A note on the constancy of horizontal turbulent shearing stress in the lower layers of the atmosphere

It may readily be shown, from elementary arguments, that for steady fluid flow in circular tubes or between parallel plates, the shear stress in the direction of the mean motion varies linearly with the distance from the wall. This result is quite independent of the nature and origin of the fluid resistance and in particular is valid if the flow be turbulent. However, it has been proved by Ertel (1933) that in the lower layers of the atmosphere (1 ⩽ z ⩽ 30 m) the horizontal turbulent shearing stress defined by τ = η ( z )∂ v /∂ z where η( z ) is the vertical austausch coefficient and ∂ v /∂ z the vertical gradient of mean velocity, is to a high degree of approximation independent of the height. In view of the importance of this conclusion in atmospheric turbulence theory, and its obvious difference from the first, it has been thought worth while to examine Ertel's treatment in detail. The latter is based on the assumption of the isotropy of the coefficient of vertical turbulent momentum interchange and the final result depends on the relative smallness of the variation of wind direction with height in the surface layers. Complete neglect of this variation, however, requires a wind velocity constant with height, in which case Ertel's result would become trivial. The object of the present note is to show that the assumption of very small but plausible values for the wind direction change with height in the surface layers (of order 10 ‐6 radians/cm.) suffices to account for the observed wind velocity increase in the lower layers. It may be concluded that Ertel's analysis is sound. Furthermore it appears on the basis of the theory that the mean wind direction change with height in the lower layers is too small to be observed practically.