Short wind waves on the ocean: Wavenumber-frequency spectra

Dominant surface waves on the ocean exhibit a dispersion relation that confines their energy to a curve in a wavenumber‐frequency spectrum. Short wind waves on the ocean, on the other hand, are advected by these dominant waves so that they do not exhibit a well‐defined dispersion relation over many...

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Bibliographic Details
Published in:Journal of geophysical research. Oceans 2015-03, Vol.120 (3), p.2147-2158
Main Author: Plant, William J.
Format: Article
Language:English
Subjects:
Analogs
Butterflies
Direction
Dispersion
dispersion relation
Dispersions
Doppler sonar
Electromagnetic scattering
Energy
Frequency spectra
Frequency spectrum
Geophysics
Incidence
Integration
Long waves
Measurement
Oceans
Orbital velocity
Oscillators
Planes
probability distribution
Probability theory
Short wave
Spectra
Surface waves
Wave dispersion
Wave frequency
Wave spectra
Wavelengths
Wavenumber
Wind
Wind waves
wind-wave spectra
Yield
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Summary:Dominant surface waves on the ocean exhibit a dispersion relation that confines their energy to a curve in a wavenumber‐frequency spectrum. Short wind waves on the ocean, on the other hand, are advected by these dominant waves so that they do not exhibit a well‐defined dispersion relation over many realizations of the surface. Here we show that the short‐wave analog to the dispersion relation is a distributed spectrum in the wavenumber‐frequency plane that collapses to the standard dispersion relation in the absence of long waves. We compute probability distributions of short‐wave wavenumber given a (frequency, direction) pair and of short‐wave frequency given a (wavenumber, direction) pair. These two probability distributions must yield a single spectrum of surface displacements as a function of wavenumber and frequency, F(k,f). We show that the folded, azimuthally averaged version of this spectrum has a “butterfly” pattern in the wavenumber‐frequency plane if significant long waves are present. Integration of this spectrum over frequency yields the well‐known k−3 wavenumber spectrum. When integrated over wavenumber, the spectrum yields an f−4 form that agrees with measurement. We also show that a cut through the unfolded F(k,f) at constant k produces the well‐known form of moderate‐incidence‐angle Doppler spectra for electromagnetic scattering from the sea. This development points out the dependence of the short‐wave spectrum on the amplitude of the long waves. Key Points: Short waves on the ocean do not obey a dispersion relation Short‐wave k and f come from the probability of long wave orbital velocities The wavenumber‐frequency spectrum of short waves exhibits a butterfly pattern
ISSN:2169-9275
2169-9291
DOI:10.1002/2014JC010586