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The General Circulation of the Atmosphere and its Variability
Dennis L. Hartmann
Dynamics Seminar
October 18, 2007
Thomson 1857
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Outline of Talk
• Description of the General Circulation in classical terms
• Review of some of the advances in the past 25-40 years
• Discussion of theories of Dynamical Variability in the Atmosphere
Thomson 1857
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• Dry Dynamics, mostly.
• Momentum, mostly
• My favorite things.
• Some Old Chesnuts
Focus of Talk
Ferrel 1856
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Zonal Average Views
• Zonal Average Climatology
• Zonal Average of x = [x]
• x - [x] = x* = deviation from the zonal average
• Time average of x = x
• x - x = x’ = deviation from time average
Ferrel 1859
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Zonal Average Zonal Wind
Ferrel 1859
ERA-40
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Zonal Average Meridional Wind
Ferrel 1859
ERA-40
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Eddy Covariances
Maury 1855
[vT ]=[v][T ] +[v* T* ]Zonal Average
of Product
Product of Zonal Averages
Zonal Averageof Eddy Product
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Eddy Meridional Temp. Flux
ERA-40
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Eddy Meridional Momentum Flux
ERA-40
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Eddy Meridional Momentum FluxTransient Total
Stationary Stationary - JJA
ERA-40
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Eddy-Driven Jets
• When you see surface westerlies with westerlies above, as in midlatitudes, these westerlies are driven by large-scale eddy momentum fluxes.
• The observed mean meridional circulations export mass-averaged westerly relative angular momentum.
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Zonal-mean Momentumdu
dt= fv−
1acosϕ
∂φ∂λ
+ Drag
Expand total derivative and use continuity in p-coord.
∂u∂t+
1
acosϕ
∂
∂λ
1
2u2
⎛⎝⎜
⎞⎠⎟+
1
cosϕ
∂
∂ϕ(uv cosϕ ) +
∂
∂p(ωu) = fv −
1
acosϕ
∂φ
∂λ+ Drag
Multiply by a cosϕ and average over longitude.
∂[m]∂t
+1
a
∂[vm]
∂ϕ+∂[ωm]
∂p= f [v]a cosϕ +[Drag]a cosϕ
m =uacosϕ
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Zonal-mean Momentum
Next, integrate this over the mass of the atmosphere.
∂[m]∂t
+1
a
∂[vm]
∂ϕ+∂[ωm]
∂p= f [v]a cosϕ +[Drag]a cosϕ
m =uacosϕ
(...)dp =(...)∂
0
ps
∫
∂[m]∂
∂t+1
a
∂[vm]∑
∂ϕ+[ωm]
0
ps = f [v]∂ a cosϕ +[Drag]∑ a cosϕ
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Zonal-mean Momentum
In steady state, this term is zero, by mass continuity.
∂[m]∂
∂t+1
a
∂[vm]∑
∂ϕ+[ωm]
0
ps = f [v]∂ a cosϕ +[Drag]∑ a cosϕ
Let’s make this part of the drag’.
So in steady state,
1
a
∂[vm]∑
∂ϕ= [Drag']∑ acosϕ
∂∂ϕ
[v][u]∑ + [v *u*]∑( )cosϕ{ } = [Drag ']∑ acosϕ
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Steady, Mass-integrated Zonal-mean Momentum Equation
Mass-integrated mean zonal wind advection
Meridional eddy flux of zonal momentum
∂∂ϕ
[v][u]∑ + [v *u*]∑( )cosϕ{ } = [Drag ']∑ acosϕ
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Steady, Mass-integrated Zonal-mean Momentum Advection
m =uacosϕPeaks at around30N, so bothHadley and Ferrel Cells export relativeangular momentum
∂∂ϕ
[v][u]∑( )cosϕ{ } + . . . = [Drag ']∑ acosϕ
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Steady, Mass-integrated Zonal-mean Momentum Advection
m =uacosϕEddies and MMCexport relativeangular momentum from the tropics and the eddies import relative angular momentum into extratropics, and focus it above the surface westerlies.
∂∂ϕ
[v *u*]∑( )cosϕ{ } ≈ [Drag ']∑ acosϕ
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Conclusion: Eddies must move momentum poleward
• If we have a climate with easterlies in the tropics and westerlies in midlatitudes, and eddies dominate the circulation in between, then eddies must transport westerly momentum poleward.
∂∂ϕ
[v *u*]∑( )cosϕ{ } ≈ [Drag ']∑ acosϕ
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Role of Eddies in MomentumLorenz (1952)
Ferrel 1859
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Role of Eddies in MomentumLorenz (1967)
Ferrel 1859
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Momentum is Funny Stuff
Consider a non-divergent, barotropic fluid
∂[u]∂t= −
∂
∂y[u * v*] = [v *ζ *]
ζ =∂v∂x−∂u
∂y
∂u∂x+∂v
∂y= 0
Enstrophy Equation
∂∂t1
2ζ *2⎡
⎣⎢⎤⎦⎥+ βeff [v *ζ *] = [F *ζ *]
βeff [v *ζ *] = [F *ζ *]
Steady Enstrophy Equation
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Momentum is Funny Stuff
∂[u]∂t= −
∂
∂y[u * v*] = [v *ζ *]
βeff [v *ζ *] = [F *ζ *]
Steady Enstrophy Equation
Zonal Wind Equation
If source F* adds enstrophy, eddy vorticity flux must be up-gradient (normally northward) to maintain steady state.
That would tend to accelerate the flow in the region wherethe source of eddy enstrophy is located.
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Momentum is Funny Stuff
This can be achieved, if the eddies are able to propagate out of the source region.
If angular momentum is conserved, there must also be aneasterly acceleration somewhere else, to balance out thewesterly acceleration produced in the eddy source region.
∂[u]∂t= −
∂
∂y[u * v*] = [v *ζ *]
N.B. Wave propagation goes in the opposite direction to the momentum flux, so if waves propagate out ofregion, momentum is transported in.
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Barotropic Cartoon
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Momentum is Funny Stuff
∂[u]∂t= −
∂
∂y[u * v*] = [v *ζ *]
+-
Where is eddy source, and sink ?+ -
βeff [v *ζ *] = [F *ζ *]
+ - +
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Momentum is Funny Stuff
∂u∂t= −
∂
∂yu 'v '( ) = v 'ζ '
In quasi-geostrophic, baroclinic case,
∂u∂t+ f v* = ρ0 acosϕ( )
−1∇gF = v 'q '
Fϕ =−ρ0 acosϕ u'v'
Fz = f ρ0 acosϕ v'θ ' / θz
F =(Fϕ , Fz) = Eliassen-Palm Flux Vector
How did the eddy heat flux end up in the momentum Budget?
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How did the eddy heat flux end up in the momentum Budget?
• The eddy heat flux represents the form drag in a hydrostatic and quasi-geostrophic wave that tilts westward with height.
• ‘Easy’ to visualize by thinking in potential temperature coordinates.
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• Consider the following picture of the temperature and pressure variations on a height surface associated with a westward tilting wave.
How did the eddy heat flux end up in the momentum Budget?
LHL HW C WC
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• Add potential temperature perturbation.
How did the eddy heat flux end up in the momentum Budget?
dθθ
=dTT
−κp
dp
LHL H
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• Sketch in dz necessary to get back to a constant potential temperature surface; dz ~ -dtheta
How did the eddy heat flux end up in the momentum Budget?
LHL H
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• Now let’s focus in on the resulting form drag.
How did the eddy heat flux end up in the momentum Budget?
H L
Form Drag = p∫∂h∂x
dx onθ surface
In westward-tilting wave,atmosphereabove exertsan eastwardtorque on atmosphere below, and vice-versa.
LHL H
Height oftheta surface,material surface.
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Eliassen-Palm Cross Sections
∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )
−1∇gF
Fϕ =−ρ0 acosϕ u'v'
Fz = f ρ0 acosϕ v'θ ' / θz
F =(Fϕ , Fz) = Eliassen-Palm Flux Vector
Heat Flux partdominatesclimatology of E-P Cross-Sections
Tanaka, et al. 2006, JMSJ
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How did the eddy heat flux end up in the momentum Budget?
• In middle latitudes, baroclinic eddies have poleward heat fluxes that are associated with
• eddy energy production,
• upward wave propagation and
• huge form drag that moves momentum from the upper to the lower troposphere.
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The Residual or Lagrangian Circulation
∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )
−1∇gF
∂θ∂t+ w *
∂θ
∂z=Q
1
acosϕ∂∂ϕ
v* cosϕ( ) +1ρ0
∂∂z
ρ0w*( ) =0
w*=−1
aρ0 cosϕ∂∂ϕ
∇gF2Ωsinϕ
⎛⎝⎜
⎞⎠⎟ϕ
dz'x
∞
∫⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
Use momentum (ignore tendency) and continuity,
Mean sinking is the meridional gradient of the drag integrated down to that level. Thermo not used.
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Zonal Mean Circulations
∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )
−1∇gF
= Residual or Lagrangian Circulation
Heat Flux partdominatesclimatology of E-P Cross-Sections
v*
Tanaka, et al. 2006, JMSJ
∂θ∂t+ w *
∂θ
∂z=Q
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Stationary and Transient Driving of Lagrangian Circulation
∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )
−1∇gF
Tanaka, et al. 2006, JMSJ
v*
Transient
Stationary
Hadley Cell
Eddy-Driven Cell
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If the eddy heat flux and form drag are so dominant in the momentum budget, are lateral eddy momentum fluxes really
that important? They have to be.
• Variability of eddy-driven jets is an important part, perhaps the most important part, of extratropical variability.
• ‘Easiest’ place to see this is in the Southern Hemisphere.
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Southern Hemisphere Eddy-Driven Jet.
Lots of Ocean, not much topography,
fairly zonally symmetric, most of form drag from high wavenumbers.
Clear, almost seasonally invarianteddy-driven jet.
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Southern Hemisphere Eddy-Driven Jet.
N H
S H
Tanaka,et al. 2006
Total4-7
1-3
>8
1-3
Form Drag by Zonal Wavenumber
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Southern Hemisphere Eddy-Driven Jet.
Lots of Ocean, not much topography,fairly zonally symmetric,
most of form drag from high wavenumbers.Clear, almost seasonally invariant
eddy-driven jet.Subtropical
Jet
Eddy-DrivenJet
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Southern Hemisphere Eddy-Driven Jet.
SubtropicalJet
Eddy-DrivenJet
Lorenz & Hartmann, 2001
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Southern Hemisphere Eddy-Driven Jet.
Lots of Ocean, not much topography,fairly zonally symmetric,
most of form drag from high wavenumbers.Clear, almost seasonally invariant
eddy-driven jet.Primary mode of low-frequency variability is
North-South movement of the Eddy-Driven Jet.
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Southern Hemisphere Eddy-Driven Jet.
Hartmann and Lo, 1998
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Southern Hemisphere Eddy-Driven Jet.
Hartmann and Lo, 1998
First EOF of zonal wind almost independent of season.
Amplitude of EOF 1 is slowly varying, with most variance > 20 days
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Southern Hemisphere Eddy-Driven Jet.
Hartmann and Lo, 1998
First EOF represents N-S shift of eddy driven jet.
1.5 standard deviation of PC-1
corresponds to 10˚ latitude shift
of surface westerlies.
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Southern Hemisphere Eddy-Driven Jet.
Hartmann and Lo, 1998
Momentum Budget of Meridional Eddy-Jet Meandering
Residual Circ. Barotropic
‘Baroclinic’aka Form Drag
Drag determined as residual
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Momentum Budget of Meridional Eddy-Jet Meandering
Hartmann and Lo, 1998
Residual Circ.
Barotropic
‘Baroclinic’aka Form Drag
Drag determined as residual
Total EddyForcing
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Southern Hemisphere Eddy-Driven Jet.
Lots of Ocean, not much topography,fairly zonally symmetric,
most of form drag from high wavenumbers.Clear, almost seasonally invariant
eddy-driven jet.Primary mode of low-frequency variability is
North-South movement of the Eddy-Driven Jet.
Eddy fluxes and residual circulation adjust to new position of jet, so that net tendency is small and jet is stable in
new position.Despite being relatively small in climatology, meridional
momentum flux convergence seems to play acentral role in N-S movement of eddy-driven jet.
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Southern Hemisphere Eddy-Driven Jet.
Lots of Ocean, not much topography,fairly zonally symmetric,
most of form drag from high wavenumbers.Clear, almost seasonally invariant
eddy-driven jet.Primary mode of low-frequency variability is
North-South movement of the Eddy-Driven Jet.
Eddy fluxes and residual circulation adjust to new position of jet, so that net tendency is small and jet is stable in
new position.But, are the eddies passive or active, and do eddies add a positive feedback that adds persistence to departures of
the eddy-driven jet position?
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Positive Eddy Feedback
Lorenz & Hartmann, 2001
Focus on vertical average momentum balance and meridional wave propagation.
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Positive Eddy Feedback
Lorenz & Hartmann, 2001
Z=u M=-d/dy(u’v’)
Vertical mean zonal wind and eddy momentum forcing of first EOF (N-S shift)are coherent across a broadrange of frequencies and forcing leads wind, except for very long periods where theycome into phase.
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Positive Eddy Feedback
Lorenz & Hartmann, 2001
Z=u is red M=-d/dy(u’v’) is whiter
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Positive Eddy Feedback
Lorenz & Hartmann, 2001
Z=u M=-d/dy(u’v’)
Clues
a. M remembers Z
b. High-frequency eddies produce low-frequency forcing.
a
bsynoptic = 2-7 days
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Simple Model ofPositive Eddy Feedback
Lorenz & Hartmann, 2001
M=-d/dy(u’v’)
b. High-frequency eddies produce low-frequency forcing, because they respond to zonal flow.
dz
dt=m−
zτ
m = %m+bz
d%z
dt= %m−
%zτ
Linear System
Assume part of momentum forcing depends on zonal wind.
Choose b to explain long-term memory,then z without feedback can be computed.
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Why is Transient Eddy Feedback Positive?Are Eddy-Driven Jets Self-Sustaining?
Yu & Hartmann 1993
• Wave source is baroclinic instability, which produces wave energy nearthe surface where the meridional temperature gradient is large.
• Waves propagate upward inwesterly winds
• Form drag produces a huge downwardzonal momentum flux
• A thermally direct overturning circulationdevelops to balance the momentumbudget.
• If waves can propagate out of barocliniczone they can bring in angular momentum.
• Diabatic heating must balance heatingby overturning cell.
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Why is Transient Eddy Feedback Positive?Are Eddy-Driven Jets Self-Sustaining?
Yu & Hartmann 1993
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Why is Transient Eddy Feedback Positive?Are Eddy-Driven Jets Self-Sustaining?
Yu & Hartmann 1993
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Why is Transient Eddy Feedback Positive?Are Eddy-Driven Jets Self-Sustaining?
• So the eddy source and eddy momentum flux convergencecan just follow the jet.
• The meridional cell forced by the form drag of the growingeddies also follows the eddy source, which is the jet.
• Remaining problem is to bring along the diabatic heating thatsustains the meridional cell associated with the form drag.
• If the heating is driven by the departure from equilibrium forced by the meridional circulation, this is not a problem, theheating couplet follows the circulation.
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Are Meridional Displacements of Eddy-Driven Jets Self-Sustaining?
∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )
−1∇gF
∂θ∂t+ w *
∂θ
∂z=Q
1
acosϕ∂∂ϕ
v* cosϕ( ) +1ρ0
∂∂z
ρ0w*( ) =0
w*=−1
aρ0 cosϕ∂∂ϕ
∇gF2Ωsinϕ
⎛⎝⎜
⎞⎠⎟ϕ
dz'x
∞
∫⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
Use momentum (ignore tendency) and continuity,
To sustain jet in new location, need to move diabaticheating with wave driving.
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Are Meridional Displacements of Eddy-Driven Jets Self-Sustaining?
∂θ∂t+ w *
∂θ
∂z=Q
w*=−1
aρ0 cosϕ∂∂ϕ
∇gF2Ωsinϕ
⎛⎝⎜
⎞⎠⎟ϕ
dz'x
∞
∫⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
To sustain jet in new location, need to move diabaticheating with wave driving.
Works fine in simple models with Newtonian heating, if baroclinic zone is broad
Q =α(Tequil −T ), ....Tequil =Acos(2ϕ )
Eddy momentum driving can define shape of heating.
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Southern Hemisphere Eddy-Driven Jet.
SubtropicalJet
Eddy-DrivenJet
Lorenz & Hartmann, 2001
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SAM & Precip
Sen Gupta & England, 2006
45
30
60
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Why is Transient Eddy Feedback Positive?Are Eddy-Driven Jets Self-Sustaining?
Yu & Hartmann 1993
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Momentum Budget of Meridional Eddy-Jet Meandering
Hartmann and Lo, 1998
Residual Circ.
Barotropic
‘Baroclinic’aka Form Drag
Drag determined as residual
Total EddyForcing
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SAM & Precip
Sen Gupta & England, 2006
45
30
60
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Parameterizing EddiesLorenz (1967)
Ferrel 1859
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Conclusion
Thomson 1857
• We can explain in simple terms that eddy momentum fluxes are associated with the
growth, propagation and absorption of waves.
• It is hard to imagine a climate of Earth, in which eddies do not move momentum poleward.
• The interaction of eddies with jets and diabatic heating produces interesting variability, about which we are still learning.