The Interaction between a Mean Flow and Random Disturbances

The study of the statistical properties of ensembles of hydrodynamical systems may be called statistical hydrodynamics. It is recommended that statistical hydrodynamics be applied to certain problems which have not previously been looked upon as statistical problems.

Statistical hydrodynamics is applied to the problem of the interaction between a mean flow an a superposed disturbance, in a two-dimensional homogeneous incompressible nonviscous fluid. The ensemble of all disturbances which may individually be superposed upon a given mean flow is assumed to be random, in the sense that it is unaltered if each disturbance is subjected to a change of sign, a translation in space, or a rotation. It is found that, ensemble-average-wise, kinetic energy is transferred from the disturbances to the mean flow if the mean flow is of small variance and coarse detail and the disturbances are on the average of large amplitude and fine detail, while kinetic energy is transferred in the opposite direction if the opposite situation exists.

This result is applied to the problem of the maintenance of kinetic energy in the earth’s atmosphere against the dissipative effect of friction. There is some evidence that both the total kinetic energy and the kinetic energy of the mean flow can be maintained through the addition of new disturbances which form random ensembles, but that they can be maintained more efficiently, and probably are maintained, by the addition of new disturbances with a systematic lack of randomness.