436 
sists of two parts: the first, perpendicular to the isobaric 
surfaces, reduces to the hydrostatic buoyant force of 
Archimedes, in the absence of the geostrophic current 
u; the second, perpendicular to both the geostrophic 
current and the earth’s axis, is an inertia force due to 
the earth’s rotation. We observe that in an adiabatic 
atmosphere (© = ©) = const), where differences be- 
tween the density of the displaced particle and that 
of its environment disappear, only the second of the 
two forces remains. If in (10) we replace © — © by 
(00/dy)oy + (00/dz)oz and vz by its value in (8), we 
find, whatever the orientation of transverse axes y and 
z, provided always that the coordinate system xyz is 
rectangular and right-handed, 
My = = =e, t= 029 = 2, Cl) 
with 
ou oll 08 
= 2 Zz 2 La Ts ee ae tae 
ne of i =| dy dy 
Ou oll 08 
yz = —2 2! 2 Ay | |Mik? M SAO 
“2 (20) + 2) ay dz 
etpaeaace) X85 NMoGumer aa 
Oa eae Om az dy’ 
OU oll ae 
= 2, eS) 
Oke 2a oy =) dz dz” 
and the symmetry relation a,z = az, resulting from the 
equilibrium condition (2). 
Making use of equations (11) and (12), which give 
the components of the force t that acts on perturbed 
particle A as functions of the physical and dynamical 
state of the atmosphere at the equilibrium position 
O of A, and of the transverse components of displace- 
=> 
ment OA, it is easy to determine the nature of the 
hydrodynamic equilibrium at point O. Depending on 
whether the applied force (J, , wz) and the displace- 
ment (y, 2) are of opposite or of the same sense, this- 
force tends to return the displaced particle A toward 
its equilibrium position O or to carry it away from this 
position, that is, depending on whether —y,y — y.z > 
0 or <0, the hydrodynamic equilibrium at O is stable 
or unstable. 
We thus rederive E. Kleinschmidt’s criterion: the 
state of geostrophic motion at any point, for any transverse 
direction r, , r. at this point, is stable or unstable depend- 
ing on whether |17, 18, 33, 37] 
Qty, Tz) = Ay ty + Wyety Ts + Geet2 > or < 0, (13) 
or again, 
Q(ry, 72) = 2(r X w),(t X curl VU); 
(14) 
— (r-VO)(r-VI) > or < 0, 
where 
curl, U = 2. , 
a Ou 
eurl, U = Qwy + aan \ (15) 
ou 
eurl, U = 2w, au? 
DYNAMICS OF THE ATMOSPHERE 
are the components of the curl of the absolute geo- 
strophic velocity U = u + o XR, in which R is the 
radius of rotation of the point relative to the earth’s 
axis. As a special case, in an adiabatic atmosphere where 
no Archimedean buoyant force is possible, there is said 
to be inertial stability or imertial instability [382], de- 
pending on whether 
a 
2(r X w)z(r X curl U), > or < 0. (16) 
Next we multiply equations (9) respectively by dy/dt 
and dz/dt. By adding them we obtain, after integrating, 
the relation thus found and taking account of initial 
conditions (5), 
Vly)” + (v.)7] — WI)” + (2)'] 
= — Way, y? + 2a}. yz + adez']. 
Consequently, depending upon whether the hydrody- 
namic equilibrium at O is stable or unstable for dis- 
placement (y, z), the kinetic energy per unit mass of 
the particle displaced transversely from (y, z) decreases 
or increases. The work Hp which is performed in the 
course of a displacement (y, 2) by the transverse force 
per unit mass, applied by the surroundings to the dis- 
placed particle, will be referred to as energy of instability 
per unit mass, latent at point O. It follows that 
K = —VWlany i 2ay.Y2 ar az. |. (18) 
In the case of stability this work is negative; the sur- 
rounding medium opposes displacement (y, 2); in fact 
this displacement ceases as soon as the kinetic energy 
of the initial impulse is dissipated. Under conditions 
of instability, on the other hand, this work is positive: 
the surroundings further the displacement (y, 2); to 
the kinetic energy of the initial impulse is added the 
energy of instability Hy set free in the course of the 
displacement [35, 37]. This gives an interpretation on 
an energy basis of the quadratic form (13). 
We return now to equation (4). By multiplying 
scalarly with V, we obtain a differential expression 
which can be integrated directly. This furnishes the 
first integral of the equation of motion of the perturbed 
particle, 
Wi(V,)° + (v,)? + (@)"] + Goll + & = const. (19) 
We set S(y, z) = 14(Vz) + Ooll + &, where V, is given 
as a function of y and z by (8) and (8). After substitut- 
ing initial conditions (5) and equation (17), equation 
(19) assumes the form [18] 
S(y, 2) — S(O, 0) = lan y” + 2ay.yz + azz]. (20) 
However, we have assumed that the perturbation of 
the particle considered is small, so that the right-hand 
side of (20) constitutes in reality only the first term of 
the expansion of the difference S(y, z) — S(O, 0) ina 
Taylor series; therefore, (0S/dy)o = (0iS/dz)o = O and 
(8°S/dy")o = an, , (8 S/dydz)o = Ay. = zy, (8°S/d2")o = 
as. . We conclude that a particle A, in the interior of an 
air mass in geostrophic motion, occupies an equilibrium 
position O when the sum of 2ts longitudinal kinetic energy 
per unit mass 14(Vz), its specific enthalpy oll, and its 
(17) 
