92 



METEOROLOGY— THEORY 



to temperature deficit is equal to the ratio of humidity 

 iluctuation to humidity deficit and very nearly equal 

 to the ratio of M fliictnation to M deficit. 



6-8 GRAVITATIONAL WAVES AND 



TEMPERATURE INVERSIONS J 



It has been noted that the guided propagation of 

 microwaves is often accompanied by deep fades with 

 periods of the order of a few minutes. The sugges- 

 tion has been made that these fluctuations may be 

 associated with atmospheric wave motion which could 

 make the top of the duct an undulating surface rather 

 than a level one.^°'^' Therefore, it seems desirable to 

 discuss, from a meteorological point of view, the pos- 

 sibility of the existence of such atmospheric waves 

 and the physical characteristics of any which might 

 exist. The purpose of this paper is to review and sum- 

 marize the meteorological information whicli is avail- 

 able concerning the subject. 



A theoretical consideration of the problem indicates 

 that atmosi^heric wave motion can occur at any sur- 

 face in the atmosphere where there is a rapid change 

 in wind velocity with height and a stable stratification 

 of temperature. Such conditions are best fulfilled at 

 temperature inversions, which, it will be noted, usu- 

 ally correspond to a rapid decrease with height of the 

 index of refraction. The wind shear supplies the 

 energy to set up the wave motion, in the same way in 

 which waves are formed at the surface of the ocean. 

 Gravitation acts as a stabilizing or restoring force. 

 Hence, these waves are of a mixed shearing and 

 gravitational type. The waves may be stable or un- 

 stable, depending on their wavelength, on the density 

 and wind speed differences between the two media, 

 and on the lapse rates in the two media. For any given 

 values of the density and wind velocity differences 

 and of the lapse rates, there is a critical wavelength 

 below which wave motion is unstable; that is, it dis- 

 appears into turbulent eddies because of the shearing 

 effect. All wavelengths above this critical value will 

 remain stable because of the gra^^tational effect. 

 Hence one may speak of the former as "shearing 

 waves" and of the latter as "gravitational waves." It 

 is the stable or gravitational type with which we are 

 concerned. 



These considerations hold for wavelengths up to 

 about 500 km. For longer wavelengths the effect of 

 the earth's rotation must be considered. In this paper 



'By Lt. R. A. Craig, AAF, Weather Division. 



only the shorter wavelengths where this effect may 

 be neglected will be discussed. 



A mathematical analysis of wave motion and deter- 

 mination of the critical wavelengths involves a solu- 

 tion of the equations of motion and continuity and an 

 application of certain boundary conditions. In order 

 to derive the critical wavelengths given below, the 

 following assumptions have been made. 



1. The inversion or shearing layer may be re- 

 garded as a strict discontinuity between the air above 

 and the air below. This assumption is sufficiently ac- 

 curate provided the thickness of the layer is small 

 compared to the wavelengths which occur. 



3. The velocities associated with the wave motion 

 are small compared with the undisturbed velocities 

 of the air masses above and below the inversion. 



3. The height of the inversion above the lower 

 boundary (ground) is equal to or greater than 40 

 per cent of the wavelength which occurs. 



4. There is no friction between the two fluids. 

 Two cases may be considered. The first is the case 



where the air masses are assumed to be incompressible 

 and homogeneous. It also holds for two air masses 

 with adiabatic lapse rates. In this case the critical 

 wavelength is given by^**^ 



A crit — 



27r (u -uy T T 



{T'-T)g T'+T 



In the second case the air masses are comjaressible 

 id isotherm 

 is given by'^ 



and isothermal. For this case the critical wavelength 



A orit — 



27r {u'-uy 



T'+T 



>/■ 



(T'-Ty-+2^^iT' 

 kR 



-T) 



{u'-uy 



In these two formulas, 



T' = temperature in the upper air, 



T = temperature in the lower air, 



U' = velocity in the upper air, 



U = \elocity in the lower air, 



g = acceleration of gravity, 



k = Cj,/Cy = 1.405, 



R = gas constant for air 



= 3.87 X 10" cmVsec' degree. 

 In Table 13 the critical wavelengths in meters are 

 tabulated for various values of wind shear and tem- 

 perature difference. Values for the adiabatic case are 



