Statistical characterization of sea surface geometry for a wave slope field discontinuous in the mean square

Abstract

Statistics of two dimensional wave groups, of steep wave events, and of a cascade pattern manifested in the surface geometry in a developed sea state are derived. However, mathematical theories used to parameterize these as well as many other features of random surfaces have very limited ranges of validity. For example, high-order moments of wave spectra appearing in the calculations of wave slope statistics cannot be evaluated because of divergence of the corresponding integrals. In the present paper the restrictions are reviewed and the difficulties are shown to be due to a pseudo-fractal geometry of the sea surface whose spectrum is known only within a limited range of frequency (characterized by either the resolution of a measuring technique or the constraints of a theoretical model). An approach is presented that solves the problem: treating the surface elevation field as specified on a spatial (temporal) running grid, an averaging procedure is developed employing the Ta_ylor microscale as the mesh size. The technique is illustrated by first exposing errors in direct calculations of the effective surface impedance for a coherently reflected L band radio wave. The errors arise from the use of wave spectra whose highfrequency tail is identified with the Phillips saturation range. The technique is then employed in the study of wave groups and steep waves for a Gaussian, two-dimensional, time-varying surface. In particular, it is found that wave groups are not observable in a developed sea. Finally, the theory is applied to estimating breaking wave statistics. A comparison with field observations is presented.

A unified deep‐to‐shallow water wave‐breaking probability parameterization

“…Because these windows are defined with a nondimensional width δ , the steepness r ( f ) H r ( f ) is truly nondimensional and, thus, an appropriate measure of the wave geometry. Following Phillips [1958] and Glazman [1986], we assume that this geometry is closely related to breaking statistics. Obviously, wave directionality may also be important for breaking statistics.…”

The whitecap coverage model from breaking dissipation parametrizations of wind waves

“…The probability model of whitecap coverage established by Snyder and Kennedy [1983] in terms of an acceleration criterion is expressed as Using time averaging method [ Glazman , 1986] and their observational data, Xu et al [2000] derived m 4 as follows Taking β = 0.3 by fitting their data, Xu et al [2000] proposed the following whitecap coverage model With , the nondimensional fetch in could be converted to wave age β * , i.e., Note that C d is wind speed dependent as shown by . In order to compare with the present study, we take C d = 1.65 × 10 −3 , which is the average value of the typically measured range of (1.0–2.3) × 10 −3 for C d and is close to C d = 1.5 × 10 −3 used by Zhao and Toba [2001] for the conversion of friction velocity to 10‐m wind speed.…”

“…Since m 4 is theoretically indeterminable by the wind wave spectra characterized by the Phillips equilibrium range and the values of β are quite variable in both theory and experiment, it is difficult to apply the analytical model of Snyder and Kennedy [1983] for practical purpose. Recently Xu et al [2000] developed this analytical model into more applicable form by calculating m 4 from the mean JONSWAP spectrum [ Hasselmann et al , 1973] with the time‐averaging method [ Glazman , 1986] and determining β on the basis of fitting their data sets. The resulting formula of whitecap coverage is expressed in terms of nondimensional fetch only, which is a kind of measure for the state of wind wave development.…”

Evaluation of a Roughness Length Model and Sea Surface Properties with Data from the Baltic Sea

“…A final concern regards filtering: Glazman (1986) filtering is supposed to account for the limited curvature of the short waves. If this filtering really describes the processes on this scale, then it should also be used when estimating z 0 , not only as a tool to calculate higher spectral moments.…”