When sea spray droplets are lofted into air from relatively warmer ocean surface compared to air, they can exchange both heat and moisture. Also, as the sea spray are saline, when evaporated, they would either result in saline crystals or, as suggested by , attain a temperature and radius at which they are in a quasi-equilibrium state with their environment.

In order to model such a dynamic process, a microphysical model similar to the one suggested by is needed. However, due to the complexity and excessive computation necessary to integrate such a model within a large-scale atmosphere model, and devised approximations to compute the equilibrium temperature and radius of the evaporating sea spray droplets. With the effects of sea spray evaporation included, the total sensible and latent heat fluxes can be written as HL,T=HL,I+αQL‾HS,T=HS,I+βQS‾-(α-γ)QL‾.

Here, QS‾ and QL‾ in Eq. () are the spray-mediated “nominal” sensible and latent heat fluxes obtained from the microphysical calculation devised by , and HL,I and HS,I are the interfacial latent and sensible heat fluxes representing the interaction at the air–sea interface. Also, α,β, and γ in Eq. () are small non-negative constants obtained by statistically fitting nominal fluxes to the field observations. The nominal fluxes in Eq. () are obtained by integrating sensible and latent fluxes for all the spray droplet radius r values considered in the model: QS‾=∫r1r2QS(r)dr,QL‾=∫r1r2QL(r)dr, where r1 and r2 are the minimum and maximum radius of spray droplets considered in the microphysical computation. The QL and QS in Eq. () are calculated for each droplet radius, where latent heat flux is given as QL(r)=-ρsLv1-r(τf)r034πr33dFdr,if τf≤τr-ρsLv1-reqr034πr33dFdr,otherwise, while sensible heat flux is QS(r)=ρscps(Ts-Teq)1-exp⁡-τfτT4πr33dFdr.

In Eqs. () and (), τf, τT, and τr represent three different timescales associated with the different stages of the sea spray droplets. As per , τf is the time duration that the spray droplet remains aloft in air, while τT is the time taken by the droplet to cool, and finally, τr is the time taken by the droplet to evaporate. Ts and Teq are the temperature of sea surface and the equilibrium temperature of evaporating droplets, ρs is the density of seawater, cps is the specific heat of seawater at constant pressure, Lv is the latent heat of vaporization, and req is the radius of the droplet at which it reaches equilibrium with its environment. The term dF/dr in Eqs. () and () represents the SSGF, which is the rate at which droplets with an initial radius r0 are generated at the sea surface, while (4πr3/3)dF/dr is the total volume flux of spray generated at the sea surface.

Following , the microphysical quantities in Eqs. () and (), i.e. the temperature evolution of the spray droplet, are approximated as T(t)-TeqTs-Teq=exp⁡(-t/τT), and the radius evolution is approximated as r(t)-reqr0-req=exp⁡(-t/τr).

These microphysical quantities depend not only on the initial droplet radius and air–sea temperature difference but also on the relative humidity near the sea surface and the water salinity, as well as the sea level pressure. For the sake of brevity, readers are referred to for details regarding the computation of these terms.

Furthermore, when sea spray is present in the DEL, total surface stress τt can be partitioned into ocean-wave-induced surface stress τw, surface stress supported by sea spray droplets τsp, and the viscous stress τν at the sea surface. The total stress τt in Eq. () can be written as τt=τw+τν+τsp.

In order to obtain the sea-spray-induced stress τsp, we follow an approach similar to the one used to obtain spray-induced thermal fluxes, in which we compute the contribution of individual droplets and then integrate them over the droplet radii considered so as to obtain the total spray-induced stress. Following , the spray-induced stress can be written as τsp=4π3ρs∫r1r2usp(r)r03dFdrdr, where usp is the horizontal velocity of the spray droplet before it falls back in the ocean and is given by usp=u*κlog⁡zspz0.

Here, zsp is the height at which sea spray droplets are ejected and is defined as zsp=0.63Hs. When computing the horizontal velocity of the sea spray, the roughness length z0, friction velocity u*, and significant wave height Hs are obtained from the wave model. When computing usp using Eq. (), it is assumed that the spray droplets are ejected at the wind speed just above the water surface. As succinctly pointed out by p. 660, “this assumption misses an important part of the life cycle of spray during which droplets accelerate from the velocity they had at water surface to the wind speed”.

The sea spray droplets present in the near-surface layer can be classified into two broad categories, film and jet droplets, which are generated by means of the bursting of bubbles formed due to the air trapped by the breaking of waves, and spume droplets, which are generated by means of the tearing off of the wave crest. As mentioned in , spume droplets are generated at higher wind speeds (>9ms-1) and are usually of radii greater than 30 µm and can be as large as 500 µm, whereas the film and jet droplets have radii less than 30 µm. Furthermore, when the spray droplets are generated, they are either at the ocean wave propagation speed or at rest. These droplets, when lofted in air, get accelerated by the wind speed, thereby affecting the momentum exchange at the air–sea interface. A few possible explanations for the effects of sea spray on the momentum exchange have been provided in literature; argued that the spray droplets, when they return to the ocean surface, will result in the sheltering of short waves, responsible for carrying much of the wave stress, while , , and suggested that the presence of spray droplets will cause suppression of turbulence, due to spray droplet mass loading, and will increase the stability of the boundary layer.

From Eqs. (), (), and (), it is evident that the volume flux (or the SSGF) of spray generated has a direct influence on the spray-mediated thermal and momentum fluxes. and showed that the spray effects on momentum and thermal fluxes increase at higher wind speeds; it is implied that this is because at higher wind speeds there are large numbers of spume droplets present. However, most of the SSGFs in the literature are only valid at wind speeds below 20ms-1 and droplet radii below 30µm. Following , , and , the spectral distribution of sea spray droplets Sn(r) can be written as Sn(r)=Wf(U)fn(r), where U is the wind speed, usually taken at 10 m height, Wf is the fraction of surface covered by whitecaps, and fn is the distribution of the droplet spectrum. Whitecaps generated at the sea surface can be taken as representative of the wave energy dissipation. Recent studies by and have described two different approaches of obtaining the whitecap fraction; the former follows the approach described by , while the latter follows the method suggested in , hereafter referred to as “Kraan96”. Following and , the turbulent kinetic energy (TKE) flux ϕoc from breaking waves to the ocean is related to the dissipation source function Sds of the spectral wave model and can be given as ϕoc=ρwg∫02π∫0∞Sdsdωdθ, where θ is the wave direction. As per , it can be further assumed that the TKE flux ϕoc in Eq. () is linearly proportional to the whitecap fraction Wf as ϕoc=ϵρwgWfωpE.

Here, ωp is the angular frequency corresponding to the peak wave energy density, E is the total wave energy density, and ϵ is the average fraction of wave energy dissipated per whitecap event and is set to 0.01. In applying Eqs. () and (), it is implicitly assumed that in the equilibrium range the energy source Sin and sink terms Sds in the wave action density equation are in balance : Sin+Snl-Sds=0.

Here, Snl is the non-linear wave–wave interaction, which represents the redistribution of wave energy from large scales to smaller scales. Hence, in the integrated action balance equation, wind input Sin and energy dissipation Sds are in balance, thus permitting the usage of a dissipation source function for estimating the spray-mediated stress term given in Eq. ().

As described earlier (Sect. ), in the present study we estimate the SSGF using Eq. (), whereby the whitecap fraction is obtained via Eq. (). developed a parametric model for estimating the whitecap fraction; they applied their model to wave energy spectrum observations from National Data Buoy Center (NDBC) buoy 46 001 moored in the Gulf of Alaska at 56.3∘ N, 147.9∘ W. They also compared the results from their model to the photographic measurements of the whitecap fraction obtained in the Gulf of Alaska. It was shown that the whitecap fraction obtained from their model was comparable to the photographic observations. They also compared their results with the whitecap fraction model from , hereafter referred to as “MOM80”.

Figure  shows the comparison of the whitecap fraction obtained using , , and Eq. (); the results obtained from Eqs. () and () are comparable to those obtained by . For details on the model, hereafter referred to as “AH2016”, the processing of the wave energy spectrum from buoy measurements, and their validation procedure, readers are referred to . Figure b shows the same data (Fig. a) binned by wind speed. From Fig. , we can infer that the whitecap fractions obtained from Eqs. () and () are higher than that from the AH2016 model. We can also see that both methods show similar wind speed dependence, and at higher wind speeds, both give a lower whitecap fraction compared to the MOM80 model.

One implication of using the wave-state-dependent SSGF is the need to obtain the coefficients α,β, and γ as given in Eqs. () and (). As described in Sect. , these coefficients are obtained from the statistical fit of total fluxes (spray-mediated and interfacial) to the fluxes from field observations. From Eqs. () and () we know that the sea spray fluxes depend on the volume flux of sea spray ejected in the DEL. For the purpose of this study, we followed the procedure described in and utilized the HEXOS dataset . utilized the same HEXOS dataset together with the FASTEX dataset to obtain the constant terms for the spray flux algorithm. Using a microphysical spray model with the COARE 2.6 bulk flux algorithm, we obtained α=7.7036, β=0.0, and γ=8.3202. We also calculated the correlation coefficients of modelled fluxes to the observed fluxes, where the correlation coefficient for sensible heat was 0.93 and for latent heat was 0.89. For the sake of brevity, we do not show the plots comparing the modelled fluxes to fluxes obtained from the observation dataset. Using Eqs. () and (), the total enthalpy flux above the DEL can be written as HS,T+HL,T=HL,I+HS,I+βQS‾+γQL‾.

When viewed in the context of the enthalpy flux, it can be said that only β and γ have an effect on the heat flux transfer from the ocean to the atmosphere. The values for the constants obtained in present study imply that only the spray-mediated latent flux has a role on the heat flux transfer. This is in contrast to the results obtained by and , who found β to be a positive non-zero value. Also, it contradicts the conclusion that the spray sensible heat flux is the primary route by which spray affects the storm energy as stated in . However, we want to stress that further investigation with more observation data is needed to support our earlier statement.

When waves are present, they affect the roughness length on the water surface, which affects the wind velocities and heat flux within the surface layer of the atmosphere. In this study, the atmosphere model provided the wind velocities at the height of 10 m to the wave model; the wave model in turn provides the surface roughness length to the atmosphere model. It is important to point out that the MIKE 21 SW model does not account for the stratification of the surface layer; i.e. it assumes that the surface layer is neutrally stratified. In order to realize the coupling between the atmosphere and the wave model, within the model coupling interface, we implemented the COARE 2.6 bulk flux algorithm, which adjusts the wind velocities obtained from the atmosphere model for neutral stratification.

In this study, surface measurements of wind and wave from two NDBC

http://www.ndbc.noaa.gov/ (last access: 8 June 2016)

Data for wind speed and Hs from the blended product of three different satellites, JASON-2, CRYOSAT, and SARAL (see Fig. ), were obtained from the French Research Institute for Exploitation of the Sea (IFREMER). These satellites follow an orbit with a period of 10 days and provide along-track data with an approximate resolution of 6 km. The dataset from the satellite altimeter measurement has some limitations as given in ; the wind speed data are only reliable for 2 to 20 ms-1, and additionally, the Hs measurements are not reliable beyond 20 m.

A comparison of simulated hurricane tracks, obtained using the minimum sea level pressure, with the location of the storm centre from the best track and the model results is presented in Fig. a. The simulated storm tracks are generally consistent with the best track; the modelled storms first track southwestward and thereafter turn and move northward before making landfall. We notice that all the modelled storms are westward of the best track; however, when coupled with the wave model, the storm tracks improved. We can also see that the storm in the uncoupled model (Expt. 1) has a higher translation speed compared to the coupled atmosphere–wave model (Expt. 2–4) and best track data.

Although sea spray coupling (Expt. 3–4) does not have any appreciable effects on the model track, it does affect the translation speed of storm; in Expt. 3 (i.e. sea spray coupling with only heat fluxes) the storm moves faster compared to Expt. 4 (i.e. sea spray coupling with heat and momentum fluxes). Also, both the storms moved faster compared to Expt. 2.

The time series of minimum sea level pressure (MSLP) are compared with the MSLP from best track data in Fig. b. Comparing the results of different numerical experiments (Expt. 1–4) indicates that the uncoupled model underestimates the storm intensity while the coupled model overestimates the storm intensity. The effect of both the sea spray heat and the momentum flux (Expt. 4) have little effect compared to the atmosphere–wave coupled model (Expt. 2). If only sea spray heat fluxes are considered (Expt. 3), the storm intensity is closer to the best track data. Including the wave and sea spray coupling (Expt. 2–4), the storm intensifies earlier compared to the uncoupled model (Expt. 1).

The temporal development of maximum wind speed at 10 m for the four experiments (Exit. 1–4) is shown in Fig. c. The effects of different model couplings on maximum velocity Vmax are similar to that on the MSLP. We can see that the hurricane under-intensifies by up to ≈10ms-1 in the uncoupled (Expt. 1) model compared to best track data. Also, we note that the storm intensity improves when the atmosphere is coupled with waves (Expt. 2–4). However, in contrast to Sect. , from Fig.  it is evident that the Vmax values are better modelled in Expt. 2 and Expt. 4.

To further investigate the effects of ocean waves and sea spray, we compare the computed wind speeds with the winds measured at NDBC buoys. Figure  shows the location of the buoys considered in reference to the track of Arthur. We can see that buoy 41 002 is located offshore, while buoy 44 014 is located along the track of Arthur. Both coupled and uncoupled models in the present study overpredict the intensity of the storm at 44 014, while they perform well at buoy 41 002 (Fig. a); however, we do note that the wave coupling in the atmosphere results in improved timing of the storm at buoy 44 014 (Fig. b). We also see that coupling the sea spray (Expt. 3–4) results in lower Vmax at the buoy location, though when the spray-mediated momentum flux is applied together with spray-mediated heat fluxes (Expt. 4), both the timing as well as the buoy location are improved. Additionally, Fig.  compares the wind direction at buoy 41 002 and 44 014 obtained from the four numerical experiments. Furthermore, the computed statistics for both wind speed and wave parameters (i.e. wave height Hs and wave period T02) are given in Table . The coupling of ocean surface wave reduces (increase) the error in wind speed at buoy 44 014 (41 002); however, when sea spray fluxes are applied, this reduction in error diminishes. Moreover, when comparing both the RMSE and correlation between wave measurements and data obtained from various numerical experiments, it can be said that the coupling of the wave model improves the model results. It can also be construed from Table  that it is necessary to account for both the spray-mediated heat and the momentum flux, as when only spray-mediated heat fluxes are accounted for, there is a noticeable reduction in the correlation coefficient with an increase in RMSE.

During a hurricane, the waves are not only affected by the winds but also by the hurricane intensity and translation speed among other factors. Figures  and show the comparison of significant wave height Hs and mean wave period T02 for the four different model experiments with the buoy measurements.

The significant wave height Hs and wave period T02 were rather well estimated when the atmosphere model was coupled with waves (Expt. 2–4) compared to the uncoupled model (Expt. 1). However, the size of storm-induced wave fields in the coupled model (Expt. 2) indicates that the storm size is bigger compared to the uncoupled model and buoy measurements.

Also, when sea spray effects are included in the coupled model, the modelled peak Hs values are similar to that of measurements; the hurricane passes the buoy locations earlier than observed. We attribute the bias in timing of the storm passage to the translation speed of the storm, whereby a higher translation speed can result in an increased effective fetch, thereby giving higher wave heights.

We also compared the collocated significant wave heights from model experiments to the satellite-altimeter-derived wave heights (Fig. ). This was done by computing the closest model data point (both in space and time) to each satellite observation point. This allows us to evaluate the spatial and temporal variation of the wave field due to different processes investigated here. The computed statistics of the modelled wave heights compared to the altimeter data are given in Table . When the wave effects were included in the wave model, a higher correlation and lower scatter compared to altimeter data was observed. When sea spray was included in the coupled atmosphere–wave model runs, we can see that including the spray-mediated momentum flux improves the model results.

It is noteworthy that there are only minor differences between the output of uncoupled and coupled model experiments at lower wave heights; however, only coupled models were able to capture the higher wave heights. It is also worth mentioning that apart from modelling uncertainties, the bias in model results can arise from the differences in temporal and spatial resolution of the wave model and the satellite altimeter. Furthermore, for the comparison presented in Fig. , no spatial (or temporal) smoothing of altimeter data was carried out. This is due to the use of an unstructured grid in the present study for the wave model set-up.