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IMM 401
August 1973

Courant Institute of
Mathematical Sciences

The Zonally Symmetric Motion
of the Atmosphere

Donald A. Drew

Sponsored by Advanced Research Projects Agency,
ARPA Order No. 2069, Amendment 2.

Approved for public release; distribution unlimited.

New York University

ARPA Order NOot 2069, Amendment 2 IMM 401

August 1973

New York University
Courant Institute of Mathematical Sciences


Donald A. Drew

This research was supported by the Advanced Research
Projects Agency under Grant No. DA-AR0-D-31-124-73-G150.

Views and conclusions contained in this study should not
be interpreted as representing the official opinion or
policy of the Courant Institute of Mathematical Sciences,
or of New York University, or of ARPA.

Approved for public release; distribution unlimited.


The steady, zonally symmetric motion of a shallow
incompressible atmosphere on a rapidly rotating earth subject
to an equator-ward temperature gradient is studied. The assump-
tions made allow the thermodynamics to be treated separately
from the motions o The turbulence terms are modelled using mixing
length arguments o Ass\iming the turbulence length scale is small
compared to the earth's radius and the motion is slow compared to
the earth's rotation speed, the north-south geopotential gradient
drives the eastward winds in the "interior" temperate regions,
that is, not near the surface, equator or poles. The meridional
winds in the interior are driven by the turbulence generated by
the shear in the eastward winds. Near the equator the advective
terms become comparable with the rotational terms, but the
turbulence terms remain unimportant o The motions there show
trade winds at the equator, changing to eastward at a predicted
latitude of about 15°. The meridional motion takes the form of
Hadley cells with rising at the equator and sinking again at
about 21°. The Hadley cell and the temperate region are
connected through a vertical layer of turbulence. Near the poles
the advective and turbulent terms become comparable with the
rotation terms. Surface Ekman layers complete the picture.


1. Introduction

When viewed from outer space, the earth's atmosphere, as
evidenced by the cloud patterns, shows no signs of symmetry or
regularityo However, analysis of wind and temperature data over
long periods of time shows that on the average the atmosphere
behaves in a regular manner. Prevailing westerlies, trade winds
and the Hadley circulation are gross features of the motion
which are observed from average data.

In 1735j George Hadley initiated the idea that solar
heating at the equator forced the air there to rise, and hence
he conjectured that the air, once aloft, travelled to the poles,
where it sank back to the surface to journey toward the equator
again. Subsequent observations by Ferrel around the 1880' s
suggested that the air rising at the equator did not travel to
the poles, but sank back to the surface at about 30 north or
south latitude. In addition, the cold air sinking at the poles
rose again at about 6o north or south latitude. In between
these cells was a single cell rising at 6o and sinking at 30
in each hemisphere. The conjectures mechanism to drive the
temperature cell was friction between the adjacent thermally
driven cells; while the eastward winds therein were the result
of the rotational (Coriolis) force interacting with the meridio-
nal motions. Because of the appearance of three cells in each
hemisphere, mechanisms to explain the motions by explaining the
presence of each cell have been called tricellular theories.

The aforementioned tricellular theory of driving the
motions has fallen into disfavor, one reason being that if the

heating drives the motions in the tropical and polar cells, the
winds in the temperate cell are necessarily weaker since it is
driven by the others. But the prevailing westerlies observed in
the temperate cell are much too strong to be secondary or
frictionally forced winds.

A recent popular explanation of the driving mechanism is
the wave theory (Pfeffer, 1964). In this theory, the north-south
temperature gradient induced by solar heating drives wave
cyclones and anticyclones. The nonlinear interactions of these
waves cause a mean zonal flow. The zonal flow in turn drives the
meridional cells through the mechanism of the Coriolis force.

We give a new explanation for the general circulation in
which the north-south temperature gradient drives the zonal
motions in the temperate cell through the geopotential gradient.
The zonal motions in the tropical and polar cells are driven by
the zonal motion in the temperate cell through friction between
the cells. More precisely, the eastward motion at the poleward
edge of the tropical cell causes an eastward motion inside the
tropical cell, at least for some distance. The Coriolis effects
cause a strong meridional motion and the trade winds. The
meridional motions in the temperate cell are driven by frictional
interactions with the mean zonal flow and the tropical and polar
meridional motions.

In order to describe the motions, we shall use the fluid
mechanical and thermodynamical equations for the zonally sym-
metric (i.e. independent of longitude) flow of a shallow layer
of fluid. Moreover, following Saltzman (I968), we shall make a

Boussinesq-like approximation, neglecting density variations
except in the vertical momentum equation. The shallow atmosphere
assumption allows us to use the hydrostatic pressure equation
for vertical momentum balance. This assumption also allows the
neglect of the vertical velocity, except in terms involving
vertical derivatives, which are large. The hydrostatic pressure
equation allows us to use the pressure as the vertical coordinate,
replacing z, the height above sea level. The relevant vertical
"velocity" is aj = dp/dt, the material derivative of the pressure.
We shall refer to cu as the vertical p-velocity, and to w = dz/dt
as the vertical z-velocity. The pressure gradient terms are
expressed in terms of the geopotential, defined as = gz((f),p),

y^ ^ ^ /N

where z(


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