1. 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

2. 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

3. 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

4. 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

5. 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)

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. 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 !

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.
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”

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. 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)

12. 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)

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

14. 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

15. 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

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. 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

18. 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

19. 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

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