HYDRODYNAMIC INSTABILITY 
tropical troposphere, dw/dy is on the average negative 
and of the order of 10 ° sec *, du/dz is positive and of 
00/dy 
00/dz 
of isentropic surfaces can attain the value 10”; conse- 
quently, under these conditions, 6u/édy is positive and 
of the order of 10° to 10 * sec '. Then the vertical 
component —6éu/dy of the vorticity which the velocity 
field u produces in the isentropic surfaces is anticyclonic, 
and when it exceeds in absolute value the limit f = 10 * 
sec , the geostrophic motion becomes unstable within 
the isentropic surfaces and in their neighborhood. In 
this case the transverse tangent d to the isentropic 
surface at any point is, to a first approximation, the 
bisector of the angle between directions d’ and d’’ which 
limit the sector of unstable displacements on either 
side of this surface at the point considered [87]. The 
energy of instability released per unit mass in the d 
direction is satisfactorily given by the approximation 
(f/2)(6u/6y —f)r° deduced from (24). 
In the atmosphere we usually have 0 < 6u/dy < f 
and 00/dz > 0, and consequently the state of geo- 
strophie motion is usually stable for any transverse 
displacement. In this case a < 0 and az, > 0. If we 
admit that the state of hydrodynamic equilibrium can 
be altered gradually, it may be asked which of the two 
quantities a and a., will be the first to change its sign. 
In reply it is enough to observe that if a., = 0, we have 
a > 0. Then a vanishes before a.. , since for a and a;, 
to vanish simultaneously it would be necessary that 
Az = An = 0, that is, 00/dy = 00/dz = O, a condition 
which does not occur in the atmosphere. From this 
follows the proposition [18]: When the state of hydro- 
dynamic equilibrium is altered, the threshold of hydro- 
dynamic instability im an isentropic surface is reached 
before that of hydrostatic instability im the field of gravity. 
In other words, in the atmosphere, hydrodynamic in- 
stability m an isentropic surface will develop before 
hydrostatic instability in the vertical direction. 
In summary, in order for hydrodynamic instability 
to appear in a region—it will first appear in the isen- 
tropic surfaces—the vertical component of vorticity, 
produced by the geostrophic velocity field in the isen- 
tropic surfaces, must increase in absolute value and 
exceed the critical value f. 
In the absence of Archimedean buoyant forces, form 
Q reduces to f(f — du/dy)r;. Then there is inertial 
stability or stability depending on whether du/dy < 
or > f. In the atmosphere we generally have du/dy < 0 
in the lower troposphere, as well as in the middle and 
high troposphere north of the “jet stream”; but south 
of the “jet stream” 0 < du/dy < f [32]. The condition 
of inertial stability is thus generally attained in the 
atmosphere. We shall say that 
the order of 10 * to 10° sec *, while the slope — 
v2 = f(f — du/dy) & 10 * sec * (37) 
defines the inertial stability. 
Let us now examine more closely the criterion 
du _ du aS ou 
by ay (Ge aeaaien (38) 
439 
for hydrodynamic instability. After 00/dy in (88) is 
replaced by its value taken from (84), this criterion 
assumes the form 
elb¢-a))-» 
if we assume that static stability is min fact realized 
(v2 > 0), which is usually the case. Inequality (39) 
shows that, whatever the magnitude of the inertial 
stability v? and the static stability v;, the hydrody- 
namic instability can appear as soon as the baroclinity 
exceeds the critical value Vx = »,v;. Instability of geo- 
strophic motion is thus essentially dependent upon the 
baroclinic character of the atmosphere and can develop 
only in regions where this character 1s sufficiently pro- 
nounced, that is, in frontal zones and in the tropical 
air immediately above and in the polar air immediately 
below these zones. 
Inequality (89) suggests the introduction of the hydro- 
dynamic stability 
Ou N? 
which is always less than the inertial stability. We see 
at once that inertial instability (vi < 0 or du/dy > f) 
implies ipso facto hydrodynamic instability (va < 0); 
this is, what takes place in a region, generally of small 
extent, of the tropical troposphere south of the “jet 
stream” [22-24]. For a given static stability (g/®) 
00/dz and baroclinity —(g/@) 90/dy, the hydrody- 
namic stability is greater when the inertial stability 
is large, that is, when the vorticity —du/dy, which 
current u produces in horizontal surfaces, is strongly 
cyclonic (du/dy << 0); it is weaker when the inertial 
stability is small, that is, when the vorticity —du/dy 
is weakly cyclonic or anticyclonic (du/dy > 0). This 
latter condition occurs in the tropical troposphere south 
of the “jet stream.” In that case hydrodynamic in- 
stability can appear for a sufficiently large positive 
value of du/dy. 
The critical slope tan a@ of the isentropic surfaces, 
defined by 6u/éy — f = 0, can be expressed as 
(40) 
tan at = v2/N © 10”. (41) 
The actual slope of these surfaces is given by 
00/dy 2 
yo UY EA 42 
tan ap 36/2 /v (42) 
so that criterion (89) can also be written a» > ag. 
Thus hydrodynamic instability occurs whenever the 
slope of the isentropic surfaces exceeds its critical value 
(41). 
Let us consider the isentropic surfaces 9 = const, 
drawn at constant intervals of, for example, AO = 5°. 
To a first approximation, the baroclinity at any point 
depends on the horizontal distance between two neigh- 
boring isentropic surfaces; the static stability depends 
