Gravity-capillary Wave Spectrum Research Articles

Directional Spreading Function of the Gravity-Capillary Wave Spectrum Derived from Radar Observations

Directional spreading function of the gravity-capillary wave spectrum can provide the high-wavenumber wave energy distribution among different directions on the sea surface. The existing directional spreading functions have been mainly developed for the low-wavenumber gravity wave with buoy data. In this paper, we use radar observations to derive the directional spreading function of the gravity-capillary wave spectrum, which is expressed as the second-order Fourier series expansion. So far the standard form of the second-order harmonic coefficient has not been proposed to correctly unify the gravity and gravity-capillary wave. Our strategy is to introduce a correcting term to replace the inaccurate gravity-capillary spectral component in Elfouhaily’s directional spreading function. The second-order harmonic coefficient at L, C and Ku band calculated by the radar observation is used to fit the correcting term to obtain one at the full gravity-capillary wave region. According to our proposed the directional spreading function, there is a spectral region between the gravity and gravity-capillary range where it signifies the negative upwind–crosswind asymmetry at low and moderate speed range. And this is not reflected by the previous models, but has been confirmed by radar observations. The Root Mean Square Difference (RMSD) of the proposed second-order harmonic coefficient versus the radar-observed one at L, C band Ku band is 0.0438, 0.0263 and 0.0382, respectively. The overall bias and RMSD are −0.0029 and 0.0433 for the whole second-order harmonic coefficient range, respectively. The result verifies the accuracy of the proposed directional spreading function at L, C band Ku band.

Dynamics of Short Sea Wave Spectrum Estimated From Microwave Radiometric Measurements

This paper presents some results of the experiment CAPMOS'05 performed on an oceanographic research platform in the Black Sea. The platform, located 600 m offshore, was equipped with a set of contact and remote sensing instruments. Conventional contact sensors were used for direct measurements of atmosphere and sea boundary layer parameters (wind speed and direction, air temperature, water temperature and salinity profiles, etc.), whereas microwave and infrared band radiometers were used for remote sensing measurements of surface temperature and wave parameters. In particular, microwave Ka-band radiometer measurements were applied for gravity-capillary wave spectrum retrieval using the original techniques based on angular measurements. The spectrum component evolution under unstable wind conditions has been investigated. It has been demonstrated that the spectral components in the vicinity of the maximum of the wave curvature are the most sensitive to the wind-velocity variations.

Retrieval of gravity-capillary spectrum parameters by means of microwave radiometric techniques

The paper presents some results obtained during a series of microwave remote sensing experiments carried out on the Black Sea coast in 1999-2002. The measurements were made from a pier at the South Department of the Shirshov Institute of Oceanology with microwave radiometers operating at wavelengths of 1.5, 0.8, 0.5, and 0.3 cm. The surface wave spectrum parameters were estimated from angular radiometric measurements using a novel approach. Microwave brightness contrasts at Ka-band (37 GHz) measured over wide range of incidence angles were used for a curvature spectrum of gravity-capillary waves and a mean-square slope of gravity waves retrieval. The evolution of spectrum parameters under variable wind (ranged from 0.5-7 m/s) was investigated. The delay of brightness variation relative to wind speed as large as 1-2 h was found. The performed spectral analysis permitted to relate the observed brightness delay to short gravity-capillary surface waves, whereas longer gravity waves followed the wind practically without any delay. It is concluded that the proposed technique of gravity-capillary wave spectrum retrieval provides a useful tool for investigation of air-sea interaction and surface wave dynamics.

Bispectral Analysis of Gravity Capillary Waves

We perform a laboratory study of the nonlinear phase relation between short gravity waves with wavelengths of the order of 10 cm and their higher harmonics including “spurious” capillary ripple. The presence of higher harmonics of the basic low-frequency component in the spectrum of gravity capillary waves (GCWs) is confirmed on the basis of the methods of bispectral analysis. To determine the quantitative relations between the contributions of free and forced components in the GCW spectrum, we use the normalized bicoherence function. We show that it is correctly defined for both broadband random processes and polyharmonic random signals. The quantitative relation between the contributions from free and forced waves in the GCW spectrum at the frequency of the second harmonic is obtained.

The Mean-Square Slope of Ocean Surface Waves and Its Effects on Radar Backscatter

The mean-square slope (MSS) of the sea surface for upwind and crosswind is derived, based on Phillips’ equilibrium spectrum and the model herein on gravity–capillary wave spectrum. The MSS integrated from the above two spectra over high-frequency dissipation length (1 mm) fits the optical observations very well. The radar backscatter cross section (RBCS), calculated from specular reflection theory using the Ku-band filtered MSS, is in keeping with the empirically based Ku-band models by Brown for the GEOS-3 13.9-GHz altimeter, and by Witter and Chelton for the Geosat 13.5-GHz altimeter. Also, the RBCS, calculated using the C-band filtered MSS, is in keeping with the ERS-1/-2 scatterometer empirically based algorithms CMOD3 and CMOD4. The physics included in this model on gravity–capillary wave spectrum is also illustrated. The short-wave dissipation due to wave–drift interactions has the effect of suppressing the spectral density at high wind condition, which further influences the directional spreading rate. This effect can be denoted by c2/U210 or c2/c2p dependence of short-wave spectrum. It is suggested that the kp/k dependence observed in the range of gravity waves should not be extended to the region of short waves. The parasitic capillary wave dissipation due to molecular viscosity can be balanced by the energy supply from the underlying waves, hence it is removed from the model. The eddy viscosity is due to turbulence at the wind-drift layer, which suppresses the spectrum of high-frequency waves with wavelengths on the order of millimeters.

The Wind-Induced Wave Growth Rate and the Spectrum of the Gravity-Capillary Waves

Abstract A form of the spectrum of gravity-capillary waves is suggested, based on a balance of the wind input, the spectral flux divergence, the viscous dissipation, and the modulation from the wave-drift interaction. The spectrum in the alongwind direction is expressed as where k is the wavenumber, m and α are constants, u* is the wind friction velocity, δ is the threshold wind friction velocity, c is the short-wave phase velocity, and mo is the zeroth moment of nondirectional frequency spectrum of long-wave orbital velocity. For shorter capillary waves, the addition of dissipation due to the eddy viscosity is needed. A kinematic wave breaking criterion is applied in the calculation of the short-wave dissipation. It is found that this short-wave dissipation, due to wave-drift interactions, has the effect of suppressing the spectrum at higher wind speed and yields a good agreement with the measurements of both the wave spectrum and the wave directional spreading rate. It is also found that the fluctuation...

Nonlinear interactions of gravity-capillary waves: Lagrangian theory and effects on the spectrum

A weakly nonlinear inviscid theory describing the interactions within a continuous spectrum of gravity-capillary waves is developed. The theory is based on the principle of least action and uses a Lagrangian in wavenumber-time space. Advantages of this approach compared to the method of Valenzuela & Laing (1972) are much simplified mathematics and final equations and validity on a longer timescale. It is shown that much of the development of the spectrum under the influence of nonlinear terms can be understood without actually having to integrate the equations. To this end multiwave space, a new concept comparable with phase space, is introduced. Using multiwave space the magnitude of the nonlinear transfer is estimated and it is shown how the energy goes through the spectrum. Also it is predicted that at fixed wavenumbers, the smallest being 520 m −1 , finite peaks will arise in the spectrum. This is confirmed by numerical integrations. From the integrations it is also deduced that nonlinear interactions are at least as important to the development of the spectrum as wind growth. Finally it is shown numerically that the near-Gaussian statistics of the sea surface are unaffected by nonlinear interactions.

Backscatter of microwaves from the sea surface and the modulation of spectra of short gravity waves

Backscatter cross-sections of microwaves from the sea surface are calculated by using the facet model and are compared withJonswap '75 experimental results. The principal features obtained are: (1) asymmetry of backscatter cross-sections between upwind and downwind directions is attributable to the modulation of the short gravity-capillary wave spectrum by a larger wave, and the non-Gaussian wave slope distribution has a tendency to cancel this effect, (2) angular spreading of the energy spectrum in the higher frequency range should have a narrower band than a simple cosine distribution, (3) the facet model itself should begin to break down at a larger incident angle than previously supposed.