666 
where A is a constant at z = 0, B is the lapse rate of 
potential temperature, z is the vertical, n is the normal 
to the slope, and # is the disturbance in potential tem- 
perature caused by heat conduction in the direction n. 
We can then expect the velocity w of the stationary 
current along the slope to be a function of n only. 
The interplay of heat transfer and heat conduction 
furnishes the differential equation, 
Bsine dn 
gB8' sine + = (i) (9) 
where £ is the coefficient of expansion, g80’ sin ¢ is the 
acceleration in the upslope direction, ¢ is the angle of 
the slope, v is the coefficient of turbulent friction, and 
x is the coefficient of turbulent heat conduction. 
The usual solutions of equation (9) are 
ow = Ce”” cos ; 
and 
w=C 98K Ge sin, = (10) 
By l 
where 
ike y/ Akp 
gGB sin e’ 
and C is the temperature disturbance at the surface of 
the slope. The actual form of the solution is shown 
schematically in Fig. 11. 
Fig. 11.—Schematic profiles (normal to the slope) of the wind 
speed w and temperature 3’. (After Prandtl [62].) 
I have checked the correctness of these theoretical 
results by a detailed comparison with actually meas- 
ured average values of # and w and have found a 
splendid agreement, except for the disturbing gradient 
influences in the upper layers. Furthermore, it is note- 
worthy that the resulting values of the austausch due 
to impulse transport (virtual friction) and of that due 
to transport of the heat content (virtual heat conduc- 
tion) are of a plausible magnitude. Figure 12 presents 
the theoretical distribution of the potential temperature 
LOCAL CIRCULATIONS 
during upslope and downslope winds, respectively, for 
which no observations are available. The characteristics 
of the stratification become very apparent. 
Fic. 12.—Theoretical distribution of the potential tempera- 
ture Over a mountain slope during (a) upslope wind and (6) 
downslope wind. (After F. Defant {17].) 
The theory can be extended to include the oscillatory 
nature of the slope wind by simply multiplying, as a 
first approximation, the solutions (10) of the stationary 
case by the factor cos 2. Because of the fourth root m 
the expression for J, variations in the values of » and 
x [17, pp. 441—444] are of little influence on the solution. 
The average air transport by the slope winds can be 
roughly computed from the calculated and the observed 
average profiles of the slope wind. We can then estimate 
how long it would take until the rising heated slope air 
has replaced the air masses over the valley bottom. 
“The interest of this question becomes apparent if we 
consider that this process of warming the air in the 
valley center has an effect on the pressure gradient and 
thus on the generation of the mountain and valley 
winds. Calculations show that a slab of air 1 m thick, 
1500 m long (the width of the valley), and 1750 m 
