METEOROLOGY — THEORY 201 
by standard, or by substandard M curves, respectively. 
For instance, the positivé M deficit in Figure 2 is a 
condition necessary for the M inversion to occur, 
Specifically, if homogeneous air blows over a water 
surface of constant temperature and if the M deficit 
is positive, there is always an M inversion at the 
water surface. Whether or not this extends sufficiently 
high to be of importance in the refraction of radio 
waves depends in part on the magnitude of the M 
deficit and on the temperature excess. 
If homogeneous air blows over a water surface of 
constant temperature and if the M deficit is zero, the 
M curve necessarily remains practically standard. 
In the case of a negative M deficit a substandard 
M curve is developed. It should be noted that in this 
case (as well as in the previous one) the air is losing 
water vapor by condensation on the water surface. 
This is simply the reverse of the process with dry air. 
Neutral Equilibrium 
For simplicity the analysis which follows is limited 
to cases of positive Mf deficit. There is then a surface 
M inversion, the height of which is a convenient 
quantity to study as a dependent variable. The inde- 
pendent ones are M deficit and temperature excess 
and, as will be seen, two others. 
The first and least complicated case is the one of 
neutral equilibrium, which corresponds to a tempera- 
ture excess close to zero, say within 1 C of zero. Since 
there is no appreciable temperature gradient, the I 
curve depends only on the moisture distribution. 
Furthermore this case practically requires a vapor- 
pressure lapse at the surface, because in the lower 
part of a homogeneous layer the vapor cannot be 
saturated (see Figure 3). Hence there is always an 
M inversion at the surface. Neutral equilibrium is 
prevalent far from shore. 
2000 
1500 
HEIGHT IN FEET 
8 
(-} 
° rare i ‘ 
uo B 6 7 19 79 I Co) 20 40 60 80 
TEMPERATURE IN MIXING RATIO Men, 
DEGREES G . 
Figure 3. Probable course of modification of warm air 
over water under ideal conditions. A, initial stage. 
D, final stage. 
With neutral equilibrium frictional turbulence is 
unhindered. Mixing extends to a height roughly pro- 
portional to the wind speed; a wind of 20 mph at 
100 ft gives mixing up to about 2,000 ft. 
The intensity of mixing increases upward rapidly 
from the surface, so large vertical gradients are con- 
fined to the region of relatively little mixing close to 
the surface. The M inversion probably never extends 
above 100 ft. 
It has been well established that under neutral 
equilibrium the eddy diffusivity is directly propor- 
tional to height within the so-called turbulent. bound- 
ary layer, which forms the lower tenth of the entire 
frictional layer mentioned above. In this case wind 
speed, temperature, and vapor pressure are linear 
functions of the logarithm of elevation. 
While the details are not presented here, it is easily 
shown that these logarithmic distributions demand 
that the height of the top of the M inversion be 
i= 50 rAM 10-6, 
where a is the radius of the earth, AJ/ is the Mf deficit, 
and T is a meteorological parameter depending on 
wind speed alone in the case of complete neutral 
equilibrium. Thus the height of the Mf inversion is 
directly proportional to the J deficit with neutral 
equilibrium. 
Published data indicate that the effect of wind 
speed is not great and give an average value of T = 
0.08. This yields d/AM = 2 ft. Data obtained during 
the summer’s (1944) project agree with this result. 
Unstable Equilibrium 
The second case is the one of unstable equilibrium 
or negative temperature excess. This is similar to 
neutral equilibrium in that the I deficit is always 
positive and there is always a surface MM inversion. 
Instability adds convective mixing to the frictional 
mixing that would otherwise be present. This con- 
vective mixing is especially effective in the central 
region of the unstable layer and hence confines the 
large vertical gradients within a still thinner surface 
layer. 
The logarithmic distributions are characteristic of 
neutral equilibrium only. Consequently, in the un- 
stable cases the height of the M inversion is not simply 
proportional to M deficit but depends on M deficit in 
a more complicated manner. In spite of this the pro- 
portionality will be assumed as a useful approxima- 
tion in studying the unstable and stable cases also. 
The ratio of height of M inversion to M deficit is 
definitely less for unstable than for neutral equili- 
brium. Tentatively it may be said to range between 
0.2 ft and 2 ft. 
Stable Equilibrium 
The last case is the one of stable equilibrium. Sta- 
bility reduces the mixing with high levels, thus per- 
mitting a deeper surface layer of strong gradients to 
form (as shown in Figure 2). Thus the ratio of height 
of M inversion to M deficit may be expected to be 
always greater in stable equilibrium than in neutral 
equilibrium. 
