Wave prediction models provide reliable forecasts of the directional wave spectra that can be used to obtain the sea-state-dependent momentum and energy fluxes into the ocean. These fluxes depend on the shape of the directional wave variance spectrum F, which for deep water waves is determined by the wave energy balance equation : ∂∂t+cg⋅∇F=Sin+Snl+Sd, where F(ω,θ) depends on the wave frequency ω and direction θ, and cg is the group velocity of the waves. The wave source terms Sin, Snl, and Sd represent wave growth by wind, nonlinear transfer between wave components, and wave dissipation due to wave breaking/white capping, respectively. When the wave field is known from a wave model, the release of kinetic energy from wave breaking can be calculated from Sd as follows e.g.,: Φoc=ρwg∫02π∫0∞Sddωdθ.

If wave spectra are not available, this energy flux can be parametrized by Φoc/ρw=αu*3 , where u* is the water-side friction velocity and α a dimensionless parameter (α=100 is frequently used).

The momentum flux to the ocean column (τo) consists of the flux transferred by turbulence across the air–sea interface and a flux of momentum from waves due to wave breaking and white capping. Using the source terms in Eq. (), the effective momentum flux into the ocean may be written as e.g., τo=τa-ρwg∫02π∫0ωcκωSin+Sddωdθ, where κ is the wave number vector and τa denotes the total atmospheric stress. It is often assumed that there is a balance between wind input and dissipation for higher frequencies than the cutoff-frequency ωc , hence this is the upper limit for the integral over frequencies in Eq. ().

For deep water waves, the Stokes drift uS=uSi^+vSj^ can be calculated to second order in wave steepness from the wave spectral density e.g.,: uS=2∫02π∫0∞ωκFexp⁡(2|κ|z)dωdθ.

From this expression the Coriolis–Stokes force can be calculated according to -ρwfk^×uS, where f is the Coriolis parameter. A discussion on how this force affects the mean flow can also be found in e.g., and in . Assuming no horizontal pressure gradients and a horizontally homogeneous ocean, the horizontal Reynolds-averaged momentum equations with the Coriolis–Stokes force read ∂u¯∂t=-ν∂2u¯∂z2-∂∂zu′w′‾+f(v¯+vS),∂v¯∂t=-ν∂2v¯∂z2-∂∂zv′w′‾-f(u¯+uS), where ν is the molecular viscosity and u′w′‾ and v′w′‾ are the Reynolds shear stresses. It has been assumed that the dominant part is related to the vertical variation. Using the Boussinesq eddy viscosity assumption, the Reynolds shear stresses are determined by u′w′‾=-νt∂u¯∂z,v′w′‾=-νt∂v¯∂z.

The eddy viscosity νt is determined by the turbulence closure model, in this case the two-equation model described later on in Sect. . With the total momentum flux into the ocean given by Eq. (), the boundary condition for u¯ at the surface is given by ρwνt∂u¯∂z|z=0=τo(x),ρwνt∂v¯∂z|z=0=τo(y).

In principle, the part of the momentum flux in Eq. () that comes from wave breaking is distributed in the upper few meters of the ocean, but the explicit form of this wave breaking stress is not known, and there is no clear consensus on how it should be distributed. However, results from indicate that this has little effect on the currents, and in this study the total momentum flux is given as a boundary condition at the surface as in Eq. (). Similar to the momentum equation, the transport equation of an active tracer quantity (such as temperature and salinity) has the form ∂Θ∂t=-∂∂zw′θ‾-νθ∂Θ∂z, where Θ and θ denote mean and fluctuating quantities, respectively, and νθ is the molecular diffusivity. For the purpose of this study, the turbulent flux w′θ‾, similar to the Reynolds shear stresses, will be modeled by an eddy diffusivity νt′ determined by the turbulence closure model such that w′θ‾=-νt′∂Θ∂z.

For the temperature equation an additional term for the solar radiation is added to Eq. (), while the sum of latent, sensible, and long-wave radiation is treated as a flux boundary condition. For the salinity equation, precipitation acts as a source of freshwater flux. Further details on the salinity and temperature equations can be found in .

Including the Stokes shear production, the TKE budget for horizontally homogeneous flow becomes e.g., ∂k∂t=-u′w′‾∂u¯∂z-v′w′‾∂v¯∂z-u′w′‾∂us∂z-v′w′‾∂vs∂z+w′b′‾-∂∂z12w′(u′u′+v′v′+w′w′)‾+1ρw′p′‾-ε, where ε is the dissipation rate, and b′ and p′ denote the fluctuating parts of buoyancy and pressure, respectively. Assuming that the transport term (sixth term on the right hand side of Eq. ) can be expressed by a simple gradient transport formulation, we obtain ∂k∂t=∂∂zνtσk∂k∂z+P+PS+G-ε, where σk is the turbulent Schmidt number. The terms P, PS, and G in Eq. () represent shear production (the two first terms on the right-hand side of Eq. ), the Stokes shear production (third and fourth terms on the right-hand side of Eq. ), and the buoyancy production (fifth term on the right-hand side of Eq. ). With the Boussinesq eddy viscosity assumption, the Reynolds shear stresses in Eq. () are determined by Eq. (). Similarly, the buoyancy production term is modeled by the eddy diffusivity νt′ as follows: G=w′b′‾=-νt′N2, where N is the buoyancy frequency. The eddy viscosity and diffusivity are given by νt=cμk12l,νt′=cμ′k12l, where l is the turbulence length scale and cμ and cμ′ are the stability functions; these can either be constants or functions derived empirically or from a higher-order turbulence model. In a recent study by it is shown that it may be necessary to modify the stability functions in order to take proper account of the effects of Langmuir turbulence in a two-equation turbulence model. However, this requires two more model constants, and here we will use the more traditional stability functions by . In this study, the flux of TKE defined by Eq. () is applied as a boundary condition at the surface, thus -νtσk∂k∂z|z=0=Φocρw.

In addition to solving the TKE equation (Eq. ), we will here use a two-equation closure scheme that requires another prognostic equation to derive information about the turbulence length or time scale. Using the generic length scale (GLS) approach , the second equation is for a generic parameter ψ. Similar to , a Stokes shear production term (PS) is added, producing the equation ∂ψ∂t=Dψ+ψk(cψ1(P+PS)+cψ3G-cψ2ε), where Dψ is a gradient transport term similar to Eq. () and cψ1, cψ2, and cψ3 are model constants. The generic length scale ψ is related to the turbulence kinetic energy k and the length scale l through ψ=(cμ0)pkmln, where cμ0 is the constant value of the stability function cμ in the log layer . For appropriate choices of the exponents p, m, and n, the variable ψ can be directly identified with the classic length-scale-determining variables, such as in the k-ε, k-ω, or k-kl models e.g.,.

Following , the value of the mixing length at the surface is given by l(z=0)=Lz0, where L and z0 are constants and the source of turbulence from breaking waves has been assumed to be at z=0. When the length scale at the surface is given by Eq. (), a Dirichelet boundary condition for ψ can be derived from Eq. (). The parameter z0 is often referred to as a surface roughness length, while L is often taken to be equal to the von Kármán constant (e.g., ). However, it is pointed out by that z0=l/L at z=0 is not related to any kind of surface roughness length; rather, it is connected to the length scale of injected turbulence, which is determined by the spectral properties of turbulence at the source. Different values of L and z0 can be found in the literature. As discussed by , parametrization of wave breaking through a flux boundary condition as in Eq. () is often accompanied by a large prescribed downward diffusion (by specifying values of z0) in order for TKE to penetrate deep enough. use z0=1.6HS and a prescribed length scale l=κ(z0-z)/(1+κ(z0-z)/h), where HS is the significant wave height of the wind sea and h is the mixed layer depth.

In the context of two-equation turbulence closure models, values of z0 are usually somewhat smaller. While and use z0=HS, , who included wave breaking in the GLS approach, use a Charnok-type expression to describe z0. find the best match with observations to be z0=1.3HS. However, use data from an area with shallow water and wave heights smaller than typical open-ocean conditions (note that for the Statfjord A case we have HS up to 4.5m). Based on results from micro-structure temperature and conductivity measurements , use z0=0.2m for wave conditions where HS=3.5m, which is more representative of the conditions during the Statfjord A oil spill. For the present study, we have found that the value of z0=0.6HS, as suggested by , gives reasonable results. Following we use L=0.25.

In this study we consider buoyant particles with a constant prescribed rise velocity wr. The situation is analogous to the suspended sediments described by except that the particles here have a positive buoyancy. The vertical distribution of the particle concentration C can be described by a suspended matter equation ∂C∂t-∂∂zνt′∂C∂z-wrC=0 .

If the concentration is high enough, the mixing of particles will start to influence the TKE budget of the upper layer; in this study we will not consider such strong concentrations and instead neglect the influence of the buoyant particles on the mixing processes.

The mixing model used in our experiments is the General Ocean Turbulence Model GOTM; for a description see, modified to take account of the wave effects described in the previous sections. The momentum equations Eq. () are solved with the upper boundary conditions in Eq. (). The turbulence closure scheme is based on the solutions of Eqs. () and (), with upper boundary conditions given by Eqs. () and (). For the bottom, zero flux boundary conditions are used.

In the experiments described below, the model has been run with rise velocities of wr=50, 100, 200, and 400 m day-1, which could represent oil droplets of different sizes and/or chemical compositions. Notably, wr=100 m day-1 can also represent northeast Arctic cod eggs . For the results shown, the model has been run with the k-ω closure scheme, which performs well in the near-surface layer e.g.,. For the k-ω scheme, the exponents in Eq. () are given by p=-1,m=1/2,n=-1. Each experiment has been run as follows: no wave forcing (control); adding only the Coriolis–Stokes force (C–S); adding only the wave-breaking parametrization (TKE-injection); adding both (C–S + TKE-injection); finally, with the Stokes shear production included (all).

To provide a tool for analyzing the model results we consider classical Ekman theory valid for a constant eddy viscosity, here denoted by Az. The solutions to the classical Ekman problem are then given by u=V0exp⁡(z/DE)cos⁡(π/4+z/DE),v=V0exp⁡(z/DE)sin⁡(π/4+z/DE), where V0=u*21fAz,DE=2Azf. For Az it is frequently assumed e.g., that Az=u*2Xf, where X is a dimensionless parameter and u* is the (water-side) friction velocity. Using Eq. (), V0 and DE can be expressed as V0=Xu*,DE=2Xu*f. finds that X=200, while , who consider wave breaking, use X=32. In a steady state, Eq. () reduces to wr∂∂zC=∂∂zAz∂∂zC.

As boundary conditions we assume C(z=0)=C0,C(z→-∞)=0 . The concentration is then given by C(z)=C0exp⁡zDC, where DC=Azwr is a characteristic concentration depth scale. The relation between the particle concentration and Ekman depth scales can then be expressed as DCDE=u*wr2X, which implies that the fraction DC/DE increases with the wind speed while it decreases with increasing particle rise speed. Interestingly, if both the friction velocity and the rise speed increases a given fraction, the fraction DC/DE remains constant. Although we realize that the factor X is important for the ratio, we do not aim at a detailed analysis of the value of X in this study; we have simply introduced the factor to be able to adjust the curves such that a meaningful comparison between model results and scaling analysis can be made.

Another interesting quantity is the effective transport velocity of particles defined as uC=∫-D0(u+uS)Cdz∫-D0Cdz, where D denotes the ocean depth. For the scaling analysis we assume that uS is zero (as it is not included in our analytical expressions). Analytical expressions that relate the transport to DC and DE can be found; however, here we do not consider these expressions, as they do not provide simple insight into the dynamics of the transport velocity and direction. However, we find a useful relation between the x and y components, uC and vC, that can be described in simple terms as vC/uC=-(1+2DC/DE). We thus find that the transport of particles will have stronger veering when the particle concentration depth scale becomes larger. As an example, for a particle rise velocity of 100 m day-1 and for a surface stress of roughly 0.05Nm-2, we find that DC=DE and vC/uC increases by a factor of 3 compared to the case when all particles are at the surface. The physical interpretation is simply that for deeper distribution of particles, the transport is dominated by the currents deeper down into the Ekman layer, which in turn has stronger veering.

For the evaluation of the model results, we define the Ekman and the characteristic concentration depth scales from the model as DE(m)=∫-D0|u|dz|u(z=0)|,DC(m)=∫-D0CdzC(z=0).

The model results for these variables are shown in Fig. . Also shown as horizontal lines are the solutions predicted by Eq. () for X=32 and X=200. While the model result approaches the solution X=200 in the case when no wave effects are included, the results when wave mixing is included are closer to X=32. This difference is consistent with the previous discussion that, without wave effects, X=200 has been used , while X=32 has been used for the wave breaking case . From simple scaling theory in Eq. () we expect the velocity at the surface and the Ekman depth to vary approximately linearly with the friction velocity (or the scaled Ekman depth to be constant); however, as seen in Fig. , the solutions are far from linear. We also see that the cases with wave breaking are further from the scaling law, suggesting that the additional flux of TKE from waves break the original assumptions in the scaling theory. Figure  (right panel), which scales as the square root of the eddy viscosity, suggests that the eddy viscosity may scale differently for the case with TKE-injection as compared to the case without TKE-injection; i.e., Eq. () is not an appropriate scale relation in the case of TKE-injection, and this spills over to the relation Eq. () that is plotted on the right panel of Fig. (). The eddy viscosity is important for the derivation of velocity, and the same argument applies to the deviations in this case as well. Thus, the scaling laws and the model deviation from the scaling laws provide valuable insight into the dynamics of the system. We especially see that the cases with wave breaking are quite different from the cases without wave breaking and show very different asymptotic behavior. For the normalized characteristic particle depth (i.e., DC/DE), we find that it increases with u* or with the square root of the stress in the scaling analysis. However, taking into account that the turbulent diffusivity is parabolic with depth in a non-stratified ocean, we do expect that the model DC/DE should lie below the scaling law when DC/DE is smaller than unity, and lie above the scaling law when DC/DE is larger than unity. This is illustrated in Fig. . Only the cases where the waves are completely neglected and the case with all wave effects are shown. It may be noted that here we have adjusted the value of X such that the scaling law fits the model for roughly DC/DE= 1.

One interesting feature we notice is that for low rise velocities, the particles are mixed very deep, indicating that the eddy viscosity is quite high even below the region where the Reynolds stresses are important (i.e., the Ekman layer depth). This means that the weak turbulent velocities must be compensated with a large mixing length scale. However it should be noted that in more realistic conditions, which are not considered in these idealized experiments, stratification has an effect on mixing by inhibiting vertical diffusion of TKE. Here we have ignored this effect in order to compare the results with the Ekman theory. The theoretical analysis requires that the well-mixed layer should be much deeper than DE and DC. A factor of 2 can be taken as an example. For momentum, this corresponds to 0.5u*/f, i.e, about 2 times the value in Fig.  (right panel), or 200m for u*=0.05ms-1. For the rising particle analysis it depends strongly on the rise speed (see Eq. ); for slow rise speed we may need a factor 5 times the Ekman depth or up to 1000m deep mixed layer (thus we expect that these situations will never occur in the real ocean and this parameter regime should be considered with some caution).

From the scaling laws Eqs. () and () we expect that the mean transport velocity of the particles will veer as we increase the wind stress, and this is clearly seen in Fig. . We notice that the model does not have as strong veering as predicted by the scaling law, and this is most likely explained by the fact that the eddy viscosity is far from constant in the model. We see that results are closer for a low rise velocity than for a high rise velocity. This is in agreement with the results of the scaling depths for momentum (DE) and particles (DC) as discussed earlier. We also see that cases with all wave effects included have a stronger veering in particle drift than the model without waves, consistent with our expectations that waves mix particles deeper and create a stronger veering due to the Coriolis–Stokes force. So far we have mainly considered non-dimensionalized drift velocities, and the translation to real situations requires a dimensionalization of the results. For more accessible results, we plot the drift speed scaled by the wind speed at 10 m height and the angle θ between the drift direction and the wind direction in Figs.  and . When the waves are not included, the total momentum must be contained in the Eulerian current; when the waves are included, the Stokes drift is added. From the mean particle drift it can be seen that in all cases this relation approaches a constant value, similar to the empirical relation (Eq. ). Since we here consider the transport defined by Eq. (), which is a depth-averaged quantity, the drift speed is far less than 3 % of the wind speed that is often used in Eq. () for surface drift. It should be noted that the drift predicted by the model is also less than the 1 % of the wind speed that was observed by for wind seas over 6 m s-1. However, this may well be due to the fact that in these idealized experiments we have neglected the effect of stratification and that we only consider constant forcing, which is unrealistic for the real ocean. The effects of stratification and changing forcing conditions are included in the realistic case considered in the next section. In agreement with earlier discussion, we also see from Figs.  and that the veering increases with increasing wind speed and that cases with all wave effects has stronger veering than the standard model setup. This effect is notably stronger for the more buoyant particles.

The Statfjord A oilfield is located at approximately 61.25∘ N, 1.85∘ E, about 200 km off the west coast of Norway. The ocean depth close to the platform is about 150m, but with rather steep bottom slopes located nearby to the northeast (Fig. ). This was the site of a large oil spill in December 2007 . Starting on 12 December 2007 at 08:17 UTC and lasting 20–45 min, an estimated 4400 cubic meters of crude oil was spilled into the ocean from a ruptured loading hose near the platform. Since the wind and wave conditions were quite severe during and after the spill (see Fig. ), few observations of the oil slick exist. After a few hours the slick was estimated to be 8 km long and 1 km wide, and by late afternoon on 12 December its surface area covered an estimated 23 km2. The only observation of the oil slick from an aircraft was made 2 days later on 14 December at 13:48 UTC. Located approximately 16 km to the east–southeast of the spill site, the slick was then about 10 km long and 5 km wide. Relating the drift to the wind during the time at which the oil spill started until the observation gives an estimated speed of 0.5 % of the wind speed at an angle of about 90–120∘ to the right.

In order to simulate the net drift of oil, the model was run with a vertical resolution of 0.25m and a time step of 20s. To initialize the model it was gradually spun up from rest over a 24 h period and run with constant forcing over a sufficiently long period to ensure quasi-steady conditions; it was then restarted about 21 h prior to the accident with initial temperature and salinity profiles from measurements. For the results in this case study the model was forced with wind, wave, and radiative forcing from the ERA Interim reanalysis . In Fig.  it can be seen that the wind was essentially directed northwards at approximately 15 m s-1 for about 2 days following the accident, and the significant wave height HS was up to 4.5m. The surface Stokes drift direction and magnitude can also be seen in Fig. . From the start of the oil spill to the observation 2 days later there was a coinciding peak in momentum and energy fluxes into the ocean (Fig. ). Also shown in Fig.  are the net shortwave radiation, net heat flux, and precipitation. Since the area is located at 61.25∘ N and the oil spill occurred in December, the shortwave radiation was rather low. For a period before the oil spill there was a net cooling at the surface, leading to destabilization of the water column.

For hydrography we use observations available from the International Council for the Exploration of the Sea (ICES). There are few observations that coincide with the Statfjord A oil spill; the closest to the time before the accident was located approximately 56 km to the south on 11 December at 11:21 UTC. The first CTD measurement after the observation was close to where the oil was spilled into the ocean (approximately 3 km east) but about 18 hours after the observation by the overflight. Differences in total heat content imply that the two CTD casts sampled slightly different water masses. Although the observations do not coincide perfectly with the oil spill, they show some important characteristics of the conditions during the oil spill. The first CTD cast was used to initialize the model, and the density profile can be seen in Fig. a. Typical for the area and time of year, the water masses before the accident were quite well mixed but with a small density gradient, due to more saline water, starting at approximately 80 m depth (Fig. a). The CTD cast after the observations shows almost no stratification, and the results from the model indicate that this had already occurred by the time the oil spill had started (Fig. b). Concentration profiles for particles wtih wr=100 m day-1 and wr=400 m day-1 are also shown in Fig. . Since the oil was spilled into the ocean from a loading hose that was located below the surface, the exact depth at which the oil was spilled is unknown. However, the oil was probably released into the ocean at various depths during the spill, and we consider initializing the spill with a concentration evenly distributed over the water column (Fig. b) to be realistic. Prevailing wind and wave conditions led to deeper mixing of the particles until approximately 14 December at 00:00 UTC (Fig. b), when the calmer conditions caused more and more particles to resurface until the observation about 14 hours later (Fig. d).

In this area there is a branch of Atlantic water flowing along the bottom contours e.g.,Fig. 9. To make an estimate of this current component we have used the model hindcast archive described by . The approximate magnitude and direction is illustrated by the daily mean at 100 m depth on 13 December 2007 in Fig. . While the wind-driven current and the Stokes drift decay with depth, the topographic current extends over most of the water column; thus the relative importance of the different current components on the transport depends on the depth of the particles.

An example of how the wave effects modify the Eulerian currents can be seen in Fig. . As can be expected from previous studies e.g.,, the Coriolis–Stokes force turns the current further to the right. In all cases when wave effects are included, the sea-state-dependent momentum flux is calculated using Eq. (). The wave-breaking parametrization has a large impact on the velocities close to the surface, while the Stokes shear production in this case is less important for the upper ocean mixing. When all wave effects are included, the surface current speed is reduced by more than 50 %.

Comparing concentration profiles for the cases with and without waves (Fig. ), it can be seen that the waves increase the mixing. In general, the increased mixing by the waves leads to a higher concentration of particles deeper down in the water masses. Hence, the currents deeper down become more important for the net transport when wave-induced mixing is included. Furthermore, it can be seen that the wave effects are more important for higher rise velocities wr; for low wr the shear turbulence is sufficient to mix the particles down, hence profiles with and without waves are more similar. While the wind and wave conditions cause strong mixing at the beginning of the accident, the calmer conditions from approximately 40 hours into the spill cause more and more particles to resurface until the time of the observation (about 53 hours after the spill).

With the current u and the relative concentration profiles C(z) from the mixing model, the background current uBG estimated from the model hindcast of , and the Stokes drift uS calculated from the wave spectra, a transport velocity similar to Eq. () can be defined as u(m)=∫-D0(u+uBG+uS)C(z)dz∫-D0C(z)dz.

Using this, the transport of particles by the model can be compared with the observation, and the result is shown in Fig. . While the direction of the predicted oil spill coincides quite nicely with the observation, the modeled drift seems to be too fast. Depending on the rise velocities, the end locations from the model spread in a southwest–northeast direction similar to the observed oil slick. Also shown in Fig.  are two different end locations predicted when using the empirically based relation (Eq. ). One is the expected drift of oil at the surface with 3 % of the wind speed at an angle of 15∘ to the right , while the other is based on observations that oil mixed below the surface has a mean drift of 1 % of the wind speed with an angle of 90∘ to the right . Clearly the latter case is the more realistic for the Statfjord A oil spill since the wind speed was well above the 6 m s-1 threshold found by (Fig. ). Consistent with the qualitative analysis by , oil was probably quickly mixed below the surface and, as Fig.  indicates, large amounts remained so for the majority of the time until the observation (after a period of calmer conditions).

To illustrate the sensitivity of the model to the depth at which the particles are released, a run with the oil released at the surface is compared with a run where the oil has been released at the bottom in Fig. . Although the initial drift directions of the bottom- and surface-released particles are quite different, the strong mixing results in similar trajectories after a time (depending on the rise speed). The less buoyant particles are affected more strongly by the release depth, and the end locations of particles with rise speeds of 50 m day-1 are located approximately 4 km apart. The background current has a significant effect on both the modeled drift using Eq. () and when using the empirical relation (Eq. ). On average, the magnitude of uBG is about 50 % of the surface value of the combined wind-driven and Stokes drift components. The net drift due to the background current, i.e., when neglecting u and uS in Eq. (), is also shown in Fig. . Here we consider the background current to be the least known, but a more detailed analysis is beyond the scope of the present study. Shear dispersion is not included in these experiments. In the Statfjord A case, the difference between the average drift in the upper 5 m is about 25 % higher than the average drift between 5 and 10 m during the first 24 h following the accident. The difference in results for different rise velocities (Fig. ) also indicates to some extent the potential role of shear dispersion. We emphasize, however, that our ultimate goal is to use a similar mixing scheme in a full 3-D model, which will likely produce more realistic results. We therefore do not wish to further parametrize the effect of shear dispersion here.