components of EMLIN to the variance of EMLIN across Simulations A-L for each SGCM Suite. SGCM Suite γ = 1 γ = 2 γ = 3 cov(EMLIN, Wave-1 EMLIN)/var(EMLIN) 0.1 1.13 0.32 cov(EMLIN, Wave-2 EMLIN)/var(EMLIN) 0.90 -0.13 0.63 cov(EMLIN, Wave-1 + Wave-2 EMLIN)/var(EMLIN) 0.99 1.02 0.95

Turning to the nonlinear effects, the ratio of the covariance between EMNL and EM to the

variance of EM across Simulations A-L for all three suites is 0.09 (compared to a value of 0.91

for the ratio of the covariance between EMLIN and EM to the variance of EM). The value of this

ratio is very similar for each suite, highlighting the fact that EMLIN is the dominant contribution

to EM and that EMNL is largely unimportant.

52

In summary, Section 2.4 illustrates that EMLIN remains the dominant contribution to the

overall meridional wave heat flux response to imposed surface cooling when the SGCM

stratospheric polar vortex configuration is altered. However, the simple relationship between

EMLIN and the phasing between the wave-1 response and control state wave that was established

in Section 2.3.3 breaks down for both the weaker (γ = 1) and stronger (γ = 3) vortex

configurations. The breakdown of this relationship is related to the fact that contributions from

both wave-1 and wave-2 to EMLIN become important for the weaker and stronger vortex

configurations due to the increase in the relative amplitude of wave-2 in both the control state

wave and wave responses in the γ = 1 and γ = 3 suites. It is dynamically interesting that the γ = 2

case borders regimes with quite different behaviour. This finding was not anticipated. In a

limited sense, it appears that the γ = 2 case is qualitatively more “realistic” because the

dominance of wave-1 phasing is analogous to the observational results of Chapters 3 and 4. More

importantly, however, Section 2.4 shows that the NAM response to surface cooling is

surprisingly sensitive to subtle features of the stationary wave field and the wave responses

represented in Figs. 2.10d-i.

2.5 Conclusions

Chapter 2 illustrates ways in which some of the complexities of the zonal mean

atmospheric circulation response to extratropical surface forcing can be explained using a linear

interference analysis. Motivated by the observed connections between snow variability and

annular mode anomalies, this analysis has been applied to comprehensive and simplified GCM

simulations. The major conclusion is that the effects of linear interference between the wave

response and the control state wave, which are found by comparing wave phase and calculating

spatial correlations, control the zonal mean Northern Annular Mode (NAM) response. The

expectation is that linear interference effects dominate provided the wave response is small

compared to the climatological stationary wave.

53

FIG. 2.11. Dependence of the SGCM response on forcing location (Simulations A-L in Table 2.1) for three polar vortex configurations, γ = 1 (green), 2 (red, shown previously in Fig. 2.7) and 3 (blue). (a) the TOTAL E-P flux divergence response averaged over 40-80°N, 10-1 hPa, and days 1-22 versus the 10-1 hPa thickness response, ∆‹Zt›, averaged over the polar cap and over days 10-40. (b) ∆{‹v*›‹T*›} (EM) at 10 hPa, averaged over 40-80°N, and cumulative to day 22 versus the E-P flux divergence response. (c) the linear contribution, EMLIN, to EM, versus EM. (d) sum of wave-1 and wave-2 components of EMLIN versus EMLIN. Black lines in (c) and (d) are the 1-1 line.

54

To begin, it was shown that the comprehensive GCM, GFDL AM2, exhibits a transient

zonal mean response that consists of an initial weak positive NAM tendency in the first 20 days,

followed by a negative NAM tendency up to about day 65, and then terminating with a positive

NAM tendency. At each stage, while the surface cooling consistently generates an upward

propagating Rossby wave train into the stratosphere, the extratropical stratospheric tendencies

are driven by a wave activity anomaly that can be diagnosed in terms of the constructive or

destructive interference of the planetary scale wave response with the control state wave (Fig.

2.3). This effect was isolated by decomposing the meridional wave heat flux response into parts

that are linear and nonlinear in the ensemble mean wave response.

In Section 2.3.3, the linear interference effect was illustrated more broadly by varying the

location, strength and sign of the forcing in the SGCM. In all simulations, the surface cooling

consistently generates an upward-propagating Rossby wave (Figs. 2.1b-c and Figs. 2.4a and c,

Figs. 2.8a and c, and Fig. 2.9d all highlight this point). The phasing of the wave response with

the control state wave is the key determinant of the nature of the zonal mean response to the

forcing (Figs. 2.1a and Figs. 2.4b and d). The interference effect, whether constructive or

destructive, can be tuned by shifting the forcing location so that the response can become more

or less phase matched with the control state wave.

In addition, it was shown that the importance of linear interference, and, hence, the zonal

mean extratropical response to this surface perturbation, depends on the forcing strength (Fig.

2.9). In nature, the amplitude of interannual lower tropospheric diabatic heating anomalies likely

plays a role in determining the importance of linear interference in externally forced

stratosphere-troposphere interactions. It is important to note that as forcing strength increases,

the shift into the nonlinear regime likely occurs at a weaker forcing strength in the SGCM than it

would in nature due to the weak control state stationary waves in the SGCM. Nevertheless, it is

interesting that the forcing, while somewhat larger than what may be found in nature, is not

completely unrealistic in terms of the surface temperatures and magnitude of the regional

response, and that in the SGCM sensitivity study, the forcing generated a response that was close

to the boundary of a regime where nonlinear effects came into play. Thus, it is expected that

these considerations will be important in other modeling contexts where relatively strong

forcings are used to elicit strong signals in the extratropics. But even this large-amplitude regime

55

might be understood in a weakly nonlinear theoretical setting. For example, the phase evolution

of the ensemble mean wave responses in the SGCM is not sensitive to the strength of the forcing

during days 1-22 (not shown). This emphasizes that a transient linear model might provide

accurate predictions of the NAM response to surface forcing.

It has also been shown that the results are somewhat sensitive to the strength of the polar

vortex (Section 2.4). Although the main conclusion, i.e., the dominant role of linear interference

in the zonal-mean response to the imposed surface forcing, is robust across the three SGCM

polar vortex configurations used, the simple phasing relationship between the wave-1

components of the background climatological wave and the wave response is not. The amplitude

of the waves becomes an important factor when the background climatology supports vertically-

propagating stationary waves with relatively large amplitudes but of smaller zonal scale.

In this setting, the role of the FL term in the time mean, ensemble mean wave heat flux

decomposition turns out to be minor. The FL term represents the ensemble-mean wave fluxes

driven by waves that are independent among realizations, typically high-frequency waves like

synoptic waves in the troposphere and stratospheric transients. In this problem, the direct wave

heat flux response of such higher-frequency waves is of secondary importance to the NAM

response. The minor role of FL represents a potential simplification of the linear dynamics

needed to obtain the main features of the high-latitude zonal mean response to surface forcing.

This does not imply that high-frequency waves are unimportant in this class of problems. For

example, high-frequency waves are indirectly involved in generating and maintaining low-

frequency anomalies, such as the ensemble mean wave response to surface forcing discussed in

this study (Branstator 1992; Sobolowski et al. 2010).

By using the SGCM framework, the ability to generate large ensembles and to isolate

dynamical mechanisms is enhanced. Much of the motivation to study linear interference effects

initially came from experimentation with the simplified GCM, because the simplified GCM

yielded a NAM response of opposite sign to what was expected (Section 2.3.2). As has been

shown, the linear interference effect is also clearly operating in the comprehensive GCM

simulations, but it first emerged most starkly in the simple GCM. This highlights the value of

56

looking at examples across the model hierarchy (Held 2005) when trying to understand complex

dynamics of the kind investigated here.

Nevertheless, the SGCM framework has its limitations. For example, it was difficult to

compare the SGCM and AM2 simulations directly, because: (i) the SGCM’s control state wave

is weaker and slightly eastward shifted relative to that of AM2; (ii) the transient evolution of the

wave response is very different in the two models; (iii) stratosphere-troposphere interactions

appear to be rather weak in the SGCM; and, (iv) the NAM response in the SGCM simulations

was not statistically significant beyond day 20-30. Despite the fact that the linear interference

effects are at work in both models, many aspects of the discrepancies of the response in the two

frameworks remain unexplained. The above results nevertheless suggest that linear modeling

approaches will be useful in helping constrain some aspects of the extratropical response

problem.

57

Chapter 3

The Role of Linear Interference in Northern Annular Mode

Variability Associated with Eurasian Snow Cover Extent

3.1 Introduction

As discussed in Chapter 1, it is well established that the observational record reveals a

statistically significant relationship between autumn Eurasian snow cover anomalies and

Northern Hemisphere wintertime extratropical circulation anomalies (Watanabe and Nitta 1998,

Cohen and Entekhabi 1999, Cohen et al. 2007). Previous modeling work, including the work

presented in Chapter 2, has helped improve the dynamical understanding of this snow-circulation

connection (Gong et al. 2002, 2003; Fletcher et al. 2009a; Orsolini and Kvamsto 2009; Smith et

al. 2010; Allen and Zender 2010). However, a complete understanding is lacking, and an

important question remains regarding the timing: what accounts for the multiple-week lag

between observed Eurasian snow cover anomalies in October and the associated peak wave

activity flux in December?

At the surface, anomalously large autumn snow cover extent in Eurasia in October leads

to colder local temperatures in the subsequent winter by enhancing cold air intrusions (Foster et

al. 1983; Vavrus 2007). The shallow layer of air overlying snow cover is colder than the

surrounding air, primarily due to the increase in surface albedo (Wagner 1973; Mote 2008).

When high-latitude Eurasian October snow cover is early and more extensive, anomalously cold

surface temperatures enhance the formation of the Siberian high. Anticyclonic flow advects cold

air southward, cooling the continent in fall and winter. Early development of the Siberian high

prevents incursions of maritime air in autumn and topographic features in the region limit warm

air advection from the south. By December the Siberian high is strong enough to prevent interior

temperatures from going above freezing, keeping the snow cover relatively constant throughout

the winter months. But this surface circulation response does not explain the hemispheric-scale

and vertically deep connection between October snow and the wintertime Northern Annular

Mode (NAM; Thompson and Wallace 1998). The primary aim in Chapter 3 is to address this

58

outstanding issue with an observational analysis that focuses on the structure and phase of the

Rossby waves associated with October Eurasian snow cover anomalies. This analysis

complements and expands on the modeling and linear interference analysis presented in Chapter

2.

Using multiple linear regression, Garfinkel et al. (2010) demonstrate that the influence of

October Eurasian snow cover on the polar stratosphere is in part associated with specific

tropospheric wave patterns in December, including an eastern European high and a northwestern

Pacific low. These wave patterns amplify the climatological stationary wave field and, via linear

interference, the wave activity flux into the stratosphere. While this result is consistent with

earlier studies (Cohen et al., 2007; Hardiman et al. 2008; Orsolini and Kvamsto 2009), the

question of the multiple-week lag between October snow cover anomalies and the wintertime

NAM remains.

After describing the methods and data used (Section 3.2), the potential importance of

linear interference effects in the reanalysis data is first established by showing that coupled

stratosphere-troposphere NAM variability is generally controlled by terms in the wave activity

flux that are linear in the interannual wave anomalies (Section 3.3.1). It is then shown that the

wave anomaly associated with the observed October Eurasian snow cover that develops in fall is

initially out of phase with the climatological wave and later moves into phase with the

climatological wave (Section 3.3.2). Thus, the delay in stratospheric wave activity flux can be

attributed to initially unfavorable interference conditions between the Rossby wave train

associated with the snow cover anomalies and the climatological stationary wave. Although the

reasons for the phase shift remain unclear, this analysis highlights the key role of linear

interference in contributing to polar stratospheric variability. In Section 3.3.3, we show how this

diagnostic approach applies to case studies of individual seasons. In particular, we present a case

study of the strong negative NAM events of fall-winter 2009 (Cohen et al. 2010) in the context

of linear interference.

The secondary aim in this Chapter is to revisit the issue of the inability of current climate

models to capture the observed snow-circulation connection (Hardiman et al. 2008). Hardiman et

al. (2008) find that the suite of Coupled Model Intercomparison Project 3 (CMIP3) models does

59

not show the NAM-like correlation between October snow cover and the wintertime circulation.

Hardiman et al. (2008) propose a variety of reasons for this, for example related to the

longitudinal scale of the anomalous waves associated with October Eurasian snow cover

anomalies. As in observations, in GCMs the linear interference effects dominate wave-driven

NAM variability; but it is found that the representation of the linear interference effect coherent

with snow is not accurately captured in the models, contributing to the unrealistic behavior

(Section 3.3.4).

3.2 Methods

The relationship between observed Eurasian snow cover and the atmospheric circulation for the

September to February season for the years 1972-2009 is analyzed. Meteorological observations

are derived from the daily averaged NCEP/NCAR reanalysis dataset (Kalnay et al. 1996). The

October Eurasian snow index, OCTSNW, is a standardized anomaly index for snow cover extent

over Eurasia in October generated using the Rutgers Global Snow Lab (GSL) monthly Eurasian

snow extent product (Robinson et al. 1993; http://climate.rutgers.edu/snowcover). The Rutgers

GSL Eurasian snow extent product is based on the National Oceanic and Atmospheric

Administration’s (NOAA) satellite-derived weekly snow cover product and is considered of

good quality from 1972 onwards. The primary difference between the Rutgers GSL and the

NOAA snow cover products is the definition of the land mask which is considered more accurate

in the Rutgers GSL product. In addition, the coupled ocean-atmosphere twentieth century

(20C3M) Coupled Model Intercomparison Project 3 (CMIP3) simulations are used, with

corresponding radiative forcing, (http://www.pcmdi.llnl.gov/projects/cmip/index.php). The

length of these simulations varies from 100 years to 150 years. Linear trends have been removed

from all time series. Correlation and regression analysis is conducted between the annual

OCTSNW index and various atmospheric fields (Wilks, 2006) through the daily evolution of the

fall to winter season. Similar notation to that of Chapter 2 is used. The atmospheric fields of

interest are the geopotential height (GPH) area-averaged over the polar cap bounded by 60°N,

denoted Zpcap, which corresponds to the NAM (Cohen et al. 2002; Baldwin and Thompson

2009); the wave GPH at 60°N, Z* (where the superscript asterisk indicates the deviation from the

60

zonal mean); and the zonal mean meridional wave heat flux averaged from 40-80°N, {v*T*}

(braces indicating a zonal mean), which corresponds to the vertical component of the wave

activity flux.

In Chapter 2 a decomposition of the wave activity flux response was developed that

distinguished terms that were linear and nonlinear in the forced wave response to surface diabatic

cooling. An analogous decomposition is employed for the interannual variability of the wave

activity flux. Nishii et al. (2009) employed this decomposition to investigate the 2005-2006

stratospheric sudden warming (SSW) and the 2002 Southern Hemisphere SSW. Nishii et al.

(2010) have also recently applied this decomposition to investigate the relationship between NH

vortex intensifications and blocking. For a given day during year, j, one may define

v*j = v*

c + v*′j and T*j = T*

c + T*′j , (3.1)

where the subscript c indicates the climatological time mean and the prime indicates the

deviation from the climatological time mean, i.e., the anomaly. The climatological mean wave

heat flux in this notation is

{v*T*}c = {v*cT*

c} + {v*'T*'}c ,

and the anomalous wave heat flux on a given calendar day in year j, {v*T*}'j = {v*jT*

j}', is

{v*jT*

j}' = {v*jT*

j} - {v*jT*

j}c

= {v*′jT*′j} + {v*′jT*c} + {v*

cT*′j} + {v*cT*

c} - {v*jT*

j}c

= {v*′jT*′j} + {v*′jT*c} + {v*

cT*′j} + {v*cT*

c} - {v*cT*

c} - {v*′j T*′j}c

= NONLIN + LIN, (3.2)

where

NONLIN = {v*′jT*′j} - {v*′j T*′j}c = {v*′jT*′j}′ and LIN = {v*′jT*c} + {v*

cT*′j} . (3.3)

This decomposition highlights two terms that capture the interannual variability of {v*T*}: a term

NONLIN that is quadratic in the wave anomaly represented by v*′j and T*′j, and a term LIN that

61

consists of terms linear in the wave anomaly. Locally, i.e. prior to zonal averaging, the LIN term

is expected to dominate if the amplitude of the wave anomaly is small compared to the

climatological wave. Under the zonal average, the sign and amplitude of the LIN term will

depend in part on the degree of constructive or destructive interference between the

climatological wave and the anomalous wave (Nishii et al. 2009; Garfinkel et al. 2010, Smith et

al. 2010). The NONLIN term describes the component of the interannual variability of the total

wave activity flux intrinsic to the wave anomalies themselves.

3.3 Results

3.3.1 Linear Interference Effects in Interannual Variability of Wave Activity.

Before the {v*T*} decomposition presented in Section 3.2 is used to examine the relationship

between observed October Eurasian snow cover and the NAM, the relative contributions of the

LIN and NONLIN terms to the interannual variability of {v*T*} in the climatology is first

examined using the NCEP-NCAR reanalysis data. The month of primary interest is December as

this is the month when heat flux anomalies are most strongly correlated with October Eurasian

snow cover anomalies (Cohen et al. 2007; Hardiman et al. 2008). The temporal variance of

{v*T*} can be decomposed as follows,

var({v*T*}) = var(LIN + NONLIN) = var(LIN) + var(NONLIN) + 2cov(LIN,NONLIN), (3.4)

where var(∙) indicates the variance and cov(∙) indicates the covariance. Using daily averaged v*

and T* at 100 hPa, {v*T*}, LIN and NONLIN each day of December are calculated (taking the

meridional mean as described in Section 3.2). The December mean of this result is then

calculated, resulting in an annual time series. Finally, the variance and covariance of the annual

time series are calculated as measures of interannual variability of the wave activity flux. It is

found that var({v*T*})= 12.05 m2 K2 s-2, var(LIN) = 10.91 m2 K2 s-2, var(NONLIN) = 4.13 m2

K2 s-2, and 2cov(LIN,NONLIN) = -2.99 m2 K2 s-2. When this calculation is repeated using

December averaged v* and T* at 100 hPa as input, instead of daily data, 11.92, 10.99, 1.56, -0.64

are obtained for these terms. The LIN term, therefore, describes the majority of the interannual

62

variability in December {v*T*}. The total variance and the var(LIN) terms are similar whether

monthly or daily data are used, while the variance contributions connected with the NONLIN

terms are relatively large when daily data are used. This suggests that interannual variability in

lower stratospheric wave activity fluxes is dominated by variability in the low-frequency (quasi-

stationary) waves, while high-frequency waves dominate the NONLIN terms. This is analogous

to the behavior that has been found in the simulated wave activity flux response to surface

cooling in Chapter 2 (Smith et al. 2010). The LIN and NONLIN terms are slightly anticorrelated

(R = -0.22) from year to year but the contribution of this to the total interannual variability is

relatively small.

To quantify the relative importance of synoptic timescale versus lower frequency

variability in the interannual variability of {v*T*}, {v*T*} is decomposed further into

contributions from high and low frequency components. A low-pass filter in the form of an 11-

day running mean of v* and T* is performed and the high-pass filtered v* and T* are approximated

as

v*high

= v* - v*low, T*

high = T* - T*

low, (3.5)

where (∙)high is the high-frequency component of the waves and (∙)low is the low-frequency

component of the waves. Using Eqn. (3.5), the variance of {v*T*} then becomes,

var({v*T*}) = var({v*low T*

low} + {v*high T*

low}

+ {v*low T*

high} + {v*high T*

high}) (3.6a)

= var({v*low T*

low}) + var({v*high T*

low})

+ var({v*low T*

high}) + var({v*high T*

high}) + R, (3.6b)

where R represents the series of covariance terms in the expansion of Eqn. (3.6a). Table 3.1

shows that the variance of the time series of the December mean {v*T*} at 100 hPa is dominated

by the first term on the RHS of Eqn. (3.6b), the low-frequency component of {v*T*}. The

remaining variance terms are an order of magnitude smaller and the covariance terms

contributing to R are also relatively small (not shown). var({v*low T*

low}) can be further

decomposed into its LIN and NONLIN components (Table 3.1), as in Eqns. (3.2) and (3.3), and

63

it is found that the LIN term dominates. As expected, the interannual variability in the December

meridional wave heat flux for the high-pass waves, which is represented by var({v*high T*

high}), is

dominated by the NONLIN term (not shown), and this represents the meridional wave heat flux

associated with high-pass waves. However, var({v*high T*

high}) represents a relatively small

contribution (Table 3.1) to the 100 hPa December meridional wave heat flux.

TABLE 3.1: Variance decomposition for December mean {v*T*} at 100 hPa calculated using daily-averaged NCEP-NCAR data from 1972-2007. Bold lettering indicates the total variance and the two dominant contributions to the variance.

Variance Terms for Dec {v*T*}at 100 hPa Variance (m K s-1)2

var(v*T*) 12.05 var(v*

lowT*low) 10.14

var(v*highT*

high) 0.53

var(v*lowT*

high) 0.12

var(v*highT*

low) 0.17

COV 1.09

var(LINlow) 10.50 var(NONLINlow) 2.45

2cov(LINlow, NONLINlow) -2.81

This shows quantitatively that wintertime interannual variability in the vertical wave activity flux

into the lower stratosphere is dominated by the terms that are linear in the low-frequency wave

anomalies. The control of wintertime interannual variability in the wave activity flux by the low-

frequency component of the flow provides a useful simplification in the dynamical

understanding of NAM variability in the stratosphere and troposphere. Unfiltered daily data will

be used in the subsequent observational analysis, but these results justify the use of monthly data

as input into the analysis of model output from the CMIP3 archive (Section 3.3.4).

64

Next, the relative contributions of the LIN and NONLIN terms in transient NAM events

that propagate from the stratosphere to the troposphere are examined. As discussed in Chapter 1,

these events were first identified by Baldwin and Dunkerton (2001) and are relevant to the

discussion of the snow-NAM teleconnection. Polvani and Waugh (2004) make the point that

these events are initiated by wave activity flux anomalies propagating into the stratosphere, and

construct time-lag composites of the NAM based on the occurrence of high- or low-wave activity

flux anomaly conditions. Following a similar method to Polvani and Waugh, composites for the

polar cap GPH anomalies based on anomalous 40-day cumulative average high- and low-wave

activity flux events are constructed. Dynamically, the wave activity flux drives a tendency in the

NAM on a multiple week timescale; Polvani and Waugh use the cumulative mean heat flux

instead of a centered running mean to produce a wave activity index that is temporally correlated

with the NAM at zero lag. Events from November-January are the events of interest (when we

observe high correlations between October Eurasian snow cover and December heat fluxes) and

composites are constructed on total heat flux anomalies, {v*T*}', that exceed a threshold of 0.5

standard deviations and that are separated by at least 20 days (Fig. 3.1). The top row of Fig. 3.1

shows the composite daily time-series of {v*T*}' (black line), LIN (red line), and NONLIN (blue

line) at 100 hPa for 22 high (left) and 15 low (right) total {v*T*}' events. For these early-to-mid-

winter events, the anomalous wave activity flux events primarily consist of the LIN term; the

NONLIN term is relatively small. The second row of Fig. 3.1 shows the composite of Zpcap or

NAM anomalies that correspond to the high and low {v*T*}' as in Baldwin and Dunkerton (2001)

and Polvani and Waugh (2004). Thus, the heat flux diagnostic confirms the result of Garfinkel et

al. (2010) and demonstrates that linear interference is not only implicated in stratospheric NAM

variability but also in NAM related stratosphere-troposphere coupling.

3.3.2 Linear Interference in the Snow-NAM link

As has been previously shown (Cohen et al. 2007; Hardiman et al. 2008), October Eurasian snow

cover is significantly and positively correlated with the vertical propagation of wave activity into

the stratosphere in December and with stratosphere-troposphere NAM events in subsequent

weeks. The basic snow-NAM connection is shown in Figure 3.2a, which shows the correlation

65

between Zpcap and OCTSNW for the years 1972-2009. For years with anomalously positive

OCTSNW, a deep NAM anomaly builds from the troposphere to the stratosphere starting in mid-

December, and propagates back downward into the troposphere in February. Note the

remarkable persistence of this NAM signal, which lasts to February based on an October

predictor.

FIG. 3.1. The first row plots the time evolution of the November-December-January composite mean of the 40-day cumulative mean total meridional wave heat flux ({v*T*}′; black curve) anomalies at 100 hPa and the corresponding LIN (red curve) and NONLIN (blue curve) components for 22 high (left) and 15 low (right) anomalous {v*T*} events. Solid sections of the heat flux curves indicate times when anomalies are different from zero at the level of 95% significance. The second row plots composites of the time evolution of the standardized anomaly polar cap GPH corresponding to these anomalous {v*T*} events as a function of pressure. The GPH contour interval is [0.25, 0.5, 1.0, 1.5], warm and cold shading are positive and negative contours, and the black contour indicates pressures and times for which anomalies are different from zero at the level of 95% significance.

The temporal evolution of geopotential coherent with OCTSNW resembles the

climatological variability illustrated in the wave activity flux composites in Fig. 3.1, and so the

66

main question is to determine how OCTSNW is consistently associated with positive wave

activity flux events in winter. The connection to wave activity is shown in Fig. 3.2b, which

shows the correlation of the 40-day cumulative mean meridional wave heat flux with OCTSNW.

There is a peak correlation in the lower stratosphere in January, corresponding to the peak warm

period in Fig. 3.2a. Figures 3.2c and 3.2d show correlations analogous to Fig. 3.2b but for the

LIN term and the NONLIN term, respectively. The correlation of the LIN term with OCTSNW

is positive and significant in the troposphere in December and in the stratosphere in January and

accounts for the significant positive correlation of {v*T*}' with OCTSNW in the stratosphere.

The NONLIN term is negatively correlated with OCTSNW in the troposphere in December and

February and is not significantly correlated with OCTSNW in the stratosphere. The LIN and

NONLIN terms associated with OCTSNW mostly cancel in the troposphere in December,

resulting in no significant correlation between the total wave activity flux anomaly and

OCTSNW. Figs. 3.2e and 3.2f are similar to Fig. 3.2c except that they show the wave-1 and 2

components of LIN, respectively. The main features of Fig. 3.2c can be attributed to these two

components of LIN; the positive correlations in the troposphere in December correspond to the

wave-2 LIN flux and in the stratosphere in January correspond to the wave-1 LIN flux. Figs.

3.2c-f show that the cancellation between the LIN and NONLIN terms in the troposphere is

primarily a cancellation between the wave-2 LIN flux and the NONLIN flux.

Since the LIN term explains most of the total wave activity flux correlation with the

OCTSNW, the climatological stationary wave field and the wave field associated with the snow

index must be constructively interfering prior to the peak wave activity flux in January. But the

question of main interest is why the LIN term is relatively small in the several weeks prior to

this. A reduced LIN term might be associated with a relatively weak amplitude wave anomaly

prior to December-January or with a linear interference effect, or a combination of the two.

Hardiman et al. (2008) show that OCTSNW is significantly correlated with an upward

propagating wave train in the extratropics starting in October. Thus, the nature of the interference

must be driving the lag between the snow anomalies and the wave activity fluxes. This

conclusion is confirmed by showing explicitly how the character of the interference evolves over

time.

67

FIG. 3.2. Correlations of OCTSNW with daily (a) polar cap GPH, (b) the 40-day cumulative mean total meridional wave heat flux averaged over 40-80°N, (c) the LIN component of (b), (d) the NONLIN component of (b), (e) the wave-1 component of (c), and (f) the wave-2 component of (c). Time-axis begins on October 10. Contour interval is 0.1, warm and cold shading are positive and negative contours, and the black contour indicates pressures and times for which correlations are different from zero at the level of 95% significance.

Figures 3.3a-f show the regressions of Z* at 60°N with the OCTSNW, denoted Z*snow, and

the wave-1 and 2 components superimposed on the climatological stationary waves, denoted Z*c,

for October 16th – November 30th (ON) and for December 1st – January 15th (DJ) averages. The

averaging periods are chosen to best illustrate the evolution of the meridional wave heat flux

correlations in Fig. 3.2. The wave anomaly associated with OCTSNW, Z*snow, undergoes a

68

complicated transient evolution from ON to DJ, shifting eastward in both the troposphere and the

stratosphere and at the same time amplifying in strength. The log-pressure weighted pattern

correlations between Z*snow and the climatological stationary wave Z*

c for ON are -0.03, 0.10 and

0.64 for all waves, wave-1 and wave-2, respectively, while the pattern correlations between

Z*snow and Z*

c for DJ are 0.61, 0.94 and -0.26 for all waves, wave-1 and wave-2, respectively.

Thus, Z*snow and Z*

c are in quadrature (neutrally in-phase) in ON (Fig. 3.3a) and become strongly

phase locked in DJ (Fig. 3.3d). The wave-1 component of Z*snow also increases in magnitude

from ON to DJ (Figs. 3.3b and 3.3e) and becomes the wave component that best accounts for the

correlation between LIN and OCTSNW in the stratosphere in January (Figs. 3.2b and 3.2c).

Figure 3.3 also illustrates that the positive correlation between LIN and OCTSNW in the

troposphere in December (Fig. 3.2c) is primarily associated with the positive phasing of the

wave-2 components of Z*snow and Z*

c (Fig. 3.3c). Since the amplitude of wave-2 is relatively

small, however, the phasing of wave-1 determines the overall anomaly correlation between Z*snow

and Z*c.

Now, the seasonal evolution of the anomalous wave activity flux is examined, along with

longitudinal phase structure of the climatological and anomalous waves. Figure 3.4a shows the

evolution of the stratospheric (100 hPa) wave-1 daily averaged wave activity flux regressed on

OCTSNW, v*T*snow, and its LIN and NONLIN components, LINsnow and NONLINsnow. Note that

Fig. 3.4 plots daily v*T*snow , LINsnow and NONLINsnow variations and not the 40-day cumulative

mean variations as in Figs. 3.1 and 3.2. The snow-related meridional wave heat flux, v*T*snow,

starts increasing from near zero in about mid-December and achieves a broad peak throughout

January. LINsnow is overall the largest component throughout this time, corresponding to the peak

in the 40-day cumulative {v*T*} in the stratosphere in early January in Fig. 3.2e. In Fig. 3.4b, the

longitudinal phases of Z*snow with Z*

c at 100 hPa are shown for wave-1; the gray shading

indicates regions in which Z*snow and Z*

c are out-of-phase. Although the phase of the wave

anomaly is relatively noisy, Fig. 3.4b shows that the wave anomaly fluctuates in and out of phase

with the climatology until December when it becomes phase-locked with the climatological

wave for about a month. This persistent phase locking allows development of sufficient upward

wave activity to modify the stratospheric circulation.

69

FIG. 3.3. Covariance of Z* with OCTSNW (black contours) superimposed on Z*c (shading) at

60°N for (a)-(c) October 16th – November 30th (ON) and (d)-(e) December 1st – January 15th. (b) and (e) show the wave-1 component and (c) and (d) show the wave-2 of Z*

snow and Z*c. Black

solid and dashed contours show positive and negative values, respectively. Contour interval is 5 m.

70

FIG. 3.4. Daily time series of (a) wave-1 40-80°N averaged zonal mean wave meridional heat flux components at 100hPa regressed on the snow index (v*T*

snow – black line; LINsnow – red line; NONLINsnow – blue line) and (b) the phase of wave-1 component of Z*

c for 1972-2009 mean (solid line) and the phase of Z*

snow (dashed line) at 60°N and 100hPa. (c) and (d) as (a) and (b) but for wave-2 at 500hPa. (e) as (d) but for all wave numbers greater than wave-2. Gray shading in (a) and (c) indicates regions where Z*

c and Z*snow are out-of-phase.

In wave-2, the strongest snow-related LIN signals were found in the troposphere; thus

Figures 3.4c-d are analogous to Figs. 3.4a-b but for the 500 hPa wave-2 meridional wave heat

flux and longitudinal phase. LINsnow is again dominant although not as much as it was for wave-1

71

in the stratosphere. It begins to increase in late November, corresponding to the tropospheric

peak shown in Fig. 3.2f in mid-December. Fig. 3.4d shows that its increase is largely reflected in

the phasing between the anomalous and climatological waves. As in Fig. 3.4b, there is a period

of about three weeks where the waves are phase-locked. Finally, Fig. 3.4e shows the 500 hPa

meridional wave heat fluxes corresponding to all wave components greater than wave-2. Unlike

Figs. 3.4a and 3.4c, Fig. 3.4e shows that contributions to tropospheric heat fluxes at smaller

scales are dominated by NONLINsnow. Specifically, in early December a negative peak in

NONLINsnow and, thus, v*T*snow is observed, corresponding to the negative correlations in the

troposphere in mid-December shown in Fig. 3.2d.

In summary, the lag between the peak in snow cover anomalies in October and the peak

in the corresponding wave activity flux into the stratosphere (Fig. 3.2b) can be partially

explained by the lack of persistent constructive interference in the dominant stratospheric wave

component, wave-1, until December. Although there is a seasonal westward shift of Z*c, the

phasing is primarily determined by the zonal propagation of Z*snow (Figs. 3.4b and 3.4d). What

causes Z*snow to shift zonally remains to be explained.

Two additional lines of dynamical analysis have been pursued to attempt to explain the

eastward shift and amplification of Z*snow from October to December. First, diagnostics related to

the evolution of the form stress anomaly associated with Eurasian snow cover extent variability

are presented. In their modeling study, Fletcher et al. (2009a) describe how tropospheric

isentropes dome up as a result of the snow-induced cooling and argue that this induces an

upstream high/downstream low circulation pattern, via potential vorticity conservation, and a

corresponding positive form stress anomaly consistent with anomalous upward propagation of

wave activity. Qualitative observational support for this viewpoint is shown in Fig. 3.5, which

shows the climatological distribution of potential temperature (in solid black contours), the

climatological wave geopotential (solid colored contours), the total potential temperature

coherent with OCTSNW (climatology plus two times the regression on OCTSNW, denoted with

a subscript p, dashed black), and the total wave geopotential coherent with OCTSNW

(climatology plus two times the regression on OCTSNW, denoted with a subscript p, dashed

color). The plots are repeated for the same time periods as in Fig. 3.3. The figure shows a

persistent doming up of potential temperature surfaces coherent with OCTSNW, and a

72

circulation anomaly corresponding to the intensification of the climatological high to the west

and the climatological low to the east of the isentropic peak near 130°E from ON to DJ. In

isobaric coordinates, one may write the meridional wave heat flux anomaly as a form stress

anomaly as, ′∂

∂−=′∂∂′ }

~{}~{~}{

******

xpZx

ZpTv , where *~p denotes the perturbation

pressure of the isentropic surface, θ*, and the braces denote the zonal mean. Figure 3.5 illustrates

the two components of the form stress that contribute to LIN. The first component involves the

steepening or shallowing of perturbation isentropes relative to the climatology and the second

corresponds to the steepening or shallowing of the perturbation geopotential gradients relative to

the climatology. A more detailed analysis (not shown) reveals that in the zonal mean both

components are generally positive (corresponding to an upward LIN wave activity flux anomaly)

and that the second component dominates. Consistent with Figs. 3.3 and 3.4, the geopotential

wave anomaly shifts to the east and intensifies, but there is no obvious eastward shift of the

potential temperature distribution. The latter implies that the shift in the longitudinal phase of the

Rossby wave response to snow forcing is not associated with a shift in the location of the forcing

itself.

Second, diagnostics related to dynamical heating in the lower troposphere are presented.

Using daily data, it is found (not shown) that advective heating in the lower troposphere is

dominated by linear terms. It is shown in Figs. 3.6 and 3.7 that these linear terms undergo a

striking change as the season evolves. Figure 3.6 shows the ON zonal, meridional, vertical and

total temperature advection integrated from 925-700 hPa and filtered to retain wavenumbers 1-3

regressed on OCTSNW; the corresponding terms are denoted ZON_ADVsnow, MER_ADVsnow,

VERT_ADVsnow, and TOT_ADVsnow. In ON, the vertical advection term VERT_ADVsnow over

Eurasia is negative and statistically significant. This cooling is partially cancelled by weak

warming from ZON_ADVsnow and MER_ADVsnow, so that TOT_ADVsnow is only weakly

negative and statistically insignificant over the continent. There is also a region of negative

MER_ADVsnow north of Scandinavia, consistent with destructive interference near the western

periphery of the Siberian high in ON, weakening poleward temperature advection

(Panagiotopoulos et al. 2005).

73

FIG. 3.5. As described in text, distribution of potential temperature (black contours) and wave GPH (red contours for positive, blue for negative) at 60°N associated with climatology (solid contours) and the climatology plus two times the regression on OCTSNW (dashed contours) for (a) October 16th –November 30th (ON) and (b) December 1st – January 15th (DJ). Contour interval is 10 K for potential temperature and 30 m for wave GPH.

In DJ (Figure 3.7), the horizontal terms ZON_ADVsnow and MER_ADVsnow are dominant

and statistically significant, and correspond to a relatively strong cooling on the eastern coast of

Eurasia near 60°N. A warming center associated with vertical advection in ON and meridional

advection in DJ is present near 150°E, 40°N. Thus, the advective heating associated with

Eurasian snow cover changes from being dominated by vertical advection to horizontal

advection as the season progresses. There is also a suggestion of an eastward shift and

intensification of this heating. Classical analyses of stationary wave dynamics (e.g. Hoskins and

Karoly 1981) do not explain this behavior.

74

FIG. 3.6. October 16th – November 30th (ON) temperature advection. (a) ZON_ADVsnow, (b) MER_ADVsnow, (c) VERT_ADVsnow, and (d) TOT_ADVsnow vertically integrated from 925-700 hPa and filtered to retain wavenumbers 1-3. Contour interval of 0.03 K day-1, warm and cold shading are positive and negative contours, and the black contour indicates regions for which correlations are different from zero at the level of 95% significance.

3.3.3 Case Study: Winter 2009 – 2010

Cohen et al. (2010; hereafter C10) have argued that the strong negative NAM events of the 2009-

2010 winter season reflected snow-forced NAM dynamics; this season is investigated from the

perspective of linear interference diagnostics. Although Eurasian snow cover extent was initially

below normal in early October 2009, by the end of the month it was the greatest it has been since

its maximum observed value in 1976. C10 connect this anomalous snow cover extent to the

subsequent negative NAM events in November-December and February and demonstrate that a

statistical forecast model including October Eurasian snow cover extent captured the spatial

pattern of anomalously cold 2009-2010 winter European surface temperatures. A complementary

analysis is presented demonstrating that the two negative NAM events highlighted in C10 were

75

preceded by anomalous upward LIN wave activity fluxes resulting from constructive

interference between the anomalous wave and the background climatological stationary wave.

FIG. 3.7. December 1st – January 15th (DJ) temperature advection. (a) ZON_ADVsnow, (b) MER_ADVsnow, (c) VERT_ADVsnow, and (d) TOT_ADVsnow vertically integrated from 925-700 hPa and filtered to retain wavenumbers 1-3. Contour interval of 0.03 K day-1, warm and cold shading are positive and negative contours, and the black contour indicates regions for which correlations are different from zero at the level of 95% significance.

Figure 3.8a shows the standardized polar cap-averaged GPH anomaly, Zpcap′, from mid-

October to the end of February, analogous to Fig. 1a in C10. The two negative NAM events are

clearly visible in November-December and early February, the second being a major sudden

stratospheric warming. Figures 3.8b-d show the standardized 40-day cumulative {v*T*}′, LIN

and NONLIN over the same time period (LIN and NONLIN are standardized by the standard

deviation of {v*T*}). The two negative NAM events are associated with positive {v*T*}′, (see

also C10 Fig. 1b). The majority of {v*T*}′ is associated with LIN (Fig. 3.8c). The contribution

from NONLIN is relatively small (Fig. 3.8d). Based on the analysis in Sections 3.3.1 and 3.3.2,

76

this result suggests that {v*T*}′ is associated with an anomalous wave that constructively

interferes with the background climatological wave prior to the two negative NAM events.

FIG. 3.8. Daily standardized (a) Zpcap′, and 40-day averaged (b) 40-80°N averaged {v*T}*′, (c) the LIN component of (b) and (d) the NONLIN component of (b). X-axis begins on October 10, 2009 and ends on February 29, 2010. Contour interval is 0.2 standard deviation units and warm and cold shading are positive and negative contours.

Analysis of the time series of the phase of wave-1 Z*′ and Z*c in the stratosphere (not

shown) indicates that the waves become phase-locked approximately 2-3 weeks prior to the

appearance of both negative NAM anomalies in the stratosphere. This persistent phase-locking

leads to the {v*T*}′ in Figs. 3.8b and c. Figure 3.9a shows the anomalous wave, Z*′,

superimposed on Z*c at 60°N for November, when the LIN wave activity fluxes become positive.

The pattern correlation between these two wave fields is 0.43. The wave-1 components of the

waves are highly correlated with a correlation coefficient of 0.66 (Fig. 3.9b). In contrast, at the

end of December, when the 40-day cumulative LIN switches from positive to negative (Fig.

77

3.8c), the preceding pattern correlation between Z*′ and Z*c in December is negative (-0.36; Fig.

3.9c). For the first negative NAM event, the timing is somewhat consistent with October

Eurasian snow cover anomalies influencing the anomalous wave but the analysis presented here

does not reflect the snow-NAM connection directly. Nevertheless, this analysis shows that the

C10 result is not merely statistical but is physically based, adding confidence to the result. The

fact that the phase-locking is relatively persistent suggests that use of linear interference

diagnostics may improve predictability of winter NAM events (see Chapter 4 for further

discussion).

In summary, the dynamical evolution of the NAM in the 2009-2010 winter season is

dominated by linear contributions to the wave activity, and the two negative NAM events of that

season correspond to two constructive linear interference events. These results complement the

analysis of C10 and highlight the potential utility of linear interference diagnostics for seasonal

forecasting.

3.3.4 Linear Interference and the Snow-NAM Link in CMIP3 Models

Table 3.2 shows the contributions to the interannual variance of the terms in the decomposition

in Eq. (3.4) for the {v*T*} time series at 100 hPa for the twentieth century runs of the CMIP3

models. The interannual variance of {v*T*} is generally weak in the models, so the contribution

of the variance in LIN and in NONLIN and the covariance between the two are divided by the

variance in the total in the third, fourth and fifth columns of the table. In the models, the linear

interference effect is dominant as in NCEP, but the contributions from var(NONLIN) and

2*cov(LIN,NONLIN) are generally larger than in NCEP and less well separated from the LIN

contribution. This is due in part to the fact that the stationary waves are typically too weak in the

CMIP3 models relative to observations.

78

FIG. 3.9. Z*′ (black contours) superimposed on Z*c (shading) at 60°N for (a)-(b) November 2009

and (c)-(d) December 2009 (b) and (d) show the wave-1 component Z*′ and Z*c. Black solid and

dashed contours show positive and negative values, respectively. Contour interval is 40 m.

Table 3.3 shows the amplitude of the wave-1 component of the wintertime (DJF) Z*c at

60°N and 50 hPa for NCEP and for each model. All models except one have weaker amplitudes

than NCEP. The wave-1 amplitude is weakly positively correlated with var({v*T*}) across the

models (R2 = 0.27) and with the quantity var(LIN)/var(NONLIN) (R2 = 0.25). This suggests

that larger simulated wave-1 amplitudes are associated with larger interannual variability in wave

activity fluxes and stronger linear interference. Conversely, the bias towards weak stationary

wave amplitudes in the CMIP3 models implies that wave activity fluxes dominated by linear

interference effects might not be well estimated in these models.

79

TABLE 3.2: Variance decomposition for December mean {v*T*} at 100 hPa calculated using monthly-averaged CMIP3 model archive data for 20th century runs.

Model var(v*T*) Fraction of

var(v*T*) from

var(LIN)

Fraction of var(v*T*)

from var(NONLIN)

Fraction of var(v*T*) from

2cov(LIN,NONLIN)

NCEP 11.92 0.92 0.13 -0.05

cccma_cgcm3_1 7.70 1.28 0.22 -0.5

cccma_cgcm3_1_t63 6.48 1.26 0.21 -0.47

cnrm_cm3 4.25 0.95 0.17 -0.12

csiro_mk3_0 2.02 0.84 0.40 -0.24

gfdl_cm2_0 5.88 1.08 0.17 -0.25

gfdl_cm2_1 6.66 1.06 0.19 -0.24

giss_model_e_r 3.55 1.04 0.10 -0.14

iap_fgoals1_0_g 8.59 0.96 0.19 -0.14

inmcm3_0 9.84 1.08 0.17 -0.26

ipsl_cm4 6.74 0.96 0.21 -0.18

miroc3_2_medres 2.64 0.92 0.24 -0.17

mpi_echam5 8.87 0.98 0.20 -0.18

mri_cgcm2_3_2a 9.82 0.89 0.13 -0.02

ncar_ccsm3_0 13.23 0.89 0.35 -0.24

ukmo_hadgem1 8.82 0.82 0.19 -0.01

Hardiman et al. (2008) demonstrated that comprehensive GCMs, including the CMIP3

models, fail to reproduce the negative correlation between October Eurasian snow cover and the

wintertime NAM. They attributed this behavior primarily to the fact that the wave anomaly

associated with the snow cover in the GCMs is unrealistically small scale and cannot therefore

effectively propagate into the stratosphere. To supplement the Hardiman et al. analysis, the role

of linear interference in the snow-NAM relationship in GCMs is investigated. Calculations

analogous to those presented in Section 3.3.2 are conducted; however, the calculations are

restricted to using monthly averaged data available from the simulation archive.

80

TABLE 3.3: Amplitude of wave-1 component of December-January-February Z*c at 60°N and 50

hPa for NCEP-NCAR (1972-2007) and the CMIP3 model archive data for 20th century runs.

Model Amplitude of wave-1 Z*

c at 60N and 50 hPa (m)

NCEP 229

cccma_cgcm3_1 224

cccma_cgcm3_1_t63 172

cnrm_cm3 220

csiro_mk3_0 56

gfdl_cm2_0 137

gfdl_cm2_1 183

giss_model_e_r 207

iap_fgoals1_0_g 150

inmcm3_0 219

ipsl_cm4 81

miroc3_2_medres 114

mpi_echam5 162

mri_cgcm2_3_2a 319

ncar_ccsm3_0 194

ukmo_hadgem1 208

Figure 3.10 shows a scatter plot of the correlation between December {v*T*}' at 100 hPa

and the October Eurasian snow index (as in Hardiman et al., 2008, with a slightly different set of

models represented) versus the correlation between 40-80°N zonal mean LIN at 100 hPa and the

October Eurasian snow index for each model (OCTSNW-M). As in observations, a positive

linear relationship between these two quantities is observed, consistent with the LIN terms

dominating the wave activity flux in the simulations. However, Fig. 3.10 illustrates that the

majority of the models produce negative correlations between {v*T*} and OCTSNW-M and that

this is mostly explained (R2 = 0.63) by the negative correlation between the LIN term and

OCTSNW-M. In addition, there is a lot of spread between the models and no model captures the

81

strong correlations illustrated in Fig. 3.2 between LIN and OCTSNW in the observations: if the

observational data were plotted in Fig. 3.10, they would be located at (0.36,0.50).

Whether other factors may also explain the spread along the x-axis in Fig. 3.10 has also

been investigated. A plot analogous to Fig. 3.10 substituting the NONLIN term for the LIN term

shows no significant relationship (figure not shown), suggesting that snow-related driving of the

NONLIN term is not an important factor in explaining the spread. In addition, there is no

relationship between the magnitude of interannual October Eurasian snow cover variability

(Hardiman et al. 2008) or the mean October Eurasian snow cover and a model’s ability to

simulate the observed snow-NAM relationship.

Closer examination of two individual models reveals how inconsistently the linear

interference effect can be represented in different models. Figure 3.11 shows plots similar to Fig.

3.3 but for the GISS model (Fig. 3.11a, c) and GFDL CM2.1 (Fig. 3.11b, d), respectively. The

GISS model produces a positive correlation between December wave activity fluxes and

OCTSNW-M while GFDL CM2.1 produces a negative correlation (see Fig. 3.10). The pattern

correlations for the October-November mean (ON) and the December-January mean (DJ) for the

GISS model are 0.15 and 0.03 (Fig. 3.11a, c) while for GFDL CM2.1 they are -0.42 and -0.26

(Fig. 3.11b, d). Although LIN regressed on OCTSNW, LINsnow, is by far the dominant

component in v*T*snow for both of these models, the pattern correlations in Fig. 3.11 are

somewhat weak. Weak phasing combined with the weaker amplitude of both Z*snow and Z*

c in the

models leads to a relatively weak LINsnow compared with the observations. As illustrated in

Section 3.3.3, many factors might drive this non-robust behavior, including variations in how

surface cooling affects stratification, in the relative roles of horizontal and vertical advection, and

in the stationary wave simulation; no single factor stands out in explaining the spread at this

point.

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FIG. 3.10. Scatter plot of the correlation between December {v*T}* and OCTSNW-M and the correlation between December LIN and OCTSNW-M for each model.

3.4 Conclusions

This chapter has illustrated how linear interference plays a dominant role in describing the

wintertime interannual variability of the vertical component of the wave activity flux into the

stratosphere, represented by the zonal mean extratropical meridional wave heat flux anomaly,

{v*T*}′ . This is accomplished by decomposing {v*T*}′ into a linear interference component,

LIN, and a nonlinear component, NONLIN. It is demonstrated that the variability of the low-

frequency component of LIN accounts for the majority of the wintertime interannual {v*T*}

variance in the upper troposphere while the variance of NONLIN arises primarily from high-

frequency waves (Table 3.1). In the middle and lower troposphere, NONLIN variability

increases as high-frequency wave variability increases. Extending the work of Polvani and

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FIG. 3.11. October-November mean Z*snow (black contours) superimposed on the Z*

c (shading) at 60°N for (a) the GISS model and (b) the GFDL CM2.1 model. (c) and (d) as (a) and (b) for December-January. Contour interval is 3 m.

Waugh (2004) and Garfinkel et al. (2010), it is shown that anomalous wintertime wave activity

flux events associated with zonal mean high-latitude stratospheric variability are dominated by

contributions that are linear in the amplitude of the wave anomalies and that correspond to events

in which wave anomalies constructively or destructively interfere with the climatological wave

field.

The main novel contribution has been to examine the relationship between October

Eurasian snow cover anomalies and the NAM in the context of linear interference. The lag

between October Eurasian snow cover index (OCTSNW) and December-January wave activity

flux is shown to be related to the lack of favorable linear interference conditions prior to

December-January. Several studies have identified a regional relationship between autumn

84

Eurasian snow cover and Eurasian/Pacific sector circulation patterns. For example, Orsolini and

Kvamsto (2009) and Wu et al. (2011) highlight a connection to a Pacific-North-American

circulation pattern but Garfinkel et al. (2010) highlight a connection to an Eastern European high

and Northwestern Pacific low pattern. Although there is some inconsistency concerning which

specific wave patterns are linked to snow, the main conclusion is that in December the planetary-

scale wave train associated with OCTSNW shifts into phase with the background wave and the

vertical wave activity, represented by the meridional wave heat flux, is amplified (Fig. 3.4).

Accompanying the shift in the wave train associated with OCTSNW changes is an intensification

of the source of anomalous form stress from the troposphere and a shift in advective heating in

the lower troposphere from vertical-advection dominated to horizontal-advection dominated. A

case study is also presented showing that the two strong negative NAM events of the 2009-2010

winter (Cohen et al. 2010) were preceded by upward LIN wave activity fluxes into the

stratosphere. It is shown that the anomalous wave phase locks with the background

climatological wave 2-3 weeks preceding the NAM events, leading to strong 40-day cumulative

LIN wave activity fluxes.

Finally, the issue of the inability of current climate models to capture the snow cover-

NAM connection is revisited (Hardiman et al. 2008). Most models show a connection of

opposite sign to that observed between October Eurasian snow extent and December wave

activity and this negative correlation is shown to be a linear interference effect: in the models,

years with greater October Eurasian snow extent typically lead to a weakening of the wintertime

wave pattern. Since CMIP3 models generally reproduce the phase of the climatological

background waves fairly well (Brandefelt and Kornich, 2008), these results suggest that the wave

anomaly in the models associated with the snow cover is not evolving in the same manner as in

observations.

Although this study demonstrates that linear interference can affect the sign and timing of

the relationship between October Eurasian snow cover anomalies, {v*T*}, and the NAM, a

detailed analysis of what causes the shift in phase of the wave train associated with a relatively

stationary surface forcing, such as snow cover, remains to be done. The work provides pointers

to follow-on research required to understand the nuanced relationships operating in this aspect of

extratropical variability. For example, Fig. 3.5 suggests that the diabatic heating associated with

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snow remains relatively stationary, but that the wave anomaly associated with the snow

undergoes a much more complicated transient evolution. This implies that it would be useful to

investigate the linear transient response to stationary surface cooling. In addition, further

investigation of the transient evolution of climatological LIN events may provide insight into the

snow-NAM problem.

86

Chapter 4

Linear Interference in Extratropical Stratosphere-Troposphere

Interactions

4.1 Introduction

The focus of the previous two chapters was on establishing a better dynamical understanding of

the observed correlation between autumn snow cover anomalies over Eurasia and the Northern

Annular Mode (NAM) in the following winter. Although the motivation for these chapters

stemmed from the Eurasian snow-NAM connection, the primary finding was the dynamical

importance of linear interference in establishing anomalous upward wave activity flux into the

stratosphere and, consequently, the phase of the stratospheric NAM associated with this wave

driving. In Section 3.3.1 a brief discussion of linear interference in the Northern Hemisphere

(NH) winter climatology was introduced as a means of providing context for the role of linear

interference in the snow-NAM connection. In the present chapter, a more detailed description of

linear interference in stratosphere-troposphere interactions is presented.

Simply stated, linear interference can be described as the interaction between anomalous

waves and the background climatological wave field. There has been recent interest in the role of

linear interference in extratropical variability (Nishii et al. 2009; Garfinkel et al. 2010; Kolstad

and Charlton-Perez 2010; Smith et al. 2010; Nishii et al. 2010; Fletcher and Kushner 2011).

While these ideas are not new, their importance has only recently become appreciated in the

context of the phenomenon of stratosphere-troposphere interactions. To briefly review some of

the previous literature in this area: Branstator (1992) introduced decompositions of the vorticity

and thermodynamic energy equations, which included terms representing linear interference. He

found that the leading modes of low-frequency variability in a perpetual winter general

circulation model (GCM) simulation are maintained primarily by these types of anomalous-

climatological stationary wave interaction fluxes. Weickmann and Sardeshmukh (1994) and

Weickman et al. (1997) use a similar decomposition to show that the seasonal cycle of zonal

mean atmospheric angular momentum anomalies are associated with the interaction of Rossby

87

waves generated by tropical convection and the zonally asymmetric background flow. Watanabe

and Nitta (1998) investigate the development of a positive NAM-like flow in the winter of 1988-

1989. They find that the second largest contribution to the GPH anomaly tendency was from the

interaction between anomalies and the background climatological wave (the largest contribution

was from the interaction of anomalies with the climatological zonal mean). DeWeaver and

Nigam (2000a) demonstrate that linear interference is also important in maintaining tropospheric

zonal mean zonal wind anomalies associated with the North Atlantic Oscillation (NAO). Using a

linear stationary wave model forced by zonal-eddy coupling terms, transient fluxes and heating,

they show that the so-called "zonal-eddy coupling term", which represents the interaction

between the zonal mean and the stationary waves, is largely responsible for maintaining the

stationary wave field. The coupling term is associated with a positive feedback between the

tropospheric zonal mean and stationary wave flow components (DeWeaver and Nigam, 2000b).

This work provides a distinct perspective on the variability of the extratropical zonal-mean flow

in the troposphere, which has been described by other authors in terms of feedbacks between the

zonal-mean flow and transient, synoptic-scale waves (Robinson 2000; Lorenz and Hartmann

2003).

Much more recently, others have begun to investigate the role of linear interference in the

stratospheric circulation. Following Nakamura and Honda (2002), Nishii et al. (2009) use a

decomposition of meridional wave heat flux anomalies into a linear interference component and

a component describing the heat flux inherent to the anomalies themselves to investigate two

stratospheric sudden warming (SSW) events, one in the NH in 2005-2006 and one in the

Southern Hemisphere (SH) in 2002. They find that both wave heat flux components are

important in the NH SSW and that the fluxes associated with the anomalous wave are the

dominant source of wave driving in the SH SSW. Garfinkel et al. (2010) further these ideas by

analyzing the spatial coherence of anomalous wave patterns with the background climatological

wave field in the troposphere. They show that the variability of the NH winter stratospheric polar

vortex is anti-correlated with a tropospheric wave pattern that is coherent with a small

wavenumber approximation of the climatological stationary wave field. In particular, Garfinkel

et al. (2010) show that the two main features of the tropospheric wave pattern associated with

stratospheric variability are the Eurasian High and Aleutian Low, implying that when the

climatological Eurasian High and/or Aleutian Low is amplified (attenuated), the stratospheric

88

polar vortex weakens (strengthens). (It is worth pointing out that Kodera et al. (1996) show a

very similar tropospheric wave pattern to that found in Garfinkel et al. associated with the first

EOF of the 50 hPa NH extratropical geopotential height (GPH)). Kolstad and Charlton-Perez

(2010) also show that a similar relationship exists in the suite of CMIP3 models.

In Chapter 2, linear interference diagnostics were developed to investigate the

stratosphere-troposphere response to a zonally asymmetric extratropical surface forcing in the

NH (Smith et al. 2010). In GCM integrations in which snow forcing and surface cooling are

prescribed, it was shown that in order to achieve amplification of the wave activity into the

stratosphere, the forced wave must constructively interfere with the pre-existing climatological

stationary wave. This effect, which corresponds to wave activity flux contributions that scale

linearly with the forced wave amplitude, dominates over nonlinear contributions for sufficiently

weak forcing. The effect helps to explain the transient dynamics of snow-forced simulations of a

comprehensive GCM and the sensitivity to different configurations of surface cooling in a suite

of simplified GCM integrations. In Chapter 3, a similar linear interference effect was found in

the observed connection between autumn Eurasian snow cover and the NAM (Smith et al., in

press). Similar relationships between the wave field and NAM-like stratospheric variability are

highlighted by Ineson and Scaife (2009), Cagnazzo et al. (2009) and Fletcher and Kushner

(2011) with respect to GCM simulations with prescribed ENSO forcing, by Martius et al. (2009),

Charlton-Perez et al. (2010) and Nishii et al. (2010) with respect to NH blocking and by Grise

and Thompson (submitted) with respect to equatorial stratospheric waves. Collectively, this work

suggests that wave activity associated with the interaction between anomalous waves and the

background climatological wave, i.e. the linear contribution, explains a significant fraction of the

observed stratospheric variability in the NH but that in certain extreme cases, such as the SSW

events examined in Nishii et al. (2009), the nonlinear contribution may play a larger role.

Although several studies have now identified linear interference as being a potentially

important factor in the interpretation of GCM and observational analysis in the NH extratropics,

a comprehensive description of the climatology of this phenomenon in stratosphere-troposphere

interactions has yet to be done. The following chapter addresses this deficiency using the

linear/nonlinear decomposition of meridional wave heat fluxes introduced in Chapter 3.

Emphasis is on the NH but comparisons are made with the SH and differences between the

89

hemispheres are highlighted. In Section 4.2, the methods will be briefly outlined, with reference

to Section 3.2. Section 4.3.1 describes the seasonal cycle of the meridional wave heat flux

decomposition in the NH. In Section 4.3.2 composite analysis of anomalous heat flux events in

the NH is conducted, quantifying the relative roles of fluxes associated with the linear and

nonlinear terms in the heat flux decomposition associated with stratospheric NAM variability.

The relationship between these composites and SSWs is established in Section 4.3.3. In Section

4.3.4, a comparison of linear interference characteristics between the Northern and Southern

hemispheres is presented. Finally, in Section 4.3.5, the role of linear interference in final

warmings is discussed. Final warmings characterize the breakdown of the stratospheric polar

vortex during the transition from strong westerly winds in winter to weak easterly winds in the

summer and involve a coupling between the stratosphere and troposphere (Black et al. 2006).

4.2 Methods

The characteristics of heat flux anomalies in the extratropical atmosphere are investigated using

daily averaged NCEP/NCAR reanalysis from 1979-2009 (Kalnay et al. 1996). The analysis is

limited to the modern satellite era given the improvements in the reanalysis in the Southern

Hemisphere for this time period (Kistler et al. 2001). Linear trends have been removed from all

time series. For the SH, the year 2002, the only year on record in which a major stratospheric

warming occurred, is excluded. The atmospheric fields of interest are the GPH anomaly area-

averaged over the polar cap bounded by 60°N or 60°S and standardized by its standard deviation,

denoted S(Zpcap′), which corresponds to the annular mode index (Cohen et al. 2002; Baldwin and

Thompson 2009); the wave GPH at 60°N or 60°S, Z*, (where the superscript asterisk indicates

the deviation from the zonal mean); and the zonal mean meridional wave heat flux averaged

from 40-80°N or 40-80°S, {v*T*} (braces indicating a zonal mean), which is used as a proxy for

the vertical component of the wave activity flux (with a sign change in the SH). Daily, monthly

and 40-day averaged heat fluxes are used. For the most part, the focus is on heat fluxes at 100

hPa, i.e., heat fluxes from the troposphere to the stratosphere; however, all of the decompositions

described below have been calculated at all vertical levels. Following Eqn. (3.1), the

climatological mean of {v*T*} corresponds to

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{v*jT*

j}c = {v*cT*

c} + {v*′j T*′j}c , (4.1)

where subscript j denotes the year, subscript c denotes the climatological mean over the total

number of years and prime indicates the deviation from the climatological mean . {v*cT*

c}

represents the contribution from the climatological stationary waves while {v*′j T*′j}c represents

the contribution from the transients. The anomalous meridional wave heat flux can be

decomposed into two components,

{v*jT*

j}' = LIN + NONLIN, (4.2)

where,

LIN = {v*′jT*c} + {v*

cT*′j} and NONLIN = {v*′jT*′j} - {v*′j T*′j}c = {v*′jT*′j}′ .

The climatological mean of both LIN and NONLIN is zero. Please refer to Section 3.2 for further

details on the derivation of Eqn. (4.2). Using Eqn. (4.2), the interannual variability of {v*T*} can

be written as (the subscript j will be omitted for the remainder of the Chapter)

var({v*T*}) = var(LIN + NONLIN)

= var(LIN) + var(NONLIN) + 2cov(LIN,NONLIN) . (4.3)

The heat fluxes can also be decomposed into high- and low-frequency wave components.

The data is low-pass filtered using an 11-day running mean. The climatological mean and the

variance of the heat fluxes may be written as (see Section 3.2 and Eqn. (3.6) for further details)

{v*T*}c = {v*low T*

low} c + {v*high T*

high} c

+ {v*low T*

high} c + {v*high T*

low} c, (4.4)

and

var({v*T*}) = var({v*low T*

low} + {v*high T*

high}

+ {v*low T*

high} + {v*high T*

low}) (4.5a)

= var({v*low T*

low}) + var({v*high T*

high})

91

+ var({v*low T*

high}) + var({v*high T*

low}) + R, (4.5b)

where R represents the series of covariance terms in the expansion of Eqn. (4.5a). In addition,

each of the terms in Eqn. (4.4) and (4.5b) can be decomposed into LIN and NONLIN terms as in

Eqn. (4.3).

Composites of anomalously high and low 40-day averaged meridional wave heat flux

events are generated (Polvani and Waugh (2004)). Many methods for developing composites

employ the selection of a threshold parameter for anomalously high and low events and a

temporal separation parameter such that the same event is not counted more than once. Although

such methods are commonly used, selection of the threshold and separation parameters is

somewhat arbitrary. As an alternative simplified procedure, here the maximum or minimum 40-

day averaged standardized {v*T*}′ for each year from November-March in the NH and June-

December in the SH for the years 1979-2009 is selected (Mudryk and Kushner, in press). It has

been verified that the results are very similar when the maximum or minimum 40-day averaged

heat flux anomaly is selected rather than the standardized anomaly. In Sections 4.3.2 and 4.3.4, it

is demonstrated that this method yields similar results to previous composite methods for a range

of threshold values, but that the similarity can break down for large threshold values. To create

these threshold composites, the threshold value is varied but always maintaining the requirement

that events be separated by 40 days.

For the analysis of the SSW events, the analysis is extended into the past using the central

dates from 1958-2009 provided in Charlton-Perez and Polvani (2007) and Butler et al. (2011). A

central date is defined as the date when the zonal mean zonal wind at 60°N and 10 hPa becomes

easterly during the season of climatological westerlies (excluding the final breakdown of the

vortex in spring). There are 33 SSW events during the 1958-2009 time period. The vortex

“displacement” and “split” SSW classification of Charlton-Perez and Polvani (2007) is used to

identify 20 displacement events and 13 split events (classifications for 2002-2009 provided by

Peter Hitchcock using the method of Charlton-Perez and Polvani (2007), personal

communication). Displacement events involve displacement of the polar vortex off the pole and

project primarily onto wave-1. Split events involve a stretching and split of the vortex into two

distinct vortices and project primarily onto wave-2.

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Finally, for the analysis of final warming events, the methods of Black and McDaniel

(2007a,b) are used. Final warmings in the NH are identified as the final time that the 50-hPa

zonal-mean zonal wind at 70°N drops below zero without returning to a value of 5 m s-1 until the

following autumn (Black and McDaniel 2007a). In the SH, final warmings are identified as the

final time that the zonal mean zonal wind at 60°S drops below 10 m s-1 until the following austral

autumn (Black and McDaniel 2007b). There are 30 NH and SH final warmings in the 1979-2009

time period.

4.3 Results

4.3.1 Northern Hemisphere Seasonal Heat Flux Characteristics

The climatological mean and the variability of the NH stratosphere both have strong seasonal

cycles (Scherhag 1952; Matsuno 1971; Baldwin and Dunkerton 1999; Yoden et al. 2002;

Kushner 2010). This seasonal cycle depends on the existence of a wave guide for vertically

propagating planetary waves in winter (Charney and Drazin 1961; Matsuno 1970; Shaw et al.

2010) and consequently, is associated with the variability in upward wave activity flux (Kodera

and Chiba 1995; Waugh et al. 1999; Newman et al. 2001; Hu and Tung 2002; Polvani and

Waugh 2004). Figure 4.1a shows the climatological mean meridional wave heat flux

decomposition at 100 hPa and averaged over 40-80°N for each month following Eqns. (4.1) and

(4.4). Only the first two terms on the RHS of Eqn. (4.4) are plotted; the remaining two, which

involve the covariance between low- and high-frequencies, are quite small. The extratropical

{v*T*}c at the 100 hPa, which characterizes the wave activity flux through the lower stratosphere,

exhibits a strong seasonal cycle with a maximum in January and a minimum in July (Fig. 4.1a,

black line; Randel 1988). From November to February, the main contribution to {v*T*}c is from

the stationary waves, i.e. the {v*cT*

c} term in Eqn. (4.1) (Fig. 4.1a, red line), while the

climatological mean of the heat flux associated with the wave anomalies, {v*′ T*′}c, (Fig. 4.1a,

blue line) contributes the majority of {v*T*}c throughout the rest of the year. This reflects a

strong seasonal cycle in the {v*cT*

c} fluxes and a relatively weaker seasonal cycle in the {v*′

T*′}c fluxes. Not surprisingly, the {v*cT*

c} fluxes consist almost entirely of low-frequency waves

93

while the {v*′ T*′}c fluxes comprise approximately equal contributions from low- and high-

frequency waves (Fig. 4.1a, green and cyan lines, respectively).

FIG. 4.1. NH meridional wave heat flux decomposition at 100 hPa averaged over 40-80°N. (a) Climatological monthly mean (see Eqn. (4.1) and (4.4)). Total and selected high- and low-frequency components are plotted (see legend). (b) Monthly variance decomposition (see Eqn. (4.3)). The asterisks denote months when the correlation between LIN and NONLIN is statistically significant at the 95% level.

During polar night, the stationary waves are clearly the dominant component to the

vertical wave activity flux in the climatological mean (Fig. 4.1a, red line); however, when

considering interannual variability associated with stratosphere-troposphere interactions, it is the

heat flux anomalies, {v*T*}', that are important. The term {v*T*}' can be decomposed into the

linear interference term, LIN, i.e., the term involving the interaction between the wave anomalies

and the climatological stationary waves and, NONLIN, i.e., the term associated with the wave

94

anomalies themselves (see Eqn. (4.2)). Figure 4.1b shows the monthly {v*T*} variance

decomposition using Eqn. (4.3). The total variance (Fig. 4.1b, black line) grows steadily over the

autumn and early winter months, peaking in February; it drops sharply in March and slowly

decreases over the spring and summer, reaching a minimum in July. The variance of the LIN flux

anomalies is the largest contribution to the total variance from October to April (Fig. 4.1b, red

line). The seasonal cycle of variance shows that the peak in the LIN variance occurs in January,

while the peak in the NONLIN variance occurs in February (Fig. 4.1b, blue line). The relative

contributions of the terms in the variance decomposition (including the covariance term, which is

negative; Fig. 4.1b, green line) result in a peak in the total variance that is one month later than

the peak in the seasonal mean (Fig. 4.1a).

Notably, the covariance between the LIN and NONLIN fluxes is negative throughout

most of the year except in May (Fig. 4.1b). In this respect, the observed internal variability is

fundamentally different from the forcing simulations in Chapter 2. In Chapter 2, both the

comprehensive GCM and the simple GCM simulations showed that the NONLIN component of

the heat flux response to surface heating or cooling is positive because the wave anomaly is

surface forced and the intrinsic wave activity response is therefore upward. NONLIN in the

Chapter 2 modeling experiments is positive regardless of the sign of the linear interference (see

Figs. 2.2, 2.4 and 2.9). But in observed interannual variability, the negative sign of the

covariance in Fig. 4.1b (green line) implies that positive LIN fluxes are associated with negative

NONLIN fluxes and vice versa. The covariance is only significant at the 95% level in

September, November, May and July and is marginally significant in January, February and

April (at the 90% level). Thus, to a certain extent, LIN and NONLIN fluxes can be considered as

independent. However, at certain times of year the partial cancellation and inter-dependence of

LIN and NONLIN presents a challenge for interpreting the dynamics of anomalous vertical wave

activity fluxes.

Since the focus of this Chapter is on stratosphere-troposphere interactions, the summer

months will no longer be discussed. Figure 4.2 shows a breakdown of the high- and low-

frequency contributions to the {v*T*} variance using Eqns. (4.3) and (4.5) for September to May.

The LIN anomalies consist primarily of low-frequency waves (blue bars). The NONLIN

anomalies are also mostly low-frequency from September to May but consist of a much larger

95

contribution from high-frequency waves (red bars) during autumn and spring. Thus, winter,

which is the season when strong stratosphere-troposphere interactions are observed, is also the

season with the cleanest decomposition of var({v*T*}) in terms of its frequency and its LIN and

NONLIN components. During the winter season, the largest contribution to var({v*T*}) is from

low-frequency LIN fluxes (see also Table 3.1). The covariance consists mostly of the covariance

between low-frequency LIN and NONLIN fluxes from September to February and of the

covariance between LIN and NONLIN fluxes associated with the interaction between low- and

high- frequency waves from March to May (Fig. 4.2). During the month of May, when the

balance of the variance decomposition transitions from mainly LIN to mainly NONLIN variance,

the covariance between the two fluxes is positive and statistically significant. For most months,

the covariance shown in Figs. 4.1 and 4.2 is typically negative throughout the depth of the

atmosphere but from March-May, between 100 hPa to 10 hPa, the covariance is positive

(although only statistically significant in May). Spring is also the season when there is a

discernable increase in the variance of the high-frequency fluxes. It is unknown why the nature

of the covariance between LIN and NONLIN and the frequency of the waves contributing to the

heat fluxes change dramatically in spring; however, these changes are likely related to the

breakdown of the polar vortex at this time of year. Both the LIN and NONLIN fluxes decrease

by an order of magnitude as the vortex breaks down in spring. The interannual variability of the

heat fluxes in the spring is likely related in part to the variability in the timing of the vortex

breakdown or final warming. The final warming would affect both the LIN and NONLIN fluxes

in the same way, perhaps leading to the positive covariance in the lower stratosphere at this time

of year.

96

FIG. 4.2. Contributions of terms in Eqns. (4.2) and (4.5) to interannual variability of NH {v* T*} at 100 hPa and averaged over 40-80°N for each climatological month (in units of m2 K2 s-2). Colour scheme corresponds to different terms in Eqn. (4.5): blue – var({v*

low T*low}); red -

var({v*high T*

high}); green - var({v*low T*

high}) + var({v*high T*

low}); yellow – R. Note the different scales on the ordinate axes.

The fact that the LIN fluxes consist of mostly low-frequency waves while the NONLIN

fluxes have a greater overall contribution from high-frequency waves suggests that LIN fluxes

97

may be more persistent than NONLIN fluxes. This is confirmed in Fig. 4.3, which shows the

autocorrelation of each term in Eqn. (4.2) at 100 hPa as a function of positive lag (in days). The

autocorrelation of LIN and NONLIN includes the effect of the covariance between them and the

sum of the two explicit cross-correlation terms is also shown (green curve; Mudryk and Kushner

(2011)). The autocorrelation of {v*T*}′ decays relatively quickly. This can be partly attributed to

clear differences in the autocorrelation characteristics of the LIN and NONLIN fluxes at lags

shorter than 10 days, with the LIN fluxes being more persistent than the NONLIN fluxes. The

negative cross-correlation also contributes to the rapid decay of the {v*T*}′ autocorrelation.

Thus, linear interference appears to enhance the overall persistence of {v*T*}′ while the

NONLIN and cross-correlation components appear to reduce it.

FIG. 4.3. Heat flux anomaly autocorrelations for {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) and the cross-correlation of LIN and NONLIN (green curve) as a function of lag.

4.3.2 Northern Hemisphere Anomalous Heat Flux Composites

Polvani and Waugh (2004) demonstrate that high (low) index NAM events are highly correlated

with anomalously low (high) extratropical meridional wave heat fluxes in the lower stratosphere.

98

Time-pressure composite plots of the NAM index based on these low/high heat flux anomalies

show remarkably similar features to those based on high/low NAM events themselves. Given the

findings of the previous section, a similar composite analysis is performed with the aim of

elucidating the relative importance of LIN and NONLIN heat flux anomalies to anomalous heat

flux events and, thus, stratosphere-troposphere coupling NAM events.

FIG. 4.4. Weak vortex composite mean 40-day averaged heat flux anomaly decomposition for (a) {v*T*}′, (c) LIN and (e) NONLIN as a function of lag and pressure. (b), (d) and (f) same as (a), (c) and (e) but for the strong vortex composite. Black contour indicates region of 95% significance.

99

Using the method outlined in Section 4.2, 30 anomalously high heat flux events and 30

anomalously low heat flux events are identified. Because high heat flux events are linked to

warm polar stratospheric conditions and a weak polar vortex, and vice versa, high and low heat

flux events will be called "weak vortex" and "strong vortex" events. There is no significant

difference in the mean timing of the weak and strong vortex composites with a mean date of

January 28th with a standard deviation of 48 days for the weak vortex composite, and a mean

date of February 2nd with a standard deviation of 42 days for the strong vortex composite.

Figures 4.4a, c and e and 4.4b, d and f show time-pressure composites for the total heat flux

anomaly, {v*T*}′, LIN and NONLIN for the weak and strong vortex events, respectively. In

general, the vertical structure of {v*T*}′ is fairly coherent. The left column of Fig. 4.4 reveals

that the contributions from both the LIN and NONLIN fluxes to {v*T*}′ in the stratosphere are

roughly equal for weak vortex events. The LIN fluxes are somewhat stronger in the upper

troposphere and, although they are weak, the LIN fluxes are statistically significant in the lower

troposphere as well. The NONLIN fluxes are only statistically significant in the stratosphere and

upper troposphere. Although in interannual variability, LIN and NONLIN are marginally anti-

correlated (Fig. 4.1b), during weak and strong vortex events they are of the same sign (this point

will be discussed in greater detail below). The right column of Fig. 4.4 shows that for strong

vortex events, the LIN flux anomalies are clearly the dominant component of {v*T*}′. Similar to

the weak vortex events, NONLIN flux anomalies associated with strong vortex events are not

robust in the lower troposphere while the LIN fluxes are robust throughout the depth of the

atmosphere.

Figures 4.5a-d show the heat flux anomaly time series composited at 100 hPa as a

function of time and the corresponding composite mean standardized polar cap-averaged GPH

anomaly, S(Zpcap′), as a function of time and pressure for both the weak and strong vortex

composites. As shown in Figs. 4.4a, c and e, examination of the left column of Fig. 4.5 shows

that the contribution to {v*T*}′ from the LIN flux is slightly greater than the NONLIN flux in the

lower stratosphere. In addition, the LIN flux increases almost linearly from a lag of -40 days

while the NONLIN flux increases sharply near the zero lag. This behaviour is consistent with the

differing autocorrelation timescales of the LIN and NONLIN fluxes (Fig. 4.3). In contrast, the

right column shows again that the LIN flux is clearly the largest contribution to strong vortex

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events (see also Figs. 4.4b, d and f). Consistent with Polvani and Waugh (2004), Figs 4.5c and d

show robust negative and positive NAM-like stratosphere-troposphere coupling associated with

the weak and strong vortex composites (see also Fig. 3.1).

FIG. 4.5. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for NH (a) weak and (b) strong vortex events. Solid sections of the curves indicate 95% significance. Composite mean S(Zpcap′) for NH (c) weak and (d) strong vortex events. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5 1 1.5]. Black contour indicates 95% significance.

Similar composites for early, mid- and late winter events demonstrate that weak vortex

events display some seasonal dependence. Figure 4.6a shows the fractional contributions from

LIN and NONLIN at the zero lag for the 10 weakest and strongest vortex events for each three-

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month period, November-December-January (NDJ), December-January-February (DJF) and

January-February-March (JFM). Early and late winter weak vortex events consist of a larger LIN

contribution while mid-winter events consist of a slightly larger NONLIN contribution. As

shown in Chapter 3, NDJ, the season with the largest LIN contribution to weak vortex events, is

the season when LIN fluxes are linked with the observed connection between October Eurasian

snow cover and the NAM (Smith et al. in press). In addition, the heat flux anomalies themselves

increase in magnitude from early to late winter (Fig. 4.6b). Figures 4.4 and 4.5 reflect an average

over these early, mid- and late winter events.

FIG. 4.6. Fraction of {v*T*}′ from LIN and NONLIN for November-December-January (NDJ), December-January-February (DJF) and January-February-March (JFM) for the (a) weak and (c) strong vortex composites. {v*T*}′, LIN and NONLIN for NDJ, DJF and JFM for the (b) weak and (d) strong vortex composites.

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Unlike the weak vortex events, the strong vortex events display a much weaker seasonal

cycle in the fractional contributions of LIN and NONLIN to early, mid- and late winter events

(Fig. 4.6c) yet a similar seasonal cycle in the magnitude of {v*T*}′.

Composite mean differences in the LIN contribution to the weak and strong vortex

composites and to early, mid- and late winter events suggest that the frequency distributions of

these fluxes may differ. Taguchi and Yoden (2002) show (and confirm in simulations) that the

observed stratospheric temperature anomalies are positively skewed, which implies that {v*T*}′

are likely also positively skewed. Figures 4.7a-c show the histograms of the three terms in Eqn.

(4.2); 40-day averaged {v*T*}′, LIN and NONLIN at 100 hPa (note the logarithmic vertical

axes). The distribution of {v*T*}′ is somewhat positively skewed (Fig. 4.7a; skew = 0.34). The

positive skew originates from the distribution of NONLIN fluxes (Fig. 4.7c; skew = 1.66) while

the slight negative skew of the distribution of LIN fluxes (Fig. 4.7b; skew = -0.19) partly

compensates for the positive skew of the NONLIN distribution. Thus, the composite differences

mentioned above, i.e. the fact that the NONLIN contribution is considerably larger in the weak

vortex events compared to the strong vortex events, partially reflect the skew of the terms in the

{v*T*}′ decomposition.

FIG. 4.7. NH 40-day averaged heat flux anomaly histogram for (a) {v*T*}′, (b) LIN and (c) NONLIN.

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In light of the distinctive characteristics of the LIN and NONLIN distributions and the

fact that LIN and NONLIN events are not entirely independent, individual weak or strong vortex

events may reflect different types of {v*T*}′ events with different relative contributions of LIN to

{v*T*}′. It is thus worth testing different types of composite methods to ensure robustness, for

example to demonstrate that the maximum/minimum event per year composite method employed

above adequately captures the relative importance of the LIN fluxes. Here, the threshold

approach used by Polvani and Waugh (2004) is employed, in which heat flux events are chosen

based on exceeding a given threshold amplitude. Figure 4.8a shows the {v*T*}′, LIN and

NONLIN at 100 hPa as a function of positive threshold value (in units of heat flux standard

deviation). Note that the number of events per composite decreases roughly linearly as the

threshold increases from 64 events to three events (green curve in Fig. 4.8a).

FIG. 4.8. Sensitivity of composite mean 40-day averaged {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) at lag zero and of the number of events in each composite (green curve) to a standardized {v*T*}′ threshold value (in units standard deviation) for NH (a) weak and (b) strong vortex events.

For threshold values ranging between 0.2 and 0.55, the fraction of LIN and NONLIN to

{v*T*}′ (~0.6 and 0.4, respectively) is basically independent of the threshold value. The

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maximum event per year composite method shows similar characteristics within this range. The

weak vortex composite mean values of {v*T*}′, LIN and NONLIN at the zero lag in Fig. 4.5 are

most similar to those corresponding to a threshold value of 0.5 in Fig. 4.8a. For threshold values

in weak vortex events beyond 0.65, NONLIN increases nonlinearly and LIN decreases after a

threshold of 0.75, reflecting the nature of the distributions of these two heat flux components

(Figs. 4.7b and c). For strong vortex events, NONLIN fluxes are generally of small amplitude,

thus, the LIN fraction is basically independent of the threshold value, even for very large

threshold values (Fig. 4.8b). For both weak and strong vortex composites, interpretation of the

sensitivity of LIN fraction to threshold value at very high thresholds becomes difficult due to the

very small number of events in these composites.

Returning to the weak and strong vortex composites of Figs. 4.4 and 4.5, since these

composites consist of both LIN and NONLIN flux events, it is of interest to ask what typical LIN

and NONLIN heat flux events look like. In other words, what can be learned about the

characteristics of the weak and strong vortex composites by looking at composites of LIN and

NONLIN events separately? To address this question, the maximum/minimum event per year

composite method is used to construct composites of 30 high and 30 low LIN and NONLIN flux

events (hereafter, LIN and NONLIN weak vortex events, and LIN and NONLIN strong vortex

events; Fig. 4.9). It is important to note that these LIN and NONLIN weak/strong vortex

composites are not independent of each other and are not independent of the weak/strong vortex

composites of Figs. 4.4 and 4.5. However, they are useful in that they highlight features of LIN

and NONLIN events.

Figure 4.9 shows the LIN (left column) and NONLIN (right column) weak vortex

composites. The composites demonstrate that the principal reason why the LIN and NONLIN

fluxes are of the same sign in Fig. 4.5 (left column) is that Fig. 4.5 largely reflects sampling over

events that consist of either predominantly LIN or predominantly NONLIN fluxes. Specifically,

of the 30 weak vortex events in Figs. 4.4 and 4.5, 12 are also LIN weak vortex events and 11 are

also NONLIN weak vortex events (1 is common to both the LIN and NONLIN weak vortex

composites). Comparing Figs. 4.9a and b shows that the LIN weak vortex composite consists of

slightly larger {v*T*}′ (black curve) than the NONLIN weak vortex composite. Consequently,

the S(Zpcap′) is stronger and more robust for the LIN weak vortex events than for the NONLIN

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weak vortex events (Figs. 4.9c and d). S(Zpcap′) also extends further into the troposphere for LIN

weak vortex events. Figures 4.9a and b also show some evidence of the anti-correlation between

LIN and NONLIN in the {v*T*} variance decomposition (Fig. 4.1b). However, it appears to be

quite small and non-robust for the LIN and NONLIN weak vortex events. Also, note that the

magnitude of the slope of the NONLIN curve in Fig. 4.9b is larger than that of the LIN curve in

Fig. 4.9a, reflecting the relatively shorter timescales of the NONLIN flux anomalies (Fig. 4.3).

Comparing Figs. 4.9 and 4.5 (left column) illustrates that the two types of events, LIN and

NONLIN, combine to produce the observed features of the weak vortex composite.

FIG. 4.9. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) LIN and (b) NONLIN weak vortex events. Solid sections of the curves indicate 95% significance. Composite mean S(Zpcap′ ) for (c) LIN and (d) NONLIN weak vortex events. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5 1 1.5]. Black contour indicates 95% significance.

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Similar composites of LIN and NONLIN strong vortex events (anomalously negative

heat flux events) were constructed (not shown). The LIN and NONLIN strong vortex composites

show similar but opposite signed features to their weak vortex counterparts. Of the 30 strong

vortex events in Figs. 4.4 and 4.5, 12 are also LIN and seven are also NONLIN strong vortex

events.

For both the weak and strong vortex composites in Figs. 4.4 and 4.5, LIN heat flux

anomalies are an important feature revealing that the interaction between the anomalous wave

and the background climatological wave is a key component of anomalous heat flux generation.

As demonstrated in Chapter 2, Section 2.3.5, linear interference includes both a phasing effect

and an amplitude effect. The phasing effect is illustrated in Fig. 4.10a, which shows the wave-1

phase differences (∆θ) between the composite mean anomalous wave GPH, Z*′, and the

background climatological wave GPH, Z*c, at 60°N and for days [-30,-1] (red and blue curves,

respectively) for the original weak and strong vortex composite represented in Figs. 4.4 and 4.5.

Because {v*T*}′ used to generate the composites are 40-day averaged, a time interval preceding

the zero lag is selected to illustrate the phase differences. The waves are in-phase if the phase

difference is -90° < ∆θ < 90° and out-of-phase if it is 90° < ∆θ < 270°. For the weak vortex

events, the composite time mean phase difference varies between 20° and 40° from the mid-

troposphere into the stratosphere, while for strong vortex events, the composite time mean phase

difference varies between 150° and 170° from the mid-troposphere into the stratosphere. The

waves are close to neutrally phased in the lower troposphere and are most strongly in or out-of-

phase in the stratosphere.

Using the anomaly correlation diagnostic presented in Chapters 2 and 3, i.e., the pattern

correlation between Z*′ and Z*c, at 60°N, the transient evolution of the phasing is illustrated

separately in both the troposphere and stratosphere. For the weak vortex composite, Figs. 4.10b

and c show the tropospheric (200 hPa and below) and stratospheric (100 hPa and above)

anomaly correlation between the composite mean of Z*′ and Z*c at 60°N for the full wave field

(solid line) and for the wave-1 component of the wave field (dashed line) as a function of lag.

The anomaly correlation (primarily the wave-1 anomaly correlation) becomes highly positive

and remains highly positive for approximately 40 days before the zero lag in both the

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troposphere and stratosphere. For corresponding plots of the strong vortex composite (Fig. 4.10e

and f) the behaviour is opposite: the anomaly correlation (again, primarily the wave-1 anomaly

correlation) becomes highly negative and remains negative for approximately 40 days before the

zero lag.

Thus, both weak and strong vortex composites exhibit persistent linear interference

(constructive and destructive, respectively) preceding the zero lag. The persistent phasing and

anti-phasing is what gives rise to the persistent positive and negative LIN flux tendencies

illustrated in Fig. 4.5 beginning around a lag of -40 days (statistically significant at around -20

days). The sudden switch in the sign of the anomaly correlation at the zero lag implies a sudden

weakening or strengthening of the part of the wave anomaly that projects onto the background

climatological wave, for the weak and strong vortex composites, respectively. Figs. 4.10b-e

suggest that identification of anomalous wave-1 patterns that constructively or destructively

interfere with the background climatological wave-1 field for several weeks may assist with

seasonal prediction of extratropical winter variability. Similar phase and anti-phase-locking was

illustrated in Fig. 3.9 for the winter of 2009-2010. After the zero lag, the anomaly correlations in

the troposphere and stratosphere in Fig. 4.10 become uncoupled for both composites and the

magnitude of the LIN flux weakens (Figs. 4.5a and b).

Further analysis demonstrates that there is little coherent change in the wave-1 or wave-2

amplitudes of Z*′ at 100 hPa and 60°N preceding the zero lag for either the weak or strong vortex

composites (the one exception is a slight increase in wave-1 amplitude preceding weak vortex

events; not shown). Thus, the primary process responsible for generating the LIN fluxes in these

composites is phasing or anti-phasing between Z*′ and Z*c. Given that there is little change in

amplitude preceding the zero lag of the weak and strong vortex composites, changes in the

baroclinic structure of Z*′ must be responsible for the NONLIN flux contribution to the

composites shown in Fig. 4.5. In other words, positive and negative NONLIN fluxes are

associated with waves whose baroclinicity (either anomalous westward or eastward tilt with

height) is changing.

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FIG. 4.10. (a) Phase difference between the composite mean Z*′ and Z*c at 60°N averaged over

days [-30,-1] for the weak (red curve) and strong vortex composites (blue curve). (b) and (d) stratospheric anomaly correlation between the composite mean Z*′ and Z*

c at 60°N at 100 hPa for the full wave field (solid curve) and the wave-1 component (dashed curve) for the weak and strong vortex composites, respectively. (c) and (e) same as (b) and (d) but for the tropospheric anomaly correlation.

Since Figure 4.10 showed that the wave-1 component of the composite mean Z*′ is

strongly in- and out-of-phase with Z*c preceding the zero lag for the weak and strong vortex

composites, respectively, time-evolving changes in vertical tilt are likely attributed to other

waves (recall that weak vortex events are also preceded by a slight increase in wave-1

amplitude). To explore this, the longitude-pressure cross-section of the weak and strong vortex

composite mean wave-2 contributions to Z*′ (contours) superimposed on Z*c (shading) averaged

over days [-15,-1] is plotted in Fig. 4.11. Figure 4.11a demonstrates that for the weak vortex

composite the anomalous wave is more westward tilted with height, indicating enhanced upward

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wave activity flux while Fig. 4.11b demonstrates that for the strong vortex composite the

anomalous wave is more eastward tilted with height, indicating reduced upward wave activity

flux. Although anomalously eastward tilting wave-2 is observed in the strong vortex case, this

effect is considerably smaller than the linear interference effect (Fig. 4.5b). There are also

contributions to the NONLIN fluxes from smaller waves but these contribute very little to the

fluxes at 100 hPa.

Perlwitz and Harnik (2003, 2004), Shaw and Perlwitz (2010) and Shaw et al. (2010)

discuss the occurrence of wave reflection in the Northern Hemisphere and find it to be a

moderate contribution to stratosphere-troposphere interactions in winter. Because the above

composites are based on {v*T*}′ rather than on the total {v*T*}, there is no direct link between

negative NONLIN fluxes and wave reflection. However, given that wave reflection would

correspond to relatively large {v*T*}′, it is likely that some of the strong vortex events with large

NONLIN fluxes correspond to wave reflection events.

In summary, the relative contributions of LIN and NONLIN fluxes to {v*T*}′ events in

the NH differ between weak and strong vortex events. The weak vortex composite has relatively

larger LIN fluxes in the lower stratosphere. This reflects the effect of combining early and late

winter weak vortex events consisting of larger contributions from the LIN fluxes and mid-winter

events consisting of larger contributions from NONLIN fluxes. In contrast, strong vortex events

primarily consist of LIN fluxes. Many of the events in the weak and strong vortex composites are

events that consist of predominantly LIN or predominantly NONLIN fluxes. Thus, LIN and

NONLIN weak and strong vortex events combine to give the features of the weak and strong

composites in Fig. 4.5.

For both composites, the sign, amplitude and timing of the LIN fluxes is dominated by

the relative phase of the wave anomaly and the climatological stationary wave; changes in wave

anomaly amplitude have relatively little effect. The timescale of the wave anomaly phase is

relatively long preceding the zero lag, which suggests that monitoring the phase of the wave

anomaly Z*′ (relative to that of the climatological wave Z*c) may help improve wintertime

seasonal prediction. Finally, NONLIN fluxes are primarily associated with anomalous vertical

tilt of wave-2 rather than with changes in wave amplitude.

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FIG. 4.11. Composite mean Z*′ (contours) and Z*c (shading) at 60°N averaged over days [-15,-1]

for the (a) weak and (b) strong vortex composites. Contour interval is 5 m.

4.3.3 Stratospheric Sudden Warming Events and Linear Interference

The most dramatic and identifiable stratospheric NAM events are SSWs, which are defined as a

reversal of the zonal mean zonal wind at 10 hPa and 60°N. In this analysis, SSW events are

identified based on the zonal mean circulation and not on the wave heat flux anomalies as in the

previous section. Thus, these SSW events represent a distinct set of events from the weak and

strong vortex events identified in Section 4.3.2.

In this Section, the relative contributions of LIN and NONLIN fluxes to SSWs are

investigated. For the time period over which the weak vortex events of Section 4.3.2 were

selected there were 19 SSWs; 12 of these SSWs correspond to weak vortex events according to

the previous classification. That is, these SSWs were the largest weak vortex events in their

respective years. Because the selection method for anomalous heat flux events can include only

one event per year, two of the seven SSWs that are not detected by this method are simply

missing due to the fact that the winters of 1987-88 and 1998-99 both had two SSWs. The left

column in Fig. 4.12 shows the composite mean daily {v*T*}′, LIN and NONLIN fluxes (Fig.

4.12a, d and g) for 33 SSWs from 1958-2009 and the corresponding composite mean S(Zpcap′) as

a function of time and pressure (Fig. 4.12j). Daily rather than 40-day averaged {v*T*}′ are used

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in order to directly compare this analysis with other published work on SSWs (Charlton-Perez

and Polvani 2007; Nishii et al. 2009; Cohen and Jones, in press). SSWs are associated with

downward-propagating positive S(Zpcap′), i.e., negative NAM anomalies (Fig. 4.12j). At first

glance, SSWs also appear to be preceded by roughly equal contributions from both positive LIN

and NONLIN fluxes (Fig. 4.12a, d and g). Based on these figures, the heat flux characteristics of

SSWs are similar to those of the weak vortex composite in Figs. 4.4a-c and 4.5a and c. Like the

weak and strong vortex composites in the NH (Section 4.3.2), SSWs also consist of distinct LIN

and NONLIN events.

This is shown by further classifying SSWs into displacement (D) and split (S) events.

Charlton-Perez and Polvani (2007) demonstrated that D and S events are preceded by anomalous

wave-1 and wave-2 heat fluxes, respectively. In addition, they showed that the magnitude of the

heat flux anomalies preceding D events is weaker but more persistent than the stronger and more

pulse-like heat flux anomalies preceding S events (their Fig. 8). The dominance of wave-1 in

LIN fluxes and the tendency for wave-2 fluxes to be mostly NONLIN found in the Section 4.3.2

(see Figs. 4.10 and 4.11), suggests that the nature of the heat flux anomalies associated with D

and S events may differ. The middle column shows the composite mean {v*T*}′, LIN (wave-1

only) and NONLIN terms for the 20 D events and the right column shows the corresponding

terms for the 13 S events (LIN for S events includes all wave numbers and NONLIN is wave-2

only; Fig. 4.12i). D and S events are clearly distinguishable by the nature of {v*T*}′ preceding

them; D events are preceded by an increase in wave-1 LIN heat fluxes (Fig. 4.12e) and S events

are preceded by a pulse of wave-2 NONLIN heat fluxes (Fig. 4.12i). Fig. 4.12e also

demonstrates the persistence of the LIN fluxes preceding the D events relative to the NONLIN

fluxes preceding the S events in Fig. 4.12f. Thus, the heat flux anomaly decomposition provides

evidence for differing processes preceding displacement and split SSW events.

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FIG. 4.12. SSW composite mean daily heat flux anomaly decomposition for (a) {v*T*}′, (d) LIN and (g) NONLIN. (b), (e) and (h) and (c), (f) and (i) same as (a), (d) and (g) but for D SSWs (LIN fluxes are wave-1 only) and S SSWs (NONLIN fluxes are wave-2 only). (j)-(l) shows the composite mean S(Zpcap′ ) for SSWs, displacement (D) SSWs and split (S) SSWs, respectively. Black contour indicates 95% significance.

These results are consistent with Martius et al.’s (2009) description of wave-1

constructive interference preceding displacement events but differ somewhat from their

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conclusion that split SSWs are associated with constructive interference of wave-2. Instead, this

result suggests that instrinsic wave activity associated with wave anomalies themselves drives

split SSWs. It is notable that Cohen and Jones (in press) illustrate that displacement events are

preceded by a zonally asymmetric tropospheric circulation pattern that is consistent with the

enhancement of LIN fluxes preceding these events.

Fig. 4.12 also demonstrates different processes driving the suppression of wave activity

flux following the different types of SSW events. D events are proceeded by a suppression of

primarily NONLIN fluxes while S events are proceeded by strong LIN fluxes (i.e. strong

negative interference). The corresponding composite mean S(Zpcap′) anomaly for D and S events

are shown in Figs. 4.12k and l. As pointed out by Charlton-Perez and Polvani (2007), there is a

weakening of the vortex prior to the wind reversal for D events. This analysis shows that the

weakening can be attributed to the persistent phase locking between the anomalous wave and the

background wave prior to the displacement event. Figs. 4.12k and l also demonstrate that the

warming associated with the D SSWs is somewhat weaker than that of the S SSWs; however,

this warming initially appears to extend further into the troposphere. This is consistent with the

greater extension of S(Zpcap′) into the troposphere for LIN weak vortex events (Fig. 4.9c). A

greater number of events are required to confirm whether there really is a significant difference

in the downward propagation of GPH anomalies between the two types of warmings.

4.3.4 Comparison between Northern and Southern Hemisphere

In this section, the role of linear interference in SH stratosphere-troposphere interactions is

explored. Although the SH has markedly weaker stationary waves than the NH and stratosphere-

troposphere interactions are also weaker, the heat flux characteristics are similar in many ways in

the two hemispheres. Figure 4.13a shows the climatological mean meridional wave heat flux

decomposition at 100 hPa averaged over 40-80°S for each month following Eqns. (4.1) and

(4.4). For ease of comparison with the NH, the sign of the heat fluxes throughout this section are

such that positive heat fluxes are poleward rather than northward. As for the NH case in Section

4.3.1, only the first two terms on the RHS of Eqn. (4.4) are shown. The {v*T*}c peaks in October

(austral spring) and reaches a minimum in January (austral summer). Relative to the NH, the

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peak {v*T*}c is somewhat smaller while the minimum is greater, resulting in a weaker seasonal

cycle in the SH {v*T*}c. There is a secondary maximum in {v*T*}c in late fall/early winter,

consistent with greater planetary wave amplitudes at this time of year (see also Randel 1988;

Plumb 1989). Throughout the year, the main contribution to {v*T*}c is from the climatological

mean of the heat flux associated with the wave anomalies, {v*′ T*′}c, except during October when

the climatological mean heat flux from the stationary waves, {v*cT*

c}, is slightly greater. The

lack of topographic and land-sea features in the SH relative to the NH partially accounts for the

weaker {v*cT*

c} fluxes. The strong polar vortex in winter also inhibits vertical wave propagation

(Charney and Drazin 1961) resulting in weaker {v*cT*

c} fluxes. As in the NH, the {v*cT*

c} fluxes

consist almost entirely of low-frequency waves. The {v*′ T*′}c fluxes consist of contributions

from both low- and high-frequency waves with the latter contributing slightly more than double

throughout the year; this is in contrast to the NH, where the low- and high-frequency

contributions to {v*′ T*′}c are comparable. As in the NH, the seasonal cycle in the {v*′ T*′}c

fluxes is weaker than in the {v*cT*

c} fluxes.

Figure 4.13b shows the monthly {v*T*} variance decomposition using Eqn. (4.3). The

total variance grows slowly over the austral autumn and winter, but there is a doubling in the

variance from August to September with a peak in October. Unlike the NH, where the peak in

interannual variability follows the peak in {v*T*}c by one month, the peak in interannual

variability in the SH coincides with the peak in the {v*T*}c. Late winter and spring are when

stratosphere-troposphere interactions are most frequent in the SH (Thompson et al. 2005). In late

spring and summer, the variance decreases sharply, reaching a minimum in February (see also

Randel 1988). From August to November, the variance of the LIN fluxes is the largest

contribution to the total variance. The variance of the NONLIN fluxes dominates from December

to July, except in June when the LIN variance is slightly greater. In June, the larger LIN

contribution to the variance reflects the weaker stratospheric winds in the early winter season

allowing for greater amplitude and vertical propagation of the stationary wave field than in July

(Plumb 1989; Yoden 1990; Scott and Haynes 2002). This is also shown in the {v*cT*

c} fluxes in

Fig. 4.13a. The covariance between LIN and NONLIN is typically negative although it is not

statistically significant and weaker in the SH relative to the NH. It is negative in the troposphere

throughout the year but it is positive in the stratosphere during fall and winter, particularly in

June.

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FIG. 4.13. SH meridional wave heat flux decomposition at 100 hPa averaged over 40-80°S. (a) Climatological monthly mean (see Eqns. (4.1) and (4.4)). Total and high- and low-frequency components are plotted (see legend). (b) Monthly variance decomposition (see Eqn. (4.3)). No points on green curve in (b) are statistically significant at the 95% level.

As in the NH, the LIN fluxes primarily consist of low-frequency waves, while the

NONLIN fluxes consist of both low- and high-frequency wave contributions, the latter being

greater in summer and fall (not shown). This is also reflected in the autocorrelation functions of

LIN and NONLIN as in the NH (not shown). Thus, despite the greater contribution to the mean

heat flux from the ANOMc fluxes throughout the year in the SH, when stratosphere-troposphere

interactions are most frequent, the {v*T*} variance, which represents interannual variability in the

SH wave activity flux, is dominated by low-frequency LIN fluxes, similar to the NH.

How does the relative contribution of LIN to the {v*T*} variance carry over to extreme

events in the SH? Using the maximum and minimum composite method, 29 anomalously high

heat flux events and 30 anomalously low heat flux events (hereafter, weak and strong vortex

events, respectively) are identified in the period 1979-2009 (the high heat flux event of 2002 was

the major SSW there; this event is excluded). The mean date of the weak vortex events is

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November 10th with a standard deviation of 66 days; the mean date of the strong vortex events is

October 21st with a standard deviation of 73 days. Similar to Fig. 4.5, Figs. 4.14a and b show

composites for {v*T*}′, LIN and NONLIN for weak and strong vortex events in the SH,

respectively (recall positive {v*T*}′ correspond to poleward {v*T*}′). For both composites, it is

observed that the NONLIN fluxes contribute slightly more to {v*T*}′. As in the NH, the LIN

fluxes increase linearly towards the zero lag while the NONLIN fluxes increase in a pulse-like

way very close to the zero lag. The corresponding S(Zpcap′) in Figs. 4.14b and d show robust

negative and positive stratospheric Southern Annular Mode (SAM) signatures, respectively. The

composite mean {v*T*}′ for both composites is slightly weaker in the SH than the NH, which

may partially explain the somewhat weaker S(Zpcap′). Also, given the stronger and less variable

climatological polar vortex in the SH it is not surprising that the stratosphere-troposphere

coupling is weaker.

As was done for the NH, complementary composites of maximum and minimum LIN and

NONLIN fluxes are constructed. While in the NH there were more LIN events that coincided

with the weak and strong vortex events, in the SH there are more NONLIN events that coincide

with weak and strong vortex events, 17 and 14 events, respectively. Three of the 17 NONLIN

weak vortex events are also LIN weak vortex events. Although there are events that are

predominantly LIN or predominantly NONLIN in the SH composites, the features of the weak

and strong vortex composites do not reflect a combination of the features associated with the

LIN and NONLIN weak and strong composites to the extent that they do in the NH (not shown).

Thus, individual weak and strong vortex events must have LIN and NONLIN fluxes sometimes

occurring simultaneously. This is not entirely inconsistent with Fig. 4.13b as the anti-correlation

between LIN and NONLIN in the climatology is not robust and does not necessarily represent

the behaviour during these specific extreme events.

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FIG. 4.14. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for SH (a) weak and (b) strong vortex events. Solid sections of the curves indicate 95% significance. Composite mean S(Zpcap′) for SH (c) weak and (d) strong vortex events. Contour interval is [-1.5 -1 -0.5 -0.25 0.25 0.5 1 1.5]. Black contour indicates 95% significance.

Since the LIN and NONLIN fluxes are both important in the SH, there are potentially

several processes determining weak and strong vortex events: linear interference, including both

phasing and amplitude effects, and the vertical phase tilt of the anomalous waves. The linear

interference diagnostics shown in Fig. 4.10 for the NH are very similar for the SH. The main

differences include slightly weaker-magnitude anomaly correlations between Z*′ and Z*c

preceding the zero lag, particularly in the troposphere, for both the weak and strong composites.

This is likely due to the less stationary nature of long waves in the SH (Manney et al. 1991).

However, the SH also exhibits persistent phasing and anti-phasing of up to 30 days preceding the

zero lag for weak and strong vortex events, respectively. As in the NH, it is found that the

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NONLIN fluxes are primarily wave-2 and result from changes in the baroclinicity of wave-2

rather than changes in amplitude. Thus, LIN heat flux anomaly generation appears to be similar

in the two hemispheres.

Distinct differences appear between the hemispheres when the threshold sensitivity is

tested. Fig. 4.15a shows that the contribution of LIN fluxes to weak vortex events in the SH is

essentially independent of the threshold value, whereas in the NH, for high threshold values, the

contribution from LIN decreases as the contribution from NONLIN increases nonlinearly. This is

consistent with the fact that the 40-day averaged {v*T*}′, LIN and NONLIN distributions are all

positively skewed ({v*T*}′ skew = 0.65, LIN skew = 0.34 and NONLIN skew = 0.42) while in

the NH the NONLIN fluxes are positively skewed and the LIN fluxes are slightly negatively

skewed. Moreover, Fig. 4.15b shows that for strong vortex events, the LIN fluxes are roughly

constant and the NONLIN fluxes decrease linearly over a wide range of thresholds. Unlike the

NH, the vertical shear of the strong polar vortex in the SH causes greater wave reflection and

thus, relatively larger negative {v*T*}′ (Shaw et al. 2010). This likely contributes to stronger

negative NONLIN fluxes in the SH strong vortex composites as the threshold decreases. The

LIN fluxes in the SH strong vortex composites show little threshold dependence except at very

high thresholds. This appears to be due to a trade-off between wave-1 destructive interference

and wave-2 constructive interference with decreasing threshold, although this has not been

examined in detail. This is consistent with increased anomalously eastward-tilting wave-2 as the

climatological wave-2 in the SH is generally reflective (Harnik et al. 2005). Recall that at very

high thresholds the composites comprise a very small number of events making the interpretation

of the sensitivity of LIN fraction to threshold value difficult (green curves in Figs. 4.15a and b).

In summary, despite the markedly weaker stationary waves in the SH compared to the

NH, the LIN fluxes continue to play an important role in {v*T*}′ events. The LIN and NONLIN

fluxes contribute roughly equally to {v*T*}′ events in the SH. The phasing behaviour of LIN

events is similar in the two hemispheres. Anomalous baroclinicity of wave-2 appears to play a

greater relative role in the SH as illustrated by the larger contribution from the NONLIN fluxes

in Fig. 4.14. In terms of anomalously eastward tilting waves, the SH and NH differ in that while

in the NH, negative NONLIN fluxes certainly exist, they do not play as important a role in

generating strong vortex events as in the SH.

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FIG. 4.15. Sensitivity of composite mean 40-day averaged {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) and of the number of events per composite (green curve) to a standardized {v*T*}′ threshold value (in units standard deviation) for SH (a) weak and (b) strong vortex events.

4.3.5 Stratospheric Final Warmings and Linear Interference

To conclude the discussion of linear interference in stratosphere-troposphere interactions,

stratospheric final warmings (SFWs) are investigated. SFWs have been associated with a

coupled tropospheric circulation anomaly that resembles the negative phase of the North Atlantic

Oscillation (Black et al. 2006). From a seasonal forecasting perspective, the effect of the

variability in the timing of SFWs on the tropospheric circulation is of particular interest

(Ayarzagüena and Serrano 2009; Hardiman et al. 2011). The timing of the SFW also has a

significant influence on polar stratospheric ozone concentrations (Salby and Callaghan 2007;

Hurwitz et al. 2010). In this section, the relative role of linear interference in early and late final

warmings is investigated.

For the analysis of final warmings, the data has not been detrended. The mean date of NH

SFW onset is April 20th with a standard deviation of 18 days and the mean date of SH SFW onset

is December 7th with a standard deviation of 12 days. Of the 30 SFWs in both the NH and SH,

the 10 earliest and 10 latest are chosen for “early” and “late” composites of SFWs. The mean

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dates of “early” and “late” SFW onset are April 1st with a standard deviation of 10 days and May

10th with a standard deviation of 9 days in the NH and November 23rd with a standard deviation

of 4 days and December 19th with a standard deviation of 6 days in the SH. In the SH, the

division into “early” and “late” events not only reflects the interannual variability of the

stratosphere but also reflects a trend, with “late” events tending to be in the latter part of the

climate record due to the cooling of the SH stratosphere associated with ozone depletion (Black

and McDaniel 2007b; Chapter 4 of SPARC CCMVAL 2010; Thompson et al. 2011).

FIG. 4.16. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) “early” and (b) “late” NH SFWs. Solid sections of the curves indicate 95% significance. Composite mean Zpcap′ for (c) NH and (d) SH final warmings. Contour interval is […, -40, -20, -10, -5, 5, 10, 20, 40,...]. Black contour indicates 95% significance.

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Figures 4.16a and c and b and d show the composite mean 40-day averaged {v*T*}′

(black curve), LIN (red curve) and NONLIN (blue curve) flux anomalies and Zpcap′ for “early”

and “late” NH SFWs, respectively. “Early” SFWs are associated with positive {v*T*}′

(Ayarzagüena and Serrano 2009), consisting of both LIN and NONLIN, from the troposphere to

the stratosphere (Fig. 4.16a) and an associated positive Zpcap′ anomaly (Fig. 4.16c). The

magnitudes of the heat flux anomalies are considerably weaker than for NH weak vortex events

and SSWs and are only marginally significant. It appears that the initial increase in {v*T*}′

preceding the zero lag is associated with an increase in LIN. Of the 10 “early” SFWs, six are

predominantly LIN events and only one is clearly a NONLIN event. “Late” SFWs, on the other

hand, are associated with primarily negative LIN anomalies. These differences between “early”

and “late” SFWs are consistent with the differences between the weak and strong vortex

composites in the NH, which reflect the positive skew of the NONLIN fluxes in the NH (Figs.

4.5 and 4.7).

These results are connected to the seasonal cycle of the different wave heat flux

contributions to interannual variability. As discussed in Section 4.3.1, the {v*T*} variance

decomposition reveals a positive anti-correlation between LIN and NONLIN in the stratosphere

in late spring (Figs. 4.1b and 4.2). This appears to be related to the suppression of both LIN and

NONLIN anomalies as the vortex switches from westerlies to easterlies. In addition, the variance

of {v*T*} transitions from the winter regime, characterized by the low-frequency LIN variance

term in Eqn. (4.5), to the summer regime, characterized by a relative increase in the contribution

of the high-frequency NONLIN variance term. Despite the changes in the relative contributions

to the {v*T*} variance in late spring, “late” NH SFWs are dominated by linear interference

effects.

SH SFWs consist of weaker {v*T*}′ than in the NH (Figs. 4.17a and c). Interestingly, the

“early” and “late” SFWs in the SH are both associated with predominantly LIN flux anomalies

preceding the zero lag, positive for the “early” composite and negative for the “late” composite.

The dominance of the LIN term in SH SFWs is somewhat surprising given that the SH weak and

strong vortex composites displayed approximately equal contributions from LIN and NONLIN

flux anomalies. Hurwitz et al. (2010) suggest that the late bias in the SH SFW onset date in

coupled-chemistry models (CCMs) is related to weak stationary wave amplitudes and

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insufficient westward tilt with height which, in the {v*T*}′ decomposition, would be captured in

the NONLIN flux anomalies. Figs. 4.17b and d further suggest that discrepancies in the phasing

of the wave anomalies with respect to the background climatological wave may also lead to a

late SFW onset bias in CCMs.

FIG. 4.17. Composite mean 40-day averaged heat flux anomaly decomposition at 100 hPa; {v*T*}′ (black curve), LIN (red curve) and NONLIN (blue curve) for (a) “early” and (b) “late” SH SFWs. Solid sections of the curves indicate 95% significance. Composite mean Zpcap′ for (c) NH and (d) SH final warmings. Contour interval is […, -40, -20, -10, 10, 20, 40,...]. Black contour indicates 95% significance.

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In summary, like the previous examples of stratosphere-troposphere interactions,

including weak and strong vortex events and SSWs, the variability in the timing of SFWs is

associated with significant contributions from linear interference effects. In the NH, “early”

SFWs are associated with anomalously strong heat fluxes resulting from positive LIN and

NONLIN flux anomalies while “late” SFWs are associated with anomalously weak heat fluxes

which are predominantly LIN. In the SH, both “early” and “late” SFWs are associated with

predominantly LIN flux anomalies – positive for the “early” composite and negative for the

“late” composite.

4.4 Conclusions

This chapter examines the climatology and diagnoses the importance of linear interference in NH

and SH stratosphere-troposphere interactions. In both hemispheres, LIN fluxes consist of low-

frequency waves throughout the entire year while NONLIN fluxes consist of mostly low-

frequency waves during the active stratospheric vortex season. Interestingly, LIN and NONLIN

fluxes in both hemispheres are anti-correlated, in general, although the significance of this anti-

correlation is marginal in the winter months; this reflects a tendency for LIN and NONLIN

contributions to partially cancel. Temporal auto-correlation analysis demonstrates that the LIN

fluxes enhance the persistence of heat flux anomalies via persistence of the zonal phase of the

wave anomalies.

In the NH, anomalously high heat flux events (weak vortex events) consist of a greater

contribution from NONLIN fluxes than anomalously low heat flux events (strong vortex events),

due in part to the positive and negative skew of the NONLIN and LIN flux distributions,

respectively. It is found that many of the NH weak and strong vortex events correspond to events

that consist of predominantly LIN or predominantly NONLIN fluxes. Thus, the composite mean

reflects a combination of relatively distinct LIN and NONLIN events. Composites of LIN fluxes

recover much of the stratosphere-troposphere coupling signature associated with the weak vortex

composites (Figs. 4.7 and 4.8). Linear interference diagnostics demonstrate that the time

evolution of the phasing of the wave-1 anomaly with the wave-1 climatological wave explains

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most of the time evolution of the spatial correlation of the full wave anomaly with the

climatological wave field. In addition, phasing throughout the depth of the troposphere and

stratosphere appears necessary in order to establish coherent LIN fluxes. Phase or anti-phase

locking between wave anomalies and the climatological wave begins approximately 40 days

before the anomalous heat flux event. Polvani and Waugh (2004) demonstrate that 40-day

averaged heat flux anomalies are the best predictors of stratospheric NAM events. Based on the

above analysis, the significance of a 40-day averaged heat flux anomaly appears to be partly

related to the persistence of the LIN fluxes (see Figs. 4.4b and e. 4.5a and b and 4.9b and d). The

persistence of the anomalous wave patterns associated with the LIN fluxes suggests that the

identification of multiple week trends in the phase structure of these patterns may improve

seasonal prediction.

NONLIN fluxes represent changes in the baroclinicity of the anomalous wave,

particularly wave-2, with positive NONLIN fluxes representing an anomalously westward-tilting

wave and negative NONLIN fluxes representing an anomalously eastward-tilting wave. The

connection between negative NONLIN fluxes and wave reflection is unclear from this analysis

given that the composites are based on heat flux anomalies rather than the full heat fluxes.

Quantifying the contribution to NONLIN from wave reflection warrants further investigation.

As an additional application of the above results, analysis of the anomalous heat flux

contributions to SSW events is conducted. It is shown that like the anomalous NH heat flux

events discussed in Section 4.3.2, SSW events can be separated into distinct LIN and NONLIN

events. These distinct events correspond to displacement and split SSWs, respectively. Thus, the

characteristics of the heat flux anomalies associated with vortex displacements and splits are

unique. Due to the persistence of anomalous wave patterns preceding LIN events, the work

suggests that displacement SSWs may be potentially predictable.

In the SH, weak and strong vortex events consist of roughly equal contributions from

LIN and NONLIN fluxes. There are more distinct NONLIN events in the SH composites. The

composites do not reflect a combination of distinct LIN and NONLIN events to the same extent

as they do in the NH, suggesting a degree of positive correlation between the fluxes during these

events. As in the NH, linear interference diagnostics show that phase or anti-phase locking

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begins roughly 30 days before the anomalous heat flux events. The sign of NONLIN fluxes

primarily results from anomalous vertical tilt of wave-2 rather than from wave amplitude

changes. Positive NONLIN fluxes correspond to an anomalously westward-tilting wave-2 while

negative NONLIN fluxes correspond to an anomalously eastward-tilting wave-2.

Finally, a comparison of “early” and “late” stratospheric final warming composites in

both the NH and SH reveals that these events are associated with weak heat flux anomalies

consisting of a substantial contribution from LIN, particularly in the SH.

In summary, as DeWeaver and Nigam (2000a) demonstrated for momentum fluxes and

zonal wind variability in the troposphere, Chapter 4 demonstrates that linear interference effects

are an integral part of heat flux variability and, consequently the coupled variability of the

stratosphere and troposphere. Taken together, the present study and DeWeaver and Nigam

(2000a) demonstrate that interactions between anomalies and the large-scale zonally asymmetric

circulation appear to be vitally important to Annular Mode dynamics in both the troposphere and

stratosphere. Ongoing work includes establishing a better understanding of the relationship

between negative NONLIN fluxes and wave reflection in the stratosphere and of how the

persistence of the LIN fluxes relates to the timescales of the Annular Modes. In addition, the

extent to which linear interference plays a role in tropospheric Southern Annular Mode (SAM)

variability has yet to be investigated.

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Chapter 5

Conclusions and Discussion

5.1 Summary

The motivation for this thesis stems from the potentially important role of autumn

Eurasian snow cover anomalies in influencing the wintertime NAM. In particular, this thesis asks

how the observed and modeled relationships between snow and the NAM connect to

fundamental questions about external influences on extratropical atmospheric variability. To

address this question and others, the modeled atmospheric response to a prescribed Siberian

snow cover anomaly was revisited in Chapter 2. Outstanding questions regarding the transient

evolution of the negative NAM response in an atmosphere-land GCM (Fletcher et al. 2009a;

Gong et al. 2003) were addressed using a novel decomposition of the meridional wave heat flux.

The decomposition revealed that the heat flux response is dominated by two terms: one that

represents the linear interference between the wave response and the control state wave (EMLIN)

and one that represents the heat flux inherent to the wave response itself (EMNL). Analysis of this

decomposition demonstrated that the time-evolving NAM response was related to the EMLIN

term. The linear interference term was clearly the dominant term in the ensemble mean heat flux

decomposition throughout the duration of the simulation. Further investigation using a relatively

simple GCM (SGCM) revealed that the NAM response to an extratropical surface cooling over

Siberia was of opposite sign to that of previous prescribed snow forcing simulations. Destructive

interference between the wave response and the control state wave resulted in a suppression of

wave activity flux into the stratosphere and a positive NAM response. Additional SGCM

simulations in which the location of the forcing was shifted longitudinally revealed that the sign

and amplitude of the NAM response depended sensitively on the location of the forcing via

constructive or destructive interference between the forced wave and the control state stationary

wave, rather than by changes in the forced wave itself. The dominance of the EMLIN term was a

robust feature in both the comprehensive GCM and the SGCM. It was also shown with the

SGCM that as the forcing strength was increased the EMNL term became the dominant term.

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For atmospheric states that resemble Northern Hemisphere winter conditions, and for

sufficiently small forcing, the linear interference effect is dominated by the phasing between the

wave response and the control state wave. When the control state was altered in the SGCM in

Chapter 2 by changing the strength of the stratospheric polar vortex, it was shown that the

amplitude component and not just the phasing component of the linear interference effect

becomes important. In general, for a wave response and control state stationary wave field that

are dominated by a single wave number component, the phasing of the waves is most important

for determining the strength of linear interference. However, when multiple wave number

components become important, then it appears that the situation is more complicated and details

of both amplitude and phasing of the individual components become important to diagnose the

linear terms.

In Chapter 3, linear interference diagnostics similar to those developed in Chapter 2 were

applied to the observed relationship between October Eurasian snow cover and the wintertime

NAM. It was shown that, as was found in the GCM simulations, the majority of the heat flux

associated with the October Eurasian snow index (OCTSNW) consisted of the linear interference

component (LIN). The two-month lag between the October snow cover anomalies and the

associated December heat flux is attributed to neutral phasing between the wave regressed on

OCTSNW (Z*snow) and the background climatological wave (Z*

c). Z*snow shifts into phase with Z*

c

in December. There is also a corresponding peak in the LIN heat flux regressed on OCTSNW at

this time of year. The shift in Z*snow from autumn to winter is associated with a change in

associated lower tropospheric heating from vertical to horizontal temperature advection and an

eastward shift and intensification of this heating. A decomposition of the thermodynamic

equation revealed that the linear interference terms dominate the heating changes associated with

OCTSNW, which may provide insight into why classical stationary wave dynamics theory (e.g.

Hoskins and Karoly 1981) does not explain the observed shift in Z*snow. A case study of 2009-

2010, a year in which October Eurasian snow cover was extensive, showed that the two large

negative NAM anomalies of that winter also corresponded to heat flux anomalies that consisted

of mostly LIN fluxes.

Chapter 3 also revisited the work of Hardiman et al. (2008), which finds that the CMIP3

GCMs typically show a weak but negative correlation between October Eurasian snow cover and

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December heat flux that is opposite to the observations. It was shown in Chapter 3 that this is

related to destructive interference between the wave train associated with the snow and the

background climatological wave in the models. This result suggests that discrepancies in the

time-dependent horizontal phasing of snow-related waves may contribute to poor simulation of

an important observed aspect of wintertime extratropical variability.

The important role of linear interference in the findings of both Chapters 2 and 3, as well

as related recent literature, suggested that linear interference is likely an important feature of

NAM variability more broadly. Chapter 4 of this thesis examined the role of linear interference

in extratropical stratosphere-troposphere interactions. In Chapter 4 it was shown that in both the

Northern Hemisphere (NH) and Southern Hemisphere (SH), the variance of the total

extratropical meridional wave heat flux from the troposphere to the stratosphere primarily

consists of the variance of low-frequency LIN fluxes in the season of strongest stratosphere-

troposphere interactions (winter and spring, respectively). For the remainder of the year, a large

contribution from the variance of high-frequency NONLIN fluxes is observed. An interesting

aspect of the heat flux variance decomposition is the negative covariance between the LIN and

NONLIN fluxes. This negative covariance is stronger in the NH. In the NH, warm vortex events

comprise approximately equal contributions from LIN and NONLIN fluxes while strong vortex

events comprise mostly LIN fluxes, reflecting the positive skew of NONLIN fluxes. Weak and

strong vortex events are associated with downward-propagating negative and positive NAM

anomalies. Many of the anomalously warm and strong vortex events in the NH are events that

consist of either predominantly LIN or predominantly NONLIN fluxes.

In addition, Chapter 4 revisited the characteristics of displacement (D) and split (S)

sudden stratospheric warmings (SSWs). It was shown that D SSWs, which are primarily wave-1,

are preceded by primarily LIN fluxes and S SSWs, which are primarily wave-2, are preceded by

NONLIN fluxes. This result indicates that D and S SSWs represent relatively distinct LIN and

NONLIN heat flux events. Positive LIN fluxes are associated with persistent wave anomalies

that are in-phase with the background climatological stationary wave, suggesting that D SSWs

may potentially be predictable through early identification and monitoring of such wave patterns.

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Although the stationary wave field is much weaker in the SH, weak and strong vortex

events comprise approximately equal contributions from LIN and NONLIN fluxes and are

associated with weak yet robust negative and positive Southern Annular Mode (SAM)

anomalies. This result demonstrates that low-frequency LIN fluxes still play an important role in

stratosphere-troposphere interactions in the SH. In contrast to the NH, there are fewer weak and

strong vortex events that consist of either predominantly LIN or predominately NONLIN fluxes.

It was also shown that the timing of stratospheric final warmings (SFWs) is related to LIN, with

positive and negative LIN flux anomalies preceding “early” and “late” SFWs, respectively,

particularly in the SH.

Reflecting upon the body of work contained in this thesis, several key points present

themselves as a concluding summary. First, this thesis demonstrated the importance of linear

interference in the modeled response to prescribed anomalous boundary conditions. This work

suggests that the location and strength of a prescribed surface forcing can significantly influence

the sign of the NAM response to the forcing. This result has broader significance than the

midlatitude cooling problem considered in depth here. For example, Fletcher and Kushner (2011)

demonstrate that the observed difference between the NAM associated with El Niño SST’s in the

tropical Pacific and Indian Oceans can be simulated in an atmosphere-land GCM with prescribed

SST forcing and that opposing linear interference effects of the poleward-propagating Rossby

wave responses emanating from the two regions are responsible for the opposing NAM

responses. The NAM response to the tropical Pacific SST forcing is negative and involves

enhanced upward wave activity fluxes resulting from primarily constructive interference of

wave-1. The authors show that if an analogous simulation is performed in which the Tibetan

Plateau is flattened, thus weakening the wave-1 stationary wave, the heat flux and NAM

responses are greatly attenuated. This study demonstrates a potential answer to the question of

how topography influences the NAM response to Siberian snow forcings, posed by Gong et al.

(2004a). Although this is only one example of how linear interference plays an important role in

atmospheric teleconnections, it is anticipated that such dynamics is always going to be a feature

of NAM variability and NAM sensitivity to forcing. In addition, as stated in Section 2.5, the

importance of linear interference cautions modelers against applying unrealistically strong

forcings in order to generate strong responses. This was a common modeling practice in studies

of the dynamical response to tropical SST anomalies (Trenberth et al. 1998). Instead, a more

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dynamically consistent strategy is to employ larger ensembles with realistic forcings to enhance

signal to noise.

Second, this thesis identified several situations where the phasing between the wave-1

components of the anomalous and background climatological waves was the key factor

determining the strength of the linear interference effect, and thus, the anomalous heat flux or

upward wave activity flux. It also identified the limitations of this simple diagnostic. For

example, for the transient response to prescribed snow forcing in a GCM, the relationship

between observed OCTSNW and upward wave activity flux and LIN fluxes in the extratropical

climatology all demonstrate that the phasing of wave-1 is important. While focussing on the

phasing aspect (e.g. Garfinkel et al. 2010; Kolstad and Charlton-Perez 2010) provides a simple

framework for interpreting linear interference effects, it works best when the background

climatological stationary wave and the wave anomaly are dominated by low wave number

components whose amplitudes are well-separated. Chapter 2 identified situations when the clean

relationship between phasing and the strength of the linear interference effect breaks down. The

more quantitative heat flux decompositions of Chapters 2 and 3 (see also Nishii et al. 2009) take

both phasing and amplitude into account and should be used to verify phasing relationships.

Third, this thesis demonstrated that linear interference likely plays an important role in

establishing the two-month lag between October Eurasian snow cover anomalies and the

associated enhanced upward wave activity flux, suggesting that linear interference could

potentially play a role in other seasonally lagged relationships in the climate system. For

example, using the United Kingdom Meteorological Office GCM, Ineson and Scaife (2009)

show that the planetary wave-1 associated with El Niño destructively interferes with the

background climatological wave-1 in autumn but constructively interferes in winter, leading to

upward-propagating wave activity flux into the high-latitude stratosphere and a downward-

propagating negative NAM. This result suggests that destructive linear interference may

contribute to the lag between El Niño and the NAM observed in these simulations.

Fourth, in Chapter 3 it was shown that GCMs do not adequately simulate the constructive

linear interference associated with Eurasian snow cover. This demonstrated that certain observed

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influences on the NAM involve the time-sensitive phasing of quasi-stationary waves, realistic

simulation of which may be a challenging task for model development.

Finally, in Chapter 4, a climatological analysis of stratosphere-troposphere interactions

quantified the importance of linear interference in both the NH and SH. One of the significant

findings of this chapter was that the nature of LIN fluxes is such that they are associated with

persistent anomalous wave patterns throughout the depth of the troposphere and lower

stratosphere, suggesting a degree of predictability for stratospheric AM events. Unlike the EMNL

fluxes in Chapter 2 which were always positive, Chapter 4 demonstrated an anti-correlation

between LIN and NONLIN fluxes in both hemispheres. Although it is unclear what drives this

anti-correlation and whether or not it is a fundamental characteristic of extratropical wave

activity fluxes, it is an interesting feature that warrants further study. Chapter 4 also suggests that

studies using relatively simple GCMs without a zonally asymmetric boundary condition and,

thus, no climatological stationary wave field, are missing an important aspect of stratospheric

AM dynamics. For example, Kushner and Polvani (2005) investigate the 2002 SSW in the

Southern Hemisphere using a simplified atmospheric GCM with no topography. They find that a

SSW can occur even without the existence of large planetary waves through transient baroclinic

wave-wave interactions. However, with respect to the observed 2002 SH SSW, linear

interference effects were important and actually acted to attenuate the event. The amplitude of

the 40-80°S meridional eddy heat flux anomaly at 100 hPa was reduced by ~40% due to

destructive linear interference (not shown). Even when simplified zonally asymmetric boundary

conditions are imposed such as the wave-2 topography of Gerber and Polvani (2009), caution

must be used when drawing comparisons with nature given that contributions to lower

stratospheric heat flux anomalies from LIN and NONLIN likely differ in the absence of a

dominant wave-1 stationary wave.

5.2 Future Work

One of the key questions that remains unanswered in this thesis is what determines the zonal

phase structure of the quasi-stationary wave associated with an anomalous boundary condition

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such as snow cover. The wave response to a prescribed snow forcing in the comprehensive GCM

simulations displayed a ~180° phase shift over the duration of the run. As mentioned in Chapter

2, the SGCM simulations with prescribed surface cooling displayed little shift in the wave

response. The distinct difference in the transient evolution of the wave response in the two

models as well as in the observations exemplifies the fact that the wave response to extratropical

surface diabatic heating anomalies is poorly understood.

Fletcher et al. 2009a (F09) indicate that the transient wave response to a prescribed snow

forcing, consisting of a high upstream of the forcing and a low downstream, was not consistent

with classical linear stationary wave theory (e.g. Hoskins and Karoly 1981; hereafter HK81). For

example, HK81 use scale analysis to argue that for shallow extratropical cooling, the dominant

steady state thermodynamic balance is between diabatic cooling and horizontal temperature

advection resulting in an upstream low/downstream high wave response, opposite to the transient

response of F09. The inconsistencies with classical theory may reflect the fact that the GCM

simulations in Chapter 2 are highly transient. Indeed there is evidence in Fig. 2.1c that the wave

response is approaching the upstream low/downstream high wave response predicted by the

HK81 scaling. However, there is reason to believe that nonlinear effects may also be important.

Figure 5.1 shows temperature tendency and temperature advection responses to the prescribed

Eurasian snow forcing in AM2 at 800hPa and averaged over the Eurasian region. Figure 5.1a

shows the responses using the linearized thermodynamic equation, Fig. 5.1b shows the responses

using the full thermodynamic equation and Fig. 5.1c shows the ensemble mean linear

interference component of the response. Figure 5.1a suggests that the temperature tendency

response dominates for only the first few days, after which time the horizontal temperature

advection response is the largest term balancing the diabatic cooling in the linear thermodynamic

response. However, Fig. 5.1b illustrates that when the full thermodynamic response is analyzed,

the vertical temperature advection response becomes dominant after the first few days. The sign

of the vertical temperature advection is opposite to that predicted by HK81. After approximately

day 50, the thermodynamic response becomes a balance between horizontal and vertical

temperature advection. The majority of the thermodynamic response is recovered by the

ensemble mean linear interference component of the response (Fig. 5.1c) suggesting that the

distribution of climatological temperature advection also plays an important role (note that the

black solid dashed lines in Fig. 5.1c are the same as those in Fig. 5.1b).

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FIG. 5.1. Thermodynamic response to prescribed Siberian snow forcing in AM2 at 800hPa and averaged over the Siberian region for (a) the linearized thermodynamic equation, (b) the full thermodynamic equation, and (c) the EMLIN component of the full thermodynamic equation. The black solid and dashed lines are the same in (b) and (c).

To better understand this problem, one approach is to use a transient linear version of the

SGCM. Diagnosing the transient evolution of the linear wave response to surface cooling may

help to distinguish the relative importance of linear and nonlinear dynamics. Although the

SGCM simulations tend to display little shift in the wave response over time, a few of the cases

do exhibit a shift and could be examined in greater detail with a transient linear model. One

example of this type of model is the transient, linear model of Hoskins and Rodwell (1995) who

examine the transient stationary wave response to diabatic heating and topography. Development

of a linear version of the SGCM used in Chapter 2 is ongoing.

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Further work along these lines may include applying the solutions of the surface quasi-

geostrophic (SQG) equations in the presence of heating to the modeled and observed relationship

between snow and the NAM. The SQG equations represent the reduction of the quasi-

geostrophic (QG) equations on to a horizontal boundary by setting the interior QG potential

vorticity to zero. The two-dimensional flow is then determined by the potential temperature

distribution at the surface. Thus, the SQG framework may be useful for interpreting the

circulation response to diabatic heating or cooling at the surface. Ambaum and Anthanasiadis

(2007) extend the arguments of HK81 using the SQG equations and demonstrate that while the

inhomogeneous response to heating yields a temperature anomaly that is stationary and

proportional to the Hilbert transform of the heating (yielding consistent results to those of

HK81), the homogenous solution yields a temperature anomaly that is eastward propagating.

Although SQG has its limitations for surface flows in the real atmosphere (for example, surface

friction effects lead to higher Rossby number flows that do not obey the SQG scaling), Ambaum

and Anthanasiadis (2007) applied their mathematical findings to the climatological distribution

of surface heating in the NH and found that they could reasonably reproduce the stationary

surface temperature anomaly and also identify eastward-propagating surface temperature

anomalies.

Finally, analysis of the co-spectra of heat and momentum fluxes in models and

observations may also help to indicate whether there is a shift from lower to higher phase speed

waves corresponding to the zonal phase shift of the anomalous wave. Changes in the wave phase

speed may be associated with circulation features common to both models and observations,

such as the tropospheric jet structure in the Eurasian sector (Randel and Held 1991; Simpson et

al. 2011).

A further line of investigation into the discrepancies between the AM2 and SGCM

simulations and the reanalysis in the timing of the zonal phase shift of circulation anomalies

associated with snow cover anomalies is to establish the role of an interactive ocean in

modulating the transient response to snow cover forcings in a GCM. To date there have been no

prescribed snow forcing simulations completed using a fully coupled atmosphere-land-ocean

GCM. The difference between high and low annual Eurasian snow cover simulations by G.

Henderson et al. (unpublished) using an atmosphere-land model coupled to a slab ocean shows a

135

more global circulation response relative to simulations with an atmosphere-land model with

climatological SST’s. The simulations by Henderson et al. (unpublished) suggest that coupling to

an ocean may strongly modify the circulation response to prescribed October Eurasian snow

cover anomalies. Current modeling work by the author involves conducting large ensemble

coupled and uncoupled October Eurasian snow forcing simulations using the NCAR CCSM4

model. Unlike the fixed snow forcing simulations of Fletcher et al. (2009a), the current

simulations apply a Siberian snow forcing of 60 cm at the first time step and then allow the snow

to equilibrate with the surrounding land and atmosphere. This method of applying a Siberian

snow forcing may generate a response that more closely resembles the transient evolution of the

circulation associated with snow cover in nature and may provide greater insight into what

determines the evolution of the wave response.

Although Chapter 3 provided additional insight into the dynamics of the observed

Eurasian snow-NAM connection, many questions remain regarding the thermodynamic

processes establishing the diabatic cooling associated with the snow cover in nature. In the AM2

simulations of Chapter 2, the largest change in the surface energy budget in response to the

imposed snow cover perturbation was a decrease in incoming shortwave radiation at the surface

due to the increased surface albedo (Fletcher et al. 2009a). In the NCEP reanalysis data,

however, the relationship between snow cover and surface shortwave radiation is somewhat

weak. Figure 5.2 shows the correlation between OCTSNW and the October all-sky incoming

surface shortwave radiation flux.

136

FIG. 5.2. Correlation between OCTSNW and October NCEP all-sky incoming surface short wave radiation flux for 1972-2008 over the Eurasian region. Positive and negative contours are red and blue, respectively, and gray shading indicates regions where the correlation is significant at the 95% level.

In October there are regions with correlation coefficients of up to -0.4 yet the correlation

is not robust over a large part of the Eurasian region. In November and December, correlations

over Eurasia are weak, consistent with the fact that October Eurasian snow cover anomalies are

not correlated with either November or December Eurasian snow cover anomalies (not shown).

It is important to note that there is a documented error in the NCEP reanalysis snow data such

that NOAA 1973 snow cover was assimilated for the years 1973-1994 inclusive

(http://www.cpc.ncep.noaa.gov/products/wesley/ek.snow.html). This error was not corrected as it

was shown to have only a small effect on surface temperatures. Although Rutgers snow data is

used throughout the thesis, errors in the NCEP snow cover may create inconsistencies in surface

radiation fluxes, particularly in autumn when interannual variability in snow cover is great.

Future research along these lines should address the following questions: How robust is

the negative correlation between OCTSNW and net incoming flux of shortwave radiation at the

surface? Is it evident in other reanalysis datasets or in models? Recent work by Allen and Zender

(in press) reports that the correlation between the Moderate Resolution Imaging

137

Spectroradiometer (MODIS) albedo product and autumn Eurasian snow cover is quite high

(~0.85). Yet, this strong signal is not reflected in regressions between net incoming shortwave

and Eurasian snow cover in Fig. 5.2. A second question is whether other components of the

surface energy budget are important in establishing the surface cooling associated with

OCTSNW. Finally, a reasonable question that often arises within the context of the snow-NAM

problem is whether or not there is a role for Arctic sea-ice. The annual Arctic sea-ice minimum

occurs in September, one month prior to the month of greatest Eurasian snow cover anomalies.

Could anomalous Arctic sea-ice extent potentially influence October Eurasian snow cover via

changes in precipitation? In fact, there is no correlation between September Arctic sea-ice area or

concentration and OCTSNW when detrended data are analyzed (R2 = 0.16). However, there is a

weak negative correlation when the trend is retained (Ghatak et al. 2010), i.e. decreasing sea-ice

extent in September is associated with increasing Eurasian snow cover in October. Thus, the

question then becomes whether sea-ice retreat could influence the NAM trend via enhanced

snow cover in the future and to what extent this effect could be important relative to the direct

thermodynamic effects of sea-ice retreat on the NAM (Deser et al. 2010).

With the upcoming release of Climate Model Intercomparison Project 5 (CMIP5) data,

answers to some of the above questions may be answered. CMIP5 is the latest World Climate

Research Programme (WCRP) climate model experiment which includes simulations for the

Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (AR5). There are

well known deficiencies and inconsistencies in the representation of snow processes in the

CMIP3-generation models, as was shown, for example, by the work on snow albedo feedback of

Hall and Qu (2006) and Qu and Hall (2006, 2007). Consequently, several modeling groups have

improved their representation of these processes (e.g. NCAR CCSM4, GFDL CM3). For

example, CCSM4 snow process improvements include aerosol deposition on snow and grain-size

dependent snow ageing and CM3 includes better representation of snow interception by

vegetation (http://www.cesm.ucar.edu/models/ccsm4.0/notable_improvements.html;

http://www.gfdl.noaa.gov/land-model). An important research question is then how these

improvements affect the simulation of the observed snow-NAM correlation in the twentieth

century simulations in the CMIP5 models. Future work along these lines includes conducting

similar analysis to that of Chapter 3 and Hardiman et al. (2008) on the next generation of climate

models. In the event that a robust Eurasian snow-NAM connection can be established in models,

138

this would then allow for substantial follow-on investigation. Unlike the limited time series

available for observational analysis, analysis of the CMIP5 data would consist of over 100 years

of data for each of the roughly 20 climate models participating in the project. Such long time

series may help to establish, for example, relationships between Eurasian snow cover and the

surface energy balance, Arctic sea-ice and SST’s and may also provide insight into the causes of

the zonal shift of the wave train associated with the snow cover. In addition, comparison of a

large number of models allows for identification of common and robust features of the

relationship. On the other hand, failure of the CMIP5 models to simulate the snow-NAM

connection may require a reevaluation of the observed snow-NAM relationship.

Although there has been little mention of the relationship between North American snow

cover and the NAM in this thesis, studies have shown that the relationship is weakly positive and

confined to the troposphere (Gong et al. 2003; Sobolowski et al. 2007, 2010; Klingaman et al.

2008; Allen and Zender 2010). An underlying assumption in many prescribed snow forcing

simulations is that Eurasian and North American snow cover and their relationships with the

NAM are independent. Despite the fact that October Eurasian and North American snow cover

are correlated (ρ = 0.5), a multi-variate EOF analysis of Eurasian and North American snow

cover shows that the assumption of independence of the two continental snow masses is likely

adequate. However, it also shows that the potential for a robust North American snow cover-

NAM relationship is likely poor. The first principal component (PC) explains 91% of the

variance in NH snow cover (excluding Greenland) with a loading pattern consisting of same-

signed weightings for both continents but a five times stronger weighting for Eurasia. The

second PC explains only 9% and has opposite signed weightings for the two continents and a

five times stronger weighting for North America. The mode in which North American snow

cover anomalies are more heavily weighted explains very little of the total snow cover variance,

suggesting that comparing correlations between North American snow cover and the NAM in the

observational record with those in prescribed North American snow forcing GCM simulations is

a difficult task. In addition, North American snow cover anomalies are associated with the

Pacific Decadal Oscillation (PDO), ENSO and the PNA which adds complicating factors to such

analysis (Ge and Gong 2009; Jin et al. 2006; Cayan 1996).

139

Beyond the Eurasian snow-NAM connection, the findings of this thesis concerning the

importance of linear interference in stratospheric NAM variability in the general climatology

warrant further investigation. Although DeWeaver and Nigam (2000a) established the extent to

which linear interference diagnostics are useful in describing tropospheric NAM variability, it is

unclear whether the momentum fluxes associated with linear interference are driving the NAM

changes in the zonal wind or whether they are maintaining them once they are established. They

argue for a positive feedback between the zonal-mean flow and the stationary wave components.

DeWeaver and Nigam’s (2000a) emphasis on the role of stationary waves conflicts with the the

idea that the transient, synoptic-scale waves are mainly responsible for the positive wave-zonal

flow feedback of the NAM (Lorenz and Hartmann 2003). Lag regression and cross covariance

analysis (similar to Lorenz and Hartmann 2003) using the linear interference decomposition of

wave momentum fluxes may help to better elucidate the co-evolution of tropospheric NAM

events and linear interference. In addition, future work will include expanding on the analysis of

DeWeaver and Nigam (2000a) to investigate whether the importance of linear interference in

tropospheric NAM variability exhibits a seasonal dependence. Given the weaker stationary

waves in summer, it is expected that linear interference will be less important in this season.

Comparisons between the tropospheric NAM and SAM may also reveal interesting differences in

the importance of linear interference. Preliminary analysis shows results consistent with the

findings of DeWeaver and Nigam (2000a), that the LIN momentum fluxes contribute

substantially to the interannual variability of tropospheric momentum fluxes in the NH.

Figures 5.3a-e show the variance decomposition of the NH January wave momentum

fluxes (see Eqn. (3.4)) as a function of latitude and pressure. January is representative of other

winter months. The right panel displays the difference between var(LIN) and var(NONLIN). The

variance of the LIN fluxes is clearly greater in the free troposphere in the extratropics.

Interestingly, there is also a region in the tropical upper troposphere where the variance of the

LIN fluxes is consistently greater which may be associated with equatorial planetary wave

intensification (Grise and Thompson 2011).

140

FIG. 5.3. Variance decomposition of NH January wave momentum fluxes (a) var({u*v*}), (b) var(LIN), (c) var(NONLIN), and (d) 2*cov(LIN,NONLIN) . (e) shows the difference between panels (b) and (c). Positive and negative contours are red and blue. Contour interval is 21, 22, 23, etc. Gray shading shows regions where the correlation of LIN and NONLIN is statistically significant at the 95% level.

Figure 5.3e also shows that the covariance between LIN and NONLIN momentum fluxes

is mostly negative, similar to the covariance between LIN and NONLIN heat fluxes presented in

Chapter 4. The covariance is statistically significant in the high-latitude stratosphere but only

weakly significant in the troposphere. Use of a linearized version of the SGCM as discussed

above might reveal whether this anti-correlation is a characteristic of linear wave dynamics or

whether nonlinear effects are important. In addition, analysis of long time series data from a

GCM control run may also prove useful in establishing potentially subtle lead-lag relationships

between LIN and NONLIN fluxes that may help to explain the anti-correlation. An important

follow-on question arises relating to the existence of negative NONLIN heat fluxes: to what

extent do these fluxes correspond to instances of wave reflection (see Fig. 4.9; Perlwitz and

Harnik 2003; Shaw et al. 2011)?

Several recent GCM studies have identified the importance of a well-resolved

stratosphere in accurately simulating certain features of the observed climate (Shaw and

Shepherd 2008; Shaw et al. 2009; Shaw and Perlwitz, 2010; Cagnazzo and Manzini 2009;

Marshall and Scaife 2010). In light of this fact, future work by the author will include the

comparison of the role of linear interference in stratosphere-troposphere interactions in the

control runs of a standard, low-top GCM and a stratosphere-resolving, high-top GCM. Shaw and

Perlwitz (2009) noted important improvements in the representation of stratospheric planetary

141

waves in stratosphere-resolving simulations relative to reanalysis. Differences in the

representation of stratospheric planetary waves in low- and high-top models could result in

important differences in linear interference and, consequently, stratospheric AM.

The analysis conducted in Chapter 4 employed detrended reanalysis data. Studies of the

CMIP3 and CCMVAL-2 climatological NH stationary wave field reveal differences in this field

relative to twentieth century conditions due to anthropogenic climate change (Brandefelt and

Kornich 2008; SPARC CCMVAL 2010; Wang and Kushner, in press). In the SH, there has been

dramatic climate change due to the combined effects of ozone depletion and GHG-warming. In

addition to the clear trends in the zonal-mean circulation in the SH (Thompson and Solomon

2005; Son et al., 2008, 2010), trends in the zonally asymmetric circulation associated with

changes in planetary wave structure have been observed (Hu and Fu, 2009; Lin et al. 2009; Neff

et al. 2008; Shaw et al. 2011). Thus, climate change may lead to changes in linear interference

and changes in the nature of the role of linear interference in stratospheric NAM and SAM

variability. Future work includes trend analysis of the wave heat and momentum flux

decompositions in both hemispheres to establish how these trends are related to trends in the

climatological stationary wave field and wave anomalies.

In conclusion, this thesis presents novel results related to the role of linear interference in

stratosphere-troposphere interactions. Much of this thesis work was related to understanding the

dynamics of the observed relationship between October Eurasian snow cover anomalies and the

NAM. Yet it was also shown that the important dynamical process involved in this relationship,

linear interference, is a characteristic feature of stratosphere-troposphere interactions in both the

Northern and Southern hemispheres. Linear interference plays a dominant role in Northern

Hemisphere extratropical stratospheric variability and, in many respects, provides a

simplification of the key dynamics involved in this variability. Many open questions remain and

future work in this area will help to enhance scientific understanding of extratropical variability

and may ultimately lead to valuable improvements in seasonal climate prediction.

142

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Copyright Acknowledgements

Chapter 2 is based on Smith, K. L., C. G. Fletcher, and P. J. Kushner, 2010: The role of linear

interference in the Annular Mode response to extratropical surface forcings. J. Climate, 23,

6036-6050.

Chapter 3 is based on Smith, K. L., P. J. Kushner, and J. Cohen: The role of linear interference in

Northern Annular Mode variability associated with Eurasian snow cover extent, J. Climate (in

press).