A possible modal view for understanding extratropical climate variability

A possible modal view for understanding extratropical climate variability

Masahiro Watanabe

Center for Climate System ResearchUniversity of Tokyo

hiro@ccsr.u-tokyo.ac.jp

UAW2008, 07/02/08UAW2008, 07/02/08

OutlineOutline

▶Mode in weather system (<10 days)

Purpose: to discuss the extent to which a modal view is relevant in understanding extratropical atmospheric circulation variability associated with the climate variability

baroclinic wave lifecycle

linear growth

Normal mode (eigenmode) or non-normal growth (optimal perturbation) ofthe vertically sheared flow:

The modal view of the synopticwaves is useful for understanding/forecasting weather system

OutlineOutline

▶Mode in weather system (<10 days)▶Mode in climate system (>month or season)

▸Statistical EOFs▸Dynamical mode in mean climate▸Dynamical mode in climate with weather ensemble

Purpose: to discuss the extent to which a modal view is relevant in understanding extratropical atmospheric circulation variability associated with the climate variability

East Asian Summer Climate under the Global WarmingEast Asian Summer Climate under the Global Warming

Simulated climate change in JJA

H

H

L

Kimoto (2005)

Other global warming signatures:

• El Nino-like tropical SST change• Positive AO-like NH pressure change

2xCO2 – 1xCO2

projection

Arai and Kimoto (2007)

H

L

H

Dominant climate variability in 20th C

Longer timescale “climate” variability, or the teleconnectionLonger timescale “climate” variability, or the teleconnection

( )d

dt

xLx N x x Q

( ) ( ) 0d

dt

xLx N x x N x x Q

( ) ( )d

dt

x x FAx

x NLx x N x x N x x Q

x

Dynamical equation for the atmosphere

Basic state (often assumed to be steady) satisfies

Equation for perturbation is written as

(1)

(2)

(3)

Ax F

For slow component that can ignore tendency,

(4)after Watanabe et al. (2006)

T42L20 LBM response to 1997/98 forcing

ERA40

LBM

Where variability comes from? Where variability comes from?

Sardeshmukh and Sura (2007)

Dry dynamical core forced by time-independent diabatic forcing

z’ one-point correlation, >10days

Atlantic Pacific

NCEP

Dynamical core

z500 stationary eddy

▶Response to increasing GHGs is often projected onto the dominant natural climate variability

▸Need to understand the mechanism of the natural variability

▶x’ can be reproduced with (4) when F’ given from obs.▸Forcing is the key ?

▶Nonlinear atmosphere can fluctuate with a similar structure to observations even if Q’ is time-independent

▸Crucial ingredients reside in A, but not in F’ ?

What is suggested? What is suggested? What is suggested? What is suggested?

References:North (1984), Branstator (1985), Dymnikov (1988), Branstator (1990), Navarra (1993), Marshall and Molteni (1993), Metz (1994), Bladé (1996), Itoh and Kimoto (1999),Kimoto et al. (2001), Goodman and Marshall (2003), Watanabe et al. (2002), Watanabe and Jin (2004)

Ax F (4)

▶Forcing → the phase and amplitude▶Internal dynamics → structure of the variability

▶“neutral mode” theory

Covariance matrix is calculated by operating to and taking an ensemble average ,

Neutral mode theoryNeutral mode theoryNeutral mode theoryNeutral mode theory

(5)

(6)

Ax FTx x

T T T1

C x x A F F A

Consider steady problem

T1

C A A

T F F IIn the simplest case, the forcing is assumed to be random in space, i.e., , then

(7)

If observed monthly or seasonal mean anomalies can be assumed to arise fromsteady response to spatially random forcing, what corresponds to the statisticalleading EOF is the eigenvector of ATA having the smallest eigenvalue !

Neutral mode theoryNeutral mode theoryNeutral mode theoryNeutral mode theory

Eigenfunctions of are obtained by means of the singular valuedecomposition (SVD) to ,

TA AA

T ,A UΣVwhere

1 2, , , ,i NV v v v v

1 2, , , ,i NU u u u u 1 2diag , , , ,i N Σ

: v-vector (right singular vector)

: u-vector (left singular vector)

: singular value (…)

(8)

Substituting (8) into (7) leads to

T2C VΣ Vindicating that v1 will appear as the leading EOF of the covariance matrix.

⇒　 set of v1 and u1 are called the “neutral mode”

is equivalent to the inverse eigenvalue of C and also associated with the square-root of the complex eigenvalue of A, so that ｖ 1 that determinesthe structure of the EOF1 to C is a mode closest to neutral.

(9)

EOF1 (62%)

EOF2 (33%)

EOF3(5%)

Neutral mode: example with the Lorenz systemNeutral mode: example with the Lorenz system

t

t

t

d x x y

d y xz rx y

d z xy bz

( , , )x y z x

0td x Ax

0 0

0 0

0

1z r x

y x b

Av2 (-1=0.38)

v3 ()

v1 ()

Lorenz (1963) model

For perturbation

x0 must be the time-mean statebut not the stationary state!

: basic state0 0 0 0( , , )x y zx

AO as revealed by the neutral mode AO as revealed by the neutral mode

Regression onto obs. AO(DJF mean anomaly)

Neutral singular vector(T21L11 LBM)

r = 0.68

Z300 anomaly

T925 anomaly

Watanabe and Jin (2004)

-1

mode #

Inverse singular values

Propagation of Rossby wave energyPropagation of Rossby wave energy

Linear evolution from the Atlantic anomalies of v1

shading: Z0.35 (>±10m), contour: V0.35 (c.i.=0.5m/s)

propagation of Rossby wave packets on the Asian Jet stream

seedWatanabe and Jin (2004)

Watanabe (2004)

Composite evolution from the NAO to the AO pattern

300hPa meridional wind anomaly

EOF1 to SLP anom. (>10dys) day 0 day 2 day 4 day 6

Is the EASM variability viewed as neutral mode? Is the EASM variability viewed as neutral mode?

Arai and Kimoto (2007)

H

L

H

Dominant variability in reanalysis (JJA 1979-1998)

Z500

Prcp

EOF1, 31%

Hirota (2008)

Dominant variability in linear responses to random forcing

H

L

H

drag=(20days)-

1

101066 m m22/s/s

EOF1EOF1

, 0 x Ax F Q

PC1

PC

2

d=(20days)-1

PC1

PC

2

moderate dampingd=(22days)-1

strong dampingTrajectory onTrajectory onthe EOF planethe EOF plane

ψ’ψ’ EOF patternsEOF patterns

65.2%65.2% EOF2EOF2 31.6%31.6%

courtesy of M.Mori

Dominant variability in a nonlinear barotropic model Dominant variability in a nonlinear barotropic model

, 0 x Ax F Q drag=(1000days)-1

101066 m m22/s/s

EOF1EOF1

PC1

PC

2

d=(1000days)-1

weak dampingTrajectory onTrajectory onthe EOF planethe EOF plane

ψ’ψ’ EOF patternsEOF patterns

22.1%22.1% EOF2EOF2 15.3%15.3%

courtesy of M.Mori

Are these prototype of nature?― probably not • Barotropic instability cannot occur on an isentropic climatological flow (Mitas & Robinson 2005)

• Barotropic model ignores interaction with synoptic disturbances

Dominant variability in a nonlinear barotropic model Dominant variability in a nonlinear barotropic model Dominant variability in a nonlinear barotropic model Dominant variability in a nonlinear barotropic model

high-frequency (<10days) EKE300 and (z+z’)300

Positive PNAPositive PNA Negative PNANegative PNA

Mori and Watanabe (2008)

x 50 m, 90%

Low-frequency z’300 (>10days) and the wave activity fluxes

Low-frequency PNA variabilityLow-frequency PNA variability

Synoptic eddies (part of storm tracks) are systematically modulated in association with the low-frequency pattern

State-dependent noiseState-dependent noise

0d

dt

xAx Bξ

ξ : noise vector

Linear stochastic equation

Lorenz’s attractor

Palmer (2001)

Third axis replaced with additive noise

If B=I, stochastic noise in (12) reduces to be additive

(12)

Stochastically fluctuating basic state 0+0’

If B=B(x’), stochastic noise in (12) is multiplicative, dependent on state vector

0 0 1( , ) ( , , ) ( , , )x y x y t x y ty y y¢= +Y +stochastic fast componentstochastic fast component

basic statebasic state perturbationperturbation

( ) ( )2 2 8 2,J f Ft

y y y a k y¶

Ñ + Ñ + + + Ñ Ñ =¶

An example in a barotropic vorticity equation

drag=(20days)-1

101055 m m22/s/s

EOF1EOF1

PC1

PC

2strong damping + state-dependent noiseTrajectory onTrajectory on

the EOF planethe EOF plane

ψ’ψ’ EOF patternsEOF patterns

24.8%24.8% EOF2EOF2 19.2%19.2%

courtesy of M.Mori

0 x Ax Bξ F

Dominant variability forced by the state-dependent noise Dominant variability forced by the state-dependent noise Dominant variability forced by the state-dependent noise Dominant variability forced by the state-dependent noise

d=(20days)-1

We cannot distinguish whether nonlinear dynamics or linear stochastic dynamics caused apparently chaotic trajectory !!

Stochastic ensemble and low-frequency variability Stochastic ensemble and low-frequency variability

Collaboration with Univ. of Hawaii

neutral mode, a

T21 barotropic model with SELF feedback

zonal wind, uaeigenvalues

Jin et al. (2006b)* Similar results are obtained with primitive model (Pan et al. 2006)

The neutral mode looks more like NAO!

selective excitation due to positive SELF interaction

Linear dynamical operator for the transient eddy feedback(or the state-dependent noise)

1 1 1( )fL L r Qt

Equation for the slow component of : SELF closure

SummarySummary

▶ Origin and structure of the dominant circulation variability seem to be explained with dynamical modes of mean climate

▶ Nonlinearity arising from interaction with synoptic disturbances (fast component of climate) may be represented as state-dependent noise

▶ Extension of the “dynamical mode in climate”▶ Interaction with physical processes (precip.,cloud)▶ Mode along the seasonal cycle (Frederiksen and Branstator 2001)

▶ Mode arising from coupling with ocean and/or land (more general view of the known air-sea coupled modes)

▶ Question: “well… mode is fine, and then what?”▶ Phase and amplitude do matter for prediction → Excitation problem