HYDRODYNAMIC INSTABILITY* 
By JACQUES M. VAN MIEGHEM 
University of Brussels 
Introduction 
The concept of dynamic instability appeared along 
with the mitial developments in the theory of atmos- 
pheric disturbances [2]. In the method of perturba- 
tions, which V. Bjerknes introduced into meteorology, 
a “small motion” is superimposed on a state of ‘simple 
motion” of the air. This “simple motion” is always a 
permanent motion characterized, at every point of the 
medium, by equilibrium of the forces perpendicular 
to the direction of the flow. These forces are gravity, 
the pressure gradient (hydrostatic equilibrium), the 
Coriolis force (state of geostrophic motion), and the 
centrifugal force (circular vortex). The “simple motion,” 
therefore, is evidently a state of hydrodynamic equilib- 
rium. 
The ‘small motion” is simply a nascent perturbation, 
defined by quantities small enough that the values of 
the products and squares of the quantities as well as 
of their derivatives are negligible. Furthermore, it is 
assumed to begin with that the perturbation is restricted 
to a train of plane sinusoidal waves, propagated with 
constant velocity c in the same sense as the simple 
motion. This perturbation must satisfy linearized equa- 
tions derived from Euler’s dynamical equations and 
from the equations of continuity and of the adiabatic 
process; from these the amplitude relations are ob- 
tained. In addition it is necessary to satisfy restrictions 
imposed by external boundaries (surface of the globe, 
free surface) and, eventually, by internal boundaries 
(frontal surface, tropopause). There generally results 
a relation between the quantities describing the physi- 
cal and dynamical state of the “simple motion,” the 
wave length L, and the velocity of propagation c. When 
this “dispersion equation” admits only real roots for 
c, the perturbation is said to be stable regardless of the 
value of the wave length Z; on the other hand, when 
the equation admits only imaginary roots for c, the 
motion is referred to as unstable. In general this equa- 
tion admits real roots, for certain values of Z and 
imaginary roots for others, that is, the stability or in- 
stability of a perturbation depends on its wave length. 
The condition for which the dispersion equation has 
only imaginary roots for c expresses the dynamic in- 
stability criterion in the sense of V. Bjerknes and H. 
Solberg [2]. This criterion, which depends not only on 
the state of simple motion, but also, and more signifi- 
cantly, on the wave length ZL, is thus a selective criterion 
that can be applied only to a flow pattern consisting 
of a perturbation of the type “plane sinusoidal waves” 
propagated uniformly in the direction of the simple 
motion. 
* Translated from the original French. 
434 
a 
This dynamic instability of selective character has 
been studied extensively since the first work of V. 
Bjerknes and H. Solberg, notably by B. Haurwitz, 
C. L. Godske, Z. Sekera, J. G. Charney, P. Queney, 
H. T. Eady, and others. On the other hand, the stability 
of the simple motion, upon which the nascent per- 
turbation is superimposed, was not considered until 
very much later, although logically it should have been 
studied in the very beginning. It is evident, without 
further amplification, that the stability or instability 
constitutes an essential characteristic of the simple 
motion and must appear as such in the dispersion 
equation. 
Kleinschmidt [{17, 18] was the first to show that, 
along with the hydrostatic instability of masses at rest 
in the gravity field, there exists a hydrodynamic in- 
stability of air masses whose motion obeys the law of 
geostrophic flow. Although the hydrostatic equation ~ 
amounts to an excellent approximation to the equation 
of atmospheric dynamics in a vertical direction, the 
classical criteria of stability and instability for the 
vertical distribution of the air are strictly applicable 
only in an atmosphere without wind, in which there 
are only vertical displacements and where only vertical 
forces, such as gravity and the vertical pressure gra- 
dient, come into play. In reality, on a synoptic scale, 
we must take account of the horizontal pressure gra- 
dient, the Coriolis force, and the centrifugal force, as 
well as of the vertical forces. In this case the criteria 
of stability and instability of hydrostatic equilibrium 
lose their validity, and it is expedient to substitute 
for them criteria of stability and instability of hydro- 
dynamic equilibrium. 
In general, the atmosphere is said to be stable (or 
unstable) at any point O at any instant ¢ for any direc- 
tion r when a particle displaced from this point, at this 
instant and in the direction r, acquires a relative motion 
with respect to the environment at O returning it 
toward (or carrying it away from) the equilibrium 
position O. 
The Stability of Geostrophic Motion 
Let us consider a mass of air in a state of geostrophic 
motion (hydrodynamic equilibrium in the sense of 
Kleinschmidt [17, 18]). In this case the isobaric surfaces 
P = const and the isentropic surfaces © = const of the 
air are cylindrical surfaces with horizontal generatrices 
parallel to the geostrophic current; the equipotential 
surfaces, @ = const, of the gravity field are assumed 
to be planes parallel to the horizontal plane at any 
given point O in the interior of the air mass. We refer 
the motion of the air to a right-handed, rectangular, 
Cartesian coordinate system Oxyz, at rest with respect 
