Center for Climate System Research UAW2008, 07/02/08 A possible modal view for understanding extratropical climate variability Masahiro Watanabe Center for Climate System Research University of Tokyo hiro@ccsr.u-tokyo.ac.jp

baroclinic wave lifecycle Outline Purpose: to discuss the extent to which a modal view is relevant in understanding extratropical atmospheric circulation variability associated with the climate variability Mode in weather system (<10 days) baroclinic wave lifecycle linear growth Normal mode (eigenmode) or non-normal growth (optimal perturbation) of the vertically sheared flow: The modal view of the synoptic waves is useful for understanding/ forecasting weather system

Mode in weather system (<10 days) Outline Purpose: to discuss the extent to which a modal view is relevant in understanding extratropical atmospheric circulation variability associated with the climate variability 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

East Asian Summer Climate under the Global Warming Arai and Kimoto (2007) H L Dominant climate variability in 20th C Simulated climate change in JJA 2xCO2 – 1xCO2 H projection L H Kimoto (2005) Other global warming signatures: El Nino-like tropical SST change Positive AO-like NH pressure change

Longer timescale “climate” variability, or the teleconnection Dynamical equation for the atmosphere after Watanabe et al. (2006) T42L20 LBM response to 1997/98 forcing ERA40 LBM (1) Basic state (often assumed to be steady) satisfies (2) Equation for perturbation is written as (3) For slow component that can ignore tendency, (4)

Where variability comes from? Dry dynamical core forced by time-independent diabatic forcing z’ one-point correlation, >10days z500 stationary eddy Atlantic Pacific NCEP Dynamical core Sardeshmukh and Sura (2007)

What is suggested? 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’ ? (4) Forcing → the phase and amplitude Internal dynamics → structure of the variability “neutral mode” theory 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)

Consider steady problem Neutral mode theory Consider steady problem (5) Covariance matrix is calculated by operating to and taking an ensemble average , (6) In 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 from steady response to spatially random forcing, what corresponds to the statistical leading EOF is the eigenvector of ATA having the smallest eigenvalue !

Neutral mode theory Eigenfunctions of are obtained by means of the singular value decomposition (SVD) to , (8) where : v-vector (right singular vector) : u-vector (left singular vector) : singular value (s1<s2<…) Substituting (8) into (7) leads to (9) indicating that v1 will appear as the leading EOF of the covariance matrix. s2 is equivalent to the inverse eigenvalue of C and also associated with the square-root of the complex eigenvalue of A, so that ｖ1 that determines the structure of the EOF1 to C is a mode closest to neutral. ⇒　set of v1 and u1 are called the “neutral mode”

Neutral mode: example with the Lorenz system Lorenz (1963) model EOF2 (33%) EOF1 (62%) EOF3(5%) For perturbation v2 (s-1=0.38) v1 (s-1=0.43) : basic state x0 must be the time-mean state but not the stationary state! v3 (s-1=0.07)

AO as revealed by the neutral mode Regression onto obs. AO (DJF mean anomaly) Neutral singular vector (T21L11 LBM) Z300 anomaly s-1 mode # Inverse singular values r = 0.68 T925 anomaly Watanabe and Jin (2004)

Propagation of Rossby wave energy Linear evolution from the Atlantic anomalies of v1 propagation of Rossby wave packets on the Asian Jet stream 300hPa meridional wind anomaly seed shading: Z0.35 (>±10m), contour: V0.35 (c.i.=0.5m/s) Watanabe and Jin (2004) Composite evolution from the NAO to the AO pattern EOF1 to SLP anom. (>10dys) day 0 day 2 day 4 day 6 Watanabe (2004)

Is the EASM variability viewed as neutral mode? Dominant variability in reanalysis (JJA 1979-1998) EOF1, 31% Hirota (2008) Dominant variability in linear responses to random forcing H L H Z500 L H Prcp Arai and Kimoto (2007)

Dominant variability in a nonlinear barotropic model drag=(20days)-1 EOF1 65.2% EOF2 31.6% ψ’ EOF patterns 106 m2/s Trajectory on the EOF plane strong damping PC1 PC2 moderate damping td=(22days)-1 td=(20days)-1 PC2 PC1 courtesy of M.Mori

Dominant variability in a nonlinear barotropic model drag=(1000days)-1 EOF1 22.1% EOF2 15.3% ψ’ EOF patterns 106 m2/s Trajectory on the EOF plane weak damping td=(1000days)-1 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 PC2 PC1 courtesy of M.Mori

Low-frequency PNA variability Positive PNA Negative PNA Low-frequency z’300 (>10days) and the wave activity fluxes x 50 m, 90% high-frequency (<10days) EKE300 and (z+z’)300 Synoptic eddies (part of storm tracks) are systematically modulated in association with the low-frequency pattern Mori and Watanabe (2008)

State-dependent noise Stochastically fluctuating basic state y0+Y0’ If B=I, stochastic noise in (12) reduces to be additive Linear stochastic equation (12) Lorenz’s attractor Palmer (2001) Third axis replaced with additive noise : noise vector If B=B(x’), stochastic noise in (12) is multiplicative, dependent on state vector stochastic fast component basic state perturbation An example in a barotropic vorticity equation

Dominant variability forced by the state-dependent noise drag=(20days)-1 EOF1 24.8% EOF2 19.2% ψ’ EOF patterns 105 m2/s Trajectory on the EOF plane strong damping + state-dependent noise td=(20days)-1 PC2 We cannot distinguish whether nonlinear dynamics or linear stochastic dynamics caused apparently chaotic trajectory !! PC1 courtesy of M.Mori

Stochastic ensemble and low-frequency variability Equation for the slow component of y1 : SELF closure Collaboration with Univ. of Hawaii Linear dynamical operator for the transient eddy feedback (or the state-dependent noise) neutral mode, ya T21 barotropic model with SELF feedback zonal wind, ua eigenvalues 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

Summary 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