632 
Pressure Tendency Equations 
In recent years the problem of pressure changes has 
often been approached from the standpoint of analyzing 
the components of various equations for the pressure 
variation dp/dt. Bjerknes [2] discussed the development 
of pressure changes by use of the following equation 
for the pressure change at a level h. 
op 5 dp a) 
—— oe — ——w = d.: 
(2), : a (ue + 0% i 
Sune ; (2) 
u v 
[ ge iS + =) dz + g(pw),, 
where wu, v, and w are the x-, y-, and z-components of 
the wind. Equation (2) states that the pressure change 
is determined by the horizontal advection and diver- 
gence above the level h and by the vertical advection 
at h. Sufficient evidence has been accumulated with the 
advective integral [15] to show that it is of the same 
order of magnitude as the pressure tendency. On the 
other hand the divergence integral and the vertical 
motion term are usually one order of magnitude greater 
than the pressure tendency. It follows then that the 
contribution of the last two terms of equation (2) is 
the small difference between two large quantities. This 
difference is expressed analytically when the tendency 
equation is written in the form given by Houghton 
and Austin [18]: 
U] 
2 dp 0p 
7 i aa i ) de 
This tendency equation shows that the field of hori- 
zontal divergence and vertical motion gives rise to a 
pressure change depending upon the manner in which 
the motion changes the density distribution above the 
level h. 
Alternate forms of the tendency equation have been 
derived in various ways. The reader is referred to a re- 
view given by Godson [13]. Another example of a 
tendency equation which does not involve the integra- 
tion to infinity is given by Matthewman [20]. In recent 
years, with the availability of upper-air data, there 
have been attempts to compute the values of the factors 
which give rise to pressure changes. For example, 
Fleagle [11] has given values of the components of 
equation (4) at various levels in the atmosphere. 
Lido weil =k opie 
= EVR ng) IO say yea § 
p at p BV EGsa ah 6 dt (4) 
K = R/c,, Ris the gas constant for dry air, cp is the 
specific heat at constant pressure, @ is the potential 
temperature, V;, is the horizontal velocity vector, and 
Vn is the horizontal gradient of potential temperature. 
The significance of these terms is apparent when it is 
noted that 
MECHANICS OF PRESSURE SYSTEMS 
Op\ _ “1 dp : 
(2) a 0 pot cee (5) 
Fleagle has also presented values of the horizontal mass 
divergence. 
The observational evidence of Fleagle and others in- 
dicates that all the components of the tendency equa- 
tion are important, that all layers of the atmosphere 
contribute to the various integrals, and that the net 
value of an integral is often the small difference between 
large values of opposite signs. For this reason it is not 
possible to compute the pressure tendency accurately 
from the components of a tendency equation. 
The utility of these tendency equations will now be 
considered. A relationship as expressed by equations 
(2)—(5) gives an insight into the mechanism of pressure 
changes only as long as there is a clear understanding 
of how the components of the equations change. All 
the tendency equations involve the three-dimensional 
field of motion and it can readily be shown that, for 
the computation of the pressure variation, the velocity 
must be known to an accuracy of the order of 1 cm 
sec. Consequently a tendency equation is helpful in 
understanding pressure changes provided that the 
mechanism of wind change is known. This involves a 
knowledge of the manner in which accelerational fields 
develop in the atmosphere, and undoubtedly the local 
pressure changes themselves are of major importance 
in producing accelerations. From such considerations as 
the above it would appear that the tendency equations 
do not contribute significantly to an understanding of 
the mechanism of pressure changes. Computations of 
the magnitudes of the various terms do indicate, how- 
ever, the importance of the various processes which . 
accompany pressure changes. These computations show 
what is happening in the atmosphere while the pressure 
changes. For this reason it is necessary that the em- 
pirical work be extended as it is probable that the 
vertical distribution of such quantities as the horizontal 
divergence differs from one part of an isallobaric system 
to another. Finally it might be noted that these tend- 
ency equations have no prognostic value insofar as it 
is not possible to predict the values of the various inte- 
grals of the tendency equations. 
A number of theoretical attempts have been made to 
analyze the pressure-change mechanism by means of a 
tendency equation and an assumed wind field. For ex- 
ample, Bjerknes and Holmboe [3] discuss the distribu- 
tion of horizontal convergence and divergence at various 
levels in the atmosphere when it is assumed that the 
wind is gradient at trough lines and wedge lines. Priest- 
ley [22] also analyzes pressure changes on the assump- 
tion of gradient flow and stresses the importance of the 
path curvature on the control of atmospheric pressure. 
Schmidt [27] discusses the components of a tendency 
equation through the introduction of an approximation 
concerning the acceleration of the wind. These ap- 
proaches are subject to criticism on account of the 
assumptions concerning the wind field. Houghton and 
Austin [18] and Priestley [23] discuss aspects of the 
