474 
and the relationship 
lop _ RTxz dpo gz OT 
AOR Fay | GE a By On” (87) 
from which it follows that (34) may be written 
OT) RT x i? 
— — —— J(TM, mo) + Vz-Vi To = 0. 38 
at Tip (To, po) + Va-Vi To (38) 
If the winds are assumed geostrophic, (33) and (38) 
combine to give 
ay = —aJ (To ) Do) 
me ; (89) 
ra ey BJ (To ) po) 
where 
a= %(1 — 7), = fils 
f Tes fpo 
These are Kibel’s equations for the first approxi- 
mation. Their integration becomes especially simple 
if it is observed that in virtue of (39) 
x (aDs iL Gin) = ©, 
so that 
6 = aTy + Bpo (40) 
is independent of time, and 
te) 
= = J(po, 9) 
(41) 
OT) _ 
he J(To , 8) 
These equations imply that the fields ) and Tp are 
propagated along lines of constant @ with a velocity 
numerically equal to | V,@|. They may also be in- 
terpreted to state that the surface pressure and tem- 
perature are steered along the contour lines of the 
isobaric surface located at approximately the height 
Z = aT /(ay + g8po) and with a velocity given by the 
wind at this level multiplied by (ay + g8po)f/g. In 
practice, the constants a and 8 are determined em- 
pivically rather than from their formulas. 
In 1948 Lettau [10] observed that Kibel’s equations 
of first approximation are very similar to a set derived 
by Exner in 1906 [8] under similar assumptions. In 
place of Kibel’s tropopause condition, Exner assumed 
that there exists a level H at which the pressures do 
not change. If H is identified with Z, Exner’s equations 
become nearly identical with Kibel’s. 
Equations (89) give zero pressure tendency at a 
pressure center, where 0p0/d% = Opo/dy = 0. For an 
estimate of the tendency at a center Kibel proceeds 
to a second approximation. As the derivation is omitted 
in both Kibel’s and Isvekov’s papers, it will be supplied 
here. 
DYNAMICS OF THE ATMOSPHERE 
The geostrophic deviation 
1 fav ov dv 
D rps 
Ue (a tu + 0) 
1 fou Ou Ou 
Vw pate pao ered 
wp (+ ut + 0M) 
is further approximated by substituting the geostrophic 
wind components u, = —p f ‘(dp/dy), % = 
p f ‘(Op/dx) for w and v. Ignoring small terms and 
using (36) and (37), one obtains, when dp/dx = 
Opo/dy = 0, 
Ria i en R(T — To) a’ po ge aT 
shemale foo dxdt  f?T') dxdt 
gH” aT.\  gHRTx Apo 
— oR (%, =) 7 pm 2m Be) (42) 
Home panei are? I) Sry Glel GPM | 
FPo oyot fT dydt 
gH? aT)\  gHRTn Apo 
— od (Te, oy = f polo J To, ay 
Evaluating 0°po/dxdt and d°po/dydt from (39) and sub- 
stituting in (33) and (88) we obtain, by elimination of 
OT /dt, 
dp »| aT a6\ | ao 06 
= aT + Bpo 
2 
ye aes eso 
3) a/ (2% r Tx i) ) 
ne Yo Pout 1) 
vy Tx 
inh] 
| 
Qr 
| 
and ya is the dry-adiabatic lapse rate. Making the ap- 
proximation 6 & 98, it is possible to deduce the con- 
sequence that the surface pressure decreases when the 
6 contours at the steering level Z diverge over the 
center in the direction of the wind. Conversely, if the 
contours converge, the surface pressure increases. 
In view of the nature of the assumptions made, 
Kibel’s first and second approximations cannot serve 
as a substitute for the general advective geostrophic 
equations. There have been references to higher ap- 
proximations, but the articles unfortunately were not 
available to the writer. 
The Linearized Barotropic Model 
The nonlinearity of the quasi-geostrophic equations 
makes it difficult to study their properties. However, 
many of the essential properties of the nonlinear mo- 
tions are preserved in the linearization. Thus, for ex- 
ample, boundary conditions will often be qualitatively 
the same for both and the numerical technique of 
integration used for the one can be used for the other. 
In this manner it is often possible to obtain semi- 
