418 
The last equation becomes particularly simple if the 
fluid is autobarotropic. In this case the horizontal dis- 
placement is independent of the vertical coordinate. 
It is this property of the system of equations for quasi- 
static motion which permits the most important simpli- 
fications of which use is made, for instance, in the theory 
of ocean tides where as a rule only incompressible, 
homogeneous fluids are considered. For fluids which 
are not autobarotropic this simplification no longer 
holds, but it is in some instances possible to consider 
a fluid model which consists of a number of auto- 
barotropic layers where the approach to more realistic 
baroclinic models is effected by assuming autobarotropic 
relations which differ from layer to layer. A rather 
comprehensive discussion of quasi-static motions has 
been given by Eliassen [8]. 
The limitations of the quasi-static hypothesis espe- 
cially in its applications on the tidal theory have been 
discussed by Solberg [20]. The differential equation 
remains of the same type under the quasi-static assump- 
tion regardless of whether the oscillations are longer 
or shorter than half a sidereal day; but when the 
vertical acceleration is taken into account it changes 
from the elliptical through the parabolic to the hyper- 
bolic type as the period increases to more than twelve 
sidereal hours. Under these conditions it appears pos- 
sible that oscillations with periods longer than half a 
sidereal day are not reproduced very accurately by 
the quasi-static hypothesis, or that additional types 
of oscillations are possible which are not given by the 
quasi-static method. 
Another way in which the hydrodynamic equations 
might successfully be simplified for investigations in 
theoretical meteorology and oceanography is indicated 
by the empirically known fact that the large-scale 
motions of the free atmosphere are nearly geostrophic. 
Tt thus appears possible when attention is to be focused 
on these large-scale motions to achieve substantial 
simplifications by assuming that the wind field is very 
nearly geostrophic. Such a “geostrophic” approxima- 
tion has been developed by Charney [5] in systematic 
form from considerations of the orders of magnitude 
of the various meteorological variables. The most im- 
portant step in this ‘‘quasi-geostrophic method”’ is the 
elimination of the horizontal divergence of motion from 
the system of hydrodynamic equations before the intro- 
duction of the geostrophic approximation. This elimi- 
nation can be effected conveniently by means of the 
equation of continuity together with an equation ex- 
pressing the conservativeness of a quantity such as 
the potential temperature. While for the vertical vortic- 
ity component, for instance, it is permissible to use the 
geostrophic approximation directly, the same is not 
true for the horizontal divergence. From a considera- 
tion of the orders of magnitude of dw/dx and dv/dy 
it follows that for the large-scale motions, these terms 
are one order of magnitude larger than their sum, the 
horizontal divergence. Since the geostrophic deviation, 
that is the difference between actual and geostrophic 
wind, is also one order of magnitude smaller than the 
wind itself it follows that a computation of the hori- 
DYNAMICS OF THE ATMOSPHERE 
zontal divergence by the geostrophic wind would not 
be a satisfactory approximation. By the elimination of 
the horizontal divergence in the manner indicated 
above, a consistent system of approximate equations 
for large-scale motions is obtained. 
A linearization of this quasi-geostrophic system of 
equations has been used by Charney and Hliassen 
[6] as the basis for a numerical method to predict the 
future pressure distribution. This method and its exten- 
sion to nonlinear mathematical models is discussed 
elsewhere in this Compendium.” But it is relevant to 
the topic of the present article to state that even the 
linear models based on the assumption of small per- 
turbations have given satisfactory forecasts in a num- 
ber of cases. 
CONCLUSION 
The atmospheric perturbation equations are a tool 
of theoretical meteorology which is to be used in the 
investigation of problems of atmospheric dynamics. 
In considering future work in this field we shall there- 
fore concern ourselves with a general survey of types 
of problems and lines of attack in which the perturba- 
tion method may appropriately be used. 
As was pointed out in the discussion of the basic 
assumptions of the perturbation theory, one of the 
important problems which is appropriately treated by 
the perturbation method is that of the stability or 
instability of a given atmospheric flow pattern. It was 
explained there that the method consists in super- 
imposing a small perturbation on the basic flow pat- 
tern and investigating whether such a perturbation 
would increase with time or not. In the first case the 
basie flow pattern would be unstable, in the second 
case stable. 
It is immediately apparent that such an instability 
investigation is not necessarily complete. When it has 
been found that an originally small perturbation in- 
creases with time it can be concluded that the perturba- 
tion equations do not describe satisfactorily the de- 
velopment of the perturbation beyond a certain state 
because sooner or later the perturbation will have 
grown to such a magnitude that the terms of higher 
than the first order in the equations can no longer be 
neglected. Thus, after the perturbation has grown to 
certain dimensions its future behavior can no longer 
be predicted by the original perturbation equations. 
It is possible that the growth of the perturbation ceases 
when this state is reached and, in fact, it is generally 
observed that atmospheric perturbations do not exceed 
certain limits. The foregoing remarks are not meant 
to imply that the study of stability and instability 
based on the perturbation method is unsatisfactory. 
The point is that, by means of the perturbation method, 
we can make statements about the development of 
the perturbation only for a limited time interval. It 
would evidently be highly desirable to extend these 
investigations in such a manner that they permit us 
2. Consult ‘Dynamic Forecasting by Numerical Process’? 
by J. G. Charney, pp. 470-482. 
