472 
unit vector, is integrated from the ground to the level 
z, fixed with respect to time, we obtain 
v= vo +4 [ kx Vs In Ode (6) 
J 0 
and, by differentiating with respect to ¢t and taking the 
curl, 
oe = he 4 ve | ee , (7) 
since the variability of f may be ignored in the differ- 
entiation. To eliminate 0£,0/dt we proceed as follows: 
If we define 
fete ae 
a= se Sclatal 5 (8) 
p 
—— 02 
We 
equation (7) becomes, by integration with respect to z 
a, 0690 = 
— i — V Z 
Oh ey cOla mati a a (9) 
where 
dlné 
A=- ; 
| dz (10) 
Utilizing the relationship [2, eq. 68] 
=! = —vy-Valta + f), (11) 
which may also be derived directly by vertical integra- 
tion of (4), we eliminate 0¢,0/d¢ between (7) and (9) to 
obtain 
065 
aes ee a iy 2 
aye Vo: Vil&g + f) aia (Vi, A VA). (12) 
It now becomes convenient to replace z by p as the 
vertical coordinate. The geopotential @ = gz may then 
be introduced as a dependent variable in place of p, 
and, with slight approximation, the local and horizontal 
derivatives may be replaced by derivatives in an iso- 
baric surface. The latter are denoted by the subscript p. 
For the large-scale systems in which we are primarily 
interested (since only for these is the geostrophic ap- 
proximation valid), f is usually several times greater 
than ¢, , and the mean defined by (8) may be replaced 
by the simple average with respect to pressure. Then, 
since 
bo SF VS, 
equation (12) becomes 
» (a® z qo ee 
vi(@+4-4)-3,(ivie+s,9), (13) 
where J, is the Jacobian operator defined by 
_ da 08 _ da 08 
J p(a, B) mia ay ay ia (14) 
and 
Po 
al ak d In @ . 6 ub (15) 
DYNAMICS OF THE ATMOSPHERE 
with po the surface pressure. We now introduce the 
advective hypothesis, 
] 
ae SY —v,'V> In 6, (16) 
so that 
PO 
A == || v,-Vp no 2 
; (17) 
il pe Ob ®) 
a df dp , 
fg J - (2. E 
with the result that all quantities in (13) are expressed 
in terms of © and its derivatives. 
The numerical integration of (18) is simpler than 
that of the general quasi-geostrophic equation. One 
solves the two-dimensional Poisson’s equation 
V2.5 = Jp (18) 
and determines the field of d®/d¢ from 
- ZAG ae (19) 
There are a number of methods, both analytic and 
numerical, by which (18) can be solved. The best for 
hand calculation is probably the relaxation method of 
Southwell [14] as it is rapid and can be used with any 
type of boundary. However this method is not always 
suitable for automatic machine computation as it makes 
large memory demands. It has been found best for a 
machine with a small internal and large external mem- 
ory to use the analytic solution expressed in terms of a 
Green’s function G: 
=o = , INT J , 
8 = ff Gt, y, 2, vd alay’, (20) 
where the integral extends over the forecast area and 
J is regarded as a function of 2’ and y’. The function G 
is a solution of the homogeneous equation V;G = 0, 
satisfying the same condition as S on the boundary. 
Finally we mention that one may also use an “analogy” 
device, consisting of a physical system whose equi- 
librium state is governed by an equation having the 
same form as (18). The physical magnitude J appearing 
in (18) is varied at will and the magnitude S deter- 
mined by measurement from the resulting equilibrium 
configuration. 
' A Simplified Version. While (13) is already a consid- 
erable simplification of the quasi-geostrophic equation, 
it still presents appreciable difficulties for numerical 
integration. Further simplifications will therefore be 
considered. The first is based on the observation that 
the isolines of temperature and potential temperature 
in large-scale systems are approximately parallel at 
all levels, that is, 
A Vy n 6 = ae V, In 8 (21) 
p 
where the bar refers to a fixed isobaric surface p. If 
V,,p, 1, and @ are the velocity, density, temperature, 
