A possible modal view for understanding extratropical climate variability

1. 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

2. baroclinic wave lifecycle Normal mode (eigenmode) or non-normal growth (optimal perturbation) of the vertically sheared flow: linear growth 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) The modal view of the synoptic waves is useful for understanding/ forecasting weather system

3. 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

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

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

6. 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)

7. 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)

8. 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 ATAhaving the smallest eigenvalue !

9. 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. s2is 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”

10. 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)

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

12. seed 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 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 day 6 day 0 day 2 day 4 EOF1 to SLP anom. (>10dys) Watanabe (2004)

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

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

15. Dominant variability in a nonlinear barotropic model drag=(1000days)-1 22.1% 15.3% EOF1 EOF2 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

16. 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)

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

18. Dominant variability forced by the state-dependent noise drag=(20days)-1 24.8% 19.2% EOF1 EOF2 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

19. Linear dynamical operator for the transient eddy feedback (or the state-dependent noise) T21 barotropic model with SELF feedback neutral mode, ya zonal wind, ua eigenvalues selective excitation due to positive SELF interaction The neutral mode looks more like NAO! * Similar results are obtained with primitive model (Pan et al. 2006) Jin et al. (2006b) Stochastic ensemble and low-frequency variability Equation for the slow component of y1 : SELF closure Collaboration with Univ. of Hawaii

20. 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