Lecture 5:
Eddies and tropospheric climate(i) The observed circulation(ii) The troposphere without eddies(iii) Tropospheric eddies and waves(iv)Baroclinic instability and synoptic eddies(v) Synoptic eddy transports(vi) Variability: Annular modes
FDEPS 2010
Alan Plumb, MIT
Nov 2010

(i) The observed circulation

observed mean meridional circulation[Oort& Peixoto]

annualDJF

JJA

annual

DJF

JJA

(ii) The troposphere without eddies

The troposphere without eddies[Held & Hou, J Atmos Sci, 1980]

Q>0EQ

Q<0v=w=0-> Q=0 -> T=T-> τs=0 -> us=0POLE

The troposphere without eddies[Held & Hou, J Atmos Sci, 1980]

no Ferrel cell
Q>0EQ

Q<0v=w=0-> Q=0 -> T=T-> τs=0 -> us=0POLE

The troposphere without eddies[Held & Hou, J Atmos Sci, 1980]

no Ferrel cell
Q>0Q<0v=w=0-> Q=0 -> T=T-> τs=0 -> us=0EQ

POLE

no surface
westerlies

The troposphere without eddies[Held & Hou, J Atmos Sci, 1980]

westerly jet much too strong :temperature gradient too large

no Ferrel cellQ>0Q<0v=w=0-> Q=0 -> T=T-> τs=0 -> us=0EQ

POLEno surfacewesterlies

(iii) Tropospheric eddies and waves

Stationary Rossby waves

Typical surface pressure analysis

(iv) Baroclinic instability and synoptic eddies

Barocllinic instability

Hence
A zonal flow is stable to inviscid, adiabatic, quasigeostrophic normal mode perturbations ifa. there is no change of sign of PV gradient within the fluid andb. the system is bounded above and below by isentropic boundaries.
The Charney-Stern theorem. (does not apply to non-normal-mode growth).

ūT̄

JAN

PV gradient (not shown) ∂q̄ /∂y > 0 in interior

lower boundary is not isentropicBarocllinic instability:the Eady problem

z = +1D
2

Simplest example, and relevant to the troposphereu 0 = Λz

1. Boussinesq (ρ = constant)z = −1D
2
2. Inviscid, adiabatic flow on an f − plane (β = 0)
3. Uniform buoyancy frequency: N 2 constant
4. Rigid upper and lower boundaries at z = ± 12 D, on which w = 0.
5. Balanced background zonal flow increasing linearly with height: u 0 = Λz

→ linear latitudinal temperature gradient:

∂∂yT0
T∗

f 0 ∂u 0f 0Λ
= g= g∂z→ no basic state QGPV gradient:∂q 0f 20 ∂ 2u 0
∂2u 0
= − 2 − 2=02∂y∂yN ∂z∂ + u ∂ q ′ + v ′ ∂q 0 =0
∂t
∂x
∂y

∂ + u ∂ q′ = 00∂t∂x→q′ = 0

→ q′

f 20 ∂ 2ψ ′∂ 2ψ ′∂ 2ψ ′=++ 2= 0222∂x∂yN ∂zLook for separable modal, wave-like solutions ψ ′ = ReΦzeikx+ly−kct  thend 2Φ − N 2 κ 2Φ = 0dz2f 20
where κ =u 0 = Λz

k2 + l 2 . Then Φ ∼ exp±Nκz/f 0 , orΦz = A cosh Nκ z + B sinh Nκ zf0f0→ boundary trapped(boundary trapped).
On upper and lower boundaries z = ±D/2, w ′ = 0∂ψ ′ ∂T 0∂∂′→+ u0T +=0∂t∂x∂x ∂y→ U − c dΦ − ΛΦ = 0 .
dz
After much manuipulation, findc = ± ΛDκLκL − tanh 1 κL22

κL − coth 1 κL22

L = ND/f 0internal radius of deformationc = ± 1κLΛDκL − tanh 1 κL
22

κL − coth 1 κL
22short waves, κL < 2. 3994 : c 2 > 0 : propagating boundary waves, no growthlong waves, κL > 2. 3994 : c 2 < 0 : nonpropagating, exponential growth

0.3

Im(c)

0.2

Im(c)

c/(ΛΛD)

Re(c)c/ΛD 0.1Re(c)0.0-0.1

-0.2

01

23

κLNDκ
κ/f04

c = ± 1
κLΛDκL − tanh 1 κL
22

κL − coth 1 κL
22short waves, κL < 2. 3994 : c 2 > 0 : propagating boundary waves, no growthlong waves, κL > 2. 3994 : c 2 < 0 : nonpropagating, exponential growthIm(ω) = k Im(c)0.3maximum growth rateat κL=1.61

Im(c)

0.2

Im(c)

c/(ΛΛD)

Re(c)c/ΛD 0.1

Re(c)

0.0

-0.1

-0.2

01

2

34

κLNDκ
κ/f0

[James]Structure of fastest growing wave:

Note.:

w′ T′ > 0v′ T ′[Holton]is poleward

Baroclinic instability in the atmosphere:

Typical values in midlatitude troposphereD ≃ 10km, N ≃ 1 × 10−2s −1, f 0 ≃ 1. 0 × 10−4s −1, Λ ≃ 2. 5 × 10−3s −1.
So the fastest growth rate is 6. 5 × 10 −6s −1, → e-folding time 1. 5 × 105 s ≃ 1. 8 days.
Wavenumber of the fastest growing wave is 1. 61f 0/ND ≃ 1. 61 × 10−6m−1,giving wavelength 2π/k ≃ 3900 km. (At 450, corresponds to zonal wavenumber 7.)

(v) Synoptic eddy transport

F, ∇ ⋅ F in troposphere[Oort & Peixoto]F, ∇ ⋅ F in troposphere[Oort & Peixoto]stationary waves in winter:
upward propagating fromsurface and near-surface
sourcesF, ∇ ⋅ F in troposphere[Oort & Peixoto]transient baroclinic eddies also
upward propagating, becausev ′ T ′ is poleward′ ′F z = f v θ > 0∂θ̄ /∂z

stationary waves in winter:
upward propagating fromsurface and near-surface
sourcesF, ∇ ⋅ F in troposphere[Oort & Peixoto]transient baroclinic eddies also
upward propagating, becausev ′ T ′ is poleward′ ′F z = f v θ > 0∂θ̄ /∂z
F divergent near (and at) surface;
generally convergent in middle and upper
tropospherestationary waves in winter:
upward propagating fromsurface and near-surface
sourcesF, ∇ ⋅ F in troposphere[Oort & Peixoto]transient baroclinic eddies also
upward propagating, becausev ′ T ′ is poleward′ ′F z = f v θ > 0∂θ̄ /∂z
note equatorward propagation
in upper tropospherestationary waves in winter:
upward propagating fromsurface and near-surface
sourcesannual mean v ′ T ′ :transient eddies dominate, but
stationary waves contribute innorthern hemisphere (especially winter)[Oort& Peixoto]annual mean u ′ v ′ :transient eddies dominate
ALL EDDIES[Oort& Peixoto]

TRANSIENTSStorm tracks – northernhemisphere1 u ′2 + v ′2 2w′T′

Storm tracks – southernhemisphere1 u ′2 + v ′2 2w′T′

EP, heat and momentum fluxes

F(arrows show F)y′ ′= −ρu v ; Fz

′ θ′v= ρf∂θ̄ /∂z

TROPOPAUSE(a)v'T' > 0z

u' v' = 0

homogeneous case u ′ v′ = 0

SURFACE(b)localized baroclinic zone on β-plane:wave activity spreads out symmetrically;u ′ v′ ≠ 0u' v' < 0u' v' > 0v'T' > 0POLEEQ(c)u' v' > 0localized baroclinic zone on the sphere:
wave activity spreads out asymmetrically;u ′ v′ predominantly poleward

u' v' < 0v'T' > 0EQPOLE

u ′ v′

Maintenance of surface westerlies

column-integrated momentum budget:

→but

1 ∂τ
− fv̄ = − ∂ u ′ v ′ − ρ∂y∂z∞∞− f ∫ ρv̄ dz = − ∂ ∫ ρ u ′ v′ dz + τ 0∂y 00∞− f ∫ ρv̄ dz =0→0

τ
EQUATOR

POLE∞
∂τ0 =
ρ u ′ v′ dz∫∂y 0uτ0 = − τ 0drag→∞∂u 0 = − τ dragρ u ′ v ′ dz∫∂y 0surface westerlies in middle
latitudes (where momentum flux isconvergent)(a)u' v'Wτslocally,v̄ = 1 ∂ u ′ v ′f ∂y(b)(u'v')y balanced by fv

WARMING COOLING→ Ferrel cellEQPOLE

fv balanced by surface stress(c)heat transport by Ferrel cell opposes(but does not overcome) effects ofeddy heat flux
→ net poleward heat transportCOOLINGEQWARMING

v'T'POLE

heat fluxes reduce verticalshearzmomentum fluxes drive
surface westerlies

uWhether eddies enhance or reduce upper tropospheric westerlies
depends on external factors, such as ratio of thermal relaxation rate tosurface drag coefficient [Robinson, J Atmos Sci, 1991]

annualDJF

JJA

(vi) Variability: Annular modes

Annular Modes• Leading patterns of variability
in extratropics of eachhemisphere• Strongest in winter but visible
year-round in troposphere;
present in “active seasons” instratosphere[Thompson and Wallace, 2000]

Lorenz & Hartmann
J. Atmos. Sci (2001); J. Clim (2003)

References
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Holton, J. R., 1992: An Introduction to Dynamic Meteorology Third Edition,Academic Press, 511pp.
Lorenz, D.J. and Hartmann, D.L, 2001: Eddy--Zonal Flow Feedback in theSouthern Hemisphere, J. Atmos. Sci., 58, 3312-3327
James, I. N., 1995: Introduction to circulating atmospheres, Cambridge
University Press, 448pp.
Robinson, W. A., 1991: The dynamics of the zonal index in a simple modelof the atmosphere, Tellus A, 43, 295-305
Peixoto, J. P. and Oort, A. H., 1992: Physics of Climate. American Institute ofPhysics, 520pp.
Held, I. M. and Hou, A. Y., 1980: Nonlinear Axially Symmetric Circulations ina Nearly Inviscid Atmosphere, J. Atmos. Sci., 37, 515-533
Thompson, D. W. J. and Wallace, J. M. 2000: Annular modes in theextratropical circulation. Part I: Month-to-month variability, J. Climate, 13,
1000-1016
NCEP,
http://www.cpc.noaa.gov/
Atmosphere and Ocean in a Laboratory,
http://www.gfd-dennou.org/library/gfd_exp/index.htm