Inverse Modeling of the Action-Balance Equation. Part I: Source Expansion and Adjoint-Model Equations

Abstract

In this paper a series of numerical experiments is defined to explore the inverse modeling of the action-balance equation governing the evolution of the surface gravity wave field, using the adjoint data-assimation model-optimization procedure of Thacker and Long. We begin by exploiting power series, functional power series, and a variety of physical and mathematical considerations to derive a systematic expansion of the source terms in this equation for the deep-water case. This expansion, which naturally incorporates a Thacker representation for the nonlinear transfer from wave?wave interaction, defines a set of dimensionless expansion coefficients to be determined by the inverse modeling and identifies the simplified cases to be investigated in the numerical experiments. Dimensional analysis determines a natural scaling for each term in this expansion and suggests a general form for the whitecap dissipation term, which includes as a special case the form proposed by Hasselmann, determining the first-order contribution to his unknown spectrum-dependent coefficient to within a multiplicative spectrum-independent constant. A general discussion of the evolution of the simplified cases reveals a striking tendency to concentrate action in a single band when whitecap dissipation has the Hasselmann form and nonlinear transfer is ignored. A derivation of the adjoint-model equations is included for one of the simplified cases and a general discussion of the model-optimization procedure is given. In these equations. nonlinear transfer is mirrored by a term of similar form, with Thacker's nonlinear transfer coefficients replaced by a related set of adjoint coefficients and the triple product of spectral intensities replaced by a product of two spectral intensities and a Lagrange multiplier.