LOCAL WINDS 
proportional to sin dw, naturally yields a continuous 
chain of circulations. If a number of such circulations 
with variable / resulting from the Fourier series are 
superimposed on one another, we obtain a single cir- 
culation of definite horizontal extent. Thus, this extent 
becomes solely a function of the form of the land-water 
temperature difference. When the influence of friction 
(Guldberg-Mohn’s friction equation, where o is the 
coefficient of friction) and the Coriolis parameter 
(f =2Q sin ¢) are taken into consideration, the equations 
of motion become 
ol = fv — ou ee 
ot p 0x’ 
> = —fu — ov, (1) 
ow 1 
0@ : 
OL = ow a az + yo. 
In addition to these equations we have the continuity 
equation in the form: 
and the heat transfer equation which, with the omission 
of negligible factors, assumes the form: 
ao ao , oo 
ay ee me (e 1F = 6) 
In these equations the y-axis is parallel to the coast, 
and the x-axis perpendicular to it, with the positive 
direction toward land. The letter # stands for the 
potential temperature; y = ga, where a is the coefficient 
of heat expansion; 6 is the vertical gradient of the 
potential temperature; « is the coefficient of turbulent 
heat conduction; and, as before, Q = 7.292 * 10-5 
sec! expresses the day frequency. According to Ray- 
leigh [64], @ is a quantity that fixes the perturbation 
pressure. 
The solution may be assumed in the form: 
u = u(z)e™ cos Ia, ® = o(z)e'** sin Ix, 
d(z)e sin la. 
ll 
(4) 
vy = v(z)e™ cos Ia, oO) 
w= wee sin la, 
Then, by substitution in (1), (2), and (8), differential 
equations are obtained for u, v, w, @, and & which are 
functions of z only and can be solved by exponential 
functions: 
—b; 
w = Aet* + Be™, 
(5) 
0 = Cet” + De™, 
and similar expressions for u, v, and @. 
The boundary conditions are 
z=0, w = 0, and o = M, (6) 
and for large values of z the circulation becomes negli- 
gibly small; then, the constants A, B, C, and D are 
661 
fixed whereas a and b are determined by the day fre- 
quency and the other values given above. 
The solutions for w and # are: 
y= pete, “haz —b2z] 12t 
u = @— [ae™* + be “Je” cos Ia, 
w= alt ~ lew* — e “Je” sin Ia, 
= a) 
f (7) 
ae CEG A” 
Lane b —s nee iy: iQt + 
v= Mie oF ape —e }lesinlz. 
The factors r and s can also be expressed by the con- 
stants given above. It should be noted that all these 
quantities, including a and 6, are complex numbers and 
that with e* a separation of the real and imaginary 
terms would require extensive calculations. 
When the solution (7) is worked out for latitude 
@ = 45° and various values of the friction coefficient 
o, it yields circulation systems of the land and sea 
breezes that are in full agreement with observations. 
Table III lists the basic factors of the circulations for; 
different values of c with and without consideration of 
the Coriolis parameter. As is usually the case, the 
phases are referred to a maximum land-water temper- 
ature difference at 1200. It can be seen from Table III 
that the height of the land or sea breeze lies at roughly 
400 m and rises with increasing friction from 320 m 
to 500 m. It is interesting to note that this altitude is 
somewhat reduced under the effect of the Coriolis 
force. Naturally, friction diminishes the velocity of the 
land and sea breezes to a considerable extent, namely 
from about 5.5 m sec to 2 m sec for every centigrade 
degree of temperature difference. For average friction 
conditions and a maximum land-water temperature dif- 
ference of 5C over the distance L/2, we arrive at a 
maximum sea-breeze intensity of about 10 m sec, 
which agrees quite well with observations. Under these 
conditions the vertical velocities reach maximum values 
of about 2 cm sec per centigrade degree of temper- 
ature difference, which is also a reasonable value. 
Of special interest is the phase of the land and sea 
breezes. If friction and the Coriolis force are neglected, 
the phase shift between the maximum temperature 
difference and the maximum intensity of the sea breeze 
is 4.7 hr. This shift decreases rapidly to 1.4 hr with 
increasing friction. In the case of ¢ = 2.5 X 10° sec 
the maximum intensity of the sea breeze follows the 
maximum temperature difference closely, as is borne 
out by most observations. The influence of the Coriolis 
force on the component perpendicular to the coast is 
small, as is to be expected. Naturally, a cross velocity v 
appears instead, whose phase shift with the velocity 
perpendicular to the coast is 10.9 hr, or nearly 12 hr. 
This phase shift stays constant up to the height of the 
zero layer and then changes sign. Figure 6 is a vector 
diagram for the case ¢ = 2.5 X 10-4 sec"! and f = 
1.031 X 10-4 sec (6 = 45°). The figure shows clearly 
