68 



METEOROLOGY— THEORY 



It lias l)C(.'ii well cstaljlished that iimler iieuti-al 

 equilibrium the eddy diifusivity 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 21 inversion be 



d = ^a rAMlO-^ 



where a is the radius of the earth, Ail/ is the M deficit, 

 and r is a nictenroldgical parameter depending on 

 wind speed alone iu the case of complete neutral 

 equilibrium. Thus the height of the Jil inversion is 

 directly proportional to the ^1/ deficit with neutral 

 equilibrium. 



Pulilished data indicate that the effect of wind 

 speed is not great and give an average value of r = 

 0.08. This yields (7/ Ail/ = 2 ft. Data obtained during 

 the summer's (19-1-1) project agree with this result. 



**^"* Unstable Equilibrium 



The second case is the one of unstable equililnium 

 or negative temperature excess. This is similar to 

 neutral equilibrium in that the M deficit is always 

 positive and there is always a surface M inversion. 



Instability adds convective mixing to the frictional 

 mixing that would otherwise be present. This con- 

 vective mixing is especially eifective in the central 

 region of the unstable layer and hence confines the 

 large vertical gradients within a still thinner surface 

 layer. 



The logarithmic distrilmtions are characteristic of 

 neutral equilibrium only. Consequently, in the un- 

 stable cases the height of the il/ 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 



fiinii (as shown in Figure 2). Thus the ratio of height 

 !)[ M inversion to M deficit may be expected to be 

 always greater in stable equilibrium than in neutral 

 equilibrium. 



Stability reduces the mixing to such an extent that 

 the air is progressively modified during a long over- 

 water trajectory. It is therefore necessary to introduce 

 a fourth independent variable, length of over-water 

 trajectory, to supplement M deficit, temperature ex- 

 cess, and wind speed. 



Under ideal conditions there is reason to believe 

 that the modification would pursue the course sketched 

 in Figure 3. The final state would be an essentially 

 homogeneous layer capped by a temperature inversion 

 at the level already mentioned for the top of the fric- 

 tionally i)roduced turlnilence iu neutral equilibrium. 

 The temperature of the layer would follow an adia- 

 batic lapse rate from the water surface to the top. 

 The water vapor would he saturated at the top of the 

 layer, specific humidity being nearly constant through- 

 out the layer except for a strong lapse at the surface. 

 Intermediate stages in the formation of this final state 

 are indicated qualitatively in Figure 3. 



It should be noted that the later stages have a 

 transitional or S-shaped M curve and that qualitative 

 theoretical considerations do not reveal which. The 

 initial stage is, however, characterized by simple sur- 

 face trapping, and it is this stage only for which data 

 are presented below. 



The soundings have been studied to determine em- 

 pirically how the ratio of height of il/ inversion to il/ 

 deficit depends on temperature excess, wind speed, 

 and length of trajectory. To eliminate complex M 

 curves the analysis has been limited to cases con- 

 forming closely to the following ideal conditions. 



1. Initially homogeneous air. 



2. Constant surface-water temperature along the 

 air trajectory. 



3. Constant wind (wind not changing with time 

 following a parcel). 



The ratio of height of M inversion to il/ deficit is 

 found to increase with length of over-water trajec- 

 tory quite markedly in the first 10 miles. From 10 

 miles to 30 miles there is not much further increase. 

 Beyond 30 miles the preliminary analysis reveals no 

 general information. 



Figure 4 gives some tentative results based on 

 various sources of information. This includes the cases 

 of neutral and unstable equilibrium in addition to 

 stable equilibrium. Within the stated range of over- 



