A note on the logarithmic -3 law of atmospheric energy

Abstract

Observations show that the atmospheric kinetic energy on average obeys a relation saying that its dependence on the longitudinal wave number is n^-3 in an interval with wave numbers larger than the wave number with maximal conversion of eddy available potential energy to eddy kinetic energy and the wave numbers dominated by frictional dissipation. Numerical experiments with a barotropic model have indicated that the power law for the kinetic energy is n^-1 for the small wave numbers. Such relations are a result of the atmospheric forcing due to the diabatic heating, the cascade processes of kinetic energy due to nonlinear interactions and the frictional dissipation. The paper presents a study of a homogeneous fluid with a free surface forced by adding and subtracting fluid. For a given forcing the steady state may be determined although the model is nonlinear. For a special forcing the kinetic energy spectrum shows the -1 and the -3 relations. An important question is whether the potential energy also obeys a -3 power law. Other cases of the forcing are investigated in order to determine the sensitivity of the results to the forcing.