HYDRODYNAMIC INSTABILITY 
where 
U Ou U oll 00 
= 9 — }{ 2w, — — + — —-—, 
"8 («. a all oy au z) dy dy 
U Ou U co) 0 Wolo) 
= =3 Dp. y ime =a ) 
on = 20+ Be) 260 45 — Re) ay 
(76) 
u Ou U oll 06 
=-—2 = = ||| My, = == SE — |) = == 
ag («, ( may a z) dz dy” 
U Ou U oll 08 
eas 2 (a — F243 =o l-e az’ 
with a,, = a.,. We note that in the differential quo- 
tients on the right-hand sides of (76), dy and dz repre- 
sent elements of are along the coordinate lines « and 
o and that these elements of are are not in general 
exact differentials [41]. 
There is an obvious analogy between the quadratic 
form (75) and equation (13) which holds in the case 
of geostrophic motion. In the case of isobaric motion, 
this form retains the dynamical significance and the 
interpretation on an energy basis of Kleinschmidt’s 
form (see pp. 434-487). Moreover, we can apply the 
discussion of Kleischmidt’s quadratic form to the 
sign of this form. If we assume the vertical distribu- 
tion of mass to be stable in the field of gravity, the 
state of isobaric motion is then stable at any point, 
whatever the transverse displacements from the poimt, 
with the exception of displacements close to the isen- 
tropic surface when 
Ve - curl U = gO curl, U + ge curl, U < 0, 
oy 0z 
where 
U 
curl, U = 2w,, =o 
(63) R. 
curl, U = 20, + OU 
Oz 
Ou U 
curl, U = 20, — — + —; 
ur (6) ay RP 
y 
U = W + u being the absolute velocity of the air. 
When the coordinate surfaces c° = const are equipo- 
tential surfaces in the field of gravity, the z-axis coin- 
cides with the zenith direction at the origin. If we 
assume that the equigeopotential surfaces are spherical 
and concentric with the earth, R. = —r, where r 
stands for the distance from the earth’s center to the 
point under consideration. In this case, R, represents 
the radius of curvature at O of the horizontal projec- 
tion of the isobar obtained by varying o'; in synoptic 
meteorology R, is “the radius of curvature R;, of the 
isobar” at this point, (R, = R;,). We note that R;, > 
0 or < 0, depending on whether the circulation along 
the isobar is cyclonic or anticyclonic at the point con- 
sidered. Since 2w, + wu/r is negligible compared to 
du/dz, it is evident that the isobaric motion is un- 
stable in the neighborhood of the isentropic surfaces 
when 
(77) 
447 
where 6u/dy has been defined by (85). This formula 
shows that the threshold value f + w/R;; of the in- 
stability of isobaric motion is higher or lower than 
the threshold value f of the instability of geostrophic 
motion (R;; = «) depending on whether the curva- 
ture of the isobar is positive or negative. Anticyclonic 
circulation thus favors hydrodynamic instability [41). 
When the isobar at O coincides with the parallel 
through this point, R;, = r ctn ¢, where ¢ is the latitude 
of O; here the limit of hydrodynamic instability is 
given by the approximate expression [22, 41] 
HH ou cing + tne 
oy r 
By virtue of the instability criterion (77) for a per- 
manent, isobaric, horizontal current, cyclonic currents 
as a whole are more stable than anticyclonic currents. 
Let us assign an arbitrary value (6w/dy) to the trans- 
verse isentropic gradient 6u/dy of the isobaric current 
u. Let us suppose that it satisfies the condition most 
generally encountered, that is, 6u/éy < f. The domain 
of stable currents will then be separated from the 
domain of unstable currents by an anticyclonic current 
of the curvature 1/R7, = (1/u)[(6u/éy) — f] < 0. In 
this case, the geostrophic current (R;, = ©), the 
limit between the domains of cyclonic and anticyclonic 
currents, will be located within the domain of stable 
currents composed of all cyclonic currents and of 
weakly anticyclonic currents (| Ri, | > | Ris. |). The 
domain of unstable flow contains only anticyclonic 
currents with a radius of curvature less than | Ri, | . 
Let us now admit, following Wippermann [46], the 
existence of a transverse isentropic exchange of air par- 
ticles. When the current wu is stable, this exchange can 
maintain itself only at the expense of the surrounding 
medium, and the displaced particles acquire an anti- 
cyclonic circulation with respect to the current wu (see 
p. 442), thereby attenuating the curvature of the cur- 
rent wu when it is cyclonic or reinforcing it when it is 
weakly anticyclonic (| Rj,| > |R%|). In this case, 
the current u, whether cyclonic (R;,; > 0) or weakly 
anticyclonic (Ri; < 0 with|R;,| > |R%|) will be 
transformed progressively into an anticyclonic current 
whose radius of curvature will approach the limit 
| R*.| = 10° m. On the other hand, if the current wu 
were unstable (R;, < 0 and | R;, | < | Ri: | ), the trans- 
versely displaced particles in the isentropic surfaces 
would acquire a cyclonic circulation (see p. 443) with 
respect to the current w, releasing energy of instability. 
The anticyclonic curvature of the unstable current wu 
would thereupon diminish, that is, the current w would 
spontaneously become a current within the domain of 
stability. Summarizing, we may say that an anti- 
cyclonic current whose radius of curvature | R;; | > 
| R%, | approaches the limit | R?.| would thus appear 
to be a current of some persistence [46]. 
Inasmuch as hydrodynamic instability is a transi- 
tory state of ephemeral duration, it will not manifest 
itself unless its cause is of even less duration (7.e., 
produced instantaneously); according to Kleinschmidt 
(78) 
