DIFFICULTIES OF LOW-LEVET, DIFFUSION PROBLEMS 



65 



of slrtbilily and wind velocity. If a logarilliiuic law 

 lield, tlieii K is a function of tliese same quantities. 



3. In 1934, Best* analyzed data which was meas- 

 ured at seven elevations between 2 cm and 5 m. He 

 concluded that the velocity variation was best repre- 

 sented by a logarithmic function of the form 



M^l0g(3-C) 



where G is a constant. 



Furthermore he found that the power law could be 

 applied only to shallow layers and even then m varies 

 quite considerably with height, wind velocity, and 

 vertical temperature gradient. 



4. In 1936, Sutton,^ who in 1933 suggested the 

 power law variation, definitely favored the logarithmic 

 variation. Furthermore he showed how one could 

 handle the problem of varying stability. Sutton an- 

 alyzed different sets of data, ranging to 30 m in 

 elevation, to support the logarithmic variation. 



5. In 1936, Sverdrup" criticized Sutton's logarith- 

 mic law and favored a power law in regions of stability. 

 As evidence he introduced Eossby and Montgomery's' 

 analysis as well as his own data. 



6. In 1937, Sutton** quite satisfactorily met Sver- 

 drup's criticism and pointed out that all experimental 

 evidence suggested the logarithmic variation rather 

 than the power law variation. 



This represents only a cross section of opinion and 

 perhaps may be summarized as follows : 



1. In an indifferent or unstable atmosphere the 

 logarithmic law is generally accepted. 



2. In a stable atmosphere there is more support 

 for the logarithmic law than for the power law. 



One writer summarized the situation very aptly 

 when he said that all modern mathematical studies 

 on atmospheric turbulence are inexact and depend 

 on certain wide assumptions. An appeal to experiment 

 is therefore essential. 



^'^■^ Conclusion 



The question now arises, should one assume a 

 power law variation, or is the true wind variation 

 better represented by a logarithmic law? Certainly 

 the experimental evidence tends to favor a logarithmic 

 variation. The advantages and disadvantages of either 

 assumption may be summarized briefly as follows : 



1. Power law variation. 



a. m varies with stability, wind velocity, rough- 

 ness, and elevation. 



b. The mathematical analysis is too complicated 

 for practical use. 



3. Logarithmic law variation. 



a. Agrees reasonably well with experimental 

 data. 



b. Agrees with von Karnian's logarithmic law. 

 von Karman has shown that this law covers 

 an exceedingly wide range of turbulence. 



c. K, like m, varies with stability, wind velocity, 

 roughness, and elevation. 



d. If the logarithmic law holds, E is then a 

 linear function of height. With this relatively 

 simple expression for K it should be much 

 easier to handle the diffusion equation than 

 in the case of a power law variation. 



e. Provided the integration of the diffusion equa- 

 tion is not too complicated, one should l)e able 

 to reconstruct the temperature and vapor 

 pressure curves. Consequently the exact shape 

 of the M curve can be calculated. 



In conclusion it should be borne in mind that 

 theoretical discussion is futile. At best we can only 

 make certain assumptions and derive a result. If this 

 result agrees with observational data then the original 

 assumptions are justified. Furthermore, practical con- 

 siderations demand that the final solution be simple 

 enough for a23plication in the field. 



It seems certain that over a wide range of elevation, 

 say 300 ft, the true wind variation cannot be uniquely 

 defined by one specific logarithmic law or one specific 

 power law. The most desirable procedure may then 

 be an analysis of observations in as simple a manner 

 as possible but yet flexible enough to take care of the 

 most important changes. Consequently it is suggested 

 that experimental data be analyzed on the assumjjtion 

 that K varies linearly with elevation, i.e., 



dT d ( dT\ dT d f dT\ 



— = — I A' — I or u — = — ( 7^' — ) , 



dt dz\ dzj dx dz\ dz/ 



where K = pz -\- q, and x is the distance measured 

 horizontally. If accuracy is not seriously affected it 

 is further suggested that approximations be intro- 

 duced in order to facilitate the application of the 

 results for field use. 



62 DIFFICULTIES OF LOW-LEVEL 



DIFFUSION PROBLEMS'^ 



The effect of a temperature inversion is largely a 

 secondary one in that by reducing the coefficient of 

 diifusion it favors the formation of large humidity 



°By Lt. Comdr. F. L. Westwater, Naval Meteorological 

 Service, Royal Navy. 



