Results of a theoretical model study of some of the expected effects of spherical geometry on laboratory simulations of the type of geophysical flow that dominates the general circulation of the earth's troposphere are reported

Fowlis, W. W.

Geisler, J. E.

English

NASA Technical Reports Server

PaperNo.40
A STUDYOFTHEEXPECTEDEFFECTSOFLATITUDE-DEPENDENT
ROTATIONRATEONLABORATORYGEOPHYSICALF OWEXPERIMENTS
J.E.Geisler, Rosenstiel School of Marine and Atmospheric Science, University
of Miami, Miami, Fla. 33149, W. W. Fowlis, Space Sciences Laboratory,
Marshall Space Flight Center, Alabama 35512
ABSTRACT
This paper reports results of a theoretical model study
of some of the expected effects of spherical geometry on
laboratory simulations of the type of geophysical flow
that dominates the general circulation of the earth's
troposPhere.
INTRODUCTION
The traditional device for laboratory simulation of the
general circulation of the atmosphere is an annular container
of liquid situated on a rotating turntable (Hide, 1958; Fowlis
and Hide, 1965). Fluid motion is driven by heating the outer
wall of the annulus and cooling the inner wall, a crude analog
of the observed fact that the earth's troposphere is heated in
the tropics and is cooled in high latitudes. In the real
atmosphere and in the laboratory annulus, the function of the
motion field is to transport heat laterally from the heat source
to the heat sink.
The annulus suffers from the defect that its geometry is
far different from that of our spherical-shell atmosphere.
Attempts to design a rotating spherical laboratory device have
been frustrated by the presence of gravity, which acts vertically
downward in the laboratory. This influence would be absent in a
satellite. Atmospheric general circulation simulations in a
spherical container with an electrostatic radial force field
acting as "gravity" on a fluid with radially dependent dielectric
constant have been proposed for Spacelab by Fowlis and Fichtl
(1977). In this paper we report some results of a simple mathe-
matical model study which isolates some of the expected differ-
ences between annulus flows and spherical shell flows.
THE MODEL
Consider a spherical shell of fluid rotating with constant
angular velocity around a polar axis. The local vertical com-
ponent of this rotation vector varies as the sine of the latitude,
being zero for an equatorial observer and a maximum for a polar
observer. As is generally well known, this latitudinal variation
241
https://ntrs.nasa.gov/search.jsp?R=19790012502 2020-03-20T18:25:12+00:00Z
of the local vertical component of rotation plays a significant
role in the dynamics of the class of motions constituting the
general circulation of the atmosphere (and laboratory simulations
thereof). The simplest mathematical model which incorporates
this effect is one in which the relevant equations are written
in a Cartesian (rectangular) coordinate system rotating with
respect to the vertical axis. This rotation rate is taken to be
constant in the equations, except when it is differentiated with
respect to the north-south coordinate. This local tangent-plane
formulation of fluid motion on a sphere is generally known as
the B-plane. If rigid walls are erected at two arbitrary lati-
tude_ in this model, we have a B_-plane channel.
In its treatment of the dynamical effects of rotation, the
B-plane channel is sort of midway between an annulus and a sphere.
In our mathematical model study, we have subjected the B-plane
channel to different degrees of cross-channel heating and differ-
ent rotation rates, and noted the character of the resulting
motion field. In particular, following the traditional descrip-
tion of the analogous flow patterns in the annul us, we distinguish
between axially-symmetric motions and wave motions.
RESULTS
Figure l, shows a so-called re--diagram (after Fowlis and
Hide, 1965) showing the types of motlon occuring in a typical
annulus experiment.
I
Ro i0_I
i0-_
10-3 _ , ,_,,,,I
10-1 1
(I239Ta
lo6 lo7 lo8
/<RREGULAR-
SY___
101 102 lO3
nZs6c-2
Figure 1
Experimentally determined regime diagram
for annulus flow
242
The horizontal axis in the figure is the rotation rate squared;
the vertical axis in the figure is a "thermal Rossby number",
which varies directly with the imposed thermal contrast across
the annulus and inversely as the square of rotation rate. We
wish only to call attention to the fact that in this parameter
space the region occupied by waves is bounded on the left by a
convex, "knee-shaped" curve which we refer to as the stability
boundary.
Figure 2 shows results of a simple mathematical model simu-
lation of annulus flow (after Barcilon, 1964).
1 0 -2
• I I ! I I I I 1 Ih=acoseo I | I I I I I I
d=h 1--
3--
I
0 2 10 3 I0 'i
E2] 
Figure 2
Theoretically determined regime diagram for
annulus flow
The parameter space in this figure can be shown to be essentially
identical to the parameter space in Figure I. Each curve is the
stability boundary for the zonal wavenumber shown. (Zonal wave-
number is formally equivalent to quantum number; that is, zonal
wavenumber l means that the horizontal wavelength of the wave is
exactly equal to the circumference of the annulus). The stability
boundary for the system is the envelope of these curves. It is
seen that this theoretically-obtained stability boundary has
243
essentially the same shape as the experimentally determined one
(Figure l).
Figure 3 shows results from our B-plane channel model when
the dimensions of the channel are the same as the annulus in
Figure 2.
$
10-1
1 0 -2
"' S'YY ......
ill I I;,_,,l _ , I I ' ,, J
0 2 0 3 10 4
(2/6
Figure 3
Theoretically determined regime diagram for
B-plane channel flow
Again, the envelope of all these curves is by definition the
stability boundary. In comparing this with Figure 2, it can be
seen that the stability boundary has a considerably different
shape and that it extends somewhat farther to the left. The con-
clusion here is that the B-effect (that is, the variation of
local vertical rotation rate with latitude) results in more of
parameter space being occupied by waves. Another interesting
result, not discussed here, is that the B-effect produces a
marked change in the vertical structure of the waves present and
it also greatly changes the dependence of the results on the
presence of a rigid lid upper boundary condition.
We have repeated these model simulations for a wide range of
244
channel dimensions and compared these results with simulations
in annular geometry. These simulations support the general
conclusion that the B-effect drives the stability boundary to
the left in the parameter space diagram.
REFERENCES
Barcilon, V., 1964: Role of Ekman layers in the stability of
the symmetric regime obtained in a rotating annulus.
_. Atmos. Sc____i.,21, 291-299.
Fowlis, W. W. and Fichtl, G. H.: Geophysical Fluid Flow Model
Experiments in Spherical Geometry. Proceedings of the
Third NASA Weather and Climate Program Science Review.
NASA Conference Publication 2029, Paper No. 32, 1977,
p. 177.
Fowlis, W. W. and R. Hide, 1965: Thermal convection in a
rotating annulus of liquid: effects of viscosity on the
transition between axisymmetric and non-axisymmetric flow
regimes. J. Atmos. Sci. 541-558.
Hide, R. 1958: An experimental study of thermal convection in
a rotating liquid. Phil. Trans. Roy. Soc. Lond. A,
250, 441-478.
245


A study of the expected effects of latitude-dependent rotation rate on laboratory geophysical flow experiments

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server