JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 88, NO. C12, PAGES 7597-7606, SEPTEMBER 20, 1983 A Non-Gaussian Statistical Model for Surface Elevation of Nonlinear Random Wave Fields NORDEN E. HUANG AND STEVEN R. LONG NASA Goddard Space Flight Center, Wallops Flight Facility CHI-CHAO TUNG AND YELl YUAN North Carolina State University LARRY F. BLIVEN Oceanic Hydrodynamics, Inc. Probability density function of the surface elevation of a nonlinear random wave field is obtained. The wave model is basedon the Stokesexpansion carried to the third order for both deep water waves and waves in finite depth. The amplitude and phase of the first-order componentof the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively. The probability densityfunction for the deep water casewas found to dependon two parameters: the root- mean-square surface elevation and the significant slope. For water of finite depth, an additional parameter, the nondimensional depth, is also required. An important differencebetween the present result and the Gram-Charlier representation is that the present probabilitydensityfunctions are always nonnegative. It is also found that the 'constant' term in the Stokes expansion, usually neglected in deterministic studies, plays an important role in determining the details of the density function. The results compare well with laboratory and field experiment data. 1. INTRODUCTION Wind waves are always random. The randomness is the result of the generating.forces as well as the consequence of the dynamic processes in wave evolution which induce different kinds of instabilities. Consequently, the only mean- ingful descriptions of wind-generated wave fields are the various statistical measures; among them, the probability density function of surface elevation is the most basic one. Limited by available theory, past studies of the statistical properties of the wind wave field are mostly based on Gaussian statistics for linear wave fields as summarized by Kinsman [1965] and Phillips [1977]. The wave field described by the Gaussian model can be envisioned as consistingof denumerably many independent pure sinusoidalcomponents. That is, the surface profile is given by [(x,t) = • aicosxi (1) i As the number of the components increases, by the central limit theorem the probability density function of the surface given by (1) becomes Gaussian. Although this simplified model offers a good approximation to a great number of problems, there are also difficulties.To begin with, indepen- dent pure sinusoidal componentsonly satisfy the first-order equationsof motion but not the complete set. To the limit, the first-order approximation can only hold true when the wave slope ak approaches zero with a, the amplitude of the wave, and k, the wave number. For many real wave fields, the values of ak may be indeed small when compared with This paper is not subjectto U.S. copyright. Published in 1983by the American GeophysicalUnion. Paper number 3C0795. unity, but nevertheless finite. When the waves approachthe breaking limit, the value of ak could reach one half. Even at a fraction of the limiting value, the nonlinear effect of the wave could still be too pronouncedto be ignored. A more accurate solution of the equationsof motion for each single componentwould be, in deep water, 1 a2k [(x,t) =-a2k+ acosx + 2 2 3a3k 2 ßcos 2X + • cos 3X + ß '' (2) 8 according to Stokes [1847], which contains contributions from higher-orderapproximation. One of the most obvious effects of the nonlinearity is the distortion of the surface causedby the harmonic components.The wave profile will show increasingly sharperpeaks and shallowertroughswith higher ak values. It is no longer symmetric with respect to the mean water level. Consequently, the surface elevation distributionwill not be Gaussiananymore. Laboratory data by Huang and Long [1980] and field observations by Forris- tall [1978] and Earle [1975] also clearly show that surface elevation deviates from Gaussian. This non-Gaussian distributioncaused by nonlineardistor- tion was first modeled by Longuet-Higgins [1963], using Edgeworth'sform of the type A Gram-Charlier series.Later study by Huang and Long [1980] showed that a truncated four-term Edgeworthexpression well repressents the proba- bility density function of surface elevation. The use of the Gram-Charlier seriesfor probability density function, how- ever, has certain drawbacks. First, the Gram-Charlier ap- proximationgives negativedensity values for somerange of elevation, especially in the cases of steep waves. This obviously violates the definition of a probability density function. Second,the Gram-Charlier approximation callsfor 7597