Fast and Slow: The
Dynamics of Superrotation
Phenomena in Planetary
Atmospheres
III. Super-rotation in
simplified systems
Peter Read
University of Oxford, UK

29/11/23

FDEPS2023

1

Super-rotation in simplified systems• Laboratory experiments and demonstrations• Illustrating mechanisms and basic principles, scaling etc.

• Simplified and intermediate complexity numerical models• Circulation regimes & parameter sweeps• Zonally symmetric forcing• Equatorial jets
• Thermal tides & tidally locked forcing

29/11/23

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2

Laboratory experiments:“Dishpan” convection• Early experiments on rotating
convection at University of
Chicago [Fultz et al. 1956]• Open cylinder (“dishpan”), heatedfrom below near the outer edge
and cooled at the centre• Rotated about its axis at Ω rad s-1• Produces circulations analogous toaxisymmetric “Hadley” andbaroclinic wavey “Rossby”
regimes, depending upon Ω.
29/11/23

Dave Fultz
Oscillating
plunger

FDEPS2023

3

Laboratory experiments:“Dishpan” convection• Early experiments on rotating
convection at University of
Chicago [Fultz et al. 1956]• Axisymmetric flow profile shows
evidence of super-rotation?

Oscillating
plunger• ℛ = 2.66
• s > 0 near outer rim of tank
• m ~ constant for 0.2 < r/b < 0.7• Viscous AM transfer or eddies….?

29/11/23

FDEPS2023

4

Laboratory experiments:
Rossby wave mixing• Generation of Rossby waves in a 2
m dia. rotating cylindrical tank
[Whitehead 1975]• Free surface deforms centrifugally
to a parabolic interface• ℎ = 2.934 + 0.00126𝑟 ! (cm)• Imposes a “planetary vorticity
gradient”
• 𝛽=

!" $#

Mean flow generated by circulation on
a /&plane:
Oscillating
An analogy with the moving flame plunger
experiment
By JOHN A. WHITEHEAD, JR., Department of Physical Oceanography, Woods Hole
Oceanographic Institution, Woods Hole, Mass. 02543, US-4

# $%(Manuscript received May 31; revised version September 26, 1954)• Various localized forcing
mechanisms• Vertically oscillating plunger• Bubbles• Radially oscillating plate

29/11/23

ABSTRACTA number of laboratory experiments are described in which water with a curved
upper surface in a rotating basin exhibited prograde flows when stirred by stirrers
which put no azimuthal torque upon the fluid. I t is suggested that the flows weregenerated by Reynolds stresses of the circulating fluid, and that this is a generalconsequence of circulations on a 8-plane. This is reinforced by an analogy between theequations of the moving flame experiment and the equations of flow on a 8-plane.
Implications upon atmospheric and oceanic flows are mentioned.

FDEPS2023

5

YAKOV D. AFALaboratory experiments: Rossby wave mixing• Generation of Rossby waves in a 2
m dia. rotating cylindrical tank
[Whitehead 1975]• Free surface deforms centrifugally
to a parabolic interface

Oscillating
plunger• ℎ = 2.934 + 0.00126𝑟 ! (cm)• Imposes a “planetary vorticity
gradient”
• 𝛽=

r

!" $#
# $%

• Various localized forcing
mechanisms• Vertically oscillating plunger• Bubbles• Radially oscillating plate

29/11/23

Figure 8.1 Sketch of the experimental setup. A tank filled with water
is on a table rotating with angular velocity ⌦. A video camera and alight source co-rotate with the tank.

FDEPS2023

6

a
p
b
i
T
r
T
t
a
a
c
d
t
t
b
fi
t
r

8

Laboratory experiments:
Rossby wave mixing• Generation of Rossby waves in a
2 m dia. rotating cylindrical tank
[Whitehead 1975]• Various localized forcing
mechanisms• Vertically oscillating plunger• Forms a prograde jet at radius of
the plunger• Weaker retrograde zonal flow
either side
29/11/23

FDEPS2023

MEAN FLOW QENEBATED BY CIBCULATION ON A /%PLANE

from the center and h is depth ofcentimeters).
The first stirrer consisted of
disk 20 cm in diameter connected
yoke of variable amplitude driven b
speed motor. When the disk was
am from the center of the turntab
lated up and down in the 6.72 cm
a t periods from 4 to 40 seconds,
variety of amplitudes, a pronounc
flow waa observed due west of
between a radius of 60 to 60 c
center (the marked circles are 10
This jet sometimes stretched compthe tank. A retrograde flow wasother regions of the tank.Such a f
in the dye-streak photos in Fig. 1.
measure this flow more carefully
was covered with sheets of vinylid(saranwrap), with one slitlike ope
Fig. 2. Dye streak
photographsof progradeinwestof the plunger
so thatflow
a satura
the rotating fluidwithgentleagitation(5- bof potaasium permanganate could
0.0148) and with the tank covered. Pictures are 32
seconds apart. a flow tracer. This covering r
retrograde flow to one revolution
Tellus XXVII (1976),
4
revolutions,which was approxi
order of magnitude slower thanflows measured. Fig. 2 shows a slow
such conditions. Velocity
was m7
timing the traverse time for dy

Laboratory experiments:
Rossby wave mixing• Generation of Rossby waves in a
2 m dia. rotating cylindrical tank
[Whitehead 1975]• Forms a prograde jet at radius of .
2
the plunger
362

J. A. WEJTEHEAD, JR
I

f

0.03

f

.

x+

+

/t

+I
I

+I *

O.O'

• Weaker retrograde zonal flow
either side1

. + ...
X

/

i

0.003
0.001

I.

7
+

0.1

0.01

fpl""g*r

numberfunction Roasby number the plunger for experimente• Strength of jet (𝜀!"# = 𝑢$%& /Ω𝐿) coveredRossby
om, +dashed line with atank. Amplitude (peak to peak): X
cm,
for comparison.
scales roughly with 𝜀'()*+", atis relatively insensitive to the plunger speed.
small amplitude
these experiments it was
During the course
Fig. 3.

of 2 :1 is inserted• Saturates at larger amplitude

29/11/23of the flow as aof
- 1.75ofobvious that in this latter case there was extensive turbulence in the fluid, while in theformer case there was virtually no turbulence.
FDEPS2023
The
exact nature of the flow field around thisplunger is poorly understood and a more de-- 3.5

with adopeof
5.20 cm. A

8

Experimental concept(Turlab – Torino, Italy)
Observe waves
Plan viewand jets
downstreamof forcing comb

Forcing comb laser
sheetsMeasure velocity
fields via PIV in
2 overlapping
camera views

BridgeSlopingbottom29/11/23

FDEPS2023

9Chaotic tracer patterns indicate
zonal jets, Rossby waves andcompact vortices

29/11/23

FDEPS2023

10

Rb ≈ 2.25

Rb ≈ 2.20

Rb ≈ 2.18

𝒌.𝛁×𝒖'(𝛀Potential vorticity 𝑞 =*29/11/23

FDEPS2023

11

Zonal mean flow: time variationstR (s)
30.3• Width of mean
zonal flow increases
with tR – scales with
Rhines scale?
40.4• Timescale ofFluctuations also
Increases with tR

61.0
119.0
29/11/23

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12

Laboratory experiments:
Internal wave streaming• Generation of a mean flow by
dissipation of vertically
propagating internal gravity
waves [Semin et al. (2016,2018)]• A single, progressive or standing
gravity wave excited at theupper boundary of a cylindricaltank of salt-stratified fluid• Generate a mean azimuthal flow
as the wave propagates
downwards and dissipates
29/11/23

Benoit Semin Francois Petrelis. Stephan Fauve
PHYSICS OF FLUIDS 28, 096601 (2016)

Generation of a mean flow by an internal wave
B. Semin, G. Facchini, F. Pétrélis, and S. Fauve

Laboratoire de Physique Statistique, École Normale Supérieure, PSL Research University,
Université Paris Diderot Sorbonne Paris-Cité, Sorbonne Universités UPMC Univ Paris 06,
CNRS, 24 Rue Lhomond, 75005 Paris, France(Received 14 March 2016; accepted 1 September 2016; published online 22 September 2016)
We experimentally study the generation of a mean flow by a two-dimensional
progressive internal gravity wave. Due to the viscous damping of the wave, a nonvanishing Reynolds stress gradient forces a mean flow. When the forcing amplitude islow, the wave amplitude is proportional to the forcing and the mean flow is quadratic
in the forcing. When the forcing amplitude is large, the mean flow decreases the wave
amplitude. This feedback saturates both the wave and the mean flow. The profilesof the mean flow and the wave are compared with a one-dimensional analyticalmodel. Decreasing the forcing frequency leads to a wave and a mean flow localized
on a smaller height, in agreement with the model. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4962937]
FDEPS2023
13

Laboratory experiments:
Internal wave streaming
Benoit Semin Francois Petrelis. Stephan Fauve••<

29/11/23

FDEPS2023µ ·14

Laboratory experiments:
Internal wave streaming
Benoit Semin Francois Petrelis. Stephan Fauve
096601-6

29/11/23

FDEPS2023

Semin et al.

Phys. Fluids 2

FIG. 4. Amplitude of the horizontal wave component u 0 normalised by the forcing amplitude M , as az. Steady state, forcing period T = 38 s, Brunt–Väisälä frequency: N = 1.44 rad s 1. · · •: M = 1 mm,⇤: M = 4 mm,
5: M = 6 mm,: M = 8 mm,⇤: M = 10 mm, and 15⇧: M = 12 mm. Inse
M  6 mm only.

Laboratory experiments:
Internal wave streaming
0 sin(!

( )=0

096601-7+ ') + ¯

Semin et al.

Phys. Fluids 28, 096601¯'
Mean flow u

1

10

0

Velocity (mm.s−1)

50

z (mm)

100
150
200
250
300
350

0

10

−1

10

¯

−2

10

∝ aM
∝ bM2

400
−40

−20

0

x (mm)
=

= .

·

=)
=)
=) 29/11/23

20

40

mean ﬂow u
wave u'

−3

10

0

10

M (mm)

=

1

10

=
= .

·

=

FIG. 5. Amplitude of the mean flow ū normalized by M 2, as a function of height z. The amplitude is measuresteady state. Forcing period T = 38 s, Brunt–Väisälä frequency: N = 1.44 rad s 1. · · •: M = 1 mm,⇥: M = 2 m
M = 2 mm,⇤: M = 4 mm,
5: M = 6 mm,: M = 8 mm,⇤: M = 10 mm, and⇧: M = 12 mm.

FDEPS2023 B. Generation of a mean flow without feedback on the wave

16

Laboratory experiments: Internal wave
streaming•⇢✓= ( 0 + ¯)•@
+ ( · r)
@

◆

=

r +⌘

+⌫

@ ¯
@

+⇢

+ ( 0)

•
@¯
=
@

@
@

¯

⌫

=

0 0

0

•
29/11/23¯ / ( 0)FDEPS2023

17

Laboratory experiments: Internal wave
streaming•

Wave u’ (mm.s−1)

1.2

feedback

1
0.8

(⇠) =

0.6
0.4

t=150sSteady state
Linear fit

0.2
0
0

5

=

=)
=)

= .

=
=

10

M (mm)=

29/11/23( )⇥exp·

FDEPS2023

0

!

=
⇠

=

¯

=

✓

Z ⇠

(

¯)

⇠

◆

0

⌫
¯

=

18

Laboratory experiments: Internal wave
streaming@¯=@( ) ⇥ exp

(⇠) =

@ ¯
@⇠

@
+⌫
@⇠
✓

Z ⇠

¯

⇠

¯)

(

◆

( )
−3

−3

Exp data
Fit

2

1
0.5

2

4

z/d
=

6
=

x 10

Exp data
Model

2.5

1.5

0
0

29/11/23

3

Wave u' (m.s-1)

Mean flow u (m. s-1)

2.5

x 10

2
1.5
1
0.5
0
0

8
= .

FDEPS2023

2

4

z/d

6

8

·

19

Laboratory experiments: Jets driven by[baroclinic] instabilitiesz) temperature fields measured by a set of six traversing thermistors averaged over the duration of eacho the average measured temperature. The experiments are arranged by increasing the rotation rate from left
m top to bottom (Table 2). The meridional locations of the six thermistor profiles are shown by the dotteden stretched vertically to show detail). The temperature fields are shown relative to the bounding surfaces: thee topographic b plane (black), and the fluid free surface (blue line).

2014• Baroclinically unstable flow in a S
differentially heated, rotating annulus

V, we identify four-jet regimes.
wavenumber notation, these arelar, prograde jet (GT09, GT01,’ 2, one retrograde and two
d GT04); kjet ’ 3–4, two pros and eddy-forced prograde jets
6, GT05, and GT10); and kjet ’mber of weaker jets (GT10 and
pectra (Fig. 8) support this deion of jet regimes.
nd GT08 (kjet ’ 1) are eachprograde jet. Eddies of approxlus gap width are spawned fromnd cause long time-scale meanno obvious periodicity. Because
tching’’ behavior in the motion,
egular flow or geostrophic tur1984; Ketchum 1972). Experihave the lowest value for b andblance to the regimes identified
dies, although it is difficult to
ct of the small b that is present
l experiment.
2 rad s21, the main prograde jet
ents GT07 and GT04, the flow isde jets, each hugging a boundingrade jet between them (kjet ’ 2).
ower RoT (GT06), the wall jets
ely to the boundaries,
and two
29/11/23
ar in between them. GT12 hasd Ta as GT06, but a lower RoT.

SMITH ET AL.

EPTEMBER

Further experiments were carried out at evenhigher values of V (GT10 and GT11). Because of theextreme deformation of the free surface (Fig. 3b), thelight sheet was not entirely immersed in the fluid, anda full meridional transect of velocities could not betaken. However, the streak images show qualitatively
that the progression to higher numbers of jets seems
to continue with the movement to more turbulent
regimes.

• Sloping bottom and free surface create
topographic b-effect• Large diameter (1m) annulus [Smith et al.
2014]
b. Eddy zonal-mean interaction• Sources of Rossby-like wave in arotating, stratified fluid
As the system moves into the turbulent regime, baroclinic instability produces eddy fluxes that, in zonal• So horizontal momentum flux 𝑢! 𝑣′ ≠ 0• Drives alternating prograde andretrograde mean flows

FDEPS2023

20

FIG. 3. Schematics of the tank geometry. (a) Plan view showing the annular domain, theLaboratory experiments: Jets driven by[baroclinic] instabilities
2282• Baroclinically unstable flow in adifferentially heated, rotating annulus

JOURNAL OF PHYSICAL OCEANOGRAPHYSEPTEMBER 2014

VOLUME 442283

SMITH ET AL.

• Sloping bottom and free surface create
topographic b-effect• Large diameter (1m) annulus [Smith et al. 𝑢
2014]• Sources of Rossby-like wave in arotating, stratified fluid• Horizontal momentum flux 𝑢! 𝑣′ ≠ 0•" $𝜵. 𝑭 ≈# $#𝑢! 𝑣′ drives alternating
prograde and retrograde mean flows

29/11/23

𝜁FIG. 6. Streak images of different regimes. The experiments are arranged from left to right in order of increasing rotation rate V andfrom top to bottom in order of decreasing DT (mirroring the placement of the black boxes in Fig. 1). At high rotation rates (GT10 andGT11), the light sheet penetrated the free surface, causing the overexposed band near the inner wall. (Animated version in supplementarymaterial.)

average, redistribute the momentum to drive the jets

bordering the prograde jets are weaker in magnitude.
Consequently, there is less eddy driving of the retrograde
The velocity fields can be separated into zonally av- jets, which are correspondingly much weaker than in theeraged and residual components
GT07 case. Here, the retrograde jets are only visible aslocal reductions in the mean zonal flow and do not result~u 5 hui 1 u(17) in negative Eulerian velocities. This momentum flux
FIG. 7. Time-averaged fields derived from PIV data plotted for severalprofile
experiments.
(top to bottom)
Speed
(mm
s21), vorticity(s21), andis consistent
with theeddy
drivingmechanismfor
FDEPS2023
21
and kinetic
into time-averaged
perturbation
*.
Even
with
the
considerable
eddy
energy (mm2 s22and). Theexperiments components
are ordered (left tothe
right)
bydecreasing
Ro
and
L
T
numerical experiments ofbPanetta (1993).
meandering visible in the streak animations, one can see the two-jet structure in GT07 and the 3–4-jet structure in GT06.
Even with the decreased velocities of GT12, a pattern
0<KE> (Vallis 2006).

ℛ =0.39

0.061

0.024

0.00960.0061

Laboratory experiments: Jets driven by[baroclinic] instabilities
2284

JOURNAL OF PHYSICAL OCEANOGRAPHYℛ =0.39• Baroclinically unstable flow in adifferentially heated, rotating annulus• Sloping bottom and free surface create
topographic b-effect• Large diameter (1m) annulus [Smith et al.
2014]

0.024

• Sources of Rossby-like wave in arotating, stratified fluid• Horizontal momentum flux 𝑢! 𝑣′ ≠ 0" $• 𝜵. 𝑭 ≈𝑢! 𝑣′ drives alternating# $#prograde and retrograde mean flows

0.0096

FIG. 8. Meridional FFT of time-averaged zonal velocity forseveral experiments. Note how the peak in meridional wavenumber (i.e., number of jets) increases from GT09, to GT07, toGT06, to GT12.

Following Vallis and Maltrud [1993, their Eq. (2.11)],
we plot the transition wavenumbers:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffib/Urms cos3/2 u ,
(22a)
kR
x 5
kR
y 5

29/11/23

VOLUME 44

qffiffiffiffiffiffiffiffiffiffiffiffiffiffib/Urms cos1/2 u sinu ,0.0054(22b)where uFDEPS2023
5 arctan(ky/kx) together with energy (Fig. 12).
The transition represents a rough boundary between
smaller-scale turbulence and larger-scale Rossby wave

22

Laboratory experiments: Moving flames andthermal tides?

• Possible sources of vertically propagatingwaves or tilted convection• So vertical momentum flux 𝑢! 𝑤′ ≠ 0• Drives a retrograde mean flow close to forcing• Inspired by the moving flame experiment[Fultz et al. 1956]?

29/11/23

FDEPS2023

23

in liquid mercury which are four times steady-state flow was established af
which is approximately proportional toas great as the speed of the moving about 5 minutes (about one-third t
the l 5 / 4 power of the inverse Prandtl
flame.
time torrequired
the flamet o conumberPrandtlfornumbers
between
The experimental apparatus (Fig. 1)1 andplete
however,
0 . 1 , a isrotation):ok the orderof unitythefor meconsisted of a cylindrical annulus ofPrandtl
cury nunlbers
was rotating
rapidlyof thesoorder10.'.in arectangular cross section. The bottom rection counter to that of the flamof the channel was an aluminum disk about 4 mm/sec, that it complet
almost two revolutions over the flamin this time.
The motion of a large number
vapor bubbles floating at the mercur
water interface showed that the flo
was uniform over a large portion
the cylindrical annulus. However, imediately above the flame, and insmall wake behind it, the motion w
disorganized. An indication of the v
locity below the surface of the mecury was obtained by observing a b
bearing 1 c n ~in diameter moving
most as fast as the smaller ones (tspeed of the large bearing was witha few percent of that of the smalones). With water as the working flu
velocities that were negligibly sm
compared with the flame speed we
imparted to the liquid.
The high angular velocities observ
in the experiment with liquid mercu
are the resultof therapidity of therm
Fig. 1. Schematic diagram of the apparatus
Prandtlnumberfor the moving flame experiment.
diffusion as compared with viscous d
Flg. 2. Ratio crf the mean velocity of thethermalAurct to the speed of the forcing
FDEPS2023
24
w'lve plotted aga~n\tPlandtl number. Fluidof a cylindrical vessel filled with water.
Within the fluid a net angular moment ~ l mwas established in the sense opposite to the motion of the flame. Stern(3) carried out a similar series of experiments in which the water was confined within a cylindrical annulus inorder to reduce the effects of radialconvection. The water acquired anaverage rotation in a direction counterto that of the flame which was 0.1 to
1.0 percent of the rate of rotation of
the flame. A linearized analysis (4) of
So vertical momentum
flux 𝑢! 𝑤′≠ 0induced in
the two-dimensionalmotionsa horizontal layer of fluid by traveling
Drives a retrogrademean flow close tosinusoidal temperature perturbations
forcing
applied at the boundaries demonstrates
how the Reynolds stress associatedwith fluctuations in the induced velocity supports the mean shear of the fluid.
This analysis is valid only if the speedof the traveling thermal wave is verymuch greater than the mean velocity
ind~lcedin the fl~lid.
These experiments show that a moving source of heat can impart angularmomentum to a fluid in a sense opposite to the motion of the source. However, the fluid rotates with an angularspeed several orders of magnitude
smaller than the rate of rotation of theLaboratory experiments : Moving flames andthermal tides?
• Possible sources of vertically propagatingwaves or tilted convection••• Inspired by the moving flame experiment[Fultz et al. 1956]?
• Extended by Schubert & Whitehead[1969] using mercury as working fluid

29/11/23

3 JANUARY 1969

Laboratory experiments : Moving flames andthermal tides?
• Possible sources of vertically propagatingwaves or tilted convection

Motion due to a moving internal heat source

47 1

A.c. supply
Solid

Segmented electrode• So vertical momentum flux 𝑢! 𝑤′ ≠ 0• Drives a retrograde mean flow close toforcingOutercoolingbath• Inspired by the moving flame experiment[Fultz et al. 1956]?
• Extended by Douglas et al. [1972] to use
a moving internal heat source

-___
I
r=O

I

).=a

zzzB
\
r-b

cooling bath~~~~

Thermally and electricallv
insulating baseSectional view

Plan view

FIGURE
I. Schematic diagram of the apparatus showing the position of the two electrodes.

The fluid was heated by passing an alternating current between the electrodes, and cooledby passing water through the two cooling baths. The working fluid occupies the regiona < r < b and -0-5d < z < 0.5d.

29/11/23

FDEPS2023

2. Apparatus

25

2.1. Convection chamber

Laboratory experiments : Moving flames andthermal tides?
http://journals.cambridge.org• Possible sources of vertically
propagating waves or tilted
convection• So vertical momentum flux 𝑢! 𝑤′ ≠ 0• Drives a retrograde mean flow close
to forcing
Downloaded: 24 Aug 2009

• Inspired by the moving flame
experiment [Fultz et al. 1956]?
• Extended by Douglas et al. [1972]
to use a moving internal heatsource3N

29/11/23

IP address: 131.11• Clearly demonstrates tilted
convection cells

W

FDEPS2023

26

STANDARDSAND SOURCES

Simplified theoretical andnumerical models: thermal tides
from theRIGHT SOURCEFor over twenty years, IPL has
offered an extensive line ofstandards and sources for use
by the nuclear industry.
• Alpha, beta & gamma
standards
Standardized & bulk
solutions• Marinelli beaker standards• Filter paper standards• Multinuclide standards• Krypton, Xenon & Tritium
Gas• Custom designed
standards

Stephen B. Fels

• Idealise a thermal tide as the excitation of internal gravity waves by amoving heat source J’tivity measurements assurance
program conducted by the
National Institute of Standards
and Technology (formerly
NBS), in cooperation with the
U.S. Council for Energy Awareness.

• Fels & Lindzen [1974]

JERRY MAHLMANNOAA Geophysical Fluid Dynamics
Laboratory
Princeton, New Jersey

Harry J. White• Inspired by moving flame experiments of Fultz et al. [1956] and later
work AND observations of Venus’s super-rotation• Aims to explain and quantify:
For more
information
please callor write:

ISOTOPE PRODUCTS
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Harry James White died on 14 November 1988 in Carmel, California.
He was a pioneer in studying theelectrical breakdown in gases, in designing radar and high-voltage equipment and, especially, in investigating
electrostatic precipitation and airpollution-control devices.
White was born on 29 July 1905 in
Fremont, Nebraska. He received a
BS in electrical engineering from the
University of California at Berkeley
in 1928, then pursued graduate workin physics at the same university.
Working with his thesis adviser, Ernest O. Lawrence, White did seminal
research on formative time lags of
spark discharges in gases, using Kerr
cells and transmission lines to measure nanosecond time intervals. He
received his PhD in 1933.
In 1935 White began a productive
and happy relationship with the Research Corporation. (The Research
Corporation is a nonprofit, researchsupporting institution whose income,until 1958, came primarily from the
manufacture and sales of electrostatic precipitators.) His research there
on electrostatic precipitation continued until 1960, interrupted only byWhite's war work at the MIT Radi-• Acceleration of mean zonal flow in opposite direction to movement of heatsource/diurnal heating• Magnitude of acceleration and its dependence on atmospheric parameters
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PHYSICS TODAY

JUNE 1990

16 November 2023 12:00:19

Atmospheric and Oceanic Sciences),and he joined the permanent staff of
the NOAA Geophysical Fluid Dynamics Laboratory in 1974.
Soon after arriving at GFDL Fels
began his pivotal research on atmospheric radiative transfer and its
interaction with dynamical processes. He developed a computercode that calculates with great accuracy the transmission functions foratmospheric infrared radiation. By
considering each spectral line inturn, the code produces atmosphericinfrared cooling rates whose accuracy is limited only by the accuracy
with which the absorption rates forindividual spectral lines are measured in the laboratory. This detailed code is now the fundamental
standard for the worldwide "Intercomparison of Radiation Codes in
Climate Models," an effort of theWorld Climate Research Programme
to compare all the radiative-transfer
codes in the world's major mathematical models of greenhouse-gasinduced climate change.
The physical insight gained fromthe program's detailed results enabled Fels to develop approximation
methods that allowed for accurate
radiative calculations which were1000 times faster than those achievedby the traditional methods. Thesemethods are now incorporated into
weather and climate models at major
research and operational centers in
America and abroad. Most significantly, the radiation code is now
running as part of the Medium-Range
Forecasting Model at the National
Meteorological Center, where it hascontributed significantly to well-recognized improvements in five-day
weather forecasting. The code has
also been incorporated into the operational forecast model at the Austra-lian Numerical Meteorology Research Centre.
Fels studied the interaction of radiative and dynamical processes insuch phenomena as stratospheric circulation, stratospheric climatechange and the Antarctic "ozone
hole." He also investigated the atmospheric circulation of Venus and explored whether its thermal tides wererelated to the strong superrotation
seen in its upper atmosphere.
In addition to doing research, Fels
was a lecturer with the rank ofprofessor at Princeton. His talentand dedication as a teacher gained
him great popularity among the students. He combined an infectious
enthusiasm for science with a passionate insistence on clear and quantitative
thinking.
His quick wit enlivRichard
Lindzenened conversations with students andcolleagues alike.
Those of us who were privileged toknow and work with Stephen Fels are
grateful for the all-too-brief time hewas with us. We miss his intellectual
vitality, his immense scientific andpersonal integrity and his widely
appreciated sense of humor.

27

STANDARDSAND SOURCES

Simplified theoretical andnumerical models: thermal tides
from theRIGHT SOURCEFor over twenty years, IPL has
offered an extensive line ofstandards and sources for use
by the nuclear industry.
• Alpha, beta & gamma
standards
Standardized & bulk
solutions• Marinelli beaker standards• Filter paper standards• Multinuclide standards• Krypton, Xenon & Tritium
Gas• Custom designed
standards• Generalisation of EP theorems• cf Andrews & McIntyre 1976,1978]

tivity measurements assurance
program conducted by the
National Institute of Standards
and Technology (formerly
NBS), in cooperation with the
U.S. Council for Energy Awareness.
For more
information
please callor write:

ISOTOPE PRODUCTS
LABORATORIES
1800 N. Keystone Street
Burbank, CA91504(818)843-7000
FAX (818) 843-6168
Telex 312429 IPL UD$• Outside source region
𝜌' 𝑢& 𝑤 & = 0 but momentum$(
has to come from somewhere – i.e. from 𝑢2 !
• So within the source region, ⟹ eastward c wave produces
westward acceleration of 𝑢2 (and vice-versa)

Our valued distributors:

TAIWANI INITPn

Lintek07 7090387
Telex: 22317 FLYLIN
John Caunt

Scientific
UNIIEUQ 8 6 5 880 4 7 9KINGDOM Telex: 838852 JCS G

29/11/23

FDEPS2023

HONG
KONG

Patrick Trading
5 266601
Telex: 75472 PTCOR HX

SWEDEN

Alnor Instruments
46 155 68050
Telex: 64211 ALNOR S

Circle number 57 on Reader Service Card

94

PHYSICS TODAY

JUNE 1990

JERRY MAHLMANNOAA Geophysical Fluid Dynamics
Laboratory
Princeton, New Jersey

Harry J. White

16 November 2023 12:00:19

𝑤 & 𝑝& = −𝜌' 𝑢2 − 𝑐 𝑢& 𝑤′• 𝑤 & 𝑝& ~ Vertical energy flux
• A wave with eastward c (relative to 𝑢2 ) carries eastward
momentum upwards
𝜕𝑢2
𝜕
𝜅𝜌'
&
&
−𝜌'
=
𝜌' 𝑢 𝑤 = −𝐷 & 𝐽′𝜕𝑡 𝜕𝑧𝑢2 − 𝑐• A thermally excited, propagating IGW will carry energy
away from the source region

Stephen B. Fels

Atmospheric and Oceanic Sciences),and he joined the permanent staff of
the NOAA Geophysical Fluid Dynamics Laboratory in 1974.
Soon after arriving at GFDL Fels
began his pivotal research on atmospheric radiative transfer and its
interaction with dynamical processes. He developed a computercode that calculates with great accuracy the transmission functions foratmospheric infrared radiation. By
considering each spectral line inturn, the code produces atmosphericinfrared cooling rates whose accuracy is limited only by the accuracy
with which the absorption rates forindividual spectral lines are measured in the laboratory. This detailed code is now the fundamental
standard for the worldwide "Intercomparison of Radiation Codes in
Climate Models," an effort of theWorld Climate Research Programme
to compare all the radiative-transfer
codes in the world's major mathematical models of greenhouse-gasinduced climate change.
The physical insight gained fromthe program's detailed results enabled Fels to develop approximation
methods that allowed for accurate
radiative calculations which were1000 times faster than those achievedby the traditional methods. Thesemethods are now incorporated into
weather and climate models at major
research and operational centers in
America and abroad. Most significantly, the radiation code is now
running as part of the Medium-Range
Forecasting Model at the National
Meteorological Center, where it hascontributed significantly to well-recognized improvements in five-day
weather forecasting. The code has
also been incorporated into the operational forecast model at the Austra-lian Numerical Meteorology Research Centre.
Fels studied the interaction of radiative and dynamical processes insuch phenomena as stratospheric circulation, stratospheric climatechange and the Antarctic "ozone
hole." He also investigated the atmospheric circulation of Venus and explored whether its thermal tides wererelated to the strong superrotation
seen in its upper atmosphere.
In addition to doing research, Fels
was a lecturer with the rank ofprofessor at Princeton. His talentand dedication as a teacher gained
him great popularity among the students. He combined an infectious
enthusiasm for science with a passionate insistence on clear and quantitative
thinking.
His quick wit enlivRichard
Lindzenened conversations with students andcolleagues alike.
Those of us who were privileged toknow and work with Stephen Fels are
grateful for the all-too-brief time hewas with us. We miss his intellectual
vitality, his immense scientific andpersonal integrity and his widely
appreciated sense of humor.

Harry James White died on 14 November 1988 in Carmel, California.
He was a pioneer in studying theelectrical breakdown in gases, in designing radar and high-voltage equipment and, especially, in investigating
electrostatic precipitation and airpollution-control devices.
White was born on 29 July 1905 in
Fremont, Nebraska. He received a
BS in electrical engineering from the
University of California at Berkeley
in 1928, then pursued graduate workin physics at the same university.
Working with his thesis adviser, Ernest O. Lawrence, White did seminal
research on formative time lags of
spark discharges in gases, using Kerr
cells and transmission lines to measure nanosecond time intervals. He
received his PhD in 1933.
In 1935 White began a productive
and happy relationship with the Research Corporation. (The Research
Corporation is a nonprofit, researchsupporting institution whose income,until 1958, came primarily from the
manufacture and sales of electrostatic precipitators.) His research there
on electrostatic precipitation continued until 1960, interrupted only byWhite's war work at the MIT Radi-

28

STANDARDSAND SOURCES

Simplified theoretical andnumerical models: thermal tides
from theRIGHT SOURCEFor over twenty years, IPL has
offered an extensive line ofstandards and sources for use
by the nuclear industry.
• Alpha, beta & gamma
standards
Standardized & bulk
solutions• Marinelli beaker standards• Filter paper standards• Multinuclide standards• Krypton, Xenon & Tritium
Gas• Custom designed
standards

Heating/cooling rate
tivity measurements assurance
program conducted by the
National Institute of Standards
and Technology (formerly
NBS), in cooperation with the
U.S. Council for Energy Awareness.

• Where 𝒫 is power input per m2 and 𝜆/ isvertical wavelength• Sign of acceleration depends on heating or
cooling• Example inspired by Venus conditions

For more
information
please callor write:

ISOTOPE PRODUCTS
LABORATORIES
1800 N. Keystone Street
Burbank, CA91504(818)843-7000
FAX (818) 843-6168
Telex 312429 IPL UD
Our valued distributors:
TAIWANI INITPn

Lintek07 7090387
Telex: 22317 FLYLIN
John Caunt

Scientific
UNIIEUQ 8 6 5 880 4 7 9KINGDOM Telex: 838852 JCS G

29/11/23

FDEPS2023

HONG
KONG

Patrick Trading
5 266601
Telex: 75472 PTCOR HX

SWEDEN

Alnor Instruments
46 155 68050
Telex: 64211 ALNOR S

Circle number 57 on Reader Service Card

94

PHYSICS TODAY

JUNE 1990

𝑢0 (𝑡)JERRY MAHLMANNOAA Geophysical Fluid Dynamics
Laboratory
Princeton, New Jersey

Harry J. White

16 November 2023 12:00:19• For a finite thickness layer (depth 𝐻8 )
𝜕𝑢)
𝒫𝜆9
≈
𝜕𝑡
2𝐻8

Stephen B. Fels

Atmospheric and Oceanic Sciences),and he joined the permanent staff of
the NOAA Geophysical Fluid Dynamics Laboratory in 1974.
Soon after arriving at GFDL Fels
began his pivotal research on atmospheric radiative transfer and its
interaction with dynamical processes. He developed a computercode that calculates with great accuracy the transmission functions foratmospheric infrared radiation. By
considering each spectral line inturn, the code produces atmosphericinfrared cooling rates whose accuracy is limited only by the accuracy
with which the absorption rates forindividual spectral lines are measured in the laboratory. This detailed code is now the fundamental
standard for the worldwide "Intercomparison of Radiation Codes in
Climate Models," an effort of theWorld Climate Research Programme
to compare all the radiative-transfer
codes in the world's major mathematical models of greenhouse-gasinduced climate change.
The physical insight gained fromthe program's detailed results enabled Fels to develop approximation
methods that allowed for accurate
radiative calculations which were1000 times faster than those achievedby the traditional methods. Thesemethods are now incorporated into
weather and climate models at major
research and operational centers in
America and abroad. Most significantly, the radiation code is now
running as part of the Medium-Range
Forecasting Model at the National
Meteorological Center, where it hascontributed significantly to well-recognized improvements in five-day
weather forecasting. The code has
also been incorporated into the operational forecast model at the Austra-lian Numerical Meteorology Research Centre.
Fels studied the interaction of radiative and dynamical processes insuch phenomena as stratospheric circulation, stratospheric climatechange and the Antarctic "ozone
hole." He also investigated the atmospheric circulation of Venus and explored whether its thermal tides wererelated to the strong superrotation
seen in its upper atmosphere.
In addition to doing research, Fels
was a lecturer with the rank ofprofessor at Princeton. His talentand dedication as a teacher gained
him great popularity among the students. He combined an infectious
enthusiasm for science with a passionate insistence on clear and quantitative
thinking.
His quick wit enlivRichard
Lindzenened conversations with students andcolleagues alike.
Those of us who were privileged toknow and work with Stephen Fels are
grateful for the all-too-brief time hewas with us. We miss his intellectual
vitality, his immense scientific andpersonal integrity and his widely
appreciated sense of humor.

Harry James White died on 14 November 1988 in Carmel, California.
He was a pioneer in studying theelectrical breakdown in gases, in designing radar and high-voltage equipment and, especially, in investigating
electrostatic precipitation and airpollution-control devices.
White was born on 29 July 1905 in
Fremont, Nebraska. He received a
BS in electrical engineering from the
University of California at Berkeley
in 1928, then pursued graduate workin physics at the same university.
Working with his thesis adviser, Ernest O. Lawrence, White did seminal
research on formative time lags of
spark discharges in gases, using Kerr
cells and transmission lines to measure nanosecond time intervals. He
received his PhD in 1933.
In 1935 White began a productive
and happy relationship with the Research Corporation. (The Research
Corporation is a nonprofit, researchsupporting institution whose income,until 1958, came primarily from the
manufacture and sales of electrostatic precipitators.) His research there
on electrostatic precipitation continued until 1960, interrupted only byWhite's war work at the MIT Radi-

29

Simplified/intermediate complexity GCMs
• Basic Primitive Equation dynamical core
• Numerically solves full (shallow) equations for conservationof momentum, energy and mass
• Moderate resolution [7.5ox7.5o – 1ox1o], 10-25 levels• Simple diabatic forcing#$• Linear relaxation to prescribed 𝑇!" 𝜑, 𝑧 ; #% =• Semi-gray (2-band) radiative transfer$!" &,( )$
*#$%

OR

• Dry convective adjustment• WEAK diffusion or hyperdiffusion• Simplified boundary conditions• Thermally insulating surface• Globally uniform - no topography• Linear frictional drag at surface [Time constant tfr]

29/11/23

FDEPS2023[Cf Earth in
Perpetual equinox][Wang et al. 2018]30

Simplified/intermediate complexity GCMs
• A long-established tool for atmospheric modellers• Since the early 1970s for other planets• For Venus?
• Early models produced only very weak super-rotation under
Venus-like conditions (deep atmosphere, slow rotation)• E.g. Rossow [1980]• DelGenio & Suozzo [1987]• Possible reasons?
• Too diffusive?
• Vertical diffusion and/or convective momentum transport
too strong?
• No diurnal cycle….?
• AM non-conservation by numerical schemes….?

29/11/23

FDEPS2023

31

Simplified/intermediate complexity
GCMs: A cautionary tale
• Venus SGCM intercomparison[Lebonnois et al. 2014]• 6 different models with total of 8
different dynamical cores• Includes spectral and finitedifference cores• Similar (Venus-like) conditions• Same Teq fields and trad
• Similar resolutions (not identical)• Linear surface drag29/11/23

FDEPS2023

32

Simplified/intermediate complexity
GCMs: A cautionary tale𝑢)

• Venus SGCM intercomparison[Lebonnois et al. 2014]• 6 different models with total of 8
different dynamical cores• Includes spectral and finitedifference cores• Very different results!
• LH = spectral models• RH = FD models• Last line = finite volume schemes
29/11/23

FDEPS2023

33

Simplified/intermediate complexity
GCMs: A cautionary tale𝑇) − 𝑇4(𝑧)• Venus SGCM intercomparison[Lebonnois et al. 2014]• 6 different models with total of 8
different dynamical cores• Includes spectral and finitedifference cores• Very different results!
• LH = spectral models• RH = FD models• Last line = finite volume schemes
29/11/23

FDEPS2023

34

Simplified/intermediate complexity
GCMs: A cautionary tale0Ψ• Venus SGCM intercomparison[Lebonnois et al. 2014]• 6 different models with total of 8
different dynamical cores• Includes spectral and finitedifference cores• Very different results!
• LH = spectral models• RH = FD models• Last line = finite volume schemes
29/11/23

FDEPS2023

35

Simplified/intermediate complexity GCMs: A
cautionary tale
Conclusions• Numerical models are not perfect physical analogues of atmosphericcirculation – especially for slow rotators• Results may be sensitive to resolution, choice of dynamical core,numerical advection schemes and other factors• Deep atmospheres surrounding slowly rotating planets are especiallychallenging because of sensitivity to weak sources or sinks of AM,
implicit numerical diffusion etc.
• Results may need to be treated with caution and/or tested forrobustness against changes in resolution and other factors.
29/11/23

FDEPS2023

36

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Although particular simulations may not be perfectly accurate, they
may be useful to explore trends in atmospheric circulation at different
values of key parameters• Explore possible scaling laws and other trends in heat and angularmomentum transfer e.g. as a function of planetary rotation rate andrelated parameters• Several recent studies have attempted this. Here we look at someresults using the University of Hamburg PUMA and University of
Exeter ISCA models• Wang et al. [2018 QJRMS]• Lewis et al. [2021 JAS]

29/11/23

FDEPS2023

37

Exploring parameter space with asimple [3D] climate model• Pseudo-spectral dynamical core - PUMA [Univ. ofHamburg] or ISCA [Univ. Exeter]••

Solves hydrostatic Primitive Equations for conservation of momentum,
energy and mass using spherical harmonics in horizontal, FD in vertical
T21-T170 [7.5ox7.5o – 1ox1o], 10-25 levels• Flat surface (no topography)• Simple radiative forcing••

Linear relaxation to specified T(f,z) OR
Semi-gray (2-band) radiative transfer• Linear frictional drag at surface•

Time constant tfr[Cf Earth in
Perpetual equinox]

1
Ω
≤
≤8
256 Ω!

• Vary W, trad or tfr• No moisture or oceans
29/11/23⊕
FDEPS2023[Wang et al. 2018]38

Slowly rotating
Planets (W<WE)

29/11/23

FDEPS2023

39

Planetary circulation regimes:
fast rotators[Yixiong Wang 2010]

Rapidly
Rotating
Planets(W>WE)

29/11/23

FDEPS2023

40c) ⌦ = 2k) ⌦ = 2

⇤

⇤
l)
l) ⌦
⌦⇤ =
= 11

Trends in circulation structure: decreasing
Ω
d) ⌦
d)
⌦⇤ =
=1
1

Ω = ΩEe) ⌦⇤ = 12

= 1212m) ⌦⇤⇤ =Ω = ΩE/2f ) ⌦⇤⇤ = 114f) ⌦ = 4n) ⌦⇤ = 1n) ⌦⇤ = 144

Ω = ΩE/4g) ⌦⇤ = 18

29/11/23o) ⌦⇤ = 18g) ⌦⇤ = 18o) ⌦⇤ = 181h) ⌦⇤ = 161p) ⌦⇤ = 161h) ⌦⇤ = 161p) ⌦⇤ = 16

FDEPS2023

Ω = ΩE/8
Ω = ΩE/16
41

Trends in circulation structure: increasing ΩΩ = 8ΩE

Ω = 4ΩE

Ω = 2ΩEΩ = ΩE
29/11/23

FDEPS2023

42

Order according to key planetaryparameters defining circulation regimes?
• Thermal Rossby [R.Hide] number [-Ratio of forces]<!
?@A" B
DA" B
•ℛ=
≈
= $ $
=>
B# =$ C$
= C

• 𝐻=

%&
; atmospheric scale height'• Burger number [-Ratio of lengthscales]
>$&
DA% B
• ℬ = E=$ C$ = C$

• 𝐿( =

29/11/23)*; Rossby deformation radius+

FDEPS2023

43

Order according to key planetaryparameters defining circulation regimes?
• Damping/dissipation parameters [Ratios of timescales]• Frictional Ekman number 𝐸FG = Ω𝜏FG
• 𝜏+# = 𝜏(#2'H(; [cf 𝐸 = Ω𝜏,-./ 01]*; [“spindown timescale”; Valdes & Hoskins 1988]
3+,

(I
J')*
• Radiative damping parameter 𝒜C =
=
=J'(&
J'(&

• Radiative timescale 𝜏#2( ≈/- 4.

; [transmissivity 𝜖]

1 (107)
5'&/00H(• OR “Taylor numbers”, ℑFG,GCL ~ 𝐸FG, 𝒜CHE• Others…….?

29/11/23

FDEPS2023

44

Schematic regime diagramℛ ∝ Δ𝜃MN / Ω𝑅 ([w.r.t. dimensionless parameters]

29/11/23

Neptune∝ (Ω𝜏F )2

FDEPS2023

45

Planetary parameters

29/11/23

Cf

W/W*

Q

NJ

4W2tR2

1/16

20

0.04

62

1/8

5

0.07

247

1/4

1.3

0.14

992

1/2

0.32

0.28

4000

1

0.08

1.57

1.5e4

Earth

2

0.02

3.1

6.3e4

Uranus &Neptune?

4

0.005

6.3

2.5e5

8

0.001

14.5

1.0e6

FDEPS2023

Titan

Mars[?]

Saturn &Jupiter46

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Hadley circulation width ( 𝜑B )0 )• and strength ( Ψ• Width fH scales with ℛ, much aspredicted by HH80

φH (degs)

10

kg s-1• Strength 𝜑@ also scales with ℛ

Ψ| x 10• ~ℛ "/1 for small ℛ• ~whole planet for large ℛ• ~ℛ 1/: − ℛ ;/: for ℛ ≷ 1Ω∗

29/11/23

FDEPS2023

47

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Peak equator-pole heat transport• Total heat transport ~ constant forℛ > 1/2 then decreases for smallerℛ• Zonally symmetric overturning
contribution decreases withincreasing Ω∗• ~ℛ +, for small ℛ• Dominant for ℛ > 1• Leads to weak horizontal T gradients• Eddy contribution peaks at ℛ ≈ 0.3• ~negligible for ℛ > 10
• Dominates for ℛ < 0.1• Baroclinic instabilities
29/11/23

FDEPS2023Ω∗

48

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Global super-rotation S• 3D simulations stay close to the
HH80 theory except aroundℛ~10 − 100• 𝑢) in gradient wind balance but
doesn’t quite reach full
cyclostrophic conditions

29/11/23

FDEPS2023

49

Ω*

Zonal wind fields (Isca)

1/16• Earth-sized planet• Held-Suarez relaxation forcing• pS = 1 bar
Ω*
1

1/64

1/4

1/256

29/11/23

FDEPS2023

50

Local super-rotation
fields slocal (Isca)

Ω*
1/16

Ω*
1

1/64

1/4

1/256

29/11/23

FDEPS2023

51

Mean meridional mass
streamfunction (Isca)

Ω*
1/16

Ω*
1

1/64

1/4

1/256

29/11/23

FDEPS2023

52

EP Fluxes(Isca)
Ω*

Ω*

1

1/16

Dominated by
Baroclinic (Rossby) waves
at mid-latitudes
Dominated by
Equatorial Kelvin/Rossby waves

1/64

1/4

29/11/23

FDEPS2023

53

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Some models are more
successful at reproducing strongsuper-rotation at Venus-likerotation rates• E.g. Yamamoto & Takahashi [2003]• Strong 𝑢O 𝑣′ equatorward
momentum fluxes maintain ~100m s-1 equatorial winds• Dominant wave modes include
Rossby, Kelvin and gravity waves• Role of equatorial Kelvin waves?
29/11/23

FDEPS2023

54

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Mitchell & Vallis [2010] noted that
Kelvin waves were only present
during spin-up of super-rotation• Equlibrated state was dominated by
Rossby and MRG waves with only
weak Kelvin modes – mechanism…?

• Barotropic Rossby-Kelvin instability
identified by Peng & Mitchell[2014]• Formed by a resonant interaction
between a pair of linear equatorial
Kelvin and Rossby waves• Doppler-shifted by zonal wind

29/11/23

FDEPS2023

55

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling• Mitchell & Vallis [2010] noted that
Kelvin waves were only present
during spin-up of super-rotation• Equlibrated state was dominated by
Rossby and MRG waves with only
weak Kelvin modes – mechanism…?

• Barotropic Rossby-Kelvin instability
identified by Peng & Mitchell[2014]• Formed by a resonant interaction
between a pair of linear equatorial
Kelvin and Rossby waves• Doppler-shifted by zonal wind
• Instability depends on reversal of PV
gradients and on Burger no. ℬ and
29/11/23
Froude number

FDEPS2023

56

Simplified/intermediate complexity GCMs:
Parameter sweeps and scaling – final remarks• Wave-zonal flow acceleration depends on wave dissipation, forcing ortransience• Many possible scenarios• Acceleration of 𝑢2 in direction of c for dissipating waves• Critical layer absorption limits 𝑢2 to ~c for dissipating waves• Acceleration of 𝑢2 in opposite direction to c for forced waves (e.g. thermal tides)• Strong super-rotation possible with forced waves….

• Many of these mechanisms can be (and have been) demonstrated in lab
experiments• Simple GCMs show some clear trends• But numerics need to be interpreted with caution, especially for slow rotators
29/11/23

FDEPS2023

57