Fast and Slow: The
Dynamics of Superrotation
Phenomena in Planetary
Atmospheres:
IntroductionPeter Read
University of Oxford, UK

30/11/2023

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Aims & Motivating Questions• Meaning and concept of super-rotation: How should it be defined?
• Where does it occur, and when?
• Explore almost every known planet, not just Earth!

• What dynamical processes and mechanisms can generate and maintain it?
• Is there a general scaling theory for atmospheric super-rotation?
• Which particular processes and mechanisms are responsible for theobserved super-rotation on Earth, other terrestrial planets and the gas andice giants?
• Implications for planetary climate?

• What outstanding issues remain for further research?
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Overall plan
I.

Fundamental concepts• Definitions, constraints, observations (1st look), analysis of an analogue system

II. Simple models and theories• Simple conceptual models, scaling arguments, wave-mean flow interactions

III. Super-rotation mechanisms in simple systems• Laboratory experiments, simplified numerical models

IV. Super-rotation in fast-rotating planets• Earth, Mars, Gas and Ice Giants

V. Super-rotation in slowly-rotating planets• Titan, Venus, tidally-locked exoplanets• Synthesis and discussion…..

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Fast and Slow: The
Dynamics of Superrotation
Phenomena in Planetary
Atmospheres:
I. Fundamental concepts
Peter Read
University of Oxford, UK

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Plan• What is super-rotation?
• Constraints on axisymmetric circulation: Hide’s theorem(s)• Observations of super-rotation around the Solar System• Super-rotation in an analogous system: axisymmetric flow in arotating, cylindrical annulus• A scaling theory for annulus flows

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What is super-rotation?
• Earliest reference that mentions it by
H Rishbeth (Nature 1971) on rotationof Earth’s upper atmosphere(thermosphere and ionosphere)• Based on observations in the 1960sof satellite orbits (King-Hele 1964) at
low altitude (~300 km)• Super-rotation defined in terms ofratio of angular velocities•𝛬=
30/11/2023!"!
"#$ %&' (#=1+

From Rishbeth 2002$(&"')# )*+ ,FDEPS2023

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What is super-rotation?
• Earth’s inner core is also described as“super-rotating” by Glatzmaier &Roberts (1996 Science) fromnumerical simulations of MHD
convection in the outer core
• Interpreted as a positive angularvelocity relative to the rest of theEarth• Maintained mainly via EM torques
from motion of outer core….
• But equatorial region of outer core
sub-rotates except close to inner coreboundary…..
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Meridional velocity.

Angular velocity w

7What is remarkable about positive angularvelocity?
• Doesn’t necessarily require
special forces to act, e.g. if
angular momentum isconserved.
𝐽 = 𝐼𝜔 = 𝑐𝑜𝑛𝑠𝑡.
• Example: Ice skater reduces her
moment of inertia and increases
her angular velocity w• Better to define super-rotation
with respect to angularmomentum?
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Angular momentum of an atmosphere in
Ω=Ωksolid-body co-rotation• Axial angular momentum defined as• 𝜇 = 𝒓×𝒑𝜽 . 𝒌 = 𝜌Ω 𝑎 + ℎ cos 𝜑 "r• Increases with cylindrical radius 𝑟 cos 𝜑from rotation axis
• Maximum at the equator• 𝜇#$% = 𝜌Ω 𝑎 + ℎ• ≈ 𝜌Ω𝑎" for ℎ ≪ 𝑎

"

a

h
𝜑

• Define super-rotation as• 𝜇 = 𝜌 𝑢 + Ω 𝑎 + ℎ cos 𝜑 𝑎 + ℎ cos 𝜑> 𝜇#$%• Occurs e.g. when u > 0 on the equator.
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Contours of µ9

When is angular momentum conserved?

• Consider zonal momentum equation in spherical coordinates for a shallow
atmosphere (h << a)
•

!"
"
!"
+
!#
$ %&' ( !)

+

* !"
!"
"* ,-. (/ !1+𝑤 −− 2Ω𝑣 sin 𝜑 = −
$ !(
!+
$
0$ !)

+ 𝐹2

(1)

• l is longitude, FB represents body forces (friction, viscosity, MHD etc…)• Define specific axial angular momentum (per unit mass)
• 𝑚=

• So
•

3
=
0

(2)

𝑢 + Ω𝑎 cos 𝜑 𝑎 cos 𝜑!4!"= 𝑎 cos 𝜑!#!#= − 𝑣 cos 𝜑!"!"+ 𝑤𝑎 cos 𝜑 − 𝑢𝑣 sin 𝜑 − 2Ω𝑎𝑣 cos 𝜑 sin 𝜑 − 𝐹2 𝑎 cos 𝜑
!(
!+

(3a)

• [neglecting terms in ∂/∂l i.e. assuming axisymmetry]=−
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* !4
!4
+𝑤
$ !(
!+

(3b)

+ 𝐹2 𝑎 cos 𝜑
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When is angular momentum conserved?
• Hence for a circulation that is symmetric
about the rotation axis•

!4
+ 𝒖. 𝛁𝑚 = 𝐹2 𝑎 cos 𝜑!#(4)• m is materially conserved in the absenceof body torques and friction by anaxisymmetric circulation.
• Conversely, m cannot exceed mmax = Ωa2
in an axisymmetric flow without nonaxisymmetric eddies (or body forces)• Zonal pressure torques etc…..

• Often referenced as Hide’s theorem (J.
Atmos. Sci. 1969)
Raymond Hide [1937 – 2016]30/11/2023

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How to measure/quantify super-rotation?
• Suggests a way to quantify super-rotation as the degree to which
Hide’s local limit on m (or its zonal mean 𝑚). is exceeded (ReadQJRMS 1986).
• Define a dimensionless local super-rotation index𝑠=

AB#C )#C )=D )*+ ,

(5)

#C

• Implies s ≤ 0 for axisymmetric circulations but s can be ≥ 0 for nonaxisymmetric flows under certain conditions

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Aside: compressible or incompressible?
•

EA
EF

+ 𝒖. 𝛁𝑚 = 𝐹G 𝑎 cos 𝜑
=

EA
EF

+ 𝛻. 𝒖𝑚 for either(6)• (i) incompressible flow [𝛻. 𝒖 = 0] or• (ii) compressible flow using p as vertical coordinate• For compressible flow with height z as vertical coordinate
•

E(HA)
EF

+ 𝛻. (𝒖𝜌𝑚) = 𝐹G 𝜌𝑎 cos 𝜑=30/11/2023

EI
EF+ 𝛻. (𝒖𝜇)FDEPS2023(7)
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How to measure/quantify super-rotation?
• In light of (6) or (7), if m is materially conserved everywhere [in a closed
system] then the total volume- [or mass-]integrated AM must also be
invariant.
• Hence we can also define a dimensionless global super-rotation index(Read 1986)
S=∭ "#$ % &'( ) &*+∭ $ % &'( ) &*+or∭ ,["#$ % &'( ) &]*+/'

(8)

• [Or its compressible equivalent based on (7)]• Where M0 is the total integrated (mass-weighted) AM for an atmosphere in solidbody co-rotation with the planet at Ω

• S = 0 for an atmosphere that does not exchange AM with other parts of theplanet and/or experiences no external body torques
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Where do we observe superrotation (and how strong)? T
Earth - Annual mean

[zonally averaged] circulationU[As observed - ERA40- Kallberg et al. 2004]Y
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Where do we observe super-rotation (andhow strong)?
Zonal mean wind (Annual Mean)

100

Local super-rotation (Annual Mean)

2

10
-50

-4

0
Latitude (deg)

50
10-2

Pressure (hPa)• Equatorial values of s < 0most of the time10
• Mass-weighted(compressible) global valuesof S ~ 0.010-0.017
10• Magnitudes ~ 0.01

• Earth’s stratosphere &Mesosphere

Data credit: Randal etal [2004]

10-2

Pressure (hPa)• Earth’s troposphere-4• Annual mean equatorial 𝑢5
westward except around 0.1
and 10-3 hPa• NB Eddy driven!

100

102

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0Latitude (deg)1650

Where do we observe super-rotation (andhow strong)?
• Earth’s troposphere• Equatorial values of s < 0most of the time
• Mass-weighted(compressible) global valuesof S ~ 0.010-0.017• Magnitudes ~ 0.01

• Earth’s stratosphere• Equatorial values of s range
over ±0.2 due to QuasiBiennial Oscillation andSemi-Annual oscillations• NB Eddy driven!
Credit: Read & Lebonnois (2018 Ann Rev. EPS)

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Where do we observe super-rotation (andhow strong)?
Equatorial Local Super-rotation• Equatorial values of s < 0• Mass-weighted(compressible) global valuesof S ~ 0.010-0.017• Magnitudes ~ 0.01

• Earth’s stratosphere• Equatorial values of s range
over ±0.14 due to QuasiBiennial Oscillation andSemi-Annual oscillations• NB Eddy driven!

10

100
1000
1960

1965

1970
Year

1975

1980

1

Pressure (hPa)• Earth’s troposphere

Pressure (hPa)

1

10

100
1000
1985

1990

1995

2000

2005

Year
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Where do we observe super-rotation (andhow strong)?
• Earth’s troposphere• Equatorial values of s < 0• Mass-weighted(compressible) global valuesof S ~ 0.010-0.017• Magnitudes ~ 0.01

Credit:Cornillon etal. [2019]• Earth’s stratosphere• Equatorial values of s range
over ±0.14 due to QuasiBiennial Oscillation andSemi-Annual oscillations• NB Eddy driven!

• Earth’s oceans!

• Equatorial undercurrent….

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Atmosphereof Mars

30/11/2023

Annual mean
Climatology[Read & Lewis 2004]

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Where do we observe super-rotation(and how strong)?
Zonal/annual mean u MY28-9• Mars atmosphere

-60

10.0

0
20

-40

60

40

20

60

100.0
0

0

-50

0

0
Latitude (deg)50

Zonal mean s MY28-9
-0.6
0
-0.
50

-0.80
-0.70

1.0

-0.90

40

.
-0

-0.3

0

100.0

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0
Latitude (deg)

-0.20
-0.3
0
-0.
40
-0.
-0.6 50
0
-0.7
0
-0.80
-0.90

10.0
-0.10

Pressure (Pa)

0.1

-50
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40

-2

0

20

Pressure (Pa)1.0
20

• Annual mean zonal wind
exhibits midlatitude
westerly jets and mostly
easterly flow in the tropics• Weak westerly flow on theequator below 20 km
altitude• Local super-rotation s
peaks at around 0.03
• Seasonal variations?

0.1

50
21

Sm

0.2

Where do we observe super-rotation(and how strong)?
0.1
0.0

3000

3200

3400
Time (sol)

3600

3800

Local super-rotation• Mars atmosphere

S0

Sm0

0.06
0.2
0.04
0.1
0.02
0.0
200
0.00
3000

300

MY28
3200

400
LS (deg)
3400
Time (sol)

500

600

3600

MY29 3800

700

Global super-rotation
0.06
S0• Annual mean zonal wind
exhibits midlatitude
westerly jets and mostly
easterly flow in the tropics• Weak westerly flow on theequator below 20 km
altitude• Local super-rotation s
peaks at around 0.3• Seasonal variations?
• Global super-rotation
varies from ~0 to 0.07

Global super-rotation

0.3

0.04
0.02
0.00

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300

400
LS (deg)

500

600

22 700Where do we observe super-rotation(and how strong)?
Cassini images: NASA/JPL• Jupiter’s atmosphere• Giant planet (a ~ 11.aEarth)rotates rapidly (trot = 9.926
hours)• Multiple eastward andwestward zonal jets in eachhemisphere at cloud tops
• Strong (>100 m s-1) eastward
equatorial jet• Local super-rotation at cloud
tops peaks at around smax =+0.006 at the equator
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Where do we observe super-rotation(and how strong)?
• Jupiter’s atmosphere• Giant planet (a ~ 11.aEarth)rotates rapidly (trot = 9.926
hours)• Multiple eastward andwestward zonal jets in eachhemisphere at cloud tops
• Strong (>100 m s-1) eastward
equatorial jet• Local super-rotation at cloud
tops peaks at around smax =+0.006 at the equator
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Cassini images: NASA/JPLFDEPS2023

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Where do we observe super-rotation(and how strong)?
Zonal velocity (m s-1)

• Giant planet (a ~ 11.aEarth)rotates rapidly (trot = 9.926
hours)• Multiple eastward andwestward zonal jets in eachhemisphere at cloud tops
• Strong (>100 m s-1) eastward
equatorial jet• Local super-rotation at cloud
tops peaks at around smax =+0.006 at the equator

150
100
50
0
-50
40

-40

0.0
-0.2
-0.4
-0.6
-0.8
-1.0
40

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20
0
-20
Planetographic Latitude (degs)(b) Local super-rotation index
Local Super-rotation s• Jupiter’s atmosphere(a) Zonal velocity

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20
0-20Planetographic Latitude (degs)-40
25

Where do we observe super-rotation(and how strong)?

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Zonal velocity (m s-1)

• Giant planet (a ~ 11.aEarth)rotates rapidly (trot = 9.926
hours)• Multiple eastward andwestward zonal jets in eachhemisphere at cloud tops
• Strong (>100 m s-1) eastward
equatorial jet• Local super-rotation at cloud
tops peaks at around smax =+0.006 at the equator

140
120
100
80
60
40
20
0
15

10

5
0
-5
Planetographic Latitude (degs)

-10

-15

-10

-15

(b) Local super-rotation index
Local Super-rotation s• Jupiter’s atmosphere(a) Zonal velocity

0.02
0.00
-0.02
-0.04
-0.06
-0.08
15
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10

5
0
-5
Planetographic Latitude (degs)

26

Where do we observe super-rotation(and how strong)?
(a) Cassini Thermal zonal wind

• Jupiter’s atmosphere• Up to s ~ 0.03 in thestratosphere?

Pressure (hPa)

10
100
40

-40

1
10
100
40

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20
0
-20
Planetographic Latitude (degs)(b) Cassini Local Super-rotation
Pressure (hPa)• Giant planet (a ~ 11.aEarth)rotates rapidly (trot = 9.926
hours)• Multiple eastward andwestward zonal jets in eachhemisphere at cloud tops
• Strong (>100 m s-1) eastward
equatorial jet• Local super-rotation sat
cloud tops peaks at around+0.006 at the equator

1

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20
0-20Planetographic Latitude (degs)-40
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Where do we observe super-rotation(and how strong)?
• Titan’s atmosphere• Major satellite of Saturn• (a ~ 0.4aEarth) rotates slowly(trot = 15.95 days)• Substantial atmosphererich in N2 and CH4• Strong (>100 m s-1)
eastward equatorial jet• Varies with season?

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Where do we observe super-rotation(and how strong)?
• Titan’s atmosphere• Major satellite of Saturn (a~ 0.4aEarth) rotates slowly(trot = 15.95 days)• Substantial atmosphererich in N2 and CH4• Strong (>100 m s-1)
eastward equatorial jet• Local super-rotation atcloud tops peaks ataround +15 at the equator
Cf Ω𝑎! = 3.02×10" m2 s-1
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Where do we observe super-rotation(and how strong)?
• Venus’s atmosphere

UV cloud tops(~65 km)• Earth-sized planet (a ~0.9aEarth) and rotates very
slowly (trot = 243 days
retrograde)• Massive atmosphere mainlyof CO2 with cloud decks of
H2SO4 droplets around 40-60
km altitude• Strong (>100 m s-1) eastward
flow at cloud tops (peaks ataround ~70 km altitude)

980 nm(~60 km)1.74 µm(~45 km)• Weak prograde flow in thelower atmosphere[Data from Pioneer Venus probes]
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Where do we observe super-rotation(and how strong)?
• Venus’s atmosphere

UV cloud tops(~65 km)• Earth-sized planet (a ~0.9aEarth) and rotates very
slowly (trot = 243 days
retrograde)• Massive atmosphere mainlyof CO2 with cloud decks of
H2SO4 droplets around 40-60
km altitude• Strong (>100 m s-1) eastward
flow at cloud tops (peaks ataround ~70 km altitude)

980 nm(~60 km)1.74 µm(~45 km)• Weak prograde flow in thelower atmosphere• ~ uniform with latitude
between ±50o - 60o

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Where do we observe super-rotation(and how strong)?
• Venus’s atmosphere

UV cloud tops(~65 km)• Earth-sized planet (a ~0.9aEarth) and rotates very
slowly (trot = 243 days
retrograde)• Strong (>100 m s-1) eastward
flow at cloud tops (~60 km
altitude)• Local super-rotation at cloud
tops peaks at around +52.2
at the equator

980 nm(~60 km)1.74 µm(~45 km)• Decreasing towards thesurface
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Where do we observe super-rotation(and how strong)?
80

Sounder
Day/night
North
Venera 8
Veneras 9 & 10
Venera 1260• Estimating global super-rotation60

𝑆 ≈ 7.7

Sounder
Day/night

50

North

AM density (x 108 kg m-1 s-1)

Height (km)Venera 840

20

Veneras 9 & 10

40

Venera 12

30

20

10

0
0.0
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0

0.2

0.4

0.6
0.8
Normalised
AM density
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1.0

1.2

-50

0
Latitude50

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Where do we observe super-rotation(and how strong)?
Knutson et al. (2007)• Hot Jupiter exoplanets• Jupiter-sized planets (a ~10aEarth) in orbits close to
their parent star.
• Rotation likely to be locked in
1:1 resonance with orbit,
due to gravitational tides• Phase curves from secondarytransits indicate eastward
displacement of sub-stellar
hot spot• Advected by strong (>1000 ms-1) eastward (super-rotating)flow…?

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Summary of super-rotating atmospheres
Planeta (km)

Ω (s-1)

umax (m s-1)

smax

SGlobal

Earth

6371

7.27 x 10-530-0.015 – +0.04

0.0135 ± 0.002

Mars

3396

7.09 x 10-5

30

0.16

0.04

Jupiter

69911

1.76 x 10-4

-60 - +140

0.005 – 0.016-

Saturn58232

1.64 x 10-4

350 - 430

0.035 – 0.045-

Uranus25562

1.01 x 10-4

-80

-0.03

-

Neptune24622

1.08 x 10-4

-400

-0.15

-

Pluto

1152

1.14 x 10-5

10 - 150.76 – 1.1-

Triton1353

1.24 x 10-5

5 - 100.3 – 0.6-

HD189733b

79500

3.28 x 10-5

2400

0.93

-

HD209458b

94380

2.06 x 10-5

1940

1.00

-

Titan

2576

4.56 x 10-6

100 - 180

8.5 - 152

Venus
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6052

2.99 x 10-7

100 - 120
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50 - 60

7.7

35

Super-rotation with eddies: The Hide/Starr
theorem (II)• For non-axisymmetric flows we need totake account of processes which break
AM conservation• [Molecular viscosity?]• Eddies….

and look for constraints on eddytransport of AM• Consider the zonally averaged form of (6)[or (7)] in a steady state.𝑚𝛻. 𝒖. = 𝐹G 𝑎 cos 𝜑 = ℱG
(8)
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Victor Starr [1909 – 1976]36

Super-rotation with eddies: The Hide-Starr
theorem (II)• Now integrate (8) over the toroidal annularvolume or ring of fluid enclosed by a closed
contour C in the meridional plane• If there exists a local maximum of 𝑚
Q = 𝑚0 at
point H in the meridional plane, we can take C to
be a contour of constant 𝑚
Q = 𝑚0 − δ• LHS of the integral of (8) over volume C•Q𝑚Q 𝑑𝑉 = ∬1 𝑚Q
Q 𝒖. 𝑑𝒏∭1 ∇. 𝒖C

(9a)

• [Divergence theorem]= 𝑚0 − 𝛿 ∬1 𝒖. 𝑑𝒏
=0

(9b)
(9c)

• by mass conservation [NB for steady state only!]
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Hide-Starr theorem (II): constraints on eddies• Now consider the RHS of (8). Suppose we canrepresent the torque due to eddies byℱ2 = ∇. 𝑬(10)
where E is a flux of AM due to eddies• From (9b) and (9c), therefore(11)∭1 ℱ2 𝑑𝑉 = ∬1 𝑬. 𝑑𝒏 = 0• This can only be satisfied if 𝑬. 𝛁𝑚
Q takes either sign[or E is everywhere parallel to contours of 𝑚],
Q so ingeneral…
• Eddy angular momentum fluxes E must be able totransfer 𝑚
Q up-gradient as well as down-gradient
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Hide-Starr Theorem(s): Summary• We can define specific (axial) angular momentum
𝑚 = Ω𝑎 cos 𝜑 + 𝑢 𝑎 cos 𝜑• Where a = planetary radius; Ω = rotation rate, u = zonal velocity & f = latitude• In zonal mean:𝜕𝑚8+ ∇. 𝑚8
8 𝒖∗ = ∇. 𝑬 + 𝐹𝜕𝑡mean
advection[E is ~Eliassen-Palm flux
u* is TEM meridional velocity –see later]

eddy bodystress forces/torque• NB m materially and globally conserved in frictionless, axisymmetric flows (E, F = 0)
• HENCE• Equatorial local super-rotation is impossible in purely axisymmetric, inviscid flow
• Local or global super-rotation must involve the existence of non-axisymmetric eddies
AND• Eddy angular momentum fluxes E must be able to transfer 𝑚
8 up-gradient
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Effects of viscosity?
• What role could fluid viscosity play in super-rotation?
• Analogous to “eddy viscosity”?

• Laboratory analogue of atmospheric circulation• Cylindrical geometry• Differentially heated rotating annulus
Working fluid

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T

Effects of viscosity?
• Numerical simulation of
axisymmetric flow in cylindricalannulus• Heated outer cylinder and cooled
inner• Stress-free top surface and rigid,
non-slip sidewalls• Meridional overturning circulation(thermally direct)• Super-rotation is very weakm/Ωb2

w = u/ry• s < 0.01
• S ~ 0.08

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T

Effects of viscosity?

w = u/r• The viscous term in (6) or (7) canbe written as
𝐹2 = −∇. 𝑭• Where𝑭 = −𝜈𝑟 3 ∇𝜔
3= −𝜈𝑟 ∇

"
4&

m/Ωb2

Torquey• Viscous forces/torques act totransfer AM downgradient forangular velocity• i.e. to relieve tangential stress• Note that F satisfies (11) in acting
up-gradient for m horizontally -• but is downgradient for m vertically

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Viscous AM fluxes – up- or down-gradient?
"33
𝑭 = −𝜈𝑟 ∇𝜔 = 𝜈𝑟 ∇ &4

and𝑚 = 𝑟 𝑢 + Ω𝑟
• So projection of F onto ∇𝑚 determines direction of transport
1 𝜕
3
𝑭. ∇𝑚 = −𝜈 ∇𝑚 −𝑚3𝑟 𝜕𝑟
• Hence• 𝑭. ∇𝑚 < 0 if F is down-gradient for m• 𝑭. ∇𝑚 > 0 if F is up-gradient for m

5"57• Also note thatis parallel to56
56• so the vertical component of F is always downgradient

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T

Effects of viscosity?
• The viscous term in (6) or (7) canbe written as
𝐹2 = −∇. 𝑭• Where𝑚
3
𝑭 = −𝜈𝑟 ∇ 3
𝑟• This allows AM to enter or leave
the flow through boundaries thatare not stress-free• Allows the flow to gain or lose AM
and change the global superrotation• S = 0.08 and smax~ 0.01 for this case

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m/Ωb2

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w = u/r

Torquey44Now with stress-free side boundaries
T

Effects of viscosity?

• The viscous term in (6) or (7) canbe written as
𝐹2 = −∇. 𝑭• Where𝑚
3
𝑭 = −𝜈𝑟 ∇ 3
𝑟• This allows AM to enter or leave
the flow through boundaries thatare not stress-free• Note that F satisfies (11) in acting
up-gradient for m horizontally• Much larger super-rotation withstress-free side boundaries• smax ~ 0.35
•30/11/2023
S ~ 0.36

m/Ωb2

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w =Torque
u/r

y

45

[Read 1986]

Effects of viscosity?

• The viscous term in (6) or (7) canbe written as
𝐹2 = −∇. 𝑭• Where𝑚
3
𝑭 = −𝜈𝑟 ∇ 3
𝑟• This allows AM to enter or leave
the flow through boundaries thatare not stress-free• Note that F satisfies (11) in acting
up-gradient for m horizontally• Much larger super-rotation withstress-free side boundaries• smax ~ 0.35
•30/11/2023
S ~ 0.36

How to predict the magnitude of S?
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A scaling theory for super-rotation in arotating annulus[Read1986]• Governing equations for steady axisymmetricflow in cylindrical annular cavity

30/11/2023~S

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A scaling theory for super-rotation in arotating annulus[Read1986]

30/11/2023~S

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A scaling theory for super-rotation in arotating annulus[Read1986]• Provided Ra and E-1 are >>1 then meridional
flow is mostly confined to sidewall thermal andhorizontal Ekman boundary layers with adistinct interior• Sidewall thermal boundary layer
𝑅𝑎 *+/ℓ ) ~𝐿𝜀
• Ekman layerℓ. ~𝐻𝐸+/"• Define squared ratio
• 𝑄=

ℓ# "

ℓ$

= 𝑅𝑎*+/" 𝐸 *+ 𝜀 *0/" ∝ Ω• Properties of the circulation largely depend on
the magnitude of Q
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A scaling theory for super-rotation in arotating annulus [Read1986]• Up to ~6 different dynamical regimes can be identified• Weak/very weak rotation regime dominated by sidewall thermal
layers so *+ ~𝜅𝜀 ,/.𝑅𝑎//.

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A scaling theory for super-rotation in arotating annulus• In regime (ii) get a nonlinearCoriolis balance in zonal
momentum equations which
implies angular momentum isconserved• V ~ Q and Roc ~ 1• Regime (iii) V scale depends oncurvature h and e but can begeostrophic or cyclostrophic(Roc > 1 )30/11/2023

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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//0𝜎 1//0𝑄 1/)
III. Rapid rotation (quasigeostrophic):
S ~ Q-2 ~ Ω-2
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-2

I.

II.

III.

&'/! '/!

∝ 𝑅𝑜%
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A scaling theory for super-rotation in arotating annulus• Circulation largely determinedby interactions between
boundary layers and interiorflow• Three main super-rotation
regimes, depending upon Q

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A scaling theory for super-rotation in arotating annulus• Circulation largely determined byinteractions between boundary
layers and interior flow
• Three main super-rotation regimes,
depending upon Q• Moderate rotation regime allows fortwo possibilities• Weak super-rotating regime has u ingeostrophic balance• Strong super-rotating regime has u incyclostrophic balance
𝜕 𝑢!
𝜕𝑇
≈ 𝑔𝛼
𝜕𝑧
𝜕𝑟
• Also requires 𝜎 > 𝜀 !/# 𝜂 $%/# sothermal diffusion is weak cf viscosity• 𝑆~𝑅𝑜& = 𝜀𝜂%/! 𝜎 $%/! 𝑄 $% so enhancedby large aspect ratio e and curvature h

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A numerical example• Increase aspect ratio e from 1 - 2• Increase curvature h from 2/3 to
3/2• Smax increases to 2.21 from 0.68
• Close to predicted factor of 2 2• Peak angular velocity wmax = 1.9
rad s-1, ~10 x W!

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A numerical example• Increase aspect ratio e from 1 - 2• Increase curvature h from 2/3 to
3/2• Smax increases to 2.21 from 0.68

m = Centrifugal –uv/r
C = Coriolis• Close to predicted factor of 2 2• Peak angular velocity wmax = 1.9
rad s-1, ~10 x W!
• Centrifugal acceleration starts todominate over Coriolis
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Pressure gradient0

Height (cm656Viscous AM fluxes in spherical geometry?
• For a DEEP spherical shell𝑚 = Ω𝑟 cos 𝜑 + 𝑢 𝑟 cos 𝜑• the viscous flux term can be written as
𝑭 = −𝜈𝑟 3 cos 3 𝜑∇𝜔= −𝜈𝑟 3 cos 3 𝜑∇"
4 &&'(&)• 𝑭. ∇𝑚 = −𝜈

𝑚3

a

h

O

R

• Where 𝑅 = 𝑟 cos 𝜑 is “cylindrical” radius• [Exercise: show this!]• This means that F can be up-gradient for min the R direction but not parallel to k (i.e.
parallel to Ω)
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r

𝜑

• So
J 5
3
∇𝑚 −
K 5K

Ω=Ωk

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Contours of µ57Viscous AM fluxes in spherical geometry?
• For a SHALLOW spherical shell, h << a andr ~ a so𝑚 = Ω𝑎 cos 𝜑 + 𝑢 𝑎 cos 𝜑• The viscous flux term is written as
𝑭 = −𝜈𝑎3 cos 3 𝜑∇𝜔"33= −𝜈𝑎 cos 𝜑∇ %&&'(&)

Ω=Ωk

h

r

a

• Where 𝜔 = 𝑢/ 𝑎 cos 𝜑 so that𝜑

• [Exercise: show this!]

R

1 𝜕
3
𝑭. ∇𝑚 = −𝜈 ∇𝑚 − 3
𝑚3 tan 𝜑𝑎 𝜕𝜑• This means that F can be up-gradient form in the 𝜑 direction but not in the local
verticalO

Contours of µ

• NB Even at the equator!
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Summary I.
• Super-rotation is best defined with respect to angular momentum,
not angular velocity• Specific angular momentum is conserved by axisymmetric flows in theabsence of viscosity and body forces• Super-rotation requires non-axisymmetric eddies that transfer AM upgradient• Super-rotation is widely observed in various planetary circulations,but with widely differing magnitude• Scaling arguments may be useful but probably depend on thesystem…?
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