Zonal mean vertical mean momentuma data analysis

and experimental prediction exercisemotivated by impacts

(work in progress)

Brian Mapes MPO seminar September 26, 2012

Outline• Motivation: impacts on our latitude (25N)• Background: – an old planetary physics problem– meaningful vs. arbitrary averaging

• Time series at one latitude• A full latitude-time analysis– climatology– anomalies and their budgets

• Statistical prediction of anomalies by LIM• Conclusions

iBTracs data

Barotropic zonal mean flow and TCsHigh-Low tercile <[u]>20-30°N composite (ASO)

H500 H

Zonal mean "pushes" the subtropical highs(by exerting a PV tendency)

Heating(PV source)

Eddy Z1000

no [u]

Eddy Z1000

w/ July [u]

Chen, Hoerling & Dole 2001

High-Low tercile <[u]>20-30°N composite (ASO)

Barotropic zonal mean flow and TCs

Outline• Motivation: impacts on our latitude (25N)• Background: – an old planetary physics problem– meaningful vs. arbitrary averaging

• Time series at one latitude• A full latitude-time analysis– climatology– anomalies and their budgets

• Statistical prediction of anomalies by LIM• Conclusions

Basic u equation

• use x,y instead of lon,lat for clarity• phyd coordinate in vertical (no r clutter)• "usual approximations" (no f*w)

PGF Cor Convergence of 3D flux

(by both fluid and molecular motions)

Break flux into components• and group d/dx terms:

• Zonal average on a phyd surface is []:

"mountain torque"

p surface L H L H

Mass-weighted vertical average <>

• These 4 (+1) terms must add up (physics)...• But in data (reanalysis-estimated budgets)

there is a 6th (implied) term: "residual"

mtn torque transport(meridionally

across latitude lines)

TENDENCY = + + Friction

Net Coriolis+

it is small

"Meaningful" averaging and <[u]>• The averaged quantity <[u]> obeys an equation w/

fewer, smaller, & simpler terms than u• It is thus "harder to change" than local u, so it has

an existence (and, we will see, a persisence) that is arguably more substantial than u

• This is quite unlike many averages of many quantities that we sum up in our computers!

» many are just fictions or conveniences» to blend (but dilute) information or signals, or reduce "noise"

• I'll speak of <[u]> as a wind that "advects" scalars

<[u]>, GLobal AAM and length of day• Velocity * lever arm *circumference of lat line– GLAAM = <[u]> (a cos(f)) (2p a cos(f))

• GLAAM exchanged w/ solid Earth (conserved) measurable length of day fluctuations

» semiannual, ENSO, MJO

• (much <[u]> lit. as AAM)

Zonal mean vs. “Annular modes”• NAM or “Arctic Oscillation” are EOFs of SLP– Related to NAO = pAzores-pIceland

• decomposition debate: semantic? profound?– NAM/SAM are patterns w/ max SLP’2 (variance)

• averaged over polar cap surface area– Some link to <[u]> via geostrophy, but loose

• Example of arbitrary vs. meaningful averaging– SLP’2 does not obey a physics equation that area-

averaging interestingly refines– If we end up calling it ‘annular’, why not use annuli? []

Outline• Motivation: impacts on our latitude (25N)• Background: – an old planetary physics problem– meaningful vs. arbitrary averaging

• Global picture Time series at 25N• A full latitude-time analysis– climatology– anomalies and their budgets

• Statistical prediction of anomalies by LIM• Conclusions

Data used

• NCEP-NCAR Reanalysis ("NCEP1")• 1980-2011 (32 years)

• 94 equally spaced latitudes (~2o, "Gaussian")

• Courtesy Dr. Klaus Weickman (NOAA PSD) who computed the zonal means and budget terms– for his papers over ~20 years

• in spherical, sigma coordinates rather than p

• (MERRA used for some climatology figs)

textbook: 4 jets, with meanders

u200 annual mean [u200] ann cyc

Jan -------------------------- Dec

<[u]>

climatology of <[u]>

NCEP1 sigma 1-21MERRA 1000-100 mb NCEP1 all levels

45

45

All Scales: +/- 25 m/s

Annual mean map MERRA1000-100 mb

30

30

u 50 mb annual mean and cycle

Outline• Motivation: impacts on our latitude (26N)• Background: – an old planetary physics problem– meaningless vs. meaningful averaging

• Time series at our latitude (26N)• A full latitude-time analysis– climatology– anomalies and their budgets

• Statistical prediction of anomalies by LIM• Conclusions

<[u]> at 26N

data => mean clim(doy) + anomaly(doy,y)

color = year (blue red rainbow)

data => mean clim(doy) + anomaly(doy,y)

data => mean clim(doy) + anomaly(poy,y)

Quantify seasonality: RMS (or stdev)

Winter has bigger anomalies on average than summeranomalies are strongly persistent from day to day

– (5-day data pre-averaging doesn’t reduce their square very much)» (the heavy line)

?

Winter anomalies are bigger• Midwinter (ground hog day) minimum?

Winter anomalies are bigger• Midwinter (ground hog day) minimum?

Quantify longevity of anomaliesby lagged multiplication

Quantify longevity by lagged multiplication

1980

mean of all years

1983

Quantify longevity by lagged multiplication

Mean of all M

arch 1s

mean of all March days'

means over all years

Repeat for all 30 days in March

Mean of all March 31s

Quantify longevity by lagged multiplication

Jan

mean of all months

Repeat for all months of the year

Feb

Mar

DecSummer

Magnitude difference is distracting. We already quantified that winter anomalies are larger: the lag-0 variance. So divideby that to normalize.

(this is "auto-covariance")

Quantify longevity by lagged multiplication Repeat for all months of the year

Jan

mean acov of all months

Feb

Mar

DecSummer

Jan

Feb

Mar

Dec

Apr

mean acov of all months

normalized by its

lag=0 value May

Repeat for all months of the year

Jan

Feb

Mar

Dec

Apr

IDEA: is it exponential decay C=C0 exp(-t/15d)?

...a postulated reinterpretation of:

characteristiclifetime (2/a)

~25 dMayJJA

Nov

Oct

Sep

... a solution of this ... (univariate)

?both

data => mean clim(doy) + anomaly(doy,y)

Power = | FT(autocov) |2

broad hump @ "wave period"

~2 characteristic times (50d)

1 2Period (months)

Histogram of 26N <[u]> anomalies

INCONSISTENT w/ this univariate model, at least w/Gaussian, white noise (Ornstein-Uhlenbeck process)

Outline

• Background: an old planetary physics problem• Motivation: impacts on our latitude• Our time series (26N)• A full latitude-time analysis– climatology– anomalies

• Statistical prediction of anomalies by LIM• Outstanding questions

climatology of <[u]>

NCEP1 sigma 1-21MERRA 1000-100 mb NCEP1 all levels

45

45

All Scales: +/- 25 m/s

Annual mean map MERRA1000-100 mb

30

30

Real time monitoring (Klaus Weickmann)

http://www.esrl.noaa.gov/psd/map/clim/aam.rean.shtml

JAN MAYSEP

2011 2012

25N

Characterize timescale w/autocorr.

autocov(lag,lat) = autocorr(lag,lat)*stdev2(lat)

autocov(lag,lat) = autocorr(lag,lat)*stdev2(lat)

<[u]> anomalies are more persistenton edges of jets

MERRA 1000-100 mb

Jan Dec

45

45

Annual mean map

Timescale: extend this to lag x lat

1980

mean of all years

1983

*apply with negtive weight*apply withpos. weight

etc. etc. for 32 years (optionally for seasons) and sum it all up

Composite anomalies in lag x latweighted by 26N base time series

This weighted composite is an array of regression coefficients

(slopes of linear fits onbase(t) vs. field(t+lag,lat)

scatterplots at each altitude)

same, along base lat...All seasons:

"Characteristic" time scales AND latitude evolution

Can do the same for tendency field, and torques

m/s

composite of d/dt = d/dt(composite)

Budget(composite over anomalies in all

seasons)

TEND = MTN + TRANSPORT + (FRICTION+GW) + CORIOLIS + RESIDUAL

TEND

MTN

TRANSPORT(FRICTION)

RESIDUAL

Outline

• Background: an old planetary physics problem• Motivation: impacts on our latitude• Our time series (26N)• A full latitude-time analysis– climatology– anomalies

• Statistical prediction of anomalies by LIM• Outstanding questions

Understand, schmunderstandCan we predict it?

• Statistical prediction by Linear Inverse Modeling (LIM)• Predicting <[u]> from itself only– ENSO signals etc. are already in <[u]>

• precisely to the extent that they affect it!» but mutiple signals are separately predictable only if they have

distinct spatial structure, not by time scale per se...

• IDEA: Using 32 years of observed time evolution, estimate the linear matrix B in this eqn:

Understand, schmunderstandCan we predict it?

• The vector of <[u]> at all latitudes is the x in:

• The expectation solution (...if ‘noise’ averages to nothing,

in an ensemble mean, which is what we’d want as a forecast...) is: » x(t) = exp(Bt) x(initial)

where t is the forecast lead

Understand, schmunderstandCan we predict it?

• In the univariate case, we fit such an exp(at) form based on a dx/dt = -ax + noise model.

• There we estimated a from: • cov(t) = exp(-at) cov(0) so a = ln( cov(t)/cov(0) )/t

Understand, schmunderstandCan we predict it?

• The multivariate version is the same– except that “matrix division” is on the left

» in a Matlab syntax I admire:

• univariate: a = ln( cov(t)/cov(0) )/t•multivariate: B = ln( cov(0)\cov(t) )/t• Really: B = ln( [cov(0)]-1 cov(t) )/t

Covariance of x with x

Eq. anoms

are broad

Midlat anoms

are flanked by

opposite values

Covariance of x with x

Note the diagonal is something weve seen before..

Covariance of x with (x, t days later)

25 N&S lobes hardly weaken w/ lag

asymmetric(poleward

prop.)

symmetric by construction: Not:

0d 5d 10d

15d 20d

B and cov(0)\cov(t) are inscrutable • But this is efficient, precise “information plumbing”• We know B’s character by its action:– x(t) = exp(Bt) x(initial)

1. Make this hindcast for all t, for every day in the record2. Use it to define the “noise”

– discrete version, daily

3. Make white noise by randomly re-sampling that in time4. Use randomized noise in the forward eq syn. data:

Use B and randomized noise to make syn. data:

• B knows about typical structure and timescale, as a function of latitude, including latitudinal drifts, width dependence of an anomaly's longevity, etc.

Which is real data?

LIM expected (or ensemble) hindcast using (no noise)

from initial spatial structure alone

LIM hindcasts: skill eval

• (Work in progress...)

• Make a summer B and a winter B? – each fitted from ½ as much data...• but the seasons may be different enough to justify 2?

• Why not a different B for every day (using a season centered around that day)? It’s just matrix calculations done once – cheap.

Conclusions• <[u]> correlated with TC recurvature longitude

• hyp: via "pushing" the subtropical high E/W

• <[u]> is a "meaningful" average – obeys a budget with fewer terms• It is "hard to change"

– & thus arguably has a solid "existence" (or at least persistence)

• Earth has 5-7 belts with enhanced persistence• Long predictability may be possible– 10s of days...

• Residual is large in <[u]> budget– models may be challenged? – statistical prediction appealing?