A Parametric Wave Prediction Model

Abstract

Measurements of fetch-limited wave spectra from various sources indicate an approximate invariance of the normalized spectral shape with fetch. It has been suggested from investigations of the spectral energy balance that this can be explained by the shape-stabilizing influence of nonlinear resonant wave-wave interactions, which are also responsible for the migration of the spectral peak to lower frequencies. Analyses of a series of further data sets obtained under non-uniform, non-stationary wind conditions show that the invariance of the spectral shape is not restricted to uniform-wind, fetch-limited situations, but applies generally for a growing wind sea. The observed shape invariance is exploited in a wave prediction model by projecting the full transport equation for the two-dimensional spectral continuum onto two variables characterizing the energy and frequency scales of the spectrum. Inspection of the resultant equations reveals further simplifications, enabling the system to be reduced to a single prediction equation for one scale variable, the peak frequency. This is feasible because of the rapid adjustment of the spectrum to a quasi-equilibrium level in which the atmospheric input is balanced by the nonlinear transfer of energy out of the central region of the spectrum to higher and lower frequencies. The balance occurs sufficiently rapidly to be treated as a local response process, thereby providing a relation between the energy level of the spectrum (characterized, for example, by Phillip' constant α), the peak frequency fm, and the local wind speed U (the latter two occurring only in the non-dimensional combination ?=Ufm/g). The directional distribution of the wave spectrum is also established locally and can be regarded as a given function of the non-dimensional frequency f/fm and ?. For the remaining independent scale parameter, the peak frequency, the dominant source term in the transport equation is determined by the nonlinear energy transfer, which can be computed rigorously. To lowest order, the one-parameter wave model is independent of the relative contributions of the atmospheric input and dissipation in the central region of the spectrum. However, because of lack of (consistent) direct measurements of the atmospheric input or dissipation, the quasi-equilibrium relation inferred between α and ? must be calibrated empirically, for example, by comparison with fetch-limited data. Within the scatter of the data, all data acts analyzed (with two exceptions, where the data were considered questionable) were reasonably consistent with a common α?? relation. The residual scatter of the data is thought to be associated largely with small (sub-grid) scale inhomogeneities of the wind field and may represent a natural limitation of the accuracy achievable with deterministic wave models. A complete wave model would need to combine the proposed parametric model for growing wind seas with a swell propagation model.