Download Prograde and Retrograde Motion in a Fluid Layer

VOL. 76, NO. 9
JOURNAL
OF GEOPHYSICAL
RESEARCH
MARCH 20, 1971
Prograde
andRetrograde
Motionin a FluidLayer'
Consequences
for ThermalDiffusion
in theVenusAtmosphere
GERALD SCHUBERT AND i•ICHARD •E. YOUNG
Department of Planetary and Space Science
University o] California, Los Angeles 90024
JOHN I-IIN CH
Department o] Applied Mathematics and Theoretical Physics
University o• Cambridge, Cambridge, England
Depending on the value of the Prandtl number, the average velocity imparted to a layer
of Boussinesqfluid by traveling thermal waves applied at the upper free surfaceis found, in
the linear case,to be either in the sameor oppositedirectionas that of the moving thermal
source.Since the mean flow is in the opposite direction only when the Prandtl number is
small, the 4-day retrograde zonal motion of the Venus atmospheremay be evidencethat the
effectivePrandtl numberof the upper atmosphereis muchlessthan unity.
It has beenproposedthat the observed4-day ity). The value of Prandtl number Po at which
retrograde circulationof the Venus atmosphere the mean velocity at the upper free surface
is a zonal motion of at least the upper atmos- changesfrom progradeto retrogradeis a funcphere driven by periodicsolar thermal forcing tion of the frequencyparameterS = •oh•'/v(o•
[Schubertand Whitehead,1969; Schubertand is the circular frequencyof the thermal forcing
Younq, 1970]. If this mechanismis indeedop- and h is the depth of the fluid layer). For P <
erative in the Venus atmosphere,the analysis Po(S) the mean velocity at the upper free surof the motionsinducedin a fluid layer by mov- face is retrograde,whereasfor P > Po(S) it is
ing thermal sourcescan provideinformationon prograde.In the limit S << 1, Po • 0.2. As S
the diffusivepropertiesof the atmosphere.
becomeslarge, the changein characterof the
In this paper we considerthe flow induced flow occurs at smaller values of the Prandtl
by traveling thermal wavesin a layer of Bous- number,Po(S) cc S-•/•'for S >> 1; at S - 100
sinesqfluid boundedaboveand belowby hori- (an estimatefor the Venus atmosphere,Schuzontal free and rigid surfaces,respectively.The bert and Young [1970]) Po • 0.025 for heat
thermal boundary conditionswill be of two flux boundaryconditions.Theseresultsindicate
types: prescribedtemperature fluctuationsat that the effective Prandtl number for at least
the free surfaceand fixed temperatureat the the upper regionsof the Venus atmosphereis
rigid surface; prescribedfluctuationsin heat muchlessthan unity.
flux at the free surfaceand a thermally insulatANALYSIS OF INDUCED MEAN MOTION
ing rigid surface.The heat flux boundary conditions have previously been consideredby
Consider the two-dimensional motion of a
Malkus [1970].
layer of Boussinesq
fluid definedby horizontal
The major result of our investigationis that planesseparatedby a distanceh. The linearized
the average motion of the fluid is either pro- equationsof motion and energyhave been congrade (in the direction of the moving thermal sidered in a number of related investigations
wave) or retrogradedependingon the magni- [Stern, 1959; Davey, 1967; Schubert,1969;
tude of the Prandtl number P = v/• (v is the Kelly and Vreeman,1970;Malkus,1970;Whitekinematicviscosityand r is the thermal diffusiv- head, 1970; Hinch and Schubert,1971]. The
followingare the pertinent equationsin this disCopyright ¸
1971 by the American GeophysicalUnion.
cussion:
2126
D•FFUS•0N •N THE VENUS ATMOSPHERE
(1)
is Oz=
2127
The general
solutions
for • and • areeasily
obtainablebut algebraicallycomplicated.Since
our main interest here is in the mean motion
(u), we presentthe resultsfor (u) at the upper
(2) free
surface ((u)(1)) and for the vertically
O• iSP•
(•2 --
averagedmean velocity(u•,g) in the limiting
dz= 2••'Imaginary
• &]
casesS, SP << 1, S, SP >> 1, and S-an, SP << 1.
(3) For the mean velocityat the upper free surface
where z is the •mensionless ve•ical coordinate
and for the vertically averagedmean velocity,
(z -- 0 is the lower•gid surfaceandz =. 1 is we find, in the limit $, SP << 1
theupperfreesurface),
• isthequantitykh (k
is the wave number of the the•al
forcing) as-
544P)
(u)(1)
• F•'S4(93(5!)•.(4!)
(7!)
7
sumedto be muchlessthanunity,(u}is the •mensionless
horizontal velocity averagedover a
wavelength,
and the asteriskdenotesthe complexconjugate.
The fluctuations
in the d•ensionlesstemperatureand vertical velocity are
•ven by
(6)
14!(9•7
'5 p) (7)
F•'S4
8143
with temperatureboundaryconditionsand
{i(x +
}]
FeS'(9- 56P)
respectively,
wherex isthehorizontal
coordinate.
(u)(1)
• 14(5!)e(4!)e(S•.p•.
q_f]4) (8)
The velocities are dimensionlesswith respect to
U = •/k, thespeed
ofthetherealwavetraveling
3F•'S4•9-- 46P)
in the negativex direction.The complexam-
<u•.v,>
• 16(11l)(S2p2
4-f]4)
plitudeof'the temperature
fluctuationis d•en-
(9)
sionless
with respect
to •,, the realamplitude
In obtainof the giventemperaturefluctuationat the free with heat flux boundaryconditions.
ing
equations
8
and
9
it
is
important
to replace
surfacewhen temperatureboundaw conditions
a•/az
•'
in
equations
i
and
2
by
a2/OF'
-areapplied.Whenheatfluxboundaryconditions
are used,• is dimensionless
with respectto (u) (1) and (u..,) for the heat flux boundary
conditionswould otherwisebehavein a singular
•h/K, where
• istherealamplitude
ofthegiven manner as SP .--) O.
periodicheatflux at the upperfree surfaceand
In the low-frequency
limit S, SP (( i both
the
mean
velocity
at
the
upperfreesurfaceand
parameter
F equals
g•,/U', for temperathe
bounda• conditions,
or (gh/U') (•ah/K)for the verticallyaveragedmean velocitycan be
on the
heat flux boundaryconditions,wherea is the eitherprogradeor retrogradedepending
K is the the•al
conductivity. Finally the
coc•cient of the•al
expansionand g is the
value of the Prandtl number. For the tempera-
acceleration
of gravity.Equations1-3 are solved ture boundaryconditions,(u)(1) is prograde
for each of the followingsetsof bo•dary con- whenP • 0.171(Po-- 0.171)and(u•.•)is pro-
gradewhenP • 0.291.Thusthereisa rangeof
ditions'
d•
• - dz- (u)= • = 0 atz = 0
- -- dz
d•2 -w
O,•
=
1
(4)
atz= 1
and
d•
dT
• - dz- (u)- dz= 0 atz = 0
d•'•
ary conditions.
In the high-frequency
limit S, SP )) i we
d•
t• -- dz"- O,•z = 1
(5)
Prandtl numbers,from approximately0.171 to
about0.291,whenthe verticallyaveragedmean
flow is retrogradewhile the velocity at the
upper free surfaceis prograde.For the heat
flux boundary conditions(u)(1) and
changefrom progradeto retrogradeat P
0.161 and P •. 0.196, respectively.Thus the
qualitativenature of theseresultsappearsnot
to be effectedby the particularthermalbound-
atz = 1
find
2128
SCHUBERT,
YOUNG,AND I-IINCH
F2(P2 -F 1)
4S(P- 1)•P•
1.0-
4P•/2
ß{(1•- P2)(1
•- P)-- 1} (10)
(u>x IO
• P=O.03
.01
.8-
F•
<u,,.g>
• 4(2)•/2Sa/•(p
_ 1)(P
+ 1)•P
•/•
ß{(7P + 3) -- pl/2(2p3+ 6P2 + P +
(11)
for the temperature boundary conditions,while
for the heat flux boundary conditions (u)(1)
o
-.oo4
t
-.oo2
0
.002
Both(u)(1) and(u,•) are•lwaysprograde
in
.008
I
.01
Fig. 1. Average velocity profiles with heat
flux boundary conditions for P -- 0.01 and P -0.03 at S -- 100, F -- 0.3.
the limit S, SP )) 1. However, for P sufficiently small, when SP is not large, the flow
will be retrograde.This may be seenby investigatingthe limit S-•', SP (( 1, wherein
fs (120
)
•(2)
[8820
(•avg)
• F2
7•
6•1/2
•1/2
-- 41SP
.006
<u>
and (u,•) are givenby the productsof 1/SP
with equations10 and 11, respectively(with F
properly interpreted).
.004
FeS
8!(s'ip+
ß(1260
•S1/2--13(2)
1/2
SP) (15)
(12)for heat flux boundaryconditions(the remarks
which follow equations8 and 9 are pertinent
(13) to the resultsobtainedin equations14 and 15).
The dependenceof Po on S is seenfrom equation 12 to be Po • 84.85 S-8•' for temperature
boundary conditionsand from equation14 to
be Po • 42.43 S-8•'for heat flux boundarycon2(2)
+
ditions. The vertically averagedmean velocity
changesfrom progradeto retrogradeat P •
ß S•-- (2) SP (14) 215.12 S-• for temperatureboundary conditions and P • 68.53 S-• for heat flux boundary
conditions.Thus we see again that the mass
TABLE
1. The Values of Prandtl
Number
at
flow may be retrogradewhile the mean velocity
Which (u)(1) Changesfrom Progradeto Retrograde
at the upper free surfaceis prograde.
for Heat Flux Boundary Conditions
We have numerically computedthe solution
Pc
P ,,(S-m,
of equations1-3 subject to the conditionsgiven
S
(numerical)
Pc(S ((1)
SP ((1)
in (5) and have determined Po($). These
values of' P, below which (u)(1) is retrograde
0.161
0.161
o
are given in Table I as a function of S. The
1
0.161
0.161
critical values of P, computedfrom the asymp0.147
0.161
lO
for temperatureboundary conditionsand
20
30
50
lOO
200
500
600
700
800
0.119
0.0924
0.0576
0.0252
0.0103
0.00298
0.00225
0.00185
0.00167
0.161
0.161
0.161
0.161
0.161
0.161
0.161
0.161
0.161
0.474
0.258
0.120
0.0424
0.015
0.00379
0.00289
0.00229
0.00188
totic formulas
8 and 14 are also listed in the
table. We note in particular that for $ -- 100
(an estimate for the Venus atmosphere)(u) (1)
is progradeor retrogradeaccordingto whether
P • or • 0.025,respectively.
Twomeanvelocity profiles,illustratingthe nature of the flows
when the motion at the upper surfaceis either
progradeor retrograde,are shownin Figure I
DIFFUSION
IN THE VENUS ATMOSPHERE
2129
for the case of heat flux boundary conditions describedby an effectiveeddy viscosity,a small
with
S --
100 and F
---- 0.3. In
the linear
problem we note that (u) scaleswith F •. The
formulasgivenby Malkus [1970] for (u)(1) in
Prandtl
number
indicates
that
thermal
diffu-
sion is causedby someprocessother than turbulence. In the upper atmosphereof Venus
heat transport is thus likely to be radiative.
Avduevsky et al. [1970a] have concludedthat
the limits of small and large frequencyfor the
heat flux boundary conditions(equations(2.20)
and (2.21) in that paper) are incorrect,apparently becauseof algebraicerrorsincurredin the
processof findingthe limiting formsof the gen-
the thermal
flow occur.
grademeanflowfor S •- 100.
balance above altitudes
of 40-50
km is radiative, whereasat lower altitudes heat
transport must be convective.Let us then aseral solution.
sume that the upward traveling long-wave
The physicalmechanismresponsible
for these radiative flux required to balancethe incoming
progradeand retrogradeflowsis the tilting of solar radiation can be described in terms of an
the traveling convectioncells. As can be seen effective thermal conductivity and a temperafrom the precedingresults,the downwarddiffu- ture gradient [Goody and Robinso.n,1966]. On
sionof the thermal field producesa tilt of these the basis of the data from Veneras 4, 5, and 6,
cells in the direction of the traveling thermal Avduevskyet al. [1970a, b] give a value of the
wave that corresponds
to a mean upward trans- temperature gradient approximately equal to
port of prograde momentum.Viscousdiffusion 7 X 10-5 øK/cm, densitiesthat range between
from the lower rigid surfaceyields a tilt of the 2 X 10-8 and 1.2 X 10-' g/cm8 betweenaltitudes
convection cells in the oppsite sense corre- of 50 and 70 km, and a value of the upward
spondingto a mean upward transport of retro- radiative flux above 50 km of about 150
grade momentum.The role of thermal diffusion watts/m•. Thesevaluesyield a radiative thermal
diffusivity that rangesbetween 10• to 2 X 106
is to produceprogrademean flow, whereasthat
of viscousdiffusionleads to retrograde mean cm'/sec.If we usea value for the eddy viscosity
of 104 cm•/sec [Goody and Robinson, 1966],
flow. Only where heat is well diffused(P << 1)
and where there is relatively little thermal we get a Prandtl number ranging from 0.1 to
tilting of the convectioncells can retrograde 0.005, which is in the range that gives retroHEAT AND MOMENTUM
TRANSPORT IN THE
VENUS ATMOSPHERE
Acknowledgment. Gerald Schubert and Richard
E. Young acknowledgesupport under NSF grant
GA
10167.
The resultsof the precedingdiscussion
show
that for a given frequency parameter either
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Rozhdestvensky, A tentative model of the
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coeffi-
1970a.
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also note
that
the
two
sets of
29, 137-150, 1967.
Goody, R. M., and A. R. Robinson,A discussion
boundary conditionsusedin the analysisled to
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2130
SCHUBERT,
YOUNG,AND I-IINCH
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