Generalizations of the Richardson criterion for the onset of atmospheric turbulence

The energetics of air‐parcel interchange are examined for the case in which the atmospheric potential temperature θ and background wind U are continuous but otherwise arbitrarily varying in space, and in which the direction of U is free from constraint. Interchange along a single s‐axis, inclined at angle α to the vertical, is shown to be energetically feasible if where g is the gravitational acceleration. This condition, which is in general far less stringent than that imposed by the standard Richardson criterion, is suggested as a condition portending the onset of anisotropic turbulence. An isotropic interchange throughout a sphere is shown to be energetically feasible if where the z co‐ordinate of the Cartesian x, y, z system is directed upward. This condition, which is in general somewhat less stringent than that imposed by the elementary Richardson criterion, may well portend the occurrence of isotropic turbulence. An outer time scale for the turbulence is estimated in each case, and its relevance to the occurrence of turbulence is discussed for the case when the background parameters of the system change with time.