STABILITY PROPERTIES OF LARGE-SCALE ATMOSPHERIC DISTURBANCES 
By R. FJORTOFT 
The Institute for Advanced Study* 
Introduction 
The large-scale motion of the earth’s atmosphere is 
to the first approximation a solid rotation from west to 
east. Upon this are superimposed a more or less orderly 
zonal flow in the relative motion and the large-scale 
disturbances familiar in meteorology. In this article 
causes for the creation and maintenance of these dis- 
turbances will be discussed, excluding, however, the 
more or less permanent disturbances forced upon the 
atmosphere because of the earth’s topographic inho- 
mogeneity. 
By the use of the phrase ‘‘disturbances in a zonal 
flow,” a separation is implied between two components 
of the flow. This may seem artificial since the hydro- 
dynamic equations are directly applicable to the total 
component of the flow. There are, however, several 
reasons for doing this: The immediate impression ob- 
tained by studying hemispherical weather maps is one 
of a more or less orderly zonal flow upon which are 
superimposed disturbances that behave to some degree 
as physical entities themselves. Further, the zonal flow 
and the disturbances undergo somewhat systematic 
changes, seemingly of great importance to weather. 
Therefore, by a separation of the atmospheric flow into 
some kind of orderly zonal flow and disturbances super- 
imposed thereupon, one isolates at the outset certain 
phenomena which appear to be related to actual 
weather. Besides, this separation will enable one to 
deal satisfactorily with problems in which a detailed 
knowledge of the motion is unnecessary, as exemplified 
by the stability investigations carried out in the dis- 
cussion of barotropic disturbances (pp. 460-463). 
Granted that such a separation of the atmospheric 
flow may prove useful, it becomes important for quanti- 
tative treatment to express this separation in mathe- 
matical terms. How this should be done is to some 
extent arbitrary because there may be several ways 
of defining the orderly flow, and thereby the disturb- 
ances. In this article, the orderly flow will be character- 
ized, for an arbitrary hydrodynamic element a, by the 
mean value of @ at a fixed time along latitude circles: 
2=- [ ow, 
Or 
while the corresponding element in the flow of dis- 
turbances will be defined by 
a’ =a— a. 
The flow therefore is composed of an orderly, axially 
symmetric motion and an irregular flow vanishing in 
* This article is to some extent based upon work performed 
under contract N6-ori-139, Task Order I between the Office 
of Naval Research and The Institute for Advanced Study. 
the mean along zonal circles. The degree of irregularity 
may, however, vary widely. In this article it is assumed 
that all irregularities considerably smaller than the 
smallest-scale cyclones have already been smoothed 
out in some way. The corresponding turbulent stresses 
will be entirely neglected throughout this article. 
As already pointed out above, the sum of the two 
components of flow must obey the hydrodynamic equa- 
tions of motion. There must therefore exist a coupling 
between these two components. Actually, in many 
cases this coupling is so strong that a full understand- 
ing of what happens with one component cannot be 
achieved without taking into account simultaneous 
changes of the other. 
It is an established procedure to separate the hydro- 
dynamic equations into one set valid for the orderly 
motion and one for the irregular flow. To implement 
this process the equations will first be simplified by 
suppressing certain terms of minor importance. With 
the conventional simplification in the Coriolis accelera- 
tion, the equation of motion is 
D 
por + sexy - 8] = — Vp, 
where p is the density, f is the Coriolis parameter, 
g is the acceleration of gravity, v is the velocity, and 
p is the pressure. The coordinate system has been 
selected so that the z-axis is directed east, the y-axis 
north, and the z-axis upward; i, j, and k are unit 
vectors in the x, y, and z directions, respectively. By 
elimination of Vp and substitution of x = In ? (where 
8 is the potential temperature) by means of the rela- 
tionship Vp — T'Vp = —pVx, one obtains 
vx [PP + jx v| = —V X xg. (1) 
The neglected term —Vx X [Dv/dt + fk X v] is small 
compared with the others. Equation (1) is equivalent 
to 
DN eevee (2) 
dt 
where Vy is a certain laminar vector. It is easily under- 
stood that by introducing the simplification above, 
the effects from solenoids in horizontal planes have 
been neglected. Now let Q(z) represent some standard 
distribution of density with height. The following ap- 
proximate equation will then constitute the continuity 
equation: 
wD = iy (3) 
-Qv = QV; Vi, 
V-Qv = QV: Vv), = 
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