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 inextratropics of each hemisphere(EOFs: leading eigenvectors ofcovariance matrix)•Strongest in winter but visible yearround in troposphere; present instratosphere during “active seasons”[Thompson and Wallace, 2000]

U, JJAφ AM λ, ϕ, z, t = P AM t E AM λ, ϕ, t
AM index

AM structure

Northern/southern annular mode indices: distribution

Eddy fluxes drive the mean flowchanges[Lorenz & Hartmann, J. Atmos. Sci(2001); J. Clim (2003)]

PV500500 K

surface

NAONAO

Impact of stratosphere duringwinter?
Ambaum & Hoskins (2002)

τNAO ∼ 10 d.

Projection onto annular mode index at each height:
Composites with respect to 10 hPa

AComposite of 18 Weak Vortex Events
103030hPa
10020300

1010000-90km-60

B-300Lag (Days)

3060

90Composite of 30 Strong Vortex Events
103030hPa
100km
20300

1010000-90-60-300Lag (Days)

3060[Baldwin & Dunkerton, Science, 2001]

90

Climate forcings andannular modes

Tropospheric responseto ozone depletion[Thompson & Solomon,2002]
GCM response to global warming[Kushner et al., 2001]

Response to altered stratospheric radiative state[Kushner & Polvani, J Clim, 2004]

The fluctuation-dissipation
theorem∂X + LX + NX = F∂tLinearize about climatological state:∂x + Ax = f .
∂tSteady forcing:x = A −1 f .

The fluctuation-dissipation
theorem∂X + LX + NX = F∂tLinearize about climatological state:∂x + Ax = f .
∂tSteady forcing:x = A −1 f .
 but we don’t know what A is:so how do we find out?

The fluctuation-dissipation
theoremWith no external forcing:Unknown A, stochastic f = є(r,t)

A= VΛWT ;∂X + LX + NX = F∂t

VWT = I = WT V
∂ WT x +ΛWT x = WT ε∂tLinearize about climatological state:∂x + Ax = f .
∂tSteady forcing:x = A −1 f .

C τ = xt + τx T t
G τ = C τ C −10→G t = VΓ τ WT ,Γ τ = exp−Λτ .
The vectors V are the principal
oscillation patterns (POPs)[e.g., Penland 2002]

The fluctuation-dissipation
theorem; principal
oscillation patterns (POPs)

With no external forcing:Unknown A, stochastic f = є(r,t)

A= VΛWT ;∂X + LX + NX = F∂t

VWT = I = WT V
∂ WT x +ΛWT x = WT ε∂tLinearize about climatological state:∂x + Ax = f .
∂tSteady forcing:x = A −1 f .

C τ = xt + τx T t
G τ = C τ C −10→TG
τTW̄ tτ == VΓCWxt+, τxtW〉〈Γ̄ τ = exp−Λτ . ̄C τ = exp−ΛτC 0

 Lag covariance of each principal
component decays exponentially withdecorrelation time τ = Λ-1
Baldwin and Dunkerton, Science (2001)

The fluctuation-dissipation
theorem; principal
oscillation patterns (POPs)∂X + LX + NX = F∂tSteady response to steadyforcing:

A = VΛWT

Linearize about climatological state:∂x + Ax = f .
∂t

ΛWT x = WT f→

Steady forcing:x = A −1 f .

y n = τngn

The fluctuation-dissipationtheorem; principal
oscillation patterns (POPs)∂X + LX + NX = F∂t

Steady response to steadyforcing:

A = VΛWT

Linearize about climatological state:∂x + Ax = f .
∂t→Steady forcing:x = A −1 f .
Response of PC}}

ΛWT x = WT f
y n = τngn
decorrelation time

projection of forcing

Application to forced
barotropic model[Gritsun & Branstator
J. Atmos. Sci., 2007]

Model Setup• GFDL dry dynamical core; no geography• T30 resolution• Linear radiation and friction schemes∂u + . . . . . . . . . . = F − γu + ν∇ 6 u∂t∂T +. . . . . . . . . . . = αT − Te∂t• 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

Troposphere “dynamical core” model with Held-Suarezlike forcing (Ring & Plumb 2007)
Mean and variability of control run

mean zonal windfirst 2 EOFs of mean uControl Run Time−Mean Zonal−Mean Zonal Wind

30−1−23−3−510−24−1−11−11223−30−152−211900
0

1−5−5−1−10−23−2−4−51

Pressure (hPa)2030−1−1−4 3−2 −2Pressure (hPa)15−3−6−1600

800

900

1000

−1−32−55240800

500

700

0700

1600

400
3510400

500−3425300

2−11 231−4 −5−632035
40

300

5200

1415200−2−4−1100−23−101100

EOFs 1 and 2 of Zonal−Mean Zonal Wind
225−80−60−40−20020

Latitude (degrees)

4060

80

1000

−80−60−40−20020

Latitude (degrees)

406080Behavior of a simplified GCM(no longitudinal variations inexternal conditions)(Ring & Plumb, J Atmos Sci, 2007)+1σ-1σEOF1+1σ∆

F, .F-1σ(Ring & Plumb, J Atmos Sci, 2007)

Responses to Mechanical Forcings
Torques − Trial L1

Response − Trial L1
2200

11−2400600

600−−0.750.50.5

1000

800

50.2

800.25−0400−60−300−1175
0.

Pressure (hPa)

200

23060

1000

−60−30Torques − Trial L3

03060

Response − Trial L32−30030
Latitude (degrees)

3−2−212−5−12800−1−60−15−4−0.25

00.75.5

1−.5

1000

−3−2600−550.7−0800

0.25

Pressure (hPa)4−36600−4−2400

13400

4200

−12120060

1000−60−30030
Latitude (degrees)

60

Ring and Plumb (2007)

Hypothesis: response in each EOF Un is proportional to projection offorcing onto Un

SH Forcing and Response Projections

NH Forcing and Response Projections

50404030

Projection of Response

Projection of Response

3020
100−1020

100−10−20−20−30−30−40−40−2−1.5−1−0.500.5

Projection of Forcing

11.5

2−5x 10−2−1.5−1−0.500.5

Projection of Forcing

11.5

2−5x 10

Ring and Plumb (2007)

Reference Temperature Changes
Confined to Poleward of Jet
Teq

Difference − Trial P2

Change in ∂T /∂φ for Selected Trialref

100

100

200

200

300

300
20.5

1663−820.5
1−61.5

3Pressure (hPa)

600

23.5

700
4700

−4−4−60.5
1600

−2500

4500

400
−222.52.50.5211.5−4−23.52.5

1.5

1000

−80−60−402863260230

40

Latitude (degrees)−20020408−30900−4−8−60.5
1−60

800−2341000

1.5

2.5

900

62800

0.5
1Pressure (hPa)

4400

6080

Latitude (degrees)

Ring and Plumb (2007)

Wind Changes Resulting
From Poleward Side Tref Changes[u] Diff. − Trial P2

800

−60

−3−1−3003060

1000−60−3003060[u] Diff. − Trial P4−66−2−1244−1−3−1−2−1342−15400−4400−5−4−2−1−1313−7−6−2−2−4−52200−4 −5−32−1−1−13600−3−1−31−21800

800−11000−60−30030
Latitude (degrees)

60

1000

−60−30030
Latitude (degrees)

60

10 K Warming

1200

Pressure (hPa)2800[u] Diff. − Trial P3

6 K Warming

3−4−2−0.50.5

600

1Pressure (hPa)

2600

600

−1400
−11000−2400

4 K Warming

200

12005−0.

2 K Warming[u] Diff. − Trial P1

Direct Response to Forcing−5−5−10−510−0.2

50.25−−00.7.5
50.75103060

1000−60−3003060

SF Diff. Trial P6 − Full Model

20−1−101025

150.25

−0.25

−0.
25

0.7 0.5560−5−30030
Latitude (degrees)

5.5−0−60800−5−0.75

5600

1510600−20 −25400

5400−1−10551000−60−30030
Latitude (degrees)60

4 K Cooling2000.2510Pressure (hPa)15−30−0.25

4 K Cooling520800200

1000−15600

SF Diff. Trial P6 − No Eddy Change

800

105−60−50.

800−2040050.56001000

0.2−155400−10200−0.25−0.5Pressure (hPa)SF Diff. Trial P2 − Full Model

4 K Warming

2000.25

4 K Warming

SF Diff. Trial P2 − No Eddy ChangeResponse to Forcing Including
Eddy Flux Changes

1000−53060

55−5−101020−25−150−115−5−15−25−1010800

51000

5−60

−30030Latitude (degrees)

60

1000

−60−30030
Latitude (degrees)60

4 K Cooling−20−5 1010−20600

15400

5400

5200

800

0SF Diff. Trial P6 − Full Model

200600510−30−560

1030

25015Pressure (hPa)5510−30800

SF Diff. Trial P6 − With Eddy Change

4 K Cooling

15−5−152520−60−60−15−25800−10−10600

1056001000−20−10400

4 K Warming

400−520015Pressure (hPa)SF Diff. Trial P2 − Full Model

200−54 K Warming

SF Diff. Trial P2 − With Eddy ChangeResponses to Poleward Side
Thermal Forcings
NH Projection of Response Versus Forcing

SH Projection of Response Versus Forcing

8206154Projection of Response

Projection of Response

105020−2−5−4−10−15−4−6−3−2−101Projection of Forcing

234−6x 10−8−4−3−2−101

Projection of Forcing

234−6x 10

Ring and Plumb (2007)

The fluctuation-dissipation
theorem; principal
oscillation patterns (POPs)∂X + LX + NX = F∂t

Steady response to steadyforcing:

A = VΛWT

Linearize about climatological state:∂x + Ax = f .
∂t→Steady forcing:x = A −1 f .

}}ΛWT x = WT f
y n = τngn

Governing eqs of system
L
Linearize about unforced time-mean state [U,V,Ω,Θ](φ,p)
Anomalies [u,v,ω,T, Fu,FT](φ,p,t)

u t + V ⋅∇u + v ⋅ ∇U − fv = −∇ ⋅ Fu − λu + hT t + V ⋅ ∇T − κp ΩT + v ⋅ ∇Θ − κp ωΘ = −∇ ⋅ FT − αT + q
Assume anomalous eddy fluxes depend linearly on anomalous u(and neglect time lags) + stochastic term:∇ ⋅ Fu = E u u + ε u ; ∇ ⋅ FT = E T u + ε Tx t + Cx = g + εGoverning eqs of system
LLinearize about unforced time-mean state [U,V,Ω,Θ](φ,p)
Anomalies [u,v,ω,T, Fu,FT](φ,p,t)

u t + V ⋅∇u + v ⋅ ∇U − fv = −∇ ⋅ Fu − λu + hT t + V ⋅ ∇T − κp ΩT + v ⋅ ∇Θ − κp ωΘ = −∇ ⋅ FT − αT + q
Nonlinear balance:−1ap R cos 2 ϕ T ϕ = 2 sin ϕ uU + Ωa cos ϕ p

Neglect advection of static stability anomaliesu t + Au = f + ε
wheref = h − HL −11 ∂qa ∂ϕ−= Kuo-Eliassen response

2 sinϕp ∂hM
33∂pRa cos ϕEffective Torques:
Mechanical Forcing−2015400600

800

80060901000−90

Applied Forcing − Trial L5400400

102090−600800

609051000−905−30030
Latitude (degrees)60−15−2600−4020−6030−10−2060
40

1000−90015200

800

−30

Effective Forcing − Trial L5200600

−60

−530

−505−30

5−60

5−56001000−90Pressure (hPa)20−−640020−5−10400

200
60

10Pressure (hPa)40200

Effective Forcing − Trial L4−15Applied Forcing − Trial L4−60−30030
Latitude (degrees)

6090Ring and Plumb (2007)

Effective Torques:
Thermal Forcing
Effective Forcing − Trial P4400400600600

800

800

4200

82−3003060−4−6−2132−60−3901000−90

Applied Forcing − Trial P8−6022−1 −21000−90−3004−−−63060

90

Effective Forcing − Trial P8200

400

400

600

600

800

800

−8−6−6200−4−2−2−8−30030
Latitude (degrees)

90−60−30030Latitude (degrees)

42864 68260

1000−90−2−60

2311000−90−1−4−3 −2

Pressure (hPa)684

Pressure (hPa)

6200

2

Applied Forcing − Trial P46090Ring and Plumb (2007)

The fluctuation-dissipation
theorem∂X + LX + NX = F∂t

Steady response to steadyforcing:

A = VΛWT

Linearize about climatological state:∂x + Ax = f .
∂t→Steady forcing:x = A −1 f .

}}ΛWT x = WT f
y n = τngn

Troposphere “dynamical core” model with Held-Suarezlike forcing (Ring & Plumb 2007)
Mean and variability of control run

mean zonal windfirst 2 EOFs of mean uControl Run Time−Mean Zonal−Mean Zonal Wind

30−1−23−3−510−24−1−1

1−11223−30−1

52−211900
01−5−5−1−10−23−2−4−51Pressure (hPa)2030−1−1−4 3−2 −2Pressure (hPa)15−3−6−1600

800

900

1000

−1−32−55240800

500

700

0700

1600

400
3510400

500−3425300

2−11 23

1−4 −5−632035
40

300

5200

1415200−2−4−1100−23−101100

EOFs 1 and 2 of Zonal−Mean Zonal Wind
225−80−60−40−20020

Latitude (degrees)

4060

80

1000

−80−60−40−20020

Latitude (degrees)

406080POP Spatial Patterns
8 EOFs retained – 10 day lag

1000

−0.04
−0.1
−0.06
0.060.08
0.12
0.1
0.04
0.02
0.08

20.0

0.02

−0.02

0.1

400

0.14

0.02

60

Eig. 2 of W − 8EOF − 10 Day Lag

−0.1

0.08

0.02

30

−0.02

−0.
08

00.04

4
−0.0

0.04

20.02

0.04
0.02

600
−0.

800

−0.04

0.04

800

0.02

−0.02

402

600
−0.0

Pressure (hPa)

−30

200

−0.1

60.0

−60

0.06

60

230

−0.08
−0.12
−00.02 .06 −0.02

00.1.04
−0800

−0.06

−0.1

0.10.04−0.02

4000.02−0.020.02−0.10.08

Pressure (hPa)600

Eig. 1 of W − 8EOF − 10 Day Lag

200−0.120.04−30−0.02−600.04

1000

−0.02

−0.02−0.06−0.080.06
0.02

800

40.0

1220.00.04

0.1

600

4000.02−0.

0.120.04400

200−0.04

200

Eig. 2 of V − 8EOF − 10 Day Lag−0.08

Eig. 1 of V − 8EOF − 10 Day Lag

20.0

1000

−60

−30030Latitude (degrees)

60

1000

−60−30030
Latitude (degrees)

60Ring and Plumb (2007)

Response Versus Effective Torques:(Ring & Plumb 2008)

x n = λ −1n gn = τn gn

decorrelation time of unforced mode
proj. of forcing onto mode

proj. of response onto mode

NH POP Analysis Forcing and Response Projections

SH POP Analysis Forcing and Response Projections
8060

Projection of Response on POP (m/s)

Projection of Response on POP (m/s)

6040200−20−40−60−340

200−20−40−60−2−101

Projection of Forcing on POP (m/s2)

23−5x 10−5−4−3−2−1012

Projection of Forcing on POP (m/s2)

circles indicate mechanically forced trials; squares thermally forced trials

345−5x 10

Response to altered stratospheric radiative state[Kushner & Polvani 2004]

Chan & Plumb (2009)Chan & Plumb (2009)τ = 262 d.

τ = 30 d.

Chan & Plumb (2009)τ = 262 d.

Chan & Plumb (2009)τ = 30 d.

Chan & Plumb (2009)Chan & Plumb (2009)

pressure-weighted

not pressure-weighted

POPs[Domeisen]

Anderson et al.

Conclusions

x n = λ −1n gn = τn gn• 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 theirautocorrelation times• POP analysis suggests distinct stratospheric and troposphericmodes• Hard to determine robust adjoint POPs for stratosphere-tropospheremodel to do accurate prediction from FDT• Response to tropical forcing does not fit the pattern – strong Hadleycirculation response

Kuo-Eliassen responseto observed forcing∆ (div F)[Thompson et al., J Atmos
Sci, 2006]∆Qχobserved

calculatedutChan & Plumb (2009)

Czz(τ)∂z = m̃ − D−1 z∂t̃ = m + Bz
m→ ∂z = m − τ −1 z∂t
τ −1 = D−1 − B

Cmm(τ)

Cmz(τ)(Chen & Plumb, 2009)∂z = m − τ −1 z∂t
τ −1 = D−1 − B

Chan & Plumb (2009)∂z = m − τ −1 z∂tC zz Δt = ∫C m m Δt + r − s exp r +τ s  ds dr
∫
−∞ −∞
00

(Chen & Plumb, 2009)References
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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 ofanomalous weather regimes, Science 244, 581-584
Chan, C. J. and Plumb, R. A., 2009: The Response to Stratospheric Forcing
and Its Dependence on the State of the Troposphere, J. Atmos. Sci., 66,2107-2115
Chen, G., Plumb, R. A., 2009: Quantifying the Eddy Feedback and thePersistence of the Zonal Index in an Idealized Atmospheric Model, J.
Atmos. Sci., 66, 3707-3720
Gritsun, A. and Branstator, G., 2007: Climate Response Using a ThreeDimensional Operator Based on the Fluctuation--Dissipation Theorem, J.
Atmos. Sci., 64, 2558-2575.
Kushner, P. J., Held, I. M., Delworth, T. L., 2001: Southern Hemisphere
Atmospheric Circulation Response to Global Warming, J. Climate,14, 22382249
Kushner, P. J., and L. M. Polvani, 2004: Stratosphere-troposphere coupling
in a relatively simple AGCM: the role of eddies, J. Climate, 17, 629-639.

References
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●Lorenz, David J., Dennis L. Hartmann, 2001: Eddy-Zonal Flow Feedback in
the Southern Hemisphere, J. Atmos. Sci., 58, 3312-3327
Lorenz, David J., Dennis L. Hartmann, 2003: Eddy-Zonal Flow Feedback in
the Northern Hemisphere Winter, J. Climate, 16 1212-1227
Ring, M. J. and Plumb, R. A., 2007: Forced Annular Mode Patterns in aSimple Atmospheric General Circulation Model, J. Atmos. Sci., 64, 36113626
Ring, M. J. and Plumb, R. A., 2008: The Response of a Simplified GCM to
Axisymmetric Forcings: Applicability of the Fluctuation--Dissipation
Theorem, J. Atmos. Sci., 65:12, 3880-3898
Thompson, D. W. J. and S. Solomon. 2002: Interpretation of recent
Southern Hemisphere climate change, Science, 296, 895-899
Thompson, D. W. J. and Furtado, J. C., Shepherd, T. G., 2006: On the
Tropospheric Response to Anomalous Stratospheric Wave Drag andRadiative Heating, J. Atmos. Sci., 63, 2616-2629.
Thompson, D. W. J., John M. Wallace, Gabriele C. Hegerl, 2000: Annular
Modes in the Extratropical Circulation. Part I: Month-to-Month Variability, J.
Climate, 13, 1000-1016