the general circulation of the atmosphere and its variability

The General Circulation of the Atmosphere and its Variability

Dennis L. HartmannDynamics SeminarOctober 18, 2007

Thomson 1857

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

• Dry Dynamics, mostly.

• Momentum, mostly

• My favorite things.

• Some Old Chesnuts

Focus of Talk

Ferrel 1856

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

Zonal Average Zonal Wind

Ferrel 1859

ERA-40

Zonal Average Meridional Wind

Ferrel 1859

ERA-40

Eddy Covariances

Maury 1855

[vT ]=[v][T ] +[v*T*]Zonal Average

of Product

Product of Zonal Averages

Zonal Averageof Eddy Product

Eddy Meridional Temp. Flux

ERA-40

Eddy Meridional Momentum Flux

ERA-40

Eddy Meridional Momentum FluxTransient Total

Stationary Stationary - JJA

ERA-40

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.

Zonal-mean Momentumdudt

=fv−1

acosf∂φ∂λ

+ Drag

Expand total derivative and use continuity in p-coord.

∂u∂t+

1acosϕ

∂∂λ

12u2⎛

⎝⎜⎞⎠⎟ +

1cosϕ

∂∂ϕ(uv cosϕ ) +

∂∂p(ωu) = fv −

1acosϕ

∂φ∂λ+ Drag

Multiply by a cosf and average over longitude.

∂[m]∂t

+1a∂[vm]∂ϕ

+∂[ωm]∂p

= f [v]a cosϕ +[Drag]a cosϕ

m =uacosf

Zonal-mean Momentum

Next, integrate this over the mass of the atmosphere.

∂[m]∂t

+1a∂[vm]∂ϕ

+∂[ωm]∂p

= f [v]a cosϕ +[Drag]a cosϕ

m =uacosf

(...)dp =(...)∂0

ps

∫

∂[m]∂

∂t+1a∂[vm]∑

∂ϕ+[ωm] 0

ps = f [v]∂ a cosϕ +[Drag]∑ a cosϕ

Zonal-mean Momentum

In steady state, this term is zero, by mass continuity.

∂[m]∂

∂t+1a∂[vm]∑

∂ϕ+[ωm] 0

ps = f [v]∂ a cosϕ +[Drag]∑ a cosϕ

Let’s make this part of the drag’.

So in steady state,

1a∂[vm ]∑

∂f= [Drag']∑ acosf

∂∂f

[v][u]∑ + [v *u*]∑( )cosϕ{ } = [Drag ']∑ acosϕ

Steady, Mass-integrated Zonal-mean Momentum Equation

Mass-integrated mean zonal wind advection

Meridional eddy flux of zonal momentum

∂∂f

[v][u]∑ + [v *u*]∑( )cosϕ{ } = [Drag ']∑ acosϕ

Steady, Mass-integrated Zonal-mean Momentum Advection

m =uacosfPeaks at around30N, so bothHadley and Ferrel Cells export relativeangular momentum

∂∂f

[v][u]∑( )cosϕ{ } + . . . = [Drag ']∑ acosϕ

Steady, Mass-integrated Zonal-mean Momentum Advection

m =uacosfEddies and MMCexport relativeangular momentum from the tropics and the eddies import relative angular momentum into extratropics, and focus it above the surface westerlies.

∂∂f

[v *u*]∑( )cosϕ{ } ≈ [Drag ']∑ acosϕ

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.

∂∂f

[v *u*]∑( )cosϕ{ } ≈ [Drag ']∑ acosϕ

Role of Eddies in MomentumLorenz (1952)

Ferrel 1859

Role of Eddies in MomentumLorenz (1967)

Ferrel 1859

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

∂∂t12ζ *2⎡

⎣⎢⎤⎦⎥+ βeff [v *ζ *] = [F *ζ *]

βeff [v *ζ *] = [F *ζ *]

Steady Enstrophy Equation


∂[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.


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.

Barotropic Cartoon

Momentum is Funny Stuff∂[u]∂t= −

∂∂y[u * v*] = [v *ζ *]

+-

Where is eddy source, and sink ?+ -

βeff [v *ζ *] = [F *ζ *]

+ - +

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 '

Ff =−r0 acosf u'v'

Fz =fr0 acosf v'θ '/ θζ

F =(Ff , Fζ) = Eliassen-Palm Flux Vector

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.

• Consider the following picture of the temperature and pressure variations on a height surface associated with a westward tilting wave.


LHL HW C WC

• Add potential temperature perturbation.How did the eddy heat flux end up in the momentum Budget?

dθθ

=dTT

−κpdp

LHL H

• Sketch in dz necessary to get back to a constant potential temperature surface; dz ~ -dtheta


LHL H

• Now let’s focus in on the resulting form drag.


H L

Form Drag = p∫∂h∂x

dx on θ surface

In westward-tilting wave,atmosphereabove exertsan eastwardtorque on atmosphere below, and vice-versa.

LHL H

Height oftheta surface,material surface.

Eliassen-Palm Cross Sections

∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )

−1∇gF

Ff =−r0 acosf u'v'

Fz =fr0 acosf v'θ '/ θζ

F =(Ff , Fζ) = Eliassen-Palm Flux Vector

Heat Flux partdominatesclimatology of E-P Cross-Sections

Tanaka, et al. 2006, JMSJ


• 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.

The Residual or Lagrangian Circulation

∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )

−1∇gF

∂θ∂t+ w *

∂θ∂z=Q

1acosf

∂∂f

v*cosf( )+1r0

∂∂ζ

r0w*( )=0

w*=−

1ar0 cosf

∂∂f

∇gF2Ωsinf

⎛⎝⎜

⎞⎠⎟f

dζ'x

∞

∫⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

Use momentum (ignore tendency) and continuity,

Mean sinking is the meridional gradient of the drag integrated down to that level. Thermo not used.

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*


∂θ∂t+ w *

∂θ∂z=Q

Stationary and Transient Driving of Lagrangian Circulation

∂u∂t+ f v* = v 'q ' = ρ0 acosϕ( )

−1∇gF


v*

Transient

Stationary

Hadley Cell

Eddy-Driven Cell

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.

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.


N H

S H

Tanaka,et al. 2006

Total4-7

1-3

>8

1-3Form Drag by Zonal Wavenumber


Lots of Ocean, not much topography,fairly zonally symmetric,

most of form drag from high wavenumbers.Clear, almost seasonally invariant

eddy-driven jet.Subtropical

JetEddy-Driven

Jet


SubtropicalJet

Eddy-DrivenJet

Lorenz & Hartmann, 2001




eddy-driven jet.Primary mode of low-frequency variability is

North-South movement of the 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



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.



Momentum Budget of Meridional Eddy-Jet Meandering

Residual Circ. Barotropic

‘Baroclinic’aka Form Drag

Drag determined as residual



Residual Circ.

Barotropic



Total EddyForcing






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.






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?

Positive Eddy Feedback


Focus on vertical average momentum balance and meridional wave propagation.



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.



Z=u is red M=-d/dy(u’v’) is whiter



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

Simple Model ofPositive Eddy Feedback


M=-d/dy(u’v’)

b. High-frequency eddies produce low-frequency forcing, because they respond to zonal flow.

dzdt

=m −ζτ

m = %m +βζ

d%zdt

= %m −%ζτ

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.

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.


Yu & Hartmann 1993


• 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.

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

1acosf

∂∂f

v*cosf( )+1r0

∂∂ζ

r0w*( )=0

w*=−

1ar0 cosf

∂∂f

∇gF2Ωsinf

⎛⎝⎜

⎞⎠⎟f

dζ'x

∞

∫⎧⎨⎪⎩⎪

⎫⎬⎪⎭⎪

Use momentum (ignore tendency) and continuity,

To sustain jet in new location, need to move diabaticheating with wave driving.

Are Meridional Displacements of Eddy-Driven Jets Self-Sustaining?

∂θ∂t+ w *

∂θ∂z=Q

w*=−

1ar0 cosf

∂∂f

∇gF2Ωsinf

⎛⎝⎜

⎞⎠⎟f

dζ'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 =a(Teθuiλ−T), .... Teθuiλ=Acos(2f )

Eddy momentum driving can define shape of heating.


SubtropicalJet

Eddy-DrivenJet


SAM & Precip

Sen Gupta & England, 2006

45

30

60


Yu & Hartmann 1993



Residual Circ.

Barotropic



Total EddyForcing

SAM & Precip

Sen Gupta & England, 2006

45

30

60

Parameterizing EddiesLorenz (1967)

Ferrel 1859

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.

the general circulation of the atmosphere and its variability

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