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Courant Institute of
Mathematical Sciences

Asymptotic Behavior of
Planetary-Scale Motions
of the Atmosphere

G. K. Morikawa

IMM 348
April 1966

Prepared under Contract Nonr-285(55)

with the Office of Naval Research NR 062-160

Distribution of this document is unlimited.

New York University

251 Mer«^ St. New YoH., N.YJ^(*

NR 062-160 IMM 3^8

April 1966

New York University
Courant Institute of Mathematical Sciences


G. K. Morikawa

rn.^^"^ ^^"^ UNIVERSITY
material) derivative operator, ^ ^ curl v Is the relative

vortlclty, p Is the density, and Is the potential

temperature (directly related to the entropy S = C In ,

where C Is the specific heat at constant pressure) .

The derivation of Eq. (1) requires the explicit

use of the following equations valid In the rotating

reference frame:

In the more recent literature an exposition of these
and related matters Is given by Truesdell [5] In his
treatise on vortlclty; more directly on meteorological
subjects,, we refer to Ellassen and Klelnschmldt [5].

1) Conservation of mass (or continuity equation)

(2) ll + p div V =:

2) Vorticity advection equation obtained by operating
with the curl ( ) on the inoment;Am equation ,

■,— > _> _> -.

(3) -^ + 20xv+p~ grad p + grad ^ =


dv^ -> -> -> -> "^ 1 i->i2

(4) -tt: = V, + (v • grad)v = v, - v x curl v + -p grad |v |

and p is the fluid pressure, ^ is the gravitational
potential, and subscripts Indicate partial differentiation;
the resulting vorticity advection equation is

(5) -if + (20+ O dlv V - [ (20+C) -gradlv +(grad p ^) (grad p)-0

3) Conservation of potential temperature for adiabatic

f^^ IT = ^

where the equation of state relates © , p and p,

7-1 1
(7) © = (7Po^ -P^) / P

where Pp, is the pressure at the ground, r = a, and the gas

constant 7 = C /C , the ratio of specific heats.

For our basic complete system of equations we choose
Eqs . (1), (3), (6) and (?) and the appropriate Initial
and boundary conditions on the surface of the earth and
at Infinity. A completely equivalent system of equations
Is given by replacing the potential vorticlty equation (l)
by the continuity equation (2); however, the asymptotic
behavior of the motion is obtained more succinctly by
suppressing the explicit use of Eq . (2) (cf. [2]).


3- Asymptotic Equations of Large-Scale Planetary Motions

The formal approximation procedure which we use to
obtain the asymptotic equations Is quite simple. That Is,
the rules are elementary although the complete physical
and mathematical justification and precise estimates of
error Involved In the approximation are not totally apparent
at the present time. Previously we have used this same
procedure to obtain approximate equations describing
two-dimensional geostrophlc vortex motion on a plane
tangent to the rotating earth (Ref. [2]). In Appendix I,
we derive the equivalent geostrophlc equations of motion
for an adlabatlc, compressible atmosphere over a tangent
plane, i.e. In which the horizontal and vertical components
of the earth's rotation rate vector O is assumed to be
constant and fixed to the latitude at the point of
tangency. In Appendix II we show the relationship
between the vertical structure of the atmosphere (at rest
with respect to the tangent plane) and the vertical
variation of the horizontal length scale of a singular
geostrophlc vortex line element. At the present time

Apparently these equations are similar to those first
derived by Obukhov [5] and Monln [6] by a related method.

In addition, the strength of the vortex is functionally
related to the vertical structure of the static atmosphere
and hence to the horizontal length scale.


the motivation for, and validity of, the asymptotic
equations must rest partly on the successful derivation
of approximate equations for geostrophic-motion over
the tangent plane, as derived in these Appendices. In
addition, further substantiation must await the testing
of these asymptotic equations by comparison with actual
atmospheric motions, using numerical methods aided by
electronic computers .

5.1. Asymptotic Equations Obtained by Perturbing on the
Atmosphere at Rest .

In order to proceed with the derivation we first
transform the basic equations (1), iJ>) , and (6), which are
in vector form, into equations in spherical polar
coordinates and introduce the scaled time t = et. Then
the independent variables are (0, (j), r, x) where is
longitude increasing eastward, ^ is latitude increasing
northward from the equator, and r is radius measured
from the earth's center. The velocity components corres-
ponding to the coordinates (0, ^, r) are (u, v, w) = v.
Thus the dependent variables expressed as a state vector
(u, V, w, p, p,@) for the atmosphere at rest (e = 0) is
(0, 0, 0, p^ , p^ , ) representing the lowest order
terms in the power series expansion with respect to e and
is independent of time. The potential vorticity equation

(1) is transformed Into spherical polar coordinates using
the following identities: The earth's rotation vector

(8) 20 = (0, 2 0cos (j), 20 sin (j))

and we abbreviate by calling the (f)- component, 20 cos (j) = fp,
and the r-component, 20 sin (\> = f', the relative vorticity

(9) C = curl V = (e, T],

= [^ [w^-(rv)^], ^ [(ru)^ - ^3!^] , ^ g^3 ^ [vq-(u cos


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