Fast and Slow: The
Dynamics of Superrotation
Phenomena in Planetary
Atmospheres:
II. Simple models & theories
Peter Read
University of Oxford, UK

Plan• Simple model of Hadley circulation (Held & Hou)
• Super-rotation with meridional overturning and diffusive eddies(Gierasch)• Extension with strong momentum diffusion (Yamamoto et al.)• Angular momentum mixing• Classification of waves• Wave-zonal flow interactions and non-acceleration

Simple theoretical models:
Held & Hou (1980)• Aim is to determine properties
and width of axisymmetricHadley circulation andassociated zonal flow andthermal structure• Consistent with steady thermalbalance• Hydrostatic and gradient windbalance

Isaac Held

Arthur Y. Hou

Simple theoretical models:
Held & Hou (1980)• Two-layer atmosphere• Quasi-inviscid so angularmomentum conservation applies
in the free atmosphere by theoverturning circulation• Hydrostatic and Gradient windbalance at all 𝑦 = sin 𝜑𝜕𝑢! tan 𝜑
𝑔 𝜕𝜃
𝑓𝑢 +
=−
𝜕𝑧
𝑎
𝑎𝜃" 𝜕𝜑• Weak flow 𝑢 ≈ 0 near thesurface due to frictional dragYM = sin fM

Simple theoretical models:
Held & Hou (1980)• Asssume a radiative equilibrium potential
temperature profile

Cool

𝜃!"
2
𝑧 1
= 1 − Δ$ 𝑃% sin 𝜑 + ∆&
−
𝜃#
3
𝐻 2

• 𝜃! ∆" is equator-pole potential temperature
contrast• 𝜃! ∆# is vertical potential temperature contrast
over z=0-H• 𝑃$ 𝑦 =%$3𝑦 $ − 1• Take integrated vertical average of () and ()𝑢$ tan 𝜑𝑔𝐻 𝜕𝜃̅𝑓𝑢 +=−𝑎𝑎𝜃! 𝜕𝜑2̅𝜃&' = 𝜃! 1 − Δ" 𝑃$ sin 𝜑
3

-YM

Heat

Cool

YM = sin fM

YM

Simple theoretical models:
Held & Hou (1980)• Assume meridional velocity 𝑣 = 0 in theextra-tropical free atmosphere.
• From gradient thermal wind balance
𝑢#$ = Ω𝑎 cos 𝜑• Where ℛ =𝑧
2ℛ + 1 − 1𝐻∆! )*is thermal Rossby number+" , "• Within the Hadley cell u is determined by AM
conservation, assuming u = 0 on the equator.
Ω𝑎 sin! 𝜑
𝑢%& =
cos 𝜑

Extra-tropics

Tropics

Simple theoretical models:
Held & Hou (1980)
YM = sin fM• Use gradient wind balance to infer thevertical mean temperature, so in the
Hadley cell𝜃̅*' " )('(*()• ( = !+% for |y|<YM
'

̅
𝜃#$

• Solve for YM or 𝜑! using matchingconditions)𝜃̅ 𝜑,= 𝜃̅ 𝜑,• And the thermal balance condition for
the Hadley cell.+.+
̅ cos 𝜑 𝑑𝜑
< 𝜃̅ cos 𝜑 𝑑𝜑 = < 𝜃/0
"

"

𝜑!

Simple theoretical models:
Held & Hou (1980)
YM = sin fM• Solution for yM can be expressed in termsof the root of the equation#
1𝑦!
1
1 + 𝑦!
"
4ℛ − 1 𝑦! −
−
𝑦
+
ln
=0
!
$
3
2
1 − 𝑦!
1 − 𝑦!
• Or rearranging
$
3 1
1
𝑦!
1
1 + 𝑦!
ℛ=+ $ +$ − 2𝑦 " ln 1 − 𝑦
4 3 𝑦!
1 − 𝑦!
!
!
• Which can be solved numerically• Asymptotic limits can be found for• ℛ ≪ 1:
• ℛ ≫ 1:#ℛ &/$𝑦! ≈""𝑦! ≈ 1 −(ℛySimple theoretical models:
Held & Hou (1980)

Y

YM = sin fM• Verify for continuously stratified, weakly viscous fluid (𝜈 inthe vertical only)• Note that VERTICAL viscosity can only diffuse m DOWNGRADIENT• Hence there can be no local maximum of m exceeding Ωa2• Solve time-dependent, 2D, axisymmetric primitive equations
numerically with heating/cooling via Newtonian relaxation𝐷𝜃 𝜃, − 𝜃=𝐷𝑡𝜏
• Meridional streamfunction width consistent with inviscid yM
estimate• Strength of circulation ~1/𝜈
• U ≤ 0 on/near the equator• U ~ AM-conserving for 𝜑 ≤ 𝜑• U varies smoothly → 0 from 𝜑- → 90oUySimple theoretical models:
Held & Hou (1980)• Estimate Sglobal for axisymmetric circulations by using solutions for uHC
and uET as given above.
• u = uHC for 0 ≤ f ≤ fM [p ≤ 0.8pS]• u = uET for fM < f ≤ π/2 [for all p]

𝑆=∫567 89: ;<=∫ 5>! 6 ! 89:! ;<== 𝑆?@ + 𝑆AB

where
𝑆%& =

'./01

∫' + *() 456* .7.78/+

∫'

∫'

'./01

0

2

∫'

3/*

:;4565 .7.78/+; 𝑆#$ =

3/*∫' 1 ∫2*67 456* .7.78/++
03/*∫' 1 ∫' :;4565 .7.78/+[Lewis et al. 2021]

Simple theoretical models:
Held & Hou (1980)• Estimate Sglobal for axisymmetriccirculations by using solutions for uHC
and uET as given above.
• u = uHC for 0 ≤ f ≤ fM [p ≤ 0.8pS]• u = uET for fM < f ≤ π/2 [for all p]• S depends primarily on ℛ
• Asymptotic limits can be found for

Venus
Titan
y

Mars
Earth

&
$
$
𝑆 ≈ = constant#• ℛ ≪ 1: 𝑆 ≈ ℛ• ℛ ≫ 1:

[Lewis et al. 2021]

Simple theoretical models:
Gierasch (1975)• Conceptual model to maintain a strong quasiaxisymmetric super-rotation• Angular momentum is transferred upwards andpolewards by a global scale meridional
overturning circulation• Eddies transfer AM equatorwards (up-gradient),
accelerating super-rotation in the tropics• Parameterised as an anisotropic viscosity 𝜈% , 𝜈& with 𝜈% ≫𝜈&• Diffusion of heat is neglected• Otherwise it erodes the equator-pole temperature contrast• Reduces upper level U which should be in cyclostrophic
balance for fast super-rotation

Peter GieraschSimple theoretical models:
Gierasch (1975)• Scale analysis and steady state solution of
axisymmetric primitive equations, assuming simple
spatial structures of key variables• Eddy thermal diffusivity assumed small enough to be negligible
1≪

'! %
() "

≪ 𝑅𝑖 =*"+,/+.

so temperature structure largely determined by radiaPve equilibrium
and 𝑊~𝑄/ /𝑁 0
• Strong super-rotation maintained by horizontal viscosity provided𝑎0𝐻𝐻0𝜏% (= ) ≪ 𝜏%1 (= ) ≪ 𝜏& (=
)
𝜈%
𝑊
𝜈&
• Cyclostrophic/gradient wind balance prevails if
𝜈&
𝜈% 𝐻
𝑈𝐻 0
≪1≪
≪
𝑊𝐻
𝑊𝑎0
𝑊𝑎

• If these conditions are satisfied thenNO%
𝑆~ :; ~exp P > 1
8Peter Gierasch

Gierasch/Rossow-Williams conceptual model
Summary
Prograde equatorial jet & max in m
Eddy momentum fluxes
Prograde mid-latitude jets
Transport AM up-gradient towards equator• Super-rotation
driven/maintained by
combination of• axisymmetric overturning
circulation and• up-gradient equatorward
eddy transport of AM• Rossow & Williams (1979)
Thermally• generation and
Direct Hadleydissipation of barotropic circulation
Rossby waves act as AM+transport mechanism via
Pbarotropic instability?

Angular momentum transported upwards &polewards

.

E

.

+

P

Surface friction torques exchange AM with planet
31/5-3/6/19

IVC2019

14

Simple theoretical/numerical models: Peter Gierasch
Parameter sweeps• Exploration of a 3D dimensionless parameter
space using time-dependent, 2D axisymmetricnumerical models with Newtonian
heating/cooling (cf Held & Hou 1980), solving
primitive equations on a sphere

YIY’s• Matsuda (1980; 1982)• Yamamoto et al. (2009 - YIY);• Yamamoto & Yoden (2013 - YY)• Define key external parameters to vary
• Thermal Rossby number ℛ =∆" *+(as before),# -#• Horizontal and vertical Ekman numbers
𝜈.
𝜈+
𝐸. =
;𝐸 =
Ω𝐻$ + Ω𝑎$• Other dimensionless groups fixedℛSimple theoretical/numericalmodels: Parameter sweeps

𝐸& = 2.1×1057ℛ = 0.12• Exploration of a 3D dimensionlessparameter space using time-dependent,
2D axisymmetric numerical models with
Newtonian heating/cooling (cf Held &Hou 1980), solving primitive equations on
a sphere• Matsuda (1980; 1982)• Yamamoto, Ishioka & Yoden (2009 – YIY)ℛℛ

• Define key external parameters to vary
• Thermal Rossby number ℛ =∆# 3%4" ) "(as before)• Horizontal and vertical Ekman numbers
𝜈&
𝜈%
𝐸& =
;
𝐸
=
Ω𝐻 0 % Ω𝑎0• Other dimensionless groups fixed

𝐸& = 2.1×1056ℛ = 0.12

Simple theoretical/numericalmodels: Parameter sweeps

𝐸& = 2.1×1057ℛ = 0.12• Exploration of a 3D dimensionlessparameter space using time-dependent,
2D axisymmetric numerical models with
Newtonian heating/cooling (cf Held &Hou 1980), solving primitive equations on
a sphere• Matsuda (1980; 1982)• Yamamoto, Ishioka & Yoden (2009 – YIY)

ℛ = 12ℛ• Define key external parameters to vary
• Thermal Rossby number ℛ =∆# 3%4" ) "(as before)• Horizontal and vertical Ekman numbers
𝜈&
𝜈%
𝐸& =
;
𝐸
=
Ω𝐻 0 % Ω𝑎0• Other dimensionless groups fixed

𝐸& = 2.1×1056ℛ = 0.12

Simple theoretical/numericalmodels: Parameter sweeps

𝐸& = 2.1×1057ℛ = 0.12• Exploration of a 3D dimensionlessparameter space using time-dependent,
2D axisymmetric numerical models with
Newtonian heating/cooling (cf Held &Hou 1980), solving primitive equations on
a sphere• Matsuda (1980; 1982)• Yamamoto, Ishioka & Yoden (2009 – YIY)

• Define key external parameters to vary
• Thermal Rossby number ℛ =∆# 3%4" ) "ℛ = 12ℛℛ = 1200(as before)• Horizontal and vertical Ekman numbers
𝜈&
𝜈%
𝐸& =
;
𝐸
=
Ω𝐻 0 % Ω𝑎0• Other dimensionless groups fixed

𝐸& = 2.1×1056ℛ = 0.12

Simple theoretical/numericalmodels: Parameter sweeps

𝐸& = 2.1×1057ℛ = 0.12• Exploration of a 3D dimensionlessparameter space using time-dependent,
2D axisymmetric numerical models with
Newtonian heating/cooling (cf Held &Hou 1980), solving primitive equations on
a sphere• Matsuda (1980; 1982)• Yamamoto, Ishioka & Yoden (2009 – YIY)ℛ

• Define key external parameters to vary
• Thermal Rossby number ℛ =∆# 3%4" ) "(as before)• Horizontal and vertical Ekman numbers
𝜈&
𝜈%
𝐸& =
;
𝐸
=
Ω𝐻 0 % Ω𝑎0• Other dimensionless groups fixedℛ

𝐸& = 2.1×1056ℛ = 0.12

Simple theoretical/numerical models:Parameter sweeps• Scale analysis and steady state solution of axisymmetric primitive equations withanisotropic viscosity, assuming simple spatial structures of key variables• Reduce parameter dependence to an algebraic equation in 𝑆 Y =𝑆′! + 2𝑆 Y + 𝐵𝑆′• Where 𝐴 = 𝜋 !𝜏];7 /𝜏^2 + 𝑆′
1 + 𝑆′

and 𝐵 = 20𝜋 !

𝜏];7 = radiative timescale; 𝜏% , 𝜏^ =Z[\

𝐴𝑆′ 2 + 𝑆′
+ 1 = 2ℛ
2 1 + 𝑆′!
_and: `8 `(;* % *,P( P8[Yamamoto & Yoden 2013; Lewis et al. 2021]

Simple theoretical/numerical models:Parameter sweeps• Scale analysis and steady state solution of axisymmetricprimitive equations with anisotropic viscosity, assuming
simple spatial structures of key variables• Reduce parameter dependence to an algebraic
79
equation in 𝑆 8 = 6
𝑆′0 + 2𝑆 8 + 𝐵𝑆′

2 + 𝑆′
1 + 𝑆′

0𝐴𝑆′ 2 + 𝑆′
+ 1 = 2ℛ
2 1 + 𝑆′

• Where 𝐴 = 𝜋 𝜏:)+ /𝜏& and 𝐵 = 20𝜋0𝜏:)+ = radiative timescale; 𝜏% , 𝜏& =;4 <$ <!

0and)" % ",=! =$• Asymptotic dependences• ℛ ≪ 1: 𝑆 ∝ ℛ - Geostrophic flow
• ℛ > 1: 𝑆 ∝ ℛ !/# - Cyclostrophic flow
• ℛ ≥ 10: 𝑆 ∝ ℛ !/$ - Modified cyclostrophic flow

[Yamamoto & Yoden 2013; Lewis et al. 2021]

Trends in S
(cylindrical annulus)• 3 basic regimes
I.

V. Slow rotation (~angularmomentum conserving
except in Ekman layer): S ~constant
II. Moderate rotation(cyclostrophic/gradientwind and diffusive
interior): S rises to shallow
peak(𝑆 ≈ 𝜀𝜂_/!𝜎 )_/!𝑄 )_)III. Rapid rotation (quasigeostrophic):
S ~ Q-2 ~ Ω-2
31/5-3/6/19

-2

I.

II.

III.

5;/0 ;/0∝ 𝑅𝑜>
IVC2019𝜎

22How do eddies ACTUALLY mix & transport
angular momentum?
• Which quantity to homogenize?
• Angular momentum?
• Vorticity?
• [Relative, absolute or potential?]• Angular velocity/tangential stress?
• Buoyancy?
• …..

• Depends strongly on the mechanism generating the eddies…..
• Instability?
• Direct forcing…?

How do eddies ACTUALLY mix & transport
angular momentum?
• Angular Momentum mixing, forexample, leads to (relative)
vorticity expulsion• [Gough & Lynden-Bell 1968]• This mixes towards a state in which𝑚 = (Ω𝑟 + 𝑢). 𝑟 = 𝑐𝑜𝑛𝑠𝑡.

Or𝑐𝑜𝑛𝑠𝑡.
𝜴×𝒓 + 𝒖 =𝒆.
𝑟

• Hence ∇×𝒖 = −2Ω𝒌 and the endstate is irrotational• Demonstrated (roughly!) in a
simple experiment?

• Inner beaker observed to increase its
rotation

How do eddies ACTUALLY mix & transport
angular momentum?
• More generally vorticity
expulsion in randomly stirred
flow acts locally but not globally• With background rotation, initialsolid body rotation with stirringforms intense cyclonic vortices• [e.g. McEwan 1976 Nature]• -> Local concentrations of vorticity• Without background rotation,
however, no tendency to form
vortices…..

No background rotation

With background rotation

How do eddies ACTUALLY mix & transport
angular momentum?
• Vorticity mixing is another possibility, especially for rotatingbarotropic flows• Taylor (1915); Rossby (1947); Eady (1950)• For an axisymmetric (or zonally averaged) flow, absolute vorticity 𝜉 isrelated to m by
1 𝜕𝑚
𝜕𝑢
𝜉=
= 2Ω + 𝜔 +𝑟 𝜕𝑟𝜕𝑟• Mixing to a state of uniform 𝜉 consistent with uniform angularvelocity (though not uniquely so)• Cf viscous diffusion releasing tangential strain…..?
• See also Rossow & Williams (1979)….vorticity mixing associated withbarotropic instability

How do eddies ACTUALLY mix & transport
angular momentum?
• Nonaxisymmetric eddies in a
stratified atmosphere on arotating planet are not uniformly
random motions!
• Dominated by waves, at least atlarge scales• But what kind of waves…?

Super-rotation: an alternative non-classical
GRW scenario
Eddy momentum fluxes
transport AM upwards/downwards atequator & towards poles

Prograde equatorial jet & max in m

• An alternative scenario in
which up-gradient AMtransfer takes place in thevertical• Still satisfies the Starr-Hidetheorem provided
horizontal AM transfers are Thermallydown-gradientDirect Hadley• Not like a viscosity – needs circulation
the action of vertically+P
propagating waves….?

.

E

.

+

P

Surface friction torques exchange AM with planet
31/5-3/6/19

IVC201928

How do eddies ACTUALLY mix & transport
angular momentum?
• Nonaxisymmetric eddies in a
stratified atmosphere on arotating planet are not uniformly
random motions!
• Dominated by waves, at least atlarge scales• But what kind of waves…?

Start with linearised primitive equations for a
stratified atmosphere at rest on a spherical planet
J7K

JN!

M

K

• JL − 𝑓𝑣 + 6 89: ; JO = 𝑋 K

zonal momentum

M JN!

JPK

• JL + 𝑓𝑢K + 6 J; = 𝑌′
•

M

J7K

6 89: ; JO
J # NK

S

+
K

J
J;

𝑣′ cos 𝜑meridional momentum
+

M J
5" JQ

TUK

• JQJL + 𝑁 𝑤 = ?

• With boundary conditions such as

𝜌R 𝑤′ = 0 continuity
Internal energy

'
noY
no
r$s " tYY
• np = −𝑤 nq = %at z=0 [for a flat lower boundary]𝑓 = 2Ω sin 𝜑Φ = geopotential
𝑋 8 , 𝑌′ = friction or forcing
𝐽 = diabatic heating
H = scale heightLook for separable solutions of the form$#%

D 𝜆, 𝜑, 𝑡𝑢K , 𝑣 K , ΦK = 𝑒 𝑈 𝑧 𝑢A 𝜆, 𝜑, 𝑡 , 𝑣A 𝜆, 𝜑, 𝑡 , Φ

(*)

$
#%

𝑤K = 𝑒 𝑊 𝑧 𝑤
F 𝜆, 𝜑, 𝑡• Take 𝑋 K = 𝑌 K = 𝐽K = 0 and substitute (*) into momentum equations to
obtain
V
J7
JL
J PX
JL

− 𝑓𝑣A +
+ 𝑓𝑢A +

M

W
JN

6 89: ; JO
W
M JN
6 J;

(a)

=0

(b)

=0

• Separate z-dependence via a separation constant 1/𝑔ℎ to obtain
MVJ7

6 89: ; JO

+

J
J;

𝑣A cos 𝜑

+

W
M JN
YZ JL

=0

(c)

[Equivalent depth h]Laplace Tidal Equations
V
J7

M

W
JN

JL

− 𝑓𝑣A + 6 89: ; JO = 0

J PX

W
M JN

JL

(a)
(b)

+ 𝑓𝑢A + 6 J; = 0VJ7M

6 89: ; JO

+

J
J;

𝑣A cos 𝜑

+

W
M JN
YZ JL

(c)

=0

• Together with vertical structure equations
< #[
<Q #

+

\#

M

<[

[

−
𝑊
=
0;
𝑈
=
−
#
YZ
]?
<Q
S?

• And suitable boundary conditions(d,e)

Now look for separable solutions of the form
2 𝜑 exp 𝑖 𝑠𝜆 − 2Ω𝜎𝑡𝑢: , 𝑣 : , Φ: = 𝑅𝑒 𝑢0 𝜑 , 𝑣0 𝜑 , Φ• Where s is zonal wavenumber index and 2𝜋/2Ω𝜎 is the wave period (or 𝜋/2𝜎
is the period in days).
• Substituting the solution into (a)-(c) and eliminating 𝑢0 𝜑 and 𝑣0 𝜑 leads to“Laplace’s Tidal Equation”
2 + 𝛾Φ
2 =0
ℒΦ
(f)
;</=/
• Where 𝛾 =is Lamb’s parameter and ℒ is a second order differential>?
operator in 𝜇 = sin 𝜑𝑑 (1 − 𝜇@ ) 𝑑1𝑠@𝑠(𝜎 @ + 𝜇@ )
ℒ=
− @
−
@
@
@
@
𝑑𝜇 (𝜎 − 𝜇 ) 𝑑𝜇
𝜎 −𝜇 1−𝜇𝜎(𝜎 @ − 𝜇@ )2 is bounded at the poles 𝜇 = ±1
• With boundary conditions that ΦSolve (f) as an eigenvalue problem• Prescribe s and s andsolve (f)
(C,E)
(C,E) (C,E)ℒΘA + 𝛾A ΘA = 0
for different integer valuesof n• ΘA are Hough functions• E.g. see Longuet-Higgins(1968)• Dispersion relations
computed as function of
g-1/2

Eastward waves

Westward waves

Free atmospheric waves• Several families of waves• Eastward modes include highfrequency inertia-gravity
modes and the equatorial
Kelvin mode
• Westward modes also
include high frequency
inertia-gravity waves and low
frequency planetary Rossby
waves (n’≥1) and the mixed
planetary-gravity mode (n=0)• Asymptotic limits• Low rotation 𝛾 0%/$ → ∞waves are global in extent• Fast rotation 𝛾 0%/$ → 0
waves are trapped either
near the equator (h > 0) or
the poles (h < 0)

Eastward waves

Kelvin wave

Free atmospheric waves• Several families of waves• Eastward modes include highfrequency inertia-gravity
modes and the equatorial
Kelvin mode
• Westward modes also
include high frequency
inertia-gravity waves and low
frequency planetary Rossby
waves (n’≥1) and the mixed
planetary-gravity mode (n=0)• Asymptotic limits• Low rotation 𝛾 0%/$ → ∞waves are global in extent• Fast rotation 𝛾 0%/$ → 0
waves are trapped either
near the equator (h > 0) or
the poles (h < 0)

MixedEastward
Rossby-Gravity
waves

Equatorial Rossby

Westward waves

Free atmospheric waves• Several families of waves• Eastward modes include highfrequency inertia-gravity
modes and the equatorial
Kelvin mode
• Westward modes also
include high frequency
inertia-gravity waves and low
frequency planetary Rossby
waves (n’≥1) and the mixed
planetary-gravity mode (n=0)• Asymptotic limits• Low rotation 𝛾 0%/$ → ∞waves are global in extent• Fast rotation 𝛾 0%/$ → 0
waves are trapped either
near the equator (h > 0) or
the poles (h < 0)

Eastward waves

Westward waves

Summary & remarks• For a stationary atmosphere (co-rotating with a spherical planet), there exists
a well defined spectrum of freely-evolving (neutrally stable) waves• Various combinations of stratification/buoyancy and inertial/Coriolis restoring forces• Propagate dispersively in the zonal direction
Used to classify each wave mode
• Characteristic and distinctive spatial structures(4,6)• Propagate dispersively (and anisotropically) also in the vertical with wavenumber 𝑚2• Phase and group velocities, and spatial structures, depend on Lamb parameter 𝛾 0%/$ =𝑔 ℎ /(Ω𝑎) [cf thermal Rossby ℛ and Burger ℬ numbers]• Asymptotic forms are well approximated for fast rotation 𝛾 GH/@ → 0 bytangent-plane models• E.g. equatorial or mid-latitude b-planes• Wave structures and dispersion relations are significantly modified in theK
presence of a zonally averaged shear flow [𝑈(𝜑,𝑧, 𝑡) ≠ 0], dissipation,
forcing and nonlinearity• Nevertheless, the classification and nomenclature of the linear, conservative waves arestill used to classify similar wave types in other regimes…..

How and when do waves accelerate ordecelerate zonal flows?
• Wave-zonal mean interaction theory [Andrews & McIntyre 1976, 1978]• Based on zonal mean primitive equations
•

CD
E
G
C
+ 𝑣̅
CF
H IJK L CL•

C&Q
&Q C&Q
C&Q++𝑤
P + 𝑢K
CF
H CL
CM

•

D
CU
, ̅ W&'− 𝜃𝑒 ! = 0
CM
V

•

G
C
H IJK L CL

•

D
D
CX
&Q CX
CX
G
C
+
+𝑤
P − 𝑄K = −
CF
H CL
CM
H IJK L CL

𝑢K cos 𝜑 − 𝑓 + 𝑤
P𝑣̅ cos 𝜑 +D RST L
E
𝑓+
H

G C
O% CM

CD
E
G
C
− 𝑋K = − "CM
H IJK L CL

D
G CU
G
C
+
− 𝑌K = −H CL
H IJK L CL

𝑢N 𝑣 N 𝑐𝑜𝑠 % 𝜑 −
𝑣 N%

𝜌P 𝑤
P =0
𝑣′𝜃′ cos 𝜑 −

G C
O% CM

𝜌P 𝑤 N 𝜃 N

G C
−
O% CM

G C
O% CM

𝜌P 𝑤 N 𝑢N

𝜌P 𝑤 N 𝑣 N

EN" RST L−
H

How and when do waves accelerate ordecelerate zonal flows?
• Problem: how to separate zonal mean circulation from eddies?
• Eddies also influence Eulerian zonal mean meridional flow (𝑣,̅ 𝑤)
U• Eddy heat fluxes can also implicitly influence zonal momentum• Instead, use Transformed Eulerian Mean (TEM) equations
•

/0
1
&
/
+ 𝑣∗
/2
- 456 7 /7

• 𝑢_
•
•
•

0 9:; 71𝑓+$%>&𝑢_ cos 𝜑 − 𝑓 + 𝑤 ∗/01&− 𝑋_ = −𝜵. 𝑭/8- 456 7

0
& /<
+
=𝐺
- /7

0
/<
= ̅
− 𝜃𝑒
=0
/8
+
&
/
& /
𝑣 ∗ cos 𝜑 +𝜌@ 𝑤 ∗ = 0- 456 7 /7
?' /8
0
0
0
/A
B ∗ /A/A& /BCAC/A//7
∗
_
+
+𝑤
−𝑄 =−
𝜌
+ 𝑤′𝜃′
0/2- /7/8?' /8 @-/A//8Where Residual Mean Circulation is••; A @%B ( C8∗𝑣 = 𝑣̅ −D@% A. A C/A.
;
A B ( C8 EFG H
∗
𝑤 =𝑤
'+
D
) EFG H AH
A C/A.

And Eliassen-Palm Flux F is
• 𝐹 (H) = 𝜌K 𝑎 cos 𝜑

D B ( C8
AL
− 𝑢8 𝑣′D
A. A C/A.

• 𝐹 (.) = 𝜌K 𝑎 cos 𝜑 …‡𝑓 −B ( C8𝑢s cos 𝜑 ˆ D − 𝑢8 𝑤 8 †) EFG H AH
A C/A.
;A

Remarks on TEM• Separates eddies and zonal mean circulations more “cleanly”• Residual mean circulation approximates to a Lagrangian Mean circulation undersome circumstances• The Eliassen-Palm (EP) flux –F represents an effective eddy flux of angularmomentum• So 𝜵. 𝑭 represents a convergence of AM flux, producing an eastward tendency for 𝑢T
from eddy stresses• In quasi-geostrophic flows, F is simplified [terms in blue] and 𝜵. 𝑭 = 𝑣 8 𝑞′

• EP fluxes are directly associated with the so-called Eliassen-Palm theorem(Eliassen & Palm 1961) and generalisations by Boyd (1976) and Andrews &McIntyre (1976,1978), and the Charney-Drazin non-acceleration theorem(Charney & Drazin 1961).

Generalised EP Theorem• Eliassen & Palm (1961) considered steady, linear waves on a basic zonal
flow 𝑢O (𝜑, 𝑧) with no frictional or diabatic effects (on a b-plane)• They showed that under these conditions 𝜵. 𝑭 = 0• This result was extended to non-zero X’, Y’ and Q’, and to sphericalgeometry, by Boyd (1976) and to time-varying waves by Andrews &McIntyre (1976,1978).
• A&M showed that𝜕𝐴
+ 𝜵. 𝑭 = 𝐷 + 𝑂(𝛼 [ )𝜕𝑡• A is “wave activity density” and D represents frictional and diabatic effects.
%• [𝐴 ≈ 𝜌! 𝑞 8 $ / 𝜕𝑞T /𝜕𝑦 for quasi-geostrophic problems ]$• a represents wave amplitude so this term represents higher order nonlinearity• Makes explicit the dependence of EP flux divergence on wave transience and nonconservative effects

Generalised EP Theorem• A&M showed thatJr
JL

+ 𝜵. 𝑭 = 𝐷 + 𝑂(𝛼 s )(*)• A is “wave activity density” and D represents frictional and diabatic effects.
• a represents wave amplitude so this term represents higher ordernonlinearity• Makes explicit the dependence of EP flux divergence on wave transience andnon-conservative effects• Note that (*) can also be written in the absence of friction anddiabatic effects under some conditions as
𝜕𝐴= −𝜵. (𝒄𝒈𝐴) + 𝑂(𝛼 Z)𝜕𝑡where 𝒄𝒈 is the group velocity, so 𝑭 is related to group velocity

Non-acceleration Theorem• TEM equations indicate that eddies will accelerate/decelerate zonal
flow if 𝜵. 𝑭 ≠ 0• But the Generalised EP theorem shows that 𝜵. 𝑭 = 0 for steady,linear and conservative waves• Hence, under those conditions a possible mean flow satisfying the full TEM
equations could be
'
n'
*
n(
• np = np = 𝑣 ∗ = 𝑤 ∗ = 0.

i.e. a non-accelerating solution• [Given appropriate boundary conditions]• The corollary of this is that wave transience, dissipation or forcing
may lead to 𝜵. 𝑭 ≠ 0 and an acceleration of a mean zonal flow

Y
Non-acceleration
Theorem: an illustration Z = 0.75H• TEM equations indicate that eddies
will accelerate/decelerate zonal
flow if 𝜵. 𝑭 ≠ 0• Illustrate with an analytic example
consisting of a linear superpositionof a QG finite-amplitude,baroclinic/barotropic wave andzonal flow with diabatic heating• N2 assumed uniform• Formulated to be non-interacting freemodes so 𝒗) . ∇𝑄 = 0• i.e. 𝑣 8 𝑄′ = 0 = ∇. 𝑭 even though
individual momentum and heat fluxes
are non-zero[Read 1985]

Z=0.5H

Z=0.25H

Non-acceleration
Theorem: an illustration• TEM equations indicate that eddies
will accelerate/decelerate zonal
flow if 𝜵. 𝑭 ≠ 0• Illustrate with an analytic example
consisting of a linear superpositionof a QG finite-amplitude,baroclinic/barotropic wave andzonal flow with diabatic heating• N2 assumed uniform• Formulated to be non-interacting freemodes so 𝒗) . ∇𝑄 = 0• i.e. 𝑣 8 𝑄′ = 0 = ∇. 𝑭 even though
individual momentum and heat fluxes
are non-zero[Read 1985]

𝑢s

𝑢8 𝑣′

𝑣 8 𝜕𝜓 8 /𝜕𝑧

EP Flux F𝜒̅𝜒∗Critical layer absorption (a quick summary!)• Propagation of Rossby or inertia-gravity
waves may be strongly affected in regions
where phase speed 𝑐 ≈ 𝑢^
• a CRITICAL LAYER or LINE• Group velocity cg may become small
perpendicular to critical line
• Reduced wavelength normal to criticalline -> increased dissipation andnonlinearity• E.g. breaking and overturning of wave

• Absorption and dissipation of wave
pseudo-momentum may lead toenhanced acceleration of 𝑢^ near criticallayer• Hence 𝑢^ tends to be accelerated towards
phase speed c as wave dissipates

Credit: Vallis [2017]

Summary• Held & Hou model captures many aspects of meridional overturning circulation• With some global super-rotation but without local super-rotation• Gierasch model provides overall concept for super-rotation via diffusive eddies• Extended to strongly viscous flows by Yamamoto & Yoden• can produce S ~ 10 as on Venus…..

• Leads to scaling dependence of Global super-rotation on Ω and ℛ• Vorticity and its relatives is more likely to be diffused by (QG?) eddies or waves,though upward AM transport possible• Wave-zonal flow interaction requires 𝜵. 𝑭 ≠ 0 – wave dissipation, forcing ortransience?
• Critical layers facilitate dissipation of waves and acceleration of zonal flows
towards phase speed c