DYNAMIC FORECASTING 
tendency S — A at a point on the ground requires a 
knowledge of the potential temperature advection 
throughout the entire vertical column above the point, 
in contradiction of what has just been stated; and here 
there is no compensation mechanism that effectively 
limits the spread of influences as in the case of the 
geostrophic approximation. It is thus possible that 
the advective assumption imposes a physically un- 
tenable constraint on the atmosphere. 
Computational Stability 
Once the physical problem of determining the equa- 
tions of motion and the boundary conditions has been 
solved, there arises the purely mathematical problem 
of approximating the solution of the continuous equa- 
tions by finite-difference methods. Here it may come 
as a surprising fact that no matter how small one 
chooses the space and time increments for the finite- 
difference equations, one has no guarantee that the 
finite-difference solution will approximate the contin- 
uous solution. This phenomenon, first discovered in 
1928 by Courant, Friedrichs, and Lewy [6] will be il- 
lustrated for the one-dimensional wave equation 
d~e ps de 
at 0x?” 
(54) 
whose finite-difference analogue for centered differences 
may be written 
g(a, t + At) + o(a, t — At) — 2(z, t) 
(Ai)? 
ashe) o(x + Az, t) + o(a — Aa, t) — 2¢(z, t) 
(Az)? 5 
Because of the linearity of (55) we may suppose that 
the solution is represented by a Fourier series. The 
long wave length components will be accurately ex- 
pressed by the solution of (55), but there will inevit- 
ably be a distortion of the components whose wave 
lengths are of the order of Az. This distortion will be 
harmless if Av is small compared with the wave lengths 
of the physically relevant components and if the small 
wave length components do not amplify. However, 
when such an amplification occurs, it does so with 
exponential rapidity and quickly makes nonsense of 
the entire solution. 
Let us, therefore, consider a Fourier component of 
(55) 
the form exp [7(ka — vt)]. Substitution im (55) gives 
got aL gue = 9 cA 2 ee aL gee = 9) 
(At)? (Ax)? 
hoe 4c sin? kAx 
(Ax)? ae 
awvAt 
and, if we set w = e '™', 
2 kAa 
2 of At re 
—2(1-2 = 
(3) ( Cc i sin 9 
The small error will cause no difficulties if it does not 
amplify, that is, if |w|< 1. If the two roots of (56) 
Jor =), (6) 
477 
are complex, the square of their common absolute 
value is given by their product, which by (56) is equal 
to unity. If the roots are real, one of them must exceed 
unity in absolute value. Hence the condition for com- 
putational stability is that the roots be complex, that 
is, that the discriminant of (56) be negative. This 
leads to the condition 
Ax NG 
= 
Me om 
or, since k must be presumed to have any value, 
Ax 
iG > 6 (57) 
Thus, no matter how small Az and At are chosen in- 
dependently, the solution of the finite-difference equa- 
tion will not approximate the continuous solution un- 
less (57) is satisfied. A geometrical interpretation of 
this criterion has been given by the author in a previous 
article [3]. 
Now it may be shown that whenever equations of 
motion permitting physical propagations are used, the 
condition (57) must be satisfied for c, the greatest 
propagation speed. Although Richardson [12] effectively 
excluded sound waves with the hydrostatic approxi- 
mation, he did not exclude external gravity waves 
whose speeds are nearly as great as those of sound, 
about 300 m sec '. He chose Az to be about 200 km; 
consequently his A¢ should have been smaller than 
200,000/300 see or 20 min. In point of fact he chose 
6 hr. Hence, a direct application of his method would 
inevitably have led to computational instability. 
By employing the quasi-geostrophic equations one 
avoids the difficulty of having to choose time incre- 
ments so small as to be meaningless in relation to the 
meteorologically significant motions. Nevertheless, the 
space and time mecrements cannot be chosen arbi- 
trarily. To show this, let us again consider equation 
(45). 
If we use centered differences, its finite-difference ana- 
logue may be written 
D? [= y, t + At) — &(a, y, t — Ad) 
At 
4 py Pet As yd = P(x — As, y, 2] (58) 
2ZAs 
= plas)? Pet AS Hs ) = Ol — As yf) 
2As 
where As is the space increment and the finite-differ- 
Ok 6 > 
ence operator D° is defined by 
Dy = ¥(a + As, y, t) + ¥@, y + As, t) 
dP ANS Ohh t 
+ ¥(x 8, y, t) (59) 
ae Y(x, Yh ery As, t) oa Ay (x, Y, t). 
Consider a Fourier component of the form exp 
[i(ka + wy — vt|= w exp [i(ka + py)]. Substitution in 
(58) gives 
w + 2aiw — 1 = 0, 
