DYNAMIC FORECASTING 
and potential temperature respectively at this level, 
pid hm, - OV, Sia / 9 
wait ff ape V, + ov’, (22) 
where 
v = —RTf V,né X k, (23) 
and 
Po 
a = (u(r) — u(p)/P; ule) = fm) dp. (24) 
Pp 
Similarly we find 
A 
A => 
where @ is the pressure average of u(p). If the level p 
is chosen so that u(p) = @, it is easily found that (13) 
becomes 
(ge) ‘u(p)¥,- Vp nd, 
(90) “Bb Vv," Vp In, 
ee = ihe» G Vd + f, ®) 
aM j (25) 
SE (FA ee (G V; In 8, ino) 
at this level. The variations in In 6 are determined from 
eal = 2 27, np = 5 Fottn 8,6). (26) 
The two equations above are sufficient to determine 
the motion. Their advantage lies in the fact that the 
problem has been reduced to a purely two-dimensional 
one: no vertical integrations are required and the initial 
data need consist only of the height and temperature 
distribution in the surface p. 
An empirical study of the function u(p) gave for o2 
the approximate mean value 0.14, and the value 600 mb 
for p in winter. In a specific situation o2 and p seemed 
to be somewhat influenced by the height of the tropo- 
pause, indicating that the position of the tropopause 
exerts an influence at least on the kinematics of the 
atmospheric motion. 
The Barotropic Model. The barotropic model results 
from a further simplification of (25). If one assumes 
that the thermal wind ov’ in (22) is proportional to 
Vv, , implying that the isobars are parallel at all heights, 
=(146 x Ne, (27) 
Vg 
and equation (25) becomes 
vs = J, Cretie ®) 
v /? 1 Ce 
+ a oJ G V,®, ®) 
at the level p. This equation may also be written 
0 
Bo + Kyy-Vato + % a= =i0) (29) 
where 
7 ee 
K=1 oP) Vy a, 
473 
in which form it is seen to be practically identical with 
the so-called ‘“equivalent-barotropic” equation derived 
by the author [3, p. 384]. The quantity | v/v, | 
R | V,T/V,® | has been found to have the average value 
1.25 and consequently K = 1 + (1.25)°(0.14) = 1 22, 
which agrees well with the value K = 1.25 given in 
the author’s paper just mentioned. Defining p* by 
v,(p*) = Kv, and multiplying (29) by K, we obtain 
are ; d d 
So + VE We bt + of oe ‘ob (+f) =0, (30) 
where the asterisks denote quantities at the level p*. 
Hence from (4) we see that p* is the level of nondiver- 
gence and that the vertical component of absolute 
vorticity ¢, + f is conserved at this level. The value 
of p* is about 100 mb less than that of p, or about 
500 mb. 
The foregoing remarks reveal clearly the exact sense 
in which the barotropic atmosphere may be considered 
an approximation to the real atmosphere. 
Kibel’s Method. In 1940, I. A. Kibel [9] proposed a 
numerical method for forecasting surface pressure and 
temperature changes on the basis of the geostrophic 
hypothesis’ by assuming that the lapse rate of tem- 
perature is everywhere constant, that is, 
T(x, Y, Z) mz T(x, y) med (2) (31) 
and that the temperature 7 at a fixed upper level is 
advected with the wind: 
ot + v-VTy = 0. (32) 
The latter approximation holds at the tropopause where 
it is considered to be a consequence of the property that 
the tropopause is a discontinuity surface of the second 
kind at which the lapse rate of temperature is approxi- 
mately zero. With v = v, + v’, where v’ is the geo- 
strophic deviation, the adiabatic equation at the ground 
becomes 
RT 
aL Tie J(To, po) + vo-Vil'o 
(33) 
_ RT 
CpPo 
where c, 1s the specific heat of dry air at constant pres- 
sure. Similarly, equation (82) becomes 
0 
(2 Bp + Vo: Vi po} = 0, 
0% RTx 
— — Vil = 0: 34 
Ap te I(T, pi) + Va-ViTo (34) 
Noting that 
BEY ea sana Daiih yen? ) 35) 
p ps exp ( Rae) (35 
we obtain the thermal wind equation 
ie = wits Vi Po ae ig ViTo (36) 
p Po To 
1. A somewhat simplified account of Kibel’s method by 
B. I. Isvekov has been translated into English. See ‘‘Professor 
I. A. Kibel’s Theoretical Method of Weather Forecasting.” 
Bull. Amer. meteor. Soc., 27:488-497 (1946). 
