annular modes and the role of the stratosphere
TRANSCRIPT
Annular Modes and the Role of the Stratosphere
Alan Plumb
( + Mike Ring, Cegeon Chan, Gang Chen & Daniela Domeisen)
M. I. T.
Annular Modes
• Leading patterns of variability in
extratropics of each hemisphere
(EOFs: leading eigenvectors of
covariance matrix)
• Strongest in winter but visible year-
round in troposphere; present in
stratosphere during “active seasons”
[Thompson and Wallace, 2000]
U, JJA
Northern/southern annular mode indices: distribution
φAMλ,ϕ, z, t = PAMt EAMλ, ϕ, t
AM index AM structure
Eddy fluxes drive the mean flow changes
[Lorenz & Hartmann, J. Atmos. Sci(2001); J. Clim (2003)]
Impact of stratosphere during winter?
Ambaum & Hoskins (2002)
τNAO ∼ 10 d.
NAO PV500
NAO
surf
ace
500 K
Composite of 18 Weak Vortex Events
-90 -60 -30 0 30 60 90Lag (Days)
hPa
A10
30
100
300
1000
km
0
10
20
30
Composite of 30 Strong Vortex Events
-90 -60 -30 0 30 60 90Lag (Days)
hPa
B10
30
100
300
1000
km
0
10
20
30
Projection onto annular mode index at each height:
Composites with respect to 10 hPa
[Baldwin & Dunkerton, Science, 2001]
GCM response to global warming [Kushner et al., 2001]
Climate forcings and annular modes
Tropospheric response to ozone depletion [Thompson & Solomon, 2002]
Response to altered stratospheric radiative state[Kushner & Polvani, J Clim, 2004]
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
The fluctuation-dissipation theorem
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
The fluctuation-dissipation theorem
but we don’t know what A is: so how do we find out?
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
Unknown A, stochastic f = є(r,t)
A= VΛWT ;
VWT = I = WTV
∂∂t
WTx +ΛWTx = WTε
Cτ = xt + τxTt
Gτ = CτC0−1
→ G t = VΓτWT ,
Γτ = exp−Λτ .
The fluctuation-dissipation theorem
With no external forcing:
The vectors V are the principal oscillation patterns (POPs)
[e.g., Penland 2002]
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
Unknown A, stochastic f = є(r,t)
A= VΛWT ;
VWT = I = WTV
∂∂t
WTx +ΛWTx = WTε
Cτ = xt + τxTt
Gτ = CτC0−1
→ G t = VΓτWT ,
Γτ = exp−Λτ .
Cτ = ⟨WTxt + τxtW⟩
Cτ = exp−ΛτC0
The fluctuation-dissipation theorem; principal oscillation patterns (POPs)
With no external forcing:
Lag covariance of each principal component decays exponentially with decorrelation time τ = Λ-1
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
A = VΛWT
ΛWTx = WTf
→ yn = τngn
Steady response to steady forcing:
The fluctuation-dissipation theorem; principal oscillation patterns (POPs)
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
A = VΛWT
ΛWTx = WTf
→ yn = τngn
Steady response to steady forcing:
The fluctuation-dissipation theorem; principal oscillation patterns (POPs)
Response of PC decorrelation time projection of forcing
Application to forced barotropic model [Gritsun & BranstatorJ. Atmos. Sci., 2007]
Model Setup
• GFDL dry dynamical core; no geography
• T30 resolution
• Linear radiation and friction schemes
• Held-Suarez-like reference temperature profile but modified for perpetual solstitial conditions
• Friction twice the value used by Held and Suarez (1994) to reduce decorrelation times
∂u∂t
+ . . . . . . . . . . = F − γu + ν∇6u
∂T∂t
+. . . . . . . . . . . = αTe − T
Troposphere “dynamical core” model with Held-Suarez-like forcing (Ring & Plumb 2007)
Mean and variability of control run
−80 −60 −40 −20 0 20 40 60 80
100
200
300
400
500
600
700
800
900
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
EOFs 1 and 2 of Zonal−Mean Zonal Wind
−6
−6−
5
−5−
5
−4−
4
−4
−4
−3
−3
−3
−3
−2
−2
−2
−2
−2
−1
−1
−1
−1
−1
11
1
11
1
2
2
2
2
2
3
3
3
3
44
4
5
5
−3
−3
−2
−2
−2
−1
−1
−1
−1
−1
−1
−1
−1
1
1
1
1
1
2
2
2
2
33
34
−80 −60 −40 −20 0 20 40 60 80
100
200
300
400
500
600
700
800
900
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
Control Run Time−Mean Zonal−Mean Zonal Wind
−10
−5
−5−
5
5
5
10
10
15
15
20
20
25
25
30
30
354
0
0
0
0
0
0
mean zonal wind first 2 EOFs of mean u
Behavior of a simplified GCM
(no longitudinal variations in
external conditions)
(Ring & Plumb, J Atmos Sci, 2007)
EOF1
+1σ
-1σ
+1σ
-1σ
F, .F
∆
(Ring & Plumb, J Atmos Sci, 2007)
Responses to Mechanical Forcings
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
Torques − Trial L1
−0.7
5−0.
5
−0.25
0.2
5
0.5 0.75
−60 −30 0 30 60
200
400
600
800
1000
Response − Trial L1
−2
−1
11
1
2
2
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
Latitude (degrees)
Torques − Trial L3
−0.75−0.5
−0.2
5
0.2
50.5
0.7
5
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
Response − Trial L3
−5
−5
−4
−4 −3
−3
−2
−2
−2
−2
−1
−1
−1
−11
1
1
1
2
2
2
2
3
3
4
4
5
6
Hypothesis: response in each EOF Un is proportional to projection of forcing onto Un
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 10−5
−40
−30
−20
−10
0
10
20
30
40
50
SH Forcing and Response Projections
Projection of Forcing
Pro
jection o
f R
esponse
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
x 10−5
−40
−30
−20
−10
0
10
20
30
40
NH Forcing and Response Projections
Pro
jection o
f R
esponse
Projection of Forcing
Reference Temperature Changes
Confined to Poleward of Jet
−60 −30 0 30 60
100
200
300
400
500
600
700
800
900
1000
Latitude (degrees)
Pre
ssure
(hP
a)
Teq
Difference − Trial P2
0.5
0.5
0.5
0.5
0.5
0.5
1
11
1
11
1.5
1.5
1.5
1.5
22
2
2
2.5
2.5
2.5
2.5
3
3
3
3
3.5
3.5
4
−80 −60 −40 −20 0 20 40 60 80
100
200
300
400
500
600
700
800
900
1000
Pre
ssure
(hP
a)
Latitude (degrees)
Change in ∂Tref
/∂φ for Selected Trial
−8
−8
−6
−6
−6
−4
−4
−4
−4
−2
−2
−2
−2
22
2
22
44
4
4 6
6
66 8
8
Wind Changes Resulting
From Poleward Side Tref Changes
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
[u] Diff. − Trial P1
−1
−0.5
−0.50
.5
−60 −30 0 30 60
200
400
600
800
1000
[u] Diff. − Trial P2
−4
−3
−2
−2 −
1
−1
1
1
2
2
3−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
[u] Diff. − Trial P3
−5−4
−4
−3−
3
−2
−2
−1
−1
−1−
1
−1
1
1
2
2
3
3
4
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
[u] Diff. − Trial P4
−7
−6
−6
−5−
5 −4
−4
−3
−3
−2
−2
−2
−2
−1
−1
−1
−1
−1−1
−1
1
1
2
2
3
3
4
4
5
6
2 K
Warm
ing
6 K
Warm
ing
4 K
Warm
ing
10 K
Warm
ing
Direct Response to Forcing4 K
Warm
ing
4 K
Coolin
g
4 K
Warm
ing
4 K
Coolin
g
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
SF Diff. Trial P2 − No Eddy Change
−0.7
5−
0.5
−0.5
−0.2
5
−0.2
5
0.2
5
0.25
0.2
5
0.5
0.5
0.7
51
−60 −30 0 30 60
200
400
600
800
1000
SF Diff. Trial P2 − Full Model
−20
−15 −
15
−10
−10
−5
−5
−5
−5
5
5
10
10
15
20
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
Latitude (degrees)
SF Diff. Trial P6 − No Eddy Change
−1−0.75
−0.5
−0.2
5
−0.2
5
−0.2
5
0.2
5
0.25
0.5
0.75
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
SF Diff. Trial P6 − Full Model
−25
−20
−15−10
−10
−5
−5
5
5
5
5
10
10
10
15
15
20
25
Response to Forcing Including
Eddy Flux Changes4 K
Warm
ing
4 K
Coolin
g
4 K
Warm
ing
4 K
Coolin
g
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssu
re (
hP
a)
SF Diff. Trial P2 − With Eddy Change
−15
−10
−10 −
5−5
5
510
15
20
25
−60 −30 0 30 60
200
400
600
800
1000
SF Diff. Trial P2 − Full Model
−25
−20
−15
−10
−5
−5
5
5
10
101
5
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
SF Diff. Trial P6 − With Eddy Change
−25
−20
−15
−10 −
5
−5
5
510
10
15
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
SF Diff. Trial P6 − Full Model
−25
−20
−15
−10
−10
−5
−5
5
5
55
10
10
10
15
15
20
25
Responses to Poleward Side
Thermal Forcings
−4 −3 −2 −1 0 1 2 3 4
x 10−6
−15
−10
−5
0
5
10
15
20
SH Projection of Response Versus Forcing
Projection of Forcing
Pro
jection o
f R
esponse
−4 −3 −2 −1 0 1 2 3 4
x 10−6
−8
−6
−4
−2
0
2
4
6
8
Projection of Forcing
Pro
jection o
f R
esponse
NH Projection of Response Versus Forcing
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
A = VΛWT
ΛWTx = WTf
→ yn = τngn
Steady response to steady forcing:
The fluctuation-dissipation theorem; principal oscillation patterns (POPs)
ut + V ⋅∇u + v ⋅ ∇U − fv = −∇ ⋅ Fu − λu + h
T t + V ⋅ ∇T − κp ΩT + v ⋅ ∇Θ − κ
p ωΘ = −∇ ⋅ FT − αT + q
L
Governing eqs of system
Linearize about unforced time-mean state [U,V,Ω,Θ](φ,p)
Anomalies [u,v,ω,T, Fu,FT](φ,p,t)
Assume anomalous eddy fluxes depend linearly on anomalous u (and neglect time lags) + stochastic term:
∇ ⋅ Fu = Euu + εu ; ∇ ⋅ FT = ETu + εT
xt + Cx = g + ε
ut + V ⋅∇u + v ⋅ ∇U − fv = −∇ ⋅ Fu − λu + h
T t + V ⋅ ∇T − κp ΩT + v ⋅ ∇Θ − κ
p ωΘ = −∇ ⋅ FT − αT + q
L
Governing eqs of system
Linearize about unforced time-mean state [U,V,Ω,Θ](φ,p)
Anomalies [u,v,ω,T, Fu,FT](φ,p,t)
ap−1R cos2ϕ Tϕ = 2sinϕ uU + Ωacosϕp
Nonlinear balance:
ut + Au = f + ε
where
= Kuo-Eliassen response
Neglect advection of static stability anomalies
f = h − HL−1 1a
∂q
∂ϕ − 2 sinϕ
a3 cos3ϕ
p
R
∂∂phM
Effective Torques:
Mechanical Forcing
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Pre
ssu
re (
hP
a)
Applied Forcing − Trial L4
−60
−40
−20
60
40
20
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Effective Forcing − Trial L4
−15
−10
−5
−5
20
15
10
5
5
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
Applied Forcing − Trial L5
−60
−40
−20
6040
20
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Latitude (degrees)
Effective Forcing − Trial L5
−20
−15
−10
−5−
5
20
15
10
5
5
5
Effective Torques:
Thermal Forcing
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Pre
ssu
re (
hP
a)
Applied Forcing − Trial P4
−3
−2
−1
3 21
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Effective Forcing − Trial P4
−6−6 −4−4 −2
−2
8
8
6
6
4
4
2
2
2
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
Applied Forcing − Trial P8
−3
−2
−1
3
2
1
−90 −60 −30 0 30 60 90
200
400
600
800
1000
Latitude (degrees)
Effective Forcing − Trial P8
−8
−8−
6
−6
−4
−4
−2−
2
−2
88 664
4 22
∂X∂t
+ LX + NX = F
Linearize about climatological state:
∂x∂t
+ Ax = f .
Steady forcing:
x = A−1f .
A = VΛWT
ΛWTx = WTf
→ yn = τngn
Steady response to steady forcing:
The fluctuation-dissipation theorem
Troposphere “dynamical core” model with Held-Suarez-like forcing (Ring & Plumb 2007)
Mean and variability of control run
−80 −60 −40 −20 0 20 40 60 80
100
200
300
400
500
600
700
800
900
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
EOFs 1 and 2 of Zonal−Mean Zonal Wind
−6
−6−
5
−5−
5
−4−
4
−4
−4
−3
−3
−3
−3
−2
−2
−2
−2
−2
−1
−1
−1
−1
−1
11
1
11
1
2
2
2
2
2
3
3
3
3
44
4
5
5
−3
−3
−2
−2
−2
−1
−1
−1
−1
−1
−1
−1
−1
1
1
1
1
1
2
2
2
2
33
34
−80 −60 −40 −20 0 20 40 60 80
100
200
300
400
500
600
700
800
900
1000
Latitude (degrees)
Pre
ssu
re (
hP
a)
Control Run Time−Mean Zonal−Mean Zonal Wind
−10
−5
−5−
5
5
5
10
10
15
15
20
20
25
25
30
30
354
0
0
0
0
0
0
mean zonal wind first 2 EOFs of mean u
POP Spatial Patterns
−60 −30 0 30 60
200
400
600
800
1000
Pre
ssure
(hP
a)
Eig. 1 of V − 8EOF − 10 Day Lag
−0.1
2
−0.1
−0.0
8−
0.0
6
−0.0
4
−0.0
2
−0.0
2
0.0
2
0.0
2
0.0
20.0
4
0.0
40.0
6
0.0
8
0.10.1
2
−60 −30 0 30 60
200
400
600
800
1000
Eig. 2 of V − 8EOF − 10 Day Lag
−0.12
−0.1
−0.0
8−
0.0
6
−0.0
4−0.0
4
−0.0
2
−0.0
2
0.0
2
0.0
2
0.0
2
0.0
2
0.0
4
0.0
4
0.0
4
0.04 0.0
6 0.0
8
0.1
0.1
2
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
Pre
ssure
(hP
a)
Eig. 1 of W − 8EOF − 10 Day Lag
−0.12
−0.1
−0.0
8
−0.0
6
−0.0
4
−0.0
4
−0.0
2
−0.0
2
−0.0
2
0.0
2
0.0
2
0.0
2
0.02
0.0
4
0.0
4
0.06
0.0
8
0.1
−60 −30 0 30 60
200
400
600
800
1000
Latitude (degrees)
Eig. 2 of W − 8EOF − 10 Day Lag
−0.1
2
−0.1 −
0.0
8
−0.0
6
−0.04
−0.0
2 −0.0
2
−0.02
0.0
2
0.0
2
0.0
20.02
0.0
4
0.0
4
0.0
4
0.0
6
0.0
80.1
0.1
2
0.1
4
8 EOFs retained – 10 day lag
Response Versus Effective Torques:(Ring & Plumb 2008)
−3 −2 −1 0 1 2 3
x 10−5
−60
−40
−20
0
20
40
60
80
SH POP Analysis Forcing and Response Projections
Projection of Forcing on POP (m/s2)
Pro
jection o
f R
esponse o
n P
OP
(m
/s)
−5 −4 −3 −2 −1 0 1 2 3 4 5
x 10−5
−60
−40
−20
0
20
40
60
NH POP Analysis Forcing and Response Projections
Projection of Forcing on POP (m/s2)
Pro
jection o
f R
esponse o
n P
OP
(m
/s)
circles indicate mechanically forced trials; squares thermally forced trials
xn = λn−1gn = τngn
proj. of response onto mode proj. of forcing onto mode
decorrelation time of unforced mode
Response to altered stratospheric radiative state
[Kushner & Polvani 2004]
Chan & Plumb (2009)
τ = 262 d.
τ = 30 d.
τ = 262 d.
τ = 30 d.
pressure-weighted not pressure-weighted
POPs
[Domeisen]
Conclusions
• FDT works reasonably well as a predictor of system response
• Response depends on projected effective forcing and on
autocorrelation time τ of “natural” fluctuations
• Model simulations need to have good EOFs (or POPs) and their autocorrelation times
• POP analysis suggests distinct stratospheric and troposphericmodes
• Hard to determine robust adjoint POPs for stratosphere-troposphere model to do accurate prediction from FDT
• Response to tropical forcing does not fit the pattern – strong Hadley circulation response
xn = λn−1gn = τngn
[Thompson et al., J AtmosSci, 2006]
Kuo-Eliassen response to observed forcing
∆ (div F) ∆Q
χ
ut
observed calculated
(Chen & Plumb, 2009)
Czz(τ)
Cmm(τ)
Cmz(τ)
∂z∂t
= m − D−1z
m = m + Bz
→ ∂z∂t
= m − τ−1z
τ−1 = D−1 − B
∂z∂t
= m − τ−1z
τ−1 = D−1 − B
(Chen & Plumb, 2009)
∂z∂t
= m − τ−1z
C zzΔt = ∫−∞
0 ∫−∞
0
Cmm Δt + r − s exp r + sτ ds dr
References Ambaum, M. H. P. and Hoskins, B. J., 2002: The NAO Troposphere--
Stratosphere Connection, J. Climate, 15:14, 1969-1978 Baldwin, M. P. and Dunkerton, T. J., 2001: Stratospheric harbingers of
anomalous weather regimes, Science 244, 581-584 Chan, C. J. and Plumb, R. A., 2009: The Response to Stratospheric Forcing
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