Nonlinear interactions treated by the methods of theoretical physics (with application to the generation of waves by wind)

Abstract

The ‘Feynman’ diagram method for analysing wave-wave interactions in random wave fields is generalized to include non-conservative interactions between wave fields and external fields. The particle interpretation is no longer applicable, but the transfer expressions can still be conveniently summarized in terms of ‘transfer’ diagrams, which correspond to collision diagrams in the particle picture. The method is applied to interactions between gravity waves and the turbulent atmospheric boundary layer. The complete set of lowest order transfer diagrams contains the Phillips and Miles mechanisms of wave generation and an additional set of wave-turbulence interactions, which have not been considered previously. The closure hypothesis invoked in the derivation of the transfer expressions is discussed briefly in appendix A. It is pointed out that Benney & Saffman’s recent derivation of the transfer expressions without the usual closure hypothesis contradicts the irreversibility of the transfer expressions and is valid only initially. The relevant statistical properties depend on the distinction between coarse grained and fine grained distributions. This is illustrated in appendix B by a discussion of the Gaussian property of linear, random wave fields.