Nonlinear Formulation of the Bulk Surface Stress over Breaking Waves: Feedback Mechanisms from Air-flow Separation

Historically, our understanding of the air-sea surface stress has been derived from engineering studies of turbulent flows over flat solid surfaces, and more recently, over rigid complex geometries. Over the ocean however, the presence of a free, deformable, moving surface gives rise to a more complicated drag formulation. In fact, within the constant-stress turbulent atmospheric boundary layer over the ocean, the total air-sea stress not only includes the traditional turbulent and viscous components but also incorporates surface-wave effects such as wave growth or decay, air-flow separation, and surface separation in the form of sea-spray droplets. Because each individual stress component depends on and alters the sea state, a simple linear addition of all stress components is too simplistic. In this paper we present a model of the air-sea surface stress that incorporates air-flow separation and its effects on the other stress components, such as a reduction of the surface viscous stress in the separated region as suggested by recent measurements. Naturally, the inclusion of these effects leads to a non-linear stress formulation. This model, which uses a variable normalized dissipation rate of breaking waves and normalized length of the separation bubble, reproduces the observed features of the drag coefficient from low to high wind speeds despite extrapolating empirical wave spectra and breaking wave statistics beyond known limits. The model shows the saturation of the drag coefficient at high wind speeds for both field and laboratory fetches, suggesting that air-flow separation over ocean waves and its accompanying effects may play a significant role in the physics of the air-sea stress, at least at high wind speeds.     

The Impact Of Air-Flow Separation On The Drag Of The Sea Surface

An approach that allows assessment ofthe impact of air-flow separation (AFS) fromwave breaking fronts on the sea-surface drag is presented. Wave breaking fronts are modelled by the discontinuities of the sea-surface slope. It is assumedthat the dynamics of the AFS from wave breaking crests is similar to thatfrom the backward facing step. The form drag supported by an individualbreaker is described by the action of the pressure drop distributed alongthe forward face of the breaking front. The total stress due to the AFS isobtained as a sum of contributions from breaking fronts of different scales.Outside the breaking fronts the drag of the sea surface is supported by theviscous surface stress and the wave-induced stress. To calculate the stressdue to the AFS and the wave-induced stress a physical model of the wind-wavespectrum is used. Together with the model of the air flow described in termsof surface stresses it forms a self-consistent dynamical system for the seasurface-atmosphere where the air flow and wind waves are strongly coupled.Model calculations of the drag coefficient agree with measurements. It is shownthat the dimensionless Charnock parameter (roughness length normalized onthe square of the friction velocity and the acceleration of gravity)increases with the increase of the wind speed in agreement with fieldmeasurements. The stress due to the AFS normalized on the square of thefriction velocity is proportional to the cube of wind speed. At low windsthe viscous surface stress dominates the drag. The role of the form drag,which is the sum of the stress due to the AFS and the wave-induced stress, isnegligible. At moderate and high winds the form drag dominates. At windspeeds higher than 10 m s^-1 the stress supported by the AFS becomescomparable to the wave-induced stress and supports up to 50% of the totalstress.     

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.The investigation was in part supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organization for Scientific Research (NWO).     

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.The investigation was in part supported by the Netherlands Geosciences Foundation (GOA) with financial aid from the Netherlands Organization for Scientific Research (NWO).     

Aerodynamic roughness of the sea surface at high winds

The role of the surface roughness in the formation of the aerodynamic friction of the water surface at high wind speeds is investigated. The study is based on a wind-over-waves coupling theory. In this theory waves provide the surface friction velocity through the form drag, while the energy input from the wind to waves depends on the friction velocity and the wind speed. The wind-over-waves coupling model is extended to high wind speeds taking into account the effect of sheltering of the short wind waves by the air-flow separation from breaking crests of longer waves. It is suggested that the momentum and energy flux from the wind to short waves locally vanishes if they are trapped into the separation bubble of breaking longer waves. At short fetches, typical for laboratory conditions, and strong winds the steep dominant wind waves break frequently and provide the major part of the total form drag through the air-flow separation from breaking crests, and the effect of short waves on the sea drag is suppressed. In this case the dependence of the drag coefficient on the wind speed is much weaker than would be expected from the standard parameterization of the roughness parameter through the Charnock relation. At long fetches, typical for the field, waves in the spectral peak break rarely and their contribution to the air-flow separation is weak. In this case the surface form drag is determined predominantly by the air-flow separation from breaking of the equilibrium range waves. As found at high wind speeds up to 60 m s⁻¹ the modelled aerodynamic roughness is consistent with the Charnock relation, i.e. there is no saturation of the sea drag. Unlike the aerodynamic roughness, the geometrical surface roughness (height of short waves) could be saturated or even suppressed when the wind speed exceeds 30 m s⁻¹.     

Effects of Drag Coefficients on Surface Heat Flux during Typhoon Kalmaegi (2014)

The lack of in situ observations and the uncertainties of the drag coefficient at high wind speeds result in limited understanding of heat flux through the air-sea interface and thus inaccurate estimation of typhoon intensity in numerical models. In this study, buoy observations and numerical simulations from an air-sea coupled model are used to assess the surface heat flux changes and impacts of the drag coefficient parameterization schemes on its simulations during the passage of Typhoon Kalmaegi (2014). Three drag coefficient schemes, which make the drag coefficient increase, level off, and decrease, respectively, are considered. The air-sea coupled model captured both trajectory and intensity changes better than the atmosphere-only model, though with relatively weaker sea surface cooling (SSC) compared to that captured by buoy observations, which led to relatively higher heat flux and thus a stronger typhoon. Different from previous studies, for a moderate typhoon, the coupled simulation with the increasing drag coefficient scheme outputted an intensity most consistent with the observation because of the strongest SSC, reasonable ratio of latent and sensible heat exchange coefficients, and an obvious reduction in the overestimated surface heat flux among all experiments. Results from sensitivity experiments showed that surface heat flux was significantly determined by the drag coefficient-induced SSC rather than the resulting wind speed changes. Only when SSC differs indistinctively (<0.4°C) between the coupled simulations, heat flux showed a weak positive correlation with the drag coefficient-impacted 10-m wind speed. The drag coefficient also played an important role in decreasing heat flux even a long time after the passage of Kalmaegi because of the continuous upwelling from deeper ocean layers driven by the impacted momentum flux through the air-sea interface.     

One-dimensional theory of the wave boundary layer

Results obtained in a 2-D modeling of the statistical structure of the wave boundary layer (WBL) are used for elaboration of the general approach to 1-D modeling taking into account the spectral properties of wave drag for an arbitrary wave field. In the case of the wave field described by the JONSWAP spectrum, the momentum and energy spectral density exchange, vertical profiles of the wave-induced momentum flux and dependence of total roughness parameter and drag coefficient on peak frequency are given. The reasons that the total roughness parameter increases with decreasing fetch are explained. The role of wind waves as an active element of the ocean-atmosphere dynamic system is also discussed.     

Geostrophic drag coefficients

Data on the relationship of the surface wind to the geostrophic wind at Porton Down, Salisbury Plain, are presented for various stability conditions and analysed in the light of the Rossbynumber similarity theory. For near-neutral conditions, the geostrophic drag coefficients for geostrophic wind speeds 5 to 15 m s^-1 are close to those found by other workers but at higher speeds the values are low. Comparisons of geostrophic and radar wind speeds for ⋍900-m height, suggest that undetectably small mean cyclonic curvatures of the trajectories of the air are responsible for this departure. A value of the geostrophic drag coefficient for the open sea at wind speeds around 8 m s^-1 (neutral conditions) is deduced from recent observations of the drag in relation to the surface wind, combined with the ratios of 900-mb radar wind to surface wind obtained from the North Atlantic weather ship data tabulations of Findlater et al. (1966).     

ANALYSIS OF SEA-SURFACE DRAG PARAMETERIZATIONS IN OPEN OCEAN CONDITIONS

Data from the Surface Waves and Processes Program (SWAPP) are employed to test current sea-surface drag parameterizations in open ocean conditions. General trends in the data indicate that drag increases with increasing wind speed and wave height, and decreases with wave age. However, scatter in the data limits the use of these parameters and other wave dependent parameterizations for modelling efforts. Upon close inspection, it is found that during the onset of three wind events analyzed separately, each of these parameters correlate well with the drag coefficient. However, the dependence of the drag coefficient on each of these parameters varies markedly from event to event. The disparity appears most closely linked to the turning rate of the wind, indicating that temporal and directional effects may play an important role. A temporal lag of O(4) hours between the rise of the wind and subsequent rise in the drag coefficient is also noticed, further pointing out the complexity of the wind-stress system.     

Fetch Limited Drag Coefficients

Measurements made at a tower located 2 km off the coast of Denmark inshallow water during the Risø Air Sea Experiment (RASEX) are analyzedto investigate the behaviour of the drag coefficient in the coastal zone.For a given wind speed, the drag coefficient is larger during conditions ofshort fetch (2-5 km) off-shore flow with younger growing waves than it isfor longer fetch (15-25 km) on-shore flow. For the strongest on-shorewinds, wave breaking enhances the drag coefficient. Variation of the neutral drag coefficient in RASEX is dominated byvariation of wave age, frequency bandwidth of the wave spectra and windspeed. The frequency bandwidth is proportional to the broadness of the waveheight spectra and is largest during conditions of light wind speeds. Usingthe RASEX data, simple models of the drag coefficient and roughness length are developed in terms of wind speed, wave age and bandwidth. An off-shoreflow model of the drag coefficient in terms of nondimensional fetch isdeveloped for situations when the wave state is not known.     

Impact of waves on the sea drag: measurements in the Baltic sea and a model interpretation

Measurements from the Baltic Sea and a wind-over-wave coupled model are used to study the wave impact on the sea drag. The study has been carried out for different wave conditions, namely a pure wind-sea, following-swell/ mixed sea and cross-swell/ mixed sea. Measurements reveal the fact that the sea drag is dependent on the sea-state. In stationary conditions and in the absence of severe cross-swell, swell reduces drag compared to wind-sea at the same wind speed. The cross-swell enhances the drag as compared to the following-swell case and the magnitude of the drag coefficient is increased with increasing the angle of swell propagation to the wind. It is shown that the agreement between the model results and measurements is good for pure wind-sea and stationary mixed-sea cases. Discrepancies occur at light winds, where most of the data represent pure swell conditions. During these pure swell conditions the data are characterized by a large variation of the drag coefficient. The variation is caused by mesoscale variability in the stress co-spectra, wind-cross-swell effects and nonstationarity in the wave and wind fields not represented in the model.     

A Note on Wind Speed and sea State Dependence of the Heat Exchange Coefficient

Both mean and wave-induced motions generate turbulence in the air flow above sea waves. Assuming a local balance between production of turbulent kinetic energy and its dissipation, an explicit relation for the heat exchange coefficient CH is obtained. It is shown that CH follows a square-root dependence on the drag coefficient CD. However, the proportionality coefficient appears to depend on the sea state, expressed in terms of the coupling parameter. Dependence on the sea state suppresses the CD1/2 wind-speed dependence, and results in a marginal increase of CH with increase in the wind speed.     

Aircraft Observations of Sea-Surface Turbulent Fluxes Near the California Coast

Aircraft turbulence data from the Autonomous Ocean Sampling Network project were analyzed and compared to the Coupled Ocean–Atmosphere Response Experiment (COARE) bulk parametrization of turbulent fluxes in an ocean area near the coast of California characterized by complex atmospheric flow. Turbulent fluxes measured at about 35 m above the sea surface using the eddy-correlation method were lower than bulk estimates under unstable and stable atmospheric stratification for all but light winds. Neutral turbulent transfer coefficients were used in this comparison because they remove the effects of mean atmospheric conditions and atmospheric stability. Spectral analysis suggested that kilometre-scale longitudinal rolls affect significantly turbulence measurements even near the sea surface, depending on sampling direction. Cross-wind sampling tended to capture all the available turbulent energy. Vertical soundings showed low boundary-layer depths and high flux divergence near the sea surface in the case of sensible heat flux but minimal flux divergence for the momentum flux. Cross-wind sampling and flux divergence were found to explain most of the observed discrepancies between the measured and bulk flux estimates. At low wind speeds the drag coefficient determined with eddy correlation and an inertial dissipation method after corrections were applied still showed high values compared to bulk estimates. This discrepancy correlated with the dominance of sea swell, which was a usually observed condition under low wind speeds. Under stable atmospheric conditions measured sensible heat fluxes, which usually have low values over the ocean, were possibly affected by measurement errors and deviated significantly from bulk estimates.     

Air–Sea/Land Interaction in the Coastal Zone

Atmospheric turbulence measurements made at the U.S. Army Corps of Engineers Field Research Facility (FRF) located on the Atlantic coast near the town of Duck, North Carolina during the CASPER-East Program (October–November 2015) are used to study air–sea/land coupling in the FRF coastal zone. Turbulence and mean meteorological data were collected at multiple levels (up to four) on three towers deployed at different landward distances from the shoreline, with a fourth tower located at the end of a 560-m-long FRF pier. The data enable comparison of turbulent fluxes and other statistics, as well as investigations of surface-layer scaling for different footprints, including relatively smooth sea-surface conditions and aerodynamically rough dry inland areas. Both stable and unstable stratifications were observed. The drag coefficient and diurnal variation of the sensible heat flux are found to be indicators for disparate surface footprints. The drag coefficient over the land footprint is significantly greater, by as much as an order of magnitude, compared with that over the smooth sea-surface footprint. For onshore flow, the internal boundary layer in the coastal zone was either stable or (mostly) unstable, and varied dramatically at the land-surface discontinuity. The offshore flow of generally warm air over the cooler sea surface produced a stable internal boundary layer over the ocean surface downstream from the coast. While the coastal inhomogeneities violate the assumptions underlying Monin–Obukhov similarity theory (MOST), any deviations from MOST are less profound for the scaled standard deviations and the dissipation rate over both water and land, as well as for stable and unstable conditions. Observations, however, show a poor correspondence with MOST for the flux-profile relationships. Suitably-averaged, non-dimensional profiles of wind speed and temperature vary significantly among the different flux towers and observation levels, with high data scatter. Overall, the statistical dependence of the vertical gradients of scaled wind speed and temperature on the Monin–Obukhov stability parameter in the coastal area is weak, if not non-existent.     

Wave-dependence of friction velocity,roughness length,and drag coefficient over coastal and open water surfaces by using three databases

The parameterization of friction velocity, roughness length, and the drag coefficient over coastal zones and open water surfaces enables us to better understand the physical processes of air-water interaction. In context of measurements from the Humidity Exchange over the Sea Main Experiment (HEXMAX), we recently proposed wave-parameter dependent approaches to sea surface friction velocity and the aerodynamic roughness by using the dimensional analysis method. To extend the application of these approaches to a range of natural surface conditions, the present study is to assess this approach by using both coastal shallow (RASEX) and open water surface measurements (Lake Ontario and Grand Banks ERS-1 SAR) where wind speeds were greater than 6.44 m s^-1. Friction velocities, the surface aerodynamic roughness, and the neutral drag coefficient estimated by these approaches under moderate wind conditions were compared with the measurements mentioned above. Results showed that the coefficients in these approaches for coastal shallow water surface differ from those for open water surfaces, and that the aerodynamic roughness length in terms of wave age or significant wave height should be treated differently for coastal shallow and open water surfaces.     

Turbulent Airflow and Wave-Induced Stress Over the Ocean

We examine the structure of turbulent airflow over ocean waves. Based on an analysis of wind and wave observations derived from a moored and floating Air–Sea Interaction Spar buoy during the Shoaling Waves Experiment field campaign, we show that the cospectra of momentum flux for wind–sea conditions follow established universal scaling laws. Under swell-dominant conditions, the wave boundary layer is extended and the universal cospectral scaling breaks down, as demonstrated previously. On the other hand, the use of peak wave frequency to reproduce the universal cospectra successfully explains the structure of the turbulent flow field. We quantify the wave-coherent component of the airflow and this clarifies how ocean waves affect momentum transfer through the wave boundary layer. In fact, the estimated wave-induced stresses for swell-dominant conditions explain the anomalous cospectral shapes observed near the peak wave frequency.     

The impact of the hexos programme

In HEXOS, a programme of coordinated laboratory, field and model studies, an international group of participants has extended the range of measurements of evaporation from the sea and has investigated the role of droplets in the transfer of water to the atmosphere. Predictions of a rapid rise in the evaporation coefficient at wind speeds above 15 m s^-1 have not been substantiated. Wind stress measurements showed a relationship with wave age. New methods were developed for coping with flow distortion.