where 
2 
a U B(As) /4 el sin kAs. 
ae kAs + sin? As | As 
2, 2 
The condition for stability, | w| < 1, is then found to 
be a < 1, and we obtain 
1 Do 
os > U sin kAs + (14)B(As) sin (kAs) ; 
a NAS ._, pAsS 
Sn roe  SEninR 
An upper bound for the first term on the right-hand 
side is U. The second term increases with decreasing 
k and yp, and we may evaluate it by assuming kAs and 
pAs to be small, thus obtaming the upper bound 
BAsk(k + py’) *. Let us now assume that (58) is to be 
solved for a rectangular region with sides L, and L, 
parallel to the «- and y-axes, respectively. Then k and 
pw must be greater than 27/L, and 2m/L, , respectively, 
and the stability condition becomes 
As BAs L’ 
>U+—_— (60) 
NG = 
where Z is the square harmonic mean, 
Dade + Ly yy? 
of L, and Ly, . 
If As is small compared to the dimensions of the 
rectangular area, the second right-hand term will be 
small, and we obtain the result that the ratio of the 
space to the time increment must be greater than the 
particle velocity. It is shown in reference [5] that a 
stability condition closely analogous to (60) holds for 
the general nonlinear barotropic equation. 
Integration of the Nonlinear Barotropic Equation 
In the following section a description will be given 
of a method that has been used to integrate the baro- 
tropic equation (30). 
Equation of Motion. The nondivergence level p 
is taken to be 500 mb. If the spherical surface of the 
earth is mapped conformally onto a plane and the geo- 
potential @ of the 500-mb surface is used as dependent 
variable, the correct form of (80) for a spherical earth 
becomes 
* 
2202 2 (™ vad po 
Gane Dp f Dp ? b) 
or 
Ob m 
Ve 35 = Jp ie Vie + f ®), (61) 
where V; and J, are to be interpreted as operators in 
the plane, and m is the scale factor of the mapping. 
Except for the factor m’/f, the left-hand side of (61) 
is the change in absolute vorticity and the right-hand 
side is the negative of the absolute vorticity advection. 
The equation therefore asserts that the vertical com- 
ponent of absolute vorticity is individually conserved. 
The Conformal Map. The stereographic projection 
DYNAMICS OF THE ATMOSPHERE 
has the advantage that it is the map on which hemi- 
spheric data are usually plotted and analyzed and that, 
in comparison with the Mercator projection, it has 
much less distortion at high latitudes so that a square 
lattice represents more nearly equal areas on the earth. 
If ¢ is the latitude, a the radius of the earth, and the 
radius of the equator on the map is chosen as the unit 
of distance, the scale factor for the stereographie pro- 
jection is m = 2a (1+ sing). 
Boundary Conditions. Since data are lacking for the 
entire earth and as the geostrophic approximation does 
not hold in the vicinity of the equator, the integration 
must be performed for a restricted area. The problem 
then arises of determing the boundary conditions. 
Tt is not a serious difficulty that these conditions are 
not actually known as functions of time, since we 
already know that influences from the boundary will 
not propagate inwards at a rate much greater than the 
particle velocity. One has only to choose the boundary 
sufficiently far from the area for which the forecast 
is desired. However, it is still important to assign 
artificial boundary conditions that are dynamically cor- 
rect, for experience has shown that if the conditions 
assigned are not physically possible, computational in- 
stabilities arise which propagate faster than the 
physical disturbances. 
If one wishes to solve (61) for the initial tendencies 
only, it is sufficient to prescribe d®/dt on the bound- 
aries. This may be done by first making a crude fore- 
cast of d&/dt, say by extrapolation from the observed 
12-hr height change, or, more roughly, by setting d/dt 
equal to 0. But in forecasting for a finite time mterval, 
it is not sufficient to specify d®/dt, and therefore ®, 
for all time. For this case one may show [5] that if 
fluid is flowing into the forecast region both the flux 
of vorticity and ® must be specified, whereas if fluid 
is flowing out of the region, only ® need be prescribed. 
Since the specification of & determines the normal 
velocity at the boundary, one has only to specify the 
vorticity in addition to where fluid is entering. 
Method of Solution. The method to be followed in 
the solution of (61) will depend on the types of com- 
puting instruments available. It may be of some in- 
terest to describe briefly the integration procedure used 
by the writer in collaboration with R. Fjgrtoft and 
J. von Neumann [5]. 
With the notation £ = V,® the basic equation (61) 
may be replaced by the system 
n=mf%&+ f, (62) 
d£/dt = Jp(n, ®), (63) 
Vr(db/dt) = d&/dt, (64) 
with the boundary conditions 
db/dt = 0; (65) 
dg/at = 0, if d&/ds < 0, (66) 
where s is the tangential coordinate along the bound- 
ary, directed so as to keep the interior domain always 
on the right. 
