V. K. Makin

Drag of the sea surface

It is shown how the drag of the sea surface can be computed from the wind speed and the sea state. The approach, applicable both for fully developed and for developing seas, is based on conservation of momentum in the boundary layer above the sea, which allows one to relate the drag to the properties of the momentum exchange between the sea waves and the atmosphere.The total stress is split into two parts: a turbulent part and a wave-induced part. The former is parameterized in terms of mixing-length theory. The latter is calculated by integration of the wave-induced stress over all wave numbers. Usually, the effective roughness is given in terms of the empirical Charnock relation. Here, it is shown how this relation can be derived from the dynamical balance between turbulent and wave-induced stress. To this end, the non-slip boundary conditions is assigned to the wave surface, and the local roughness parameter is determined by the scale of the molecular sublayer.The formation of the sea drag is then described for fully developed and developing seas and for light to high winds.For the Charnock constant, a value of about 0.018–0.030 is obtained, depending on the wind input, which is well within the range of experimental data.It is shown that gravity-capillary waves with a wavelength less than 5 cm play a minor role in the momentum transfer from wind to waves. Most of the momentum is transferred to decimeter and meter waves, so that the drag of developing seas depends crucially on the form of the wave spectrum in the corresponding high wavenumber range.The dependence of the drag on wave age depends sensitively on the dependence of this high wavenumbertail on wave age. If the tail is wave-age independent, the sea drag appears to be virtually independent of wave age. If the tail depends on wave age, the drag also does. There is contradictory evidence as to the actual dependence. Therefore, additional experiments are needed.

Coupled sea surface-atmosphere model: 2. Spectrum of short wind waves

A physical model of the short wind wave spectrum in the wavelength range from a few millimeters to few meters is proposed. The spectrum shape results from the solution of the energy spectral density balance equation. Special attention is paid to the description of the capillary range of the short wave spectrum. It is assumed that in this range the spectrum shape is determined mainly by the mechanism of generation of parasitic capillaries. This is described as the cascade energy transfer from the gravity to the capillary waves. Thus the capillary wave spectrum results through the balance between generation of capillaries and their viscous dissipation. The short gravity wave spectrum results through the balance between wind input and dissipation due to wave breaking. A parameterization of wind input is obtained in part 1 of the present paper. To describe the dissipation due to wave breaking, the approach developed by Phillips [1985] is used. The spectral rate of energy dissipation is presented in the form of a power dependence of the ratio of the saturation spectrum to some threshold level. It is further shown that the threshold level depends on the drift current shift in the water viscous sublayer, which affects the energy losses by wave breaking. To obtain a short wave spectrum which is valid in the whole wavenumber domain, the capillary and the short gravity wave spectra are patched in the vicinity of the wavenumber corresponding to the minimum phase velocity. This short wave spectrum is incorporated into the wind over waves coupled model developed in part 1 of the present paper. The measured statistical properties of the sea surface related to the short waves, such as the spectral shape of omnidirectional and up-wind spectra, their wind speed dependence and angular spreading, and the wind speed dependence of integral mean square slope and skewness parameters, are well reproduced by the model. Also the model well reproduces the measured wind speed dependence of the drag coefficient and of the coupling parameter.

Coupled sea surface-atmosphere model: 1. Wind over waves coupling

A wind over waves coupling scheme to be used in a coupled wind waves-atmosphere model is described. The approach is based on the conservation of momentum in the marine atmospheric surface boundary layer and allows to relate the sea drag to the properties of the sea surface and the properties of the momentum exchange at the sea surface. Assumptions concerning the local balance of the turbulent kinetic energy production due to the mean and the wave-induced motions, and its dissipation, as well as the local balance between production and dissipation of the mean wave-induced energy allow to reduce the problem to two integral equations: the resistance law above waves and the coupling parameter, which are effectively solved by iterations. To calculate the wave-induced flux, the relation of Plant [1982] for the growth rate parameter is used. However, it is shown by numerical simulations that the local friction velocity rather than the total friction velocity has to be used in this relation, which makes the growth rate parameter dependent on the coupling parameter. It is shown that for light to moderate wind a significant part of the surface stress is supported by viscous drag. This is in good agreement with direct measurements under laboratory conditions. The short gravity and capillary-gravity waves play a significant role in extracting momentum and are strongly coupled with the atmosphere. This fact dictates the use of the coupled short waves-atmosphere model in the description of the energy balance of those waves.

Experimental evidence of the rapid distortion of turbulence in the air flow over water waves

Detailed observations of the air flow velocity, pressure and Reynolds stresses above water waves in a wave flume are presented. The static pressure fluctuations induced by the waves are observed following a new procedure that eliminates acoustical contamination by the wave maker. The measurements are analysed by comparing them with numerical simulations of the air flow over waves. In these numerical simulations the sensitivity to the choice of turbulence closure is studied. We considered both first-order turbulence closure schemes based on the eddy viscosity concept, and a second-order Reynolds stress model. The comparison shows that turbulence closure schemes based on the eddy viscosity concept overestimate the modulation of the Reynolds stress in a significant part of the vertical domain. When an eddy viscosity closure is used, the overestimated modulation of the Reynolds stress gives a significant contribution to the wave growth rate. Our results confirm the conclusions Belcher & Hunt reached on the basis of the rapid distortion theory. The ratio of the wind speed to the phase speed of the paddle wave in the experiment varies between 3 and 6. The observed amplitudes of the velocity and pressure perturbation are in excellent agreement with the simulations. Comparison of the observed phases of the pressure and velocity perturbations shows that the numerical model underpredicts the downwind phase shift of the undulating flow. The sheltering coefficients for the flow over hills and the growth rates of waves that are slow compared to the wind calculated with the Reynolds stress model are in excellent agreement with the analytical model of Belcher & Hunt. Extending the calculations to fast waves, we find that the energy flux to waves travelling almost as fast as the wind is increased on going from the mixing length turbulence closure to the Reynolds stress model.

Impact Of Dominant Waves On Sea Drag

The impact of air-flow separation from breaking dominant waves is analyzed.This impact results from the correlation of the pressure drop with theforward slope of breaking waves. The pressure drop is parameterized via thesquare of the reference mean velocity. The slope of breaking waves isrelated to the statistical properties of the wave breaking fronts describedin terms of the average total length of breaking fronts. Assuming that thedominant waves are narrow and that the length of breaking fronts is relatedto the length of the contour of the breaking zone it is shown that theseparation stress supported by dominant waves is proportional to thebreaking probability of dominant waves. The breaking probability of dominantwaves, in turn, is defined by the dominant wave steepness. With thedominant wave steepness increasing, the breaking probability is increasedand so does the separation stress. This mechanism explains wave age (youngerwaves being steeper) and finite depth (the spectrum is steeper in shallowwater) dependence of the sea drag. It is shown that dominant waves support asignificant fraction of total stress (sea drag) for young seas due to theair-flow separation that occurs when they break. A good comparison of themodel results for the sea drag with several data sets is reported.

Air-sea exchange of heat in the presence of wind waves and spray

Waves extract a considerable part of the surface stress. While breaking, they eject spray into the atmosphere. Spray evaporates and influences a balance of heat and moisture above the waves. A one-dimensional model of the stratified marine surface boundary layer (MSBL) accounts directly for the impact of waves on the momentum flux and the impact of sea spray on fluxes of heat and moisture. The model is viewed as a higher order parameterization of the MSBL compared to the bulk parameterization. The model is based on the balance equations of momentum, the turbulent kinetic energy and the dissipation rate, heat, and moisture. A general experimental knowledge is used to parameterize the jet droplet concentration above the sea. That is, the surface droplet concentration is proportional to the cube of the friction velocity of the air, and the fast decay of droplet concentration with elevation above waves is parameterized by exponential decay. The exchange coefficients for heat, moisture, and momentum are computed from the wind speed and the sea state. Consistency of the dynamical part of the model is checked against measurements of the drag coefficient. Consistency of the thermodynamical part is checked against measurements of the sensible heat flux for light to moderate winds. The impact of spray is then assessed for stronger winds. It is shown that for a wind speed of about 25 m s−1 and above the impact of sea spray on heat and moisture fluxes becomes significant. The magnitude and the sign of the spray mediated heat and moisture fluxes depend on stratification of the atmosphere. To settle the issue whether or not sea spray plays an important role in exchange of heat and moisture above the sea, simultaneous direct measurements of sensible heat and moisture flux under different stratification conditions at wind speeds of about 25 ms−1 are needed.

Impact of waves on air-sea exchange of sensible heat and momentum

The impact of sea waves on sensible heat and momentum fluxes is described. The approach is based on the conservation of heat and momentum in the marine atmospheric surface layer. The experimental fact that the drag coefficient above the sea increases considerably with increasing wind speed, while the exchange coefficient for sensible heat (Stanton number) remains virtually independent of wind speed, is explained by a different balance of the turbulent and the wave-induced parts in the total fluxes of momentum and sensible heat.Organised motions induced by waves support the wave-induced stress which dominates the surface momentum flux. These organised motions do not contribute to the vertical flux of heat. The heat flux above waves is determined, in part, by the influence of waves upon the turbulence diffusivity.The turbulence diffusivity is altered by waves in an indirect way. The wave-induced stress dominates the surface flux and decays rapidly with height. Therefore the turbulent stress above waves is no longer constant with height. That changes the balance of the turbulent kinetic energy and of the dissipation rate and, hence the diffusivity.The dependence of the exchange coefficient for heat on wind speed is usually parameterized in terms of a constant Stanton number. However, an increase of the exchange coefficient with wind speed is not ruled out by field measurements and could be parametrized in terms of a constant temperature roughness length. Because of the large scatter, field data do not allow us to establish the actual dependence. The exchange coefficient for sensible heat, calculated from the model, is virtually independent of wind speed in the range of 3–10 ms-1. For wind speeds above 10 ms-1 an increase of 10% is obtained, which is smaller than that following from the ‘constant roughness length’ parameterization.

Modulation of wind ripples by long surface waves via the air flow : A feedback mechanism

The evolution of a short-wave (SW) spectrum along a long wave (LW) isstudied. The evolution of the SW spectrum variation is treated in therelaxation time approximation. The variation of the SW spectrum is caused bythe LW orbital velocities and by the variation of the wind stress along thesurface of a LW. The latter is due to the distortion of the flow by a LW, andto the variation of the roughness induced by the modulated short waves. Thisintroduces a feedback mechanism: more SWs give rise to a larger roughness,which by increasing the local stress stimulates the growth of more SWs. It isshown that this aerodynamic feedback effect dominates the modulation of theSW spectrum for moderate and strong winds. The feedback mechanism is mosteffective for SWs in the gravity-capillary range, increasing its dominancewith increasing windspeed and decreasing frequency of a LW. The maximum ofthe SW amplitude modulation is situated at the crest of a LW. The results arein agreement with laboratory and field measurements of the short-wavemodulation.

Boundary-Layer Model Results for Wind-Sea Growth

Abstract A numerical model of the boundary layer of the atmosphere above a gravity surface wave is reviewed. The model results are used to obtain an expression for the wind input, the wave growth due to the wind. This is done for wave components that propagate at an arbitrary angle to the wind. Like other purely theoretical expressions for the wind input, the wind input from the boundary-layer model is much smaller than the wind input inferred from field experiments. To study the growth of wind sea, the wind input of the third-generation wave model WAM is replaced by the wind input from the boundary-layer model. The original WAM used a wind input that was inferred from field experiments. For the wave-wave interactions the discrete-interaction approximation is used, while the dissipation is tuned to get an appropriate saturation sea state. The balance between wind input, dissipation, and wave-wave interactions in the energy-containing range of the wave spectrum in this version of the WAM is very different ...

Simplified Model Of The Air Flow Above Waves

A simplified model of the air flow over surface water waves propagating at arbitrary phase velocity as compared to the wind velocity is presented. The approach is based on the subdivision of the air flow into an outer (OR) and an inner (IR) region. In the OR the wave-induced motion experiences inviscid undulation, while in the IR it is strongly affected by the turbulent shear stress. The subdivision of the air flow into two regions considerably simplifies the solution of the problem. The critical height (the height where the wind speed and the wave phase velocity are equal) is for mostcases located inside the IR. Its singular behaviour is strongly suppressed byturbulence. The description of the wind velocities in the OR is based on anapproximate solution of the Rayleigh equation. The description of the IR isbased on the solution of the vorticity equation accounting for turbulentdiffusion. The local eddy-viscosity mixing length closure scheme is used toparameterize the turbulent shear stress. Exponential damping of the shearstress variation with height towards the OR is introduced. This dampingdescribes phenomenologically the basic feature of the wave boundary layer: arapid distortion of turbulence in the OR. Wave-induced velocity and shearstress profiles, and the wave growth rate, resulting from the model showreasonable agreement with those obtained by a two-dimensional numericalmodel based on a second-order closure scheme. Moreover, the velocityprofiles are in good agreement with laboratory measurements.