The Stochastic Parametric Mechanism for Generation of Surface Water Waves by Wind

Theoretical understanding of the generation and growth of wind driven surface water waves has been based on two distinct mechanisms; the first being stochastic excitation by wave incoherent random atmospheric pressure fluctuations unrelated to wave amplitude (Eckart 1953; Phillips 1957) and the second instability arising from wave induced coherent atmospheric pressure fluctuations proportional to wave amplitude (Helmholtz 1868; Kelvin 1871; Jeffreys 1925; Miles 1957). Incoherent wave independent forcing produces growth of surface height variance linear in time while coherent forcing proportional to wave amplitude produces exponential growth. While observed wave developments can be fit to these forms, and despite broad agreement on the underlying physical process of momentum transfer from the atmospheric boundary layer shear flow to the water waves by atmospheric pressure fluctuations, quantitative agreement between theory and field observations of wave growth has proved elusive. Accepting that the dominant contribution to wave growth results from an at least statistical proportionality and coherency in phase lag between atmospheric pressure fluctuations and surface elevation, at issue is the mechanism by which this relationship is produced and maintained. Mechanisms previously proposed by Jeffreys and Miles have been extensively examined and found to be in variance with observations notably those taken in turbulent air flow (Snyder & Cox 1966; Barnett & Wilkerson 1967; Stewart & Teague 1980). In this work an alternative mechanism is proposed that unites the wave incoherent atmospheric forcing process, with its essentially turbulent character and linear in time variance growth, with the wave induced coherent atmospheric forcing with its exponential growth. The mechanism produces exponential growth that reduces in laminar flow to that of Miles but exceeds that produced by laminar critical layer instability in turbulent air flow. The growth rate dependence on wavenumber is in better accord with observations and the long tail distribution of wave height observations in strong winds is also explained. This stochastic parametric instability is an example of the universal instability arising from the nonnormality of nearly all time-dependent flows (Farrell & Ioannou 1999).