404 
is not a variable of state. If this quantity is also to be 
considered as unknown, further additional equations 
have to be introduced for the transfer of heat by con- 
duction and radiation. In many instances in meteoro- 
logical problems it may be assumed that the changes 
of state are adiabatic, so that dh = 0. A slight gen- 
eralization of this procedure is to assume polytropic 
changes where 
dh = caT™*. (11) 
Here ¢ is a constant of the character of a specific heat. 
In most meteorological problems it can further be 
assumed that 
de = c, dT*, 
where ¢, is the specific heat at constant density, an 
assumption which holds for an ideal gas and is con- 
sequently as a rule a satisfactory assumption for the 
atmosphere. Then with the aid of (9), 
RT* 
edit = ed = oF dp*, 
since the specific heat at constant pressure for ideal 
gases 
Oy = Gy ap dite 
By integration of the preceding differential relation 
under the assumption that the specific heats are con- 
stant, which applies to ideal gases, 
T* p* K 
m= Gz) 
where K = (cp — ¢,)/(c» — c). The relation between 
pressure and specific volume consequently becomes 
(13) 
(12) 
Se SS 
= Pogo ; 
p*q* 
where \ = (c, — c)/(c, — c), the modulus of the poly- 
tropic curve. We shall confine ourselves in the follow- 
ing discussions to the special case of adiabatic changes 
when 
pee 
Cy 
and 
ou = jae (13a) 
Equation (13) or (13a) gives the variation of the specific 
volume of a fluid particle as a function of the pressure 
variation. The original pressure and density, po and 
qo will, of course, in general vary from one particle 
to another. Compared to the more general equation 
(8), equations (13) and (13a) have one variable less 
since the temperature is eliminated by the additional 
assumptions about the physical process by which the 
particle changes its density. Thus, the number of de- 
pendent variables is reduced by one. Such an equation 
for the change of state of a particle which, in addition 
to the pressure, contains only one more variable of 
state has been called by V. Bjerknes an equation of 
DYNAMICS OF THE ATMOSPHERE 
piezotropy, and a fluid whose every particle is follow- 
ing an equation of this type is called piezotropic. More 
generally, a piezotropic fluid satisfies an equation of 
the form 
i@* Gia, Aly B, 220) a 0, (14) 
where the parameters A, B, --- will in general not be 
the same as in (8). Equation (14) may also be written 
in the form 
It is often convenient to introduce the coefficient of 
piezotropy, 
_ ag" 
eS dp* (15) 
In the case of adiabatic changes, for instance, 
Aer (15a) 
ae zt 
It should be clearly understood that this coefficient of 
piezotropy describes the physical changes which a parti- 
cle undergoes, while the coefficient of barotropy depends 
on the distribution of the fields of pressure and specific 
volume. 
When an equation of piezotropy exists for the fluid, 
the system of hydrodynamic equations is complete 
because it consists now of five equations with five 
dependent variables. 
The properties of barotropy and piezotropy are en- 
tirely unconnected. The former refers to the stratifica- 
tion of a fluid, the latter to its changes of state. A fluid 
whose stratification is barotropic at a given instant 
will in general not remain so if in motion. However, 
if the appropriate state of piezotropy prevails, barotropy 
may be maintained. Such a fluid is called autobaro- 
tropic, as mentioned before. The condition for auto- 
barotropy is that the coefficients of barotropy and of 
piezotropy are identical, provided, of course, that the 
fluid was barotropic at a fixed instant. Examples of 
autobarotropic fluids are a homogeneous and incom- 
pressible fluid, or an atmosphere with adiabatic changes 
of temperature and an adiabatic lapse rate of tempera- 
ture. In such a fluid a displaced particle has always 
the same density as its environment so that indifferent 
equilibrium is maintained. Consequently, autobaro- 
tropic fluids are only a minor generalization of the 
homogeneous and incompressible fluids considered in 
classical hydrodynamics. 
The Hydrodynamic Equations. Any fluid, whether 
compressible or not, is subject to the laws of dynamics. 
For continua, the equations expressing these laws are 
more complicated than for the mass points of particle 
dynamics, because the direct effects of neighboring 
fluid particles on each other have to be taken into 
account. This interrelationship leads to the appearance 
of the terms for the viscous stresses and for the pres- 
sure-gradient forces in the hydrodynamic equations of 
motion. In many instances in classical hydrodynamics 
and in geophysical applications, however. the effects 
