11 - Scaling in geophysical fluid dynamics

Summary

Scaling laws for the atmospheric surface layer

Geophysical fluid dynamics has become in the last few decades a broad subject (see Pedlosky, 1979) with many applications in earth sciences and in engineering practice. In all branches of geophysical fluid dynamics using similarity considerations, scaling laws and self-similar solutions play an important, often decisive role. We have chosen in this chapter for demonstration's sake some topics from geophysical fluid mechanics related mainly to geophysical turbulence.

The surface layer of the atmosphere is usually modelled (see, e.g., Monin and Yaglom, 1971) by a turbulent flow that is statistically horizontally-homogeneous and stationary, and is bounded below by a horizontal plane. The shear stress τ in the surface layer is also assumed to be constant. The essential difference from the flow in the wall region considered in section 10.2 consists in the presence in the surface layer of thermal stratification – temperature inhomogeneity over the height of the layer. The stratification is stable if the temperature increases with height and unstable in the opposite case. Owing to the thermal inhomogeneity, a vertical displacement of fluid particles, produced by a vertical velocity fluctuation, is accompanied by work done against the force of gravity (or extracted, depending on whether the stratification is stable or not). This work is either taken from the turbulent energy or added to it, thus influencing the turbulence level, i.e. the transfer of heat, mass and momentum, and consequently also influencing the vertical distribution of the mean longitudinal velocity across the flow. The effectiveness of the influence of thermal stratification on the balance of turbulent energy is governed by the product of the coefficient of thermal expansion of the air and the acceleration of gravity, the so-called buoyancy parameter. The air in the atmospheric surface layer is usually considered to be a thermodynamically ideal gas, for which the coefficient of thermal expansion is equal to 1/T, where T is the absolute temperature.