Tl PES OF W CI K\ i:s 



43 



where a is the earth's radius. Similar expressions 

 are valid for the quantities having the subscript 0. 

 Multiplying out and neglecting quantities that are 



small of the second order, one obtains 



>h, + - (h — ho) = k (a 2 — da) 

 a I 



(3) 



It has become customary to introduce the modified 

 refractive index .1/ by 



n + -= 1 + M • 10" 6 , 



(4) 



whereupon Snell's law assumes the form 



(M - Mo) • 10- 6 = | {a- - a -) . (5) 



This equation indicates how the angle a between a 

 ray and the horizontal changes as a function of M 

 which, in turn, is a function of the height, both 

 explicitly by equation (4) and implicitly because n 

 is a function of the height in a stratified atmosphere. 



e.s TYPES OF M CURVES 



An M curve is a diagram in which M as abscissa 

 is plotted against the height h as ordinate. Extensive 

 experience has led to a classification of M curves 

 which is shown in Figure 1. The six types exhibited 



TRANSITIONAL 



SUBSTANDARD 



SIMPLE SURFACE 

 TRAPPING 



ELEVATED S SHAPE GROUND-8ASED S SHAPE 



l\ DUCT 



' \INVERSION LAYER 



INVERSION LAYER •< 



T 

 INVERSION, LAYER 



DUCT 



M M 



Figure 1. Types of M curves. 



comprise all cases that are of practical interest. 

 M curves of a more involved structure are rare. In 

 all cases it is assumed in accord with experience 

 that at sufficiently high elevations the M curves 

 become linear and have, or nearly have, the standard 

 slope. 



The height at which these variations in refractive 

 index occur may vary from a few feet to several 

 hundred or even a few thousand feet though they 

 are likely to be found at very low elevations in cold 

 climates and at- higher elevations in warm climates. 

 The meteorological conditions which yield these 

 curves will be dealt with extensively in Chapter 9, 

 and few indications may suffice here. Ordinarily, 

 on going aloft the temperature decreases at a slow 

 and fairly steady rate. When, instead, the tempera- 

 ture increases with increasing height, a phenomenon 

 known to meteorologists as a temperature inversion, 

 equation (9), Chapter 5, shows that n decreases with 

 increasing height. This does not necessarily imply 

 that M decreases with height since, by equation (4), 

 M contains the term h/a, which increases with 

 height. If, however, the variation of temperature is 

 sufficiently great, a decrease or inversion of M 

 results. Such an inversion produces a duct, a term 

 which refers essentially to certain meteorological 

 phenomena and whose exact significance is explained 

 below. A variation of humidity over the layer has 

 an effect essentially analogous to, but distinctly 

 more pronounced than, the effect of temperature. 

 In this case M increases with height with a decreas- 

 ing moisture content and vice versa. Variations of 

 humidity are common in the lower atmosphere, and 

 they constitute the main cause of refractive index 

 variations, with temperature variations frequently 

 a contributing factor. 



The six cases shown in Figure 1 are as follows: 

 the standard case which needs no further comment; 

 the transitional case where the moisture or tempera- 

 ture variation is not great enough to produce a true 

 inversion of the M curve but merely results in a 

 nearly constant value of M in the lowest strata; the 

 substandard case in which M increases more rapidly 

 with height than in the standard case; and three 

 cases of ducts. The simple ground-based duct or sur- 

 face trapping, consists in an M inversion immediately 

 adjacent to the ground or sea. There are two types 

 of elevated M inversions distinguished by the posi- 

 tion of the minimum value of M aloft. If this mini- 

 mum is larger than the value of M at the ground so 

 that the vertical projection from the minimum inter- 

 sects the M curve, it is considered a true elevated, 

 S-shaped duct. If this minimum is less than the 

 value of M at the ground it is an elevated M inver- 

 sion but a ground-based duct. 



In dealing with these M curves it is universally 

 assumed that the stratification is the same over the 



