DYNAMIC FORECASTING 
east, y north, and z upwards. The quantity f is the 
Coriolis parameter 2Q sin ¢, with Q the angular speed 
of the earth’s rotation and ¢ the latitude; p is the 
pressure; a, 6, and y are functions of p and its space 
derivatives; 9 is the potential temperature, and ¢, is 
the geostrophie vorticity: 
~~ LieD TP) a i, =o 
Stash & aay) ar te @) 
where Y; is the horizontal operator, p the density, and 
the symbol “=” denotes approximate equality. For 
simplicity of presentation, the curvature of the earth 
is here ignored. 
Equation (1) is elliptic or hyperbolic in the pressure 
tendency according as the factor multiplying the ¢-de- 
rivative terms is positive or negative. Since this factor 
has the same sign as the approximate potential vorticity, 
p (€ + f)@ln6/dz, we may infer that the elliptic or 
hyperbolic character of (1) is preserved at a material 
point. Thus we are assured that if (1) is everywhere 
elliptic initially it will remain so. On the other hand, if 
(1) is elliptic at some poimts, and hyperbolic at others 
it will retain this property. In the latter case the 
integration problem becomes exceedingly complicated. 
Moreover, it is not known if, or to what extent, the 
geostrophic approximation applies when the potential 
vorticity is negative. Fortunately, however, negative 
potential vorticities occur rarely, and then only locally, 
and we may therefore be justified in assuming at the 
start that equation (1) is elliptic everywhere and for 
all time. (In the more general case, where the isentropic 
surfaces are not assumed quasi-horizontal, it can also 
be proved that the pressure-tendency equation is and 
remains a second order elliptic differential equation if 
the potential vorticity is initially positive everywhere.) 
We may envisage a solution of equation (1) by the 
followmg procedure. Assuming that p is known at 
time ¢, we evaluate its coefficients and nonhomogeneous 
terms at time ¢ and solve for dp/dt by a relaxation 
method (Southwell [14]). The boundary conditions are 
that there shall be no influx or efflux of mass through 
the bottom or top of the atmosphere, translated into 
equivalent conditions on the pressure tendency. Hav- 
ing obtained dp/dt we calculate the pressure at time 
t + Aé from 
ptt + At) = p(t).+ At op/dt. 
The sequence of steps is then iterated to give p(t + 2At), 
p(t + 8A), ete. j 
As computing machinery for calculations of such 
great’ complexity have not yet been available, more 
simplified atmospheric models have been devised to 
test the validity of some of the basic assumptions. 
These will now be described. 
Advective Models 
The General Case. A first simplification is obtained 
by ignoring the effect of vertical motion on the change 
in potential temperature. Although this assumption is 
more questionable than those already made, one cannot 
ignore the fact that the advective hypothesis has been 
471 
used with a degree of success in present-day synoptic 
practice. Hence, in accordance with the plan of utilizing 
the results of synoptic experience in the construction 
of models, we shall investigate the form taken by the 
equations of motion under this hypothesis. 
First, by way of further justification, we shall show 
by means of scale considerations that the term 
wd In 6/d% + vdl1n 6/dy is usually larger in order of 
magnitude than wd In 6/dz in the adiabatic equation, 
dln 6 
din@é_dlm@ oie 0iné 
dt SHRM t. Oe aA ARISTA TAGE 
= 0, (3) 
in which u, v, and w are the 2, y, and z velocity com- 
ponents, respectively. 
From the geostrophic and thermal wind equations 
we have 
0 In é OMG ai op Ou 
Oe =v ay See 2), 
and from the vertical vorticity and continuity equa- 
tions 
1 F) F) a) Wes 
aalatud+ 3c 4 i) é 
a (4 ele 
e dx | Oy) p a dz" 
The symbol ‘‘~” denotes equality in order of magnitude 
ee a sinusoidal dependency, with wave lengths 
(he Ug 5 BIOGl Us 5 
u = Uexp [2mi(x/l. + y/ly + 2/1.)] 
v = V exp [2mi(a/l. + y/ly + 2/l2)I 
w = W exp [2ri(a/l. + y/ly + 2/12)I, 
where U ~ V and I, ~ l, , and noting that each term 
on the left-hand side of (4) has the same order of 
magnitude, we obtain 
wd In 6/dz le vs 
ud In 6/dx + vd In 6/dy iB ive 
where v; = f is the frequency of a horizontal inertial 
oscillation and v3 = gd In 6/dz is the frequency of a 
buoyancy oscillation. For the large-scale motions in 
middle latitudes U, , l, = 2 X 10° to 6 X 10° m; 1, = 
10° m to 2 X 10° m; », & 10° sec’; and »%» & 10% 
sec . Hence the above ratio varies between 0.03 and 1. 
For a typical large-scale system (l,, l, = 4 X 10° m; 
1. = 1.5 X 10’ m) its value is about 0.14. We note, 
however, that the advective hypothesis is less valid 
in the stratosphere where the buoyancy frequency % is 
several times greater than 10° sec -. 
The following derivation of the geostrophic-advective 
equations is due substantially to Fjgrtoft [5], although 
it is much in the spirit of Sutcliffe’s work [15]. 
If the thermal wind equation 
OV, J i 
— =k V;, In 0, 5 
ae j x Vi (5) 
in which vy, is the geostrophic wind and k is the vertical 
