Roberto Buizza ([email protected]) 1 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors...

Roberto Buizza ([email protected]) 1 ECMWF TC/PR/RB L2 – 5 Apr 05Introduction to singular vectors

TC/PR/RB Lecture 2 – Introduction to singular vectors

Roberto Buizza ([email protected])

European Centre for Medium-Range Weather Forecasts

http://www.ecmwf.int


Outline

Definition of singular vectors: brief summary.

Singular vector characteristics:

– geographical location;

– vertical structure;

– spectra. Metrics and singular vectors.


Total-energy metric (and norm) definition

Given two state-vectors x and y expressed in terms of vorticity , divergence D, temperature T,

specific humidity q and surface pressure , the total energy metric (and the associated norms) is

defined (<..,..> is the Euclidean inner product) as:

dp

TR

dp

dTTT

CDDyEx

yxr

rd

yxr

pyxyxTE

)lnln(

)(2

1; 1111


The adjoint operator

Given any two vectors x and y, the adjoint operator L* of the linear operator L with respect to the

Euclidean norm <..,..> is the operator that satisfies the following property:

Using the adjojnt operator L* the time-t E-norm of z’ can be written as:

LyxyxL ;;*

0*

0002

;;)( zELLzzELzLtz


Singular vector definition

Consider an N-dimensional system:

Denote by z’ a small perturbation around a time-evolving trajectory z:

The time evolution of the small perturbation z’ is described to a good degree of approximation by the

linearized system Al(z) defined by the trajectory. Note that the trajectory is not constant in time.

)(yAt

y

)(

)(

zAt

z

zzAt

zl

zl z

zAzA

)(

)(



The solution of the linearized system can be written in terms of the linear propagator L(t,0):

The linear propagator is defined by the system equations and depends on the trajectory

characteristics.

The E-norm of the perturbation at time t is given by:

0)0,()( ztLtz

002

)0,(;)0,()();()( ztELztLtzEtztz



The computation of the directions of maximum growth can be stated as ‘finding the directions in

the phase-space of the system characterized by the maximum ratio between the time-t and the

initial norms’:

The problem reduces to solving the following eigenvalue problem:

000

0*

02

0

2

;

;max

)(max

0

0

0 xEx

ELxLx

x

txx

E

Ex

2210

*210 ELELE


SVs’ geographical distributions and Eady index

The geographical distribution of the singular vectors reflect the characteristics of

the underlying basic-state flow. A measure of the baroclinic instability of the basic-

state flow is given by the Eady index:

which is the growth rate of the most unstable Eady mode (Hoskins & Valdes

1990). In this equation, the static stability N and the vertical wind shear can be

estimated using the 300- and 1000-hPa potential temperature and wind. Results

indicate that locations with maximum singular vector concentration coincide with

regions with maximum Eady index.

dz

du

N

fE 31.0


Example 1: singular vectors for 18-20 Jan 1997

This figure shows the

amplification rate (i.e. the

singular value) of the leading

30 unstable singular vectors

growing between 18 and 20

January 1997. The SVs were

computed at the resolution

T42L31 and were used to

generate the EPS initial

conditions.


Ex 1: singular vector 1 for 18-20 Jan 1997

This figure shows the most unstable

singular vector growing between 18 and

20 Jan 1997. Left (right) panels show the

SV at initial and final (i.e. +48h) time. The

top panels show the SV T at model level

18 (~500hPa, shading) and the Z500

analysis; the bottom panels the SV T at

model level 23 (~700hPa, shading). The

contour interval is 8dam for Z, and 0.01

(0.05) deg for T at initial (final) time (the

SV is normalized to have unit total energy

norm at initial time).

T=0 T=+48h

500hPa

700hPa


Ex 1: vertical cross section of SV 1 for 18-20 Jan 1997

This figure shows, for SV 1, the vertical cross

section of the T component at initial time (top,

for 36N) and of the vorticity component at final

time (bottom, for 44N). The two cross sections

have been taken along the parallel where the

SV had maximum amplitude. Note the strong

initial tilt, suggesting baroclinic instability, and

the final time more barotropic-type structure.

Note that T is shown at initial time and vor at

final time because the initial time SV has a

strong potential energy part.


Ex 1: energy distribution of SV 1 for 18-20 Jan 1997

The top figure shows, for SV 1, the vertical

distribution at initial time of the kinetic (red

dotted, x100) and total (red solid, x100)

energy, and the corresponding final time

distributions (blue). The bottom figure shows

the total energy spectrum at initial (red solid,

x100) and at final time (blue solid). Note the

upward and upscale energy

transfer/growth, and the transformation

from initial potential to mainly final

kinetic energy.


Ex 1: singular vector 4 for 18-20 Jan 1997

This figure shows the 4th SV

growing between 18 and 20 Jan

1997. Left (right) panels show the

SV at initial (final, +48h) time; top

(bottom) panels show the SV T at

model level 18 (23) and Z500

(Z700) analysis. The contour

interval is 8dam for Z, and 0.01

(0.05) deg for T at initial (final) time

(the SV is normalized to have unit

total energy norm at initial time).

T=0 T=+48h


Ex 1: vertical cross section of SV 4 for 18-20 Jan 1997

This figure shows, for SV 4, the vertical cross

section of the T component at initial time (top,

for 44N) and of the vorticity component at final

time (bottom, for 56N). The two cross sections

have been taken along the parallel where the

SV had maximum amplitude. Note the strong

initial tilt, suggesting baroclinic instability, and

the final time more barotropic-type structure.

Note that T is shown at initial time and vor at

final time because the initial time SV has a

strong potential energy part.


Ex 1: energy distribution of SV 4 for 18-20 Jan 1997

The top figure shows, for SV 4, the vertical

distribution at initial time of the kinetic (red

dotted, x100) and total (red solid, x100)



the total energy spectrum at initial (red solid,

x100) and at final time (blue solid). By

contrast to SV 1, note that this SV is

characterized by the superposition of two

different structures, growing at different

levels and with different characteristic

lengths.


Ex 1: average energy distribution for 18-20 Jan 1997

The top figure shows the SV1:25 average

vertical distribution at initial time of the kinetic

(red dotted, x100) and total (red solid, x100)



the SV1:25 average total energy spectrum at

initial (red solid, x100) and at final time (blue

solid). Note the SV typical upward and

upscale energy transfer/growth, and the

transformation from initial potential to

mainly final kinetic energy.


Ex 1: SVs’ and Eady index for 18-20 Jan 1997

The top panel shows the t+24h average

root-mean-square (rms) amplitude (in terms

of Z500) of the first 25 singular vectors

growing between 18 and 20 January 1997.

The bottom panel shows the 18-20 January

1997 average Eady index. The contour

isolines are 0.5dam for the SV’s rms

amplitude and 0.5d-1 for the Eady index.

Results indicate a good correspondence

between areas of SV concentration and of

maximum value of the Eady index.


Ex 2: NH SVs and Eady index - JFM 1997 and 1998

The top panels show the average

t+24h root-mean-square amplitude

(in terms of Z500, ci=0.3dam) of

the first 25 singular vectors during

JFM 1997 (left) and 1998 (right)

over the NH. The bottom panels

show the average Eady index

computed between 1000 and 300

hPa (ci=0.2d-1). Results indicate a

good agreement between areas of

large Eady index and high SV

concentration.


Ex 2: SH SVs and Eady index - JFM 1997 and 1998



(in terms of Z500, ci=0.3dam) of

the first 25 singular vectors during

JFM 1997 (left) and 1998 (right)

over the SH. The bottom panels

show the average Eady index

computed between 1000 and 300

hPa (ci=0.2d-1). Results indicate a

good agreement between areas of

large Eady index and high SV

concentration.


Ex 3: NH SVs & Eady index - JJA 1997 & 1998



(in terms of Z500, ci=0.3dam) of the

first 25 singular vectors during JJA

1997 (left) and 1998 (right) over the

NH. The bottom panels show the

average Eady index computed

between 1000 and 300 hPa

(ci=0.2d-1). Results confirm a good

agreement between areas of large

Eady index and high SV

concentration.


Ex 3: SH SVs & Eady index - JJA 1997 & 1998




first 25 singular vectors during JJA


SH. The bottom panels show the






concentration.


Ex 3: NH SVs & Eady index - JFM 2000 & 2001




first 25 singular vectors during JFM


NH. The bottom panels show the






concentration.


Ex 3: SH SVs & Eady index - JFM 2000 & 2001




first 25 singular vectors during JFM


SH. The bottom panels show the






concentration.


Impact of horizontal resolution on SVs

This figure shows the time evolution of the average

(10 SVs for 8 cases) total energy vertical

distribution from t=0 (x30, red dotted) to t=48h (x3,

red solid) every 12h for T21 (top right), T42 (bottom

left) and T63 (bottom right) SVs (values have been

scaled differently at each forecast step).


Impact of horizontal resolution on SVs

This figure shows the time evolution of the total

energy average (10 SVs for 8 cases) spectrum

from t=0 (x30, red dotted) to t=48h (x3, red solid)

every 12h for T21 (top right), T42 (bottom left)

and T63 (bottom right) SVs (values have been

scaled differently at each forecast step).


Seasonal variation of the NH SVs’ amplification rate

Singular vectors’ location and amplification rate depend on the atmospheric flow. This

figure shows the average of the amplification rate (i.e. singular values) of the SV1:3

localized in different NH regions for winter 98/99 (D98-JF99).

D1998-JF1999 (30 SVs)

02468

1012141618202224

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86

day from 1/12/1998

ave s

ing

-val

(1:3

)

Europe Pacific North America Atlantic



This figure shows the mean of the

average of the amplification rate of

SV1:3 for 6 different NH regions for

winter 1997/1998 (top) and 1998/1999

(bottom). During winter the amplification

rates are larger over the the Pacific and

the Atlantic regions. Comparing the two

winters, results indicate that the

amplification rates were larger during

winter 1998/99 especially over the

Pacific and the Atlantic regions.

D1997-JF1998 (30 SVs)

02468

1012141618

ave

sin

g-v

al (

1:3)

mean 9.60 8.37 9.42 12.60 9.01 10.40

mean-std 7.11 6.84 6.86 10.38 6.99 8.21

mean+std 12.09 9.90 11.98 14.82 11.03 12.59

EuropeCentral

AsiaEast Asia Pacific

North America

Atlantic

D1998-JF1999 (30 SVs)

02468

1012141618

ave

sin

g-v

al (

1:3)

mean 9.70 8.01 9.01 14.00 9.64 11.30

mean-std 7.34 6.05 5.21 11.15 6.91 9.04

mean+std 12.06 9.97 12.81 16.85 12.37 13.56

EuropeCentral Asia

East Asia Pacif icNorth

AmericaAtlantic



This figure shows the mean of the

average of the amplification rate of

SV1:3 for 6 different NH regions for

summer 1998 (top) and 1999 (bottom).

During summer the amplification rates

are not so strongly depend on the

geographical region. Compared to

winter, summer amplification rates are

smaller and less daily variable (please

note that SVs during these seasons

were computed with a dry tangent

model).

JJA 1998

0.002.004.006.008.00

10.0012.0014.00

ave

sin

g-v

al (

1:3)

mean 6.33 6.23 6.43 6.71 6.25 6.72

mean-std 5.20 5.26 5.14 5.37 4.83 5.72

mean+std 7.46 7.20 7.72 8.05 7.67 7.72

EuropeCentral

AsiaEast Asia

PacificNorth

AmericaAtlantic

JJA 1999

0.002.004.006.008.00

10.0012.0014.00

ave

sin

g-v

al (

1:3)

mean 6.19 6.35 7.18 7.18 6.64 7.32

mean-std 5.05 5.36 5.52 5.76 4.85 5.83

mean+std 7.33 7.34 8.84 8.60 8.43 8.81

EuropeCentral

AsiaEast Asia

PacificNorth

AmericaAtlantic



This figure shows the seasonal average amplification rate of SV1:3 for 4

summer and 4 winter seasons.

Average singular value of SVs (1:3)

0.002.004.006.008.00

10.0012.0014.0016.0018.00

ave

sin

g-v

alu

e (1

:3)

Europe 6.24 6.33 6.19 6.94 9.81 9.60 9.70 10.20

Central Asia 6.44 6.23 6.35 6.78 8.56 8.37 8.01 7.60

East Asia 7.12 6.43 7.18 6.94 8.87 9.42 9.01 9.08

Pacific 6.60 6.71 7.18 7.08 11.20 12.60 14.00 13.30

North America 6.52 6.25 6.64 6.96 9.10 9.01 9.64 9.05

Atlantic 7.39 6.72 7.32 7.29 11.60 10.40 11.30 11.50

NH ave 6.73 6.45 6.82 7.00 9.92 9.99 10.46 10.29

average

MJ J 97

average

MJ J 98

average

MJ J 99

average

J J A00

average

D96-J F97

average

D97-J F98

average

D98-J F99

average

D99-J F00



This figure shows the seasonal variation (with respect to the seasonal average) of the

amplification rate of SV1:3 for 4 summer and 4 winter seasons.

Seasonal relative instability (%) based on ave sing-val (1:3)

-12%-9%-6%-3%0%3%6%9%

12%

Per

cen

tag

e

Europe -2.88% -1.48% -3.66% 8.02% -0.18% -2.31% -1.30% 3.79%

Central Asia -0.16% -3.41% -1.55% 5.12% 5.22% 2.89% -1.54% -6.58%

East Asia 2.93% -7.05% 3.79% 0.33% -2.47% 3.57% -0.93% -0.16%

Pacifi c -4.24% -2.65% 4.17% 2.72% -12.33% -1.37% 9.59% 4.11%

North America -1.10% -5.20% 0.72% 5.57% -1.09% -2.07% 4.78% -1.63%

Atlantic 2.92% -6.41% 1.95% 1.53% 3.57% -7.14% 0.89% 2.68%

NH ave -0.30% -4.48% 1.09% 3.69% -2.37% -1.71% 2.86% 1.22%

J J A97 J J A98 J J A99 J J A00 D96-J F97 D97-J F98 D98-J F99 D99-J F00



Focussing over the NH and

Europe, results indicate that

summer amplification rates

were largest during 2000 for

both areas, and that winter

amplification rates were

largest during during

1999/2000 for Europe but

during 1998/1999 for the NH.

Seasonal relative instability (%) based on ave sing-val (1:3)

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

Per

cen

tag

e

Europe -2.88% -1.48% -3.66% 8.02% -0.18% -2.31% -1.30% 3.79%

NH ave -0.30% -4.48% 1.09% 3.69% -2.37% -1.71% 2.86% 1.22%

JJA97 JJA98 JJA99 JJA00D96-JF97

D97-JF98

D98-JF99

D99-JF00


SVs can be used to identify unstable regions

The average (or a

weighted average) of

the geographical

distribution of the

singular vectors’

total energy can be

used to identify

unstable regions.

1 Jan 97

10 Jan

30 Jan

20 Jan


Impact of the initial and final metrics on the SVs

The computation of the leading singular vectors reduces to solving the following eigenvalue

problem:

It is evident from this equation that singular vectors depend on the choice of the initial and final

time metrics E0 and E.

2210

*210 ELELE



Singular vectors computed with four different initial time norms are compared:

dp

TR

dp

dTTT

CDDyEx

yxr

rd

yxr

pyxyxTE

)lnln(

)(2

1; 1111

dp

dDDyEx yxyxKE

)(2

1; 1111

dp

dyEx yxVOR

)(2

1;

dp

dyEx yxPSI

)(2

1; 11



The total energy norm has been used at final time. Furthermore, singular vectors

have been computed to maximize the final time norm inside the region

A={30:80N;-30:10E}. SVs have been computed at T42L19 resolution, with a 48-h

optimization time interval, with starting date 5 December 1994.

SV type Initial time norm Final time normTE-TE TE TEKE-TE KE TEVO-TE VOR 2̂ TEPS-TE PSI 2̂ TE




streamfunction of the dominant

SV (T42L19, 48h optimization

time interval) at initial time on a

model level near 500hPa

computed with different

metrics: TE-TE (top left), KE-

TE (top right), VO-TE (bottom

left) and PS-TE (bottom right).

VO-

TE

KE-TE

PS-TE

TE-TE




streamfunction of the

dominant SV (T42L19, 48h

optimization time interval) at

final time on a model level

near 500hPa computed with

different metrics: TE-TE (top

left), KE-TE (top right), VO-

TE (bottom left) and PS-TE

(bottom right). The contour

interval is 20 times larger

than at initial time.

TE-TE KE-TE

PS-TEVO-TE



Average total energy spectrum of SV1:16

(as a function of the 2D total wave-number)

for SVs computed with diferent metrics:TE-

TE (top), VO-TE (middle) and PS-TE

(bottom). The blue lines show the spectra at

initial time and the red line at final time

scaled by 100 for TE-TE, 1 for VO-TE and

10 for PS-TE.

TE-TE

VO-TE

PS-TE



Average vertical distribution of total energy

of SV1:16 for SVs computed with diferent

metrics: TE-TE (top), VO-TE (middle) and

PS-TE (bottom). The blue lines show the

spectra at initial time and the red line at final

time scaled by 100 for TE-TE, 1 for VO-TE

and 10 for PS-TE.

TE-

TE

VO-TE

PS-TE



These results show that there is a strong dependence of singular vectors on the

choice of the metric. More precisely, the initial structure of T42L19 SVs of a

primitive equation model (without moist processes) are very sensitivity to the

initial-time metric. The comparison of SVs computed with different final-time

metrics indicate that these SVs are less sensitive to changes in the final time

norm (not shown here). The reader is referred to Palmer et al (1998) and to

Barkmeijer et al (1998-1999) for a more detailed discussion of the sensitivity of

singular vectors to the choice of the metric.


The effect of spatial scale on SV structures

To investigate the scale dependence of singular vector growth, singular vectors

have been computed applying a filter at initial and/or at final time (Hartmann et al,

1995). Denote by s(n,m) the following function:

where n is the total wave-number, m is the zonal wave and N identifies a region of

the spectral space. Denote by Sx the multiplication of each spectral component of

the vector x by the function s(n,m):

Nmnif

Nmnifmns

),(

),(

0

1),(

Nmnif

NmnifxSx

),(

),(

0


The effect of spatial scale on SV structures

Singular vectors with chosen initial and

final time spectral characteristics can be

computed by applying the spectral filter S

at initial and/or final time:

This leads to the SV generalized

eigenvector problem:

22100

***0

210 ELSESSLSE TT

xLSESxLSStx TT 0002

;)(



This figure show the total energy spectra at initial (dashed, x100 in the top 6 panels and x30

in the bottom 3 panels) and final (solid) time of T21L19 singular vectors with imposed (via S0

and ST) different initial and final time spectral characteristics.



Hartmann et al (1995) results indicate that:

SVs’ linear growth is most rapid for the smallest scales.

Disturbances (i.e. SVs) composed of sub-synoptic-scale wavenumbers may very rapidly

grow to synoptic-scale waves.

The final-time SV structure is insensitive to the the presence or absence of initial-time

large scales. This indicates that information for the final-time synoptic-scale structure is

contained in the sub-synoptic scales.

SVs constrained to be of large scale at initial time grow much more slowly than

unconstrained, smalelr-scale perturbations.


Conclusions

Singular vector characteristics have been discussed:

– SVs are localized in areas of strong barotropic and baroclinic instability;

– average SV total energy maps can be used to identify unstable regions;

– SV growth is characterized by an upscale and upward energy transfer, with initial-time potential energy converted during time evolution into kinetic energy.

Seasonal variability of SVs’ characteristics:

– winter is more unstable than summer (but note that currently SVs have been computed without moist processes in the tangent model), and during winter the Pacific and the Atlantic sectors are on average more unstable than other regions.

The choice of the initial time metric has a strong effect on the SVs.

Small-scale disturbances grow faster than large-scale disturbances, and

affect large scales at optimization time.


Bibliography

On normal modes and baroclinic instability:

– Hoskins, B. J., & Valdes, P. J., 1990: On the existence of storm tracks. J. Atmos. Sci., 46, 1854-1864.– Orr, W, 1907: Stability and instability of the steady motions of a perfect fluid. Proc. Roy. Irish Acad., 9-138.

On singular vector characteristics:

– Borges, M., & Hartmann, D. L., 1992: Barotropic instability and optimal perturbations of observed non-zonal flows. J. Atmos. Sci., 49, 335-354.

– Barkmeijer, J, Van Gijzen, J., & Bouttier, F., 1998: Singular vectors and estimates of the analysis covariance metric. Q. J. R. Meteorol. Soc., 124, 1695-1713.

– Barkmeijer, J, Buizza, R., & Palmer, T. N., 1999: 3D-Var Hessian singular vectors and their potential use in the ECMWF Ensemble Prediction System. Q. J. R. Meteorol. Soc., 125, 2333-2351.

– Buizza, R., 1998: Impact of horizontal diffusion on T21, T42 and T63 singular vectors. J. Atmos. Sci., 55, 1069-1083.

– Buizza, R., & Palmer, T. N., 1995: The singular vector structure of the atmospheric general circulation. J. Atmos. Sci., 52, 1434-1456.

– Buizza, R., Tribbia, J., Molteni, F., & Palmer, T. N., 1993: Computation of optimal unstable structures for a numerical weather prediction model. Tellus, 45A, 388-407.

– Farrell, B. F., 1982: The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci., 39, 1663-1686.– Farrell, B. F., 1989: Optimal excitation of baroclinic waves. J. Atmos. Sci., 46, 1193-1206.– Hartmann, D. L., Buizza, R., & Palmer, T. N., 1995: Singular vectors: the effect of spatial scale on linear

growth on disturbances. J. Atmos. Sci., 52, 3885-3894.– Hoskins, B., Buizza, R., & Badger, J., 2000: The nature of singular vcetor growth and structure. Q. J. R.

Meteorol. Soc., 126, 1565-1580.– Molteni, F., & Palmer, T. N., 1993: Predictability and finite-time instability of the northern winter circulation.

Q. J. R. Meteorol. Soc., 119, 1088-1097.– Palmer, T. N., Gelaro, R., Barkmeijer, J., & Buizza, R., 1998: Singular vectors, metrics, and adaptive

observations. J. Atmos. Sci., 55, 633-653.

Roberto Buizza ([email protected]) 1 ECMWF TC/PR/RB L2 – 5 Apr 05 Introduction to singular vectors...

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