It is well known that prograde equatorial jet stresses cannot occur in an axisymmetric inviscid fluid, owing to the constraints of local angular momentum conservation. For a viscous fluid, the constraints of mass conservation prevent the formation of any local maximum of absolute angular momentum (m) without a means of transferring m against its gradient (delta m) in the meridional plane. The circumstances under which m can be diffused up-gradient by normal molecular viscosity are derived, and illustrated with reference to numerical simulations of axisymmetric flows in a cylindrical annulus. Viscosity is shown to act so as to tend to expel m from the interior outwards from the rotation axis. Such an effect can produce local super-rotation even in a mechanically isolated fluid. The tendency of viscosity to result in the expulsion of m is shown to be analogous in certain respects to a vorticity-mixing hypothesis for the effects of non-axisymmetric eddies of the zonally-averaged flow. It is shown how the advective and diffusive transport of m by non-axisymmetric eddies can be represented by the Transformed Eulerian Mean meridional circulation and the Eliassen-Palm (EP) flux of Andrews and McIntyre respectively, in the zonal mean. Constraints on the form and direction of the EP flux in an advective/diffusive flow for such eddies are derived, by analogy with similar constraints on the diffusive flux of m due to viscosity

Read, P. L.

English

NASA Technical Reports Server

N87-17635
TRANSPORT OF ABSOLUTE ANGULAR MOMENTUM IN QUASI-AXISYMMETRIC
EQUATORIAL JET STREAMS
P. L. Read
Meteorological Office, U. K.
It is well known that prograde equatorial jet streams cannot occur in
an axisymmetric inviscid fluid, owing to the constraints of local
angular momentum conservation ("Hide's theorem"). For a viscous
fluid, the constraints of mass conservation prevent the formation of
any local maximum of absolute angular momentum m without a means of
transferring m against its gradient Vm in the meridional plane (e.g.,
Held and Hou, 1980, J. Atmos. Sci. 37, 515-533.) The circumstances
under which m can be diffused up-gradient by normal molecular viscos-
ity are derived, and illustrated with reference to numerical simula-
tions of axisymmetric flows in a cylindrical annulus. Viscosity is
shown to act so as to tend to expel m from the interior outwards from
the rotation axis. Such an effect can produce local super-rotation
(m>QR 2) even in a mechanically-isolated fluid (e.g. contained by
rigid stress-free boundaries), and is illustrated in a further
numerical experiment. The tendency of viscosity to result in the
expulsion of m is shown to be analogous in certain respects to a
"vorticity-mixing" hypothesis for the effects of non-axisymmetric
eddies on the zonally-averaged flow. We show how the advective and
'diffusive' transport of m by non-axisymmetric eddies can be
represented by the Transformed Eulerian Mean meridional circulation
and the '_liassen-Palm" (EP) flux of Andrews & McIntyre (1976, J.
Atmos. Sci. 27, 15-30) respectively, in the zonal mean. Constraints
on the form and direction of the EP flux in an advective/"diffusive"
flow for such eddies are derived, by analogy with similar constraints
on the diffusive flux of m due to viscosity. From a consideration of
these constraints on E, V.E, and Z*, and the properties of the super-
rotating numerical simulations discussed above, we suggest ways of
using observations of the zonal mean flows on Jupiter and Saturn to
infer the sources and sinks of m required to maintain the observed
flow. The associated form of E can also be used to infer some of the
properties of the non-axisymmetric eddies responsible for the trans-
port of m within Jupiter's equatorial jet.
210
https://ntrs.nasa.gov/search.jsp?R=19870008202 2020-03-20T12:35:53+00:00Z
EQUATORIALJETSANDANGULARMOMENTUM
It is well known from cloud-tracked wind data, obtained from ground-based
observations and the Voyager spacecraft that both Jupiter and Saturn exhibit
strong westerly (prograde) jet streams near their equators. Whenmeasured
with respect to the "interior" rotation rate (defined by their radio rotation
frequencies), these equatorial jet streams are found to be extremely intense(u>100 m s-I on Jupiter and u~500 m s-I on Saturn), and are remarkably
persistent features in the large-scale circulations of the major planets. Some
recently measuredvelocity profiles are shownschematically in Fig. i.
c_QJ7o
oJ7o
_.J
30
30
(a) (b)
\
\
|
M
M
I
I
I
I
i
I
I
i
I
0 80
60
30
0-
-30
-60
I
0
| ! ! I I I
!
/
/
f
f
s-
I I I ! i
80 160 100 200 300 400 500
Zonal Velocity (ms "I)
Figure I. Latitudinal profile of mean zonal velocity (with respect to the
measured radio rotation rate) at the cloud tops of (a) Jupiter (adapted from
the data of Ingersoll et al., 1981) and (b) Saturn (adapted from the data of
Smith et al., 1982). Also shown (dashed) are velocity profiles corresponding
to a uniform angular velocity with latitude, equivalent to the approximate
rotation period measured at the equator on each planet. The latitudes of the
maximum observed angular velocity in Jupiter's equatorial jet are indicated in
(a) by M.
211
The existence of these apparently steady equatorial jet streams poses some
intriguing dynamical questions, especially regarding the origin of their
angular momentum and the nature of the processes which maintain them. If we
consider the simplest possible configuration, in which only axisymmetric
motions (e.g., driven by a latitudinal distribution of diabatic heat sources
and sinks) are permitted (i.e., we exclude non-axisymmetric pressure
gradients, hydromagnetic effects etc.), prograde equatorial jets cannot occur
without rather special initial conditions. This is because, in inviscid
axisymmetric flow, the angular momentum per unit mass
m = (QR cos k + u) R cos k (i)
(where Q is the planetary rotation rate, R the planetocentric radius and k is
latitude) is conserved following the motion in the meridional plane, i.e.,
Dm/Dt = 0 (where DIDt = DIDt + v-V) (2)
N
Since u>0 at the equator requires that m>_R2max, (2) requires the initial flow
also to contain fluid elements with m>_R2max, i.e., with more m than any
fluid element initially at rest in the rotating frame (This is sometimes
referred to as "Hide's theorem"--see Hide 1969). The volume-integrated
angular momentum is similarly constrained to a constant value at all times.
EFFECTS OF INTERNAL VISCOSITY
Since an inviscid, axisymmetric system (initially at rest) cannot produce an
equatorial jet, it is natural to consider the effects of Newtonian (molecular)
viscosity, while still preserving an axisymmetric flow. Viscosity removes the
formal conservation of m but, for a steady flow, some constraints on the flow
can still be obtained (using arguments similar to Schneider 1977; Held and
Hou 1980) as follows. With viscosity, Eq. (I) becomes
Dm/Dt = -V-F (3)
where F is the diffusive flux of m due to viscosity which, for an isotropic
fluid, is given by
F = -vR 2 (cos 2 k) V 7 (4)
where v is the kinematic viscosity and
7 = u/(R cos k)
the local relative angular velocity.
(5)
212
If we consider any local maximum in m (necessarily, though not exclusively,
associated with an equatorial jet), m = m o (say), we may draw a closed m con-
tour in the meridional plane at m o - e (where e may be arbitrarily small; see
Fig. 2). Integrating the steady form of (3) over the toroidal volume enclosed
by the m - e contour, we get
fffv.(vm) dq; = (m o - e) _ v.n ds
= 0 (by mass conservation)
= - _ F.n ds (6)
_N
Thus, either F.n = 0 everywhere, or else F must possess both inward and outward
components in_d_fferent regions of the m contour (i.e., F must act up-gradient
with respect to Vm somewhere in the flow). Since F is _anti-) parallel to Ay,
and not to Vm (see (4) and (5) above), a necessary _ondition for the existence
of a steady viscous jet is for Vy.Vm to take either sign in different parts of
the flow. It is not necessary to invoke "negative viscosity" phenomena, which
are fundamentally associated with non-axisymmetric effects.
Height
H
\
/
\
\
/
0
N 0 S
Latitude
Figure 2. Map of m in a hypothetical flow on a sphere, with a local maximum
in m near the equator. Constraints derived for such a flow, associated with
a balance between advection (in a meridional circulation, shown by continuous
lines) and diffusion, result in a need for F to act inwards in some places and
outwards in others for a closed m contour region (shaded), with _ _._ ds = 0.
213
(a) TEMPERATURE (b)ANGULAR VELOCITY (C) ANGULAR MOMENTUM
DIFFUSIVE AM FLUX VECTORS
"rw_
(.3m
c:;"
8
°3.00 3'.95 _.90 5.B
RADIUS
; iO_
o /""
/"'l
3'.95 _'.9o : 5'.86 3 95 _, 90 5 86
RADIUS RADIUS
(d) STREAM FUNCTION (e) v,scousTORQUE
w.i"
..:-
;o
,,i"
Om
.2-
O"
8
°.3.0_
/"
3'.9,5 _'.90 5.86
RADIUS
_-_I __
IllI \ _ _,
:!i
.00 3.95 _,.90 5.86
RADIUS
Figure 3. Steady state fields shown in the (r,z) plane for thermally-driven
flow (by differential heating at the sidewalls) in a system with stress-free
boundaries. (a) Temperature T. (Contour interval [CI] 0.5 deg C); (b) Angu-
lar velocity y = u/r (CI = 0. I rad s-l); (c) m and vectors of F (CI = 0. I m/Qb2);
(d) Meridional stream function Y,. (CI = 0. I cm 2 s-l); (e) Local viscous
torque (-V-F). (CI = 0.025 cm2 s-2).
214
AN EXAMPLE IN THE CYLINDRICAL ANNULUS
We illustrate the application of these ideas in a numerical simulation of axi-
symmetric flow in a rotating cylindrical fluid annulus. The fluid is incom-
pressible and Boussinesq (for simplicity), and contained between two coaxial
cylindrical surfaces (at r=a,b) and horizontal boundaries at z = O,H. Motion
is driven by maintaining the two sidewall boundaries at different (constant)
temperatures (Ta and Tb) , and in the present example, all boundaries are rigid
and stress-free (so that Vy'_=F._ = 0, and the fluid cannot exchange m with
its environment). The model used is a grld-point finite-difference formula-
tion of conventional design, with a stretched mesh to resolve boundary layers,
and a realistic representation of molecular viscosity and thermal conductivity
(being the only sub-gridscale processes, see Hignett et al., 1985 and Read
1986a for further details).
Figure 3 shows a selection of steady state fields in the (r,z) plane. With
T a = Tb, a meridional overturning circulation is driven between the two side-
walls which is predominantly thermally-direct (cf. Figs. 3(a) and (d)), though
some smaller, indirect cells also occur owing to the effects of a diffusive
instability for Prandtl number o>>i. During the "spin-up" of the flow, m is
redistributed by the main merldional circulation (though because of the bound-
ary conditions, the total amount must remain constant). It accumulates near
the top of the inner cylinder (top left in Fig. 3), generating a local maximum
in y (see Fig. 3(b)). Since F is related to Vy (see Eq. (6)), viscous diffu-
sion transfers m outwards from the y maxlmum, tending to transfer m horizon-
tally against its local gradient Vm. In the steady state, this outward
transfer of m exactly balances advection by the meridional circulation, and
the resultant distribution of _ and m is shown in Fig. 3(c). Note that fluid
elements near r=b, z=H have now acquired m>Qb2 in a wedge-shaped region of the
flow (shaded in Fig. 3(c)), even though the total angular momentum is un-
changed from the initial rest state. Thus, viscosity has acted as a mechanism
for angular momentum expulsion towards the outer cylinder.
Since Eq. (6) is satisfied not only for a region bounded by contours of m, but
also by impermeable boundaries, it applies to Fig. 3. This can be seen in
Fig. 3(c), with F being largely up-gradlent (with respect to Vm) in the hori-
zontal, and down-gradient in the vertical. The corresponding local torque on
the fluid (-V.F) is shown in Fig. 3(e), and consists of a complicated array of
sources and sinks which, when integrated over the volume of the fluid, have a
resultant of zero.
APPLICATION TO REAL EQUATORIAL JETS
The configuration considered in Fig. 3 is analogous in some respects to an
equatorial jet in a thermally-driven flow on an entirely fluid spherical plan-
et (i.e., without a solid underlying surface), if we regard the (r,z) plane as
equivalent to the latitude, height) plane and r=b as representing the equator.
The resulting advectlve/diffusive flow is similar in some respects to some
simple models of the Jovian atmosphere (e.g. Williams and Robinson 1973; Mayr
and Harris 1983), although such models have tended to require a stress-bearing
lower boundary to generate a prograde equatorial jet. The experiment in Fig. 3
215
uses a stress-free lower boundary, yet still supports a local maximum in
m > Qb 2 in the steady state. The main criterion is that Vy.Vm<O somewhere
in the flow, implying that maxima in y cannot coincide with maxima in m.
Some evidence for this is apparent in Ju_piter'_ equatorial jet (Fig. l(a)),
which clear__ly shows_ a local minimum in u (and y) at the equator itself, and
maxima in u (and y) at the poleward extremes of the jet. Transfers of m by
molecular viscosity are unlikely to be significant for Jupiter and Saturn's
jets, however, and are probably dominated by eddy mixing (which is fundamen-
tally non-axisymmetric). Since _ is related to the gradient of (relative or
absolute) vorticity _, however, viscosity is seen to act in some respects
llke a vorticity-mixing process (tending to even out gradients in _), such as
has been suggested for a variety of eddy phenomena in fluid mechanics and
geophysics (e.g., Taylor, 1915; Rossby, 1947; Green, 1970; Read, 1986b). The
experiment in Fig. 3 can be regarded, therefore, as indicating in a general
way how an equatorial super-rotation can be driven and maintained by a
thermally-driven meridional circulation interacting with eddies which act to
mix vorticity.
Are there ani useful constraints on non-axisymmetric flows? We present an
eddy transfer theorem for closed m contours. If we form the equation for the
conservation of zonally-averaged angular momentum m, using the residual Euler-
Jan mean circulation of Andrews & Mclntyre (1976), we get (e.g., in spherical
geometry using pressure coordinates in the vertical)
D m
bml5t + v -Vm = - V.E (- V-F) (7)
where
m
v, = [ v - ( v'8'/ep)p, co + ( v'e' cos klSp)k/(R cos k] (8)
and represents the (quasi-Langrangian) mean meridional circulation associated
with diabatic heating, eddy dissipation and transience. E is related to the
so-called Eliassen-Palm flux, and given by
~ - v'O'/Op ,E = R cos k[u'v' Up
m
_0'u' + {(u COS X)x/(R cos k) - 2Q sin k} v'S'/ep] (9)
216
(cf. Andrews & Mclntyre 1978). Since _, obeys a similar continuity equation
to _, E must obey a constraint similar to Eq. (6) for closed, steady m
contours, provided we may ignore molecular viscosity. Thus,
_E.n ds = 0 (I0)
around any closed m contour.
Further aspects of the ideas presented in Sections 1-4 above are discussed in
greater detail by Read (1986a,b).
DIAGNOSTICS FOR JUPITER'S EQUATORIAL JET?
Given the framework of constraints on equatorial jets associated with a
balance between advection of m and its transfer by non-axisymmetric eddies,
it may be possible to use observations of Jupiter's equatorial jet to obtain
information concerning the processes by which the jet is maintained. We
suggest below some strategies for possible diagnostic studies of Jupiter's
atmosphere which may bear on this important problem:
a) From detailed observations of the zonal mean flow on Jupiter, we may
investigate the properties of candidate eddy processes and their associated
sources and sinks of m. The equatorially-localized structure of the jets
might suggest the relevance of equatorially-trapped Kelvin and Mixed Rossby-
Gravity (MRG) modes, for example, as studied in a simple model of Jupiter's
equatorial jet by Maxworthy (1975). Given the thermal and velocity structure
of the mean zonal flow, it is possible (subject to certain assumptions) to
calculate the properties of steady modes which are compatible with that
(steady) mean flow. An example of such a procedure was given by Plumb and
Bell (1982), who undertook a similar study in connection with the quasi-
biennial oscillation in the terrestrial stratosphere. Given the form of u(y,z)
and N 2 the mode structure can be obtained, together with quantities analogous
to E and V.E for the main Kelvin and MRG modes. The latter could then be
tested against the constraints of Eq. (I0) for consistency with the distribu-
tion of m and a suitable meridional circulation.
b) Since the merldional circulation compatible with the constraints on E and
V-E is quasi-Lagranglan, it primarily reflects the distribution of diabatic
heat sources and sinks in the Jovian atmosphere. This can also be compared,
therefore, with the known sources and sinks, e.g., due to radiative forcing
obtained from radiation budget studies, to check for consistency of the
solution, and to provide further requirements which the eddy processes need
to satisfy.
The feasibility of such strategies have yet to be demonstrated for Jupiter,
although some progress on a similar scheme for the analysis of data for Venus
(from Pioneer Venus project) has been made by Hou (1984), Hou and Goody (1985)
and Valdes (1984 and private communication).
217
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Transport of absolute angular momentum in quasi-axisymmetric equatorial jet streams

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