The criterion of turbulence in a fluid of variable density, with particular reference to conditions in the atmosphere

Abstract The energy equation of eddying motion in a fluid of variable density is derived by methods based on a treatment by Ertel, with particular reference to conditions in the atmosphere. The general form of the criterion for the growth or decay of turbulence is then deduced. The limitations inherent in the Richardson criterion of turbulence are examined and it is shown that the latter neglects terms which are potentially large. These represent respectively, the rate of dissipation of eddying energy into heat by molecular viscosity, the rate of working of the fluctuating gradients of static pressure on the eddying motion, and the eddy diffusion of eddying kinetic energy. Richardson's criterion also involves the assumption of a special functional form for the vertical heat flux due to turbulence, which may not be generally valid. If it is, however, the critical value of the Richardson number above which turbulence is damped out is probably less than unity, in contradiction to Richardson's analysis, but in agreement with a theoretical analysis by Schlichting, which has been shown to be in satisfactory agreement with laboratory experiments.