Surface Wave Effects in the NEMO Ocean Model: Forced and Coupled Experiments111Final version published in J Geophys Res: Oceans, 2015, doi:10.1002/2014JC010565.
The NEMO general circulation ocean model is extended to incorporate three physical processes related to ocean surface waves, namely the surface stress (modified by growth and dissipation of the oceanic wave field), the turbulent kinetic energy flux from breaking waves, and the Stokes-Coriolis force. Experiments are done with NEMO in ocean-only (forced) mode and coupled to the ECMWF atmospheric and wave models. Ocean-only integrations are forced with fields from the ERA-Interim reanalysis. All three effects are noticeable in the extra-tropics, but the sea-state dependent turbulent kinetic energy flux yields by far the largest difference. This is partly because the control run has too vigorous deep mixing due to an empirical mixing term in NEMO. We investigate the relation between this ad hoc mixing and Langmuir turbulence and find that it is much more effective than the Langmuir parameterization used in NEMO. The biases in sea surface temperature as well as subsurface temperature are reduced, and the total ocean heat content exhibits a trend closer to that observed in a recent ocean reanalysis (ORAS4) when wave effects are included. Seasonal integrations of the coupled atmosphere-wave-ocean model consisting of NEMO, the wave model ECWAM and the atmospheric model of ECMWF similarly show that the sea surface temperature biases are greatly reduced when the mixing is controlled by the sea state and properly weighted by the thickness of the uppermost level of the ocean model. These wave-related physical processes were recently implemented in the operational coupled ensemble forecast system of ECMWF.
Surface waves affect the ocean surface boundary layer (OSBL) through a number of processes, but perhaps most visibly through breaking waves which can be seen as whitecaps on the ocean surface (Monahan, 1971; Wu, 1979). These breaking waves enhance the turbulence in the upper part of the ocean significantly (Craig and Banner, 1994; Craig, 1996). Waves absorb energy and momentum from the wind field when they grow and in turn release it when they break (Janssen et al., 2004; Rascle et al., 2006; Ardhuin and Jenkins, 2006; Janssen, 2012). This lowers or raises the stress on the water side (i.e., the stress below the oceanic wave field) relative to the air-side stress, depending on whether the sea state is growing or decaying. Only when the wave field is in equilibrium with the energy injected by the wind will the stress on the two sides of the surface be equal.
Through the interaction with the Coriolis effect, the Stokes drift velocity associated with the wave field adds an additional term to the momentum equation. The effect was first presented by Hasselmann (1970) and has since been investigated for idealized cases by Weber (1983), Jenkins (1987), McWilliams and Restrepo (1999) and McWilliams and Sullivan (2000) among others. The force is variously known as the Stokes-Coriolis force or the Hasselmann force depending on whether it is considered to be purely an effect of the average Coriolis force acting on a particle with a Lagrangian velocity as given by the mean currents and the waves or as a tilting of the planetary vorticity (Polton et al., 2005; Broström et al., 2014). The force does not directly modify the total mass transport but it will alter the distribution of momentum over the depth of the Ekman layer (McWilliams and Restrepo, 1999; Polton, 2009).
The impact of the oceanic wave field on upper-ocean mixing and mean properties has been studied in a number of single-column mixed-layer model experiments (Craig and Banner, 1994; McWilliams and Restrepo, 1999; McWilliams and Sullivan, 2000; Burchard, 2001; Kantha and Clayson, 2004; Mellor and Blumberg, 2004; Rascle et al., 2006; Ardhuin and Jenkins, 2006; Huang et al., 2011; Janssen, 2012). Several studies have employed large eddy simulations (LES) to investigate the impact of Langmuir turbulence in the upper ocean (Skyllingstad and Denbo, 1995; McWilliams et al., 1997; Teixeira and Belcher, 2002; Polton and Belcher, 2007; Grant and Belcher, 2009), and in some cases even direct numerical simulations (DNS) have been employed (Sullivan et al., 2004). Most of these studies find that waves do indeed seem to have a rather profound impact on the upper part of the ocean, but there is still considerable disagreement about which processes are more important. So far there have been few studies of the wave impact on three-dimensional ocean circulation models or fully coupled models of the ocean, the atmosphere and the oceanic wave field although the potential impact of waves on the climate system is recognized (Babanin et al., 2009; Cavaleri et al., 2012; Fan and Griffies, 2014). Fan et al. (2009) demonstrated the importance of correctly modelling momentum and energy fluxes from the wave field to the ocean under hurricane conditions. Fan and Griffies (2014) found that the introduction of Langmuir turbulence following the parameterizations by McWilliams and Sullivan (2000) and Smyth et al. (2002) as well as the parameterization of mixing by non-breaking waves suggested by Qiao et al. (2004) significantly changed the upper-ocean temperature in long-term coupled climate integrations. This latter mixing process appears similar to the mixing due to the high Reynolds numbers of the orbital motion of non-breaking waves explored by Babanin (2006) and Babanin and Haus (2009). Using a climate model of intermediate complexity, Babanin et al. (2009) explored three wave-related mixing processes, namely injection of turbulent kinetic energy from breaking waves, Langmuir circulation and the aforementioned mixing by non-breaking waves. Like Fan and Griffies (2014) they found that all three processes contributed to the mixed layer depth and the temperature of the mixed layer. Similarly, Huang et al. (2011) coupled WAVEWATCH III (Tolman et al., 2002) to a version of the Princeton Ocean Model (Blumberg and Mellor, 1987) and demonstrated an improved summertime temperature profile using the non-breaking parameterization of Qiao et al. (2004). They found very little direct impact by the breaking waves on the temperature.
These wave-driven processes influence the vertical structure of the temperature and current fields in the mixed layer in general, and in the upper few meters in particular. This has implications for coupled models as these processes will affect the feedback between the ocean and the atmosphere (Janssen et al., 2013). However, on shorter time scales and at higher spatial resolution it is also clear that these processes will influence the drift of objects and pollutants on the sea surface or partially or wholly submerged. This has practical importance for oil spill modelling (Hackett et al., 2006), search and rescue (Breivik and Allen, 2008; Davidson et al., 2009; Breivik et al., 2013) and dispersion of biological material (Röhrs et al., 2014).
The NEMO ocean model (Madec and the NEMO team, 2012) has been coupled to the atmospheric model with wave forcing from the wave model as part of the ensemble suite of the Integrated Forecast System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF) since November 2013 (IFS Cycle 40R1). Here we describe the implementation of the three wave effects mentioned above in forced (ocean-only) integrations of NEMO using forcing from the ERA-Interim reanalysis (Dee et al., 2011) as well as their implementation in a fully coupled atmosphere-wave-ocean seasonal forecast system.
The paper is organized as follows. Sec 2 describes the processes that have been implemented and lays out their actual implementation in NEMO. Sec 3 describes the results of long ocean-only integrations and compares with control runs, observations and the ORAS4 ocean reanalysis (Balmaseda et al., 2013). Sec 4 describes the coupled atmosphere-wave-ocean coupling used for seasonal integrations and compares the results of a control run where no direct coupling exists between the wave model and the ocean model to a run where NEMO is forced with stresses, turbulent fluxes and Stokes drift from the wave model ECWAM (ECMWF, 2013). Sec 5 discusses the results and the deficiencies in the existing model setup. Sec 6 concludes and makes suggestions for further work on the investigation of wave effects in ocean-only as well as coupled atmosphere-wave-ocean models.
2 Wave effects in the Ocean Surface Boundary Layer
Introducing wave forcing in an Eulerian ocean model entails communicating the relevant forcing fields from a wave model. We start with a brief presentation of spectral wave models and how the two-dimensional wave spectrum relates to the fluxes and fields that have a bearing on the ocean surface boundary layer.
2.1 Fluxes and fields estimated from a spectral wave model
Third generation spectral wave models (Hasselmann et al., 1988; Tolman, 1991; Komen et al., 1994; Ris et al., 1999; Janssen, 2004; Tolman et al., 2002; Holthuijsen, 2007; Cavaleri et al., 2007) solve the action balance equation (see Eq (1.185) by Komen et al. (1994) and Eq (2.71) by Janssen (2004)) for the wave action density (a function of the Cartesian co-ordinate , frequency and direction ) as follows,
Here, the right-hand source terms refer to wind input (in), nonlinear transfer (nl), and dissipation due to wave breaking (ds), respectively. The dissipation term may include shallow-water effects and bottom friction. The gradient in frequency represents shoaling and refraction, and is the absolute frequency as seen by an observer standing still, whereas is the intrinsic frequency as seen by an observer moving with the current. We also note that
where Â is the wave group veloctiy vector. The wave action density is related to the wave variance density through . In deep water with no current refraction Eq (1) reduces to the energy balance equation,
2.2 The air-side stress modified by surface waves
The presence of an undulating surface affects the roughness felt by the airflow. The atmospheric momentum flux to the oceanic wave field is denoted . It is convenient to define an air-side friction velocity in relation to the total air-side stress, , as
Here, is the surface air density. Charnock (1955) was the first to relate the roughness of the sea surface to the friction velocity,
where (Bidlot, 2012) and the wave-induced stress, , is related to the wind input to the wave field as
where is von Kármán’s constant. Note that the drag coefficient as defined here is related to the 10-m neutral wind speed, . This drag coefficient is computed by ECWAM. The Charnock parameter (6) is the main coupling mechanism between the atmosphere and the wave field in IFS, in place since 1998 (Janssen, 2004).
2.3 The water-side stress modified by surface waves
As the wind increases, the wave field responds by first growing and storing more momentum. In this phase there is a net influx of momentum to the wave field. Then, as the waves mature and the breaking intensifies, the momentum flux from the wave field to the ocean starts to close on the flux from the atmosphere to the waves. This is the equilibrium state where dissipation matches wind input, also referred to as fully developed windsea (since the waves cannot become higher), see e.g. Komen et al. (1994); World Meteorological Organization (1998); Holthuijsen (2007). Finally, as the wind dies down there will be a net outflux of momentum from the wave field, almost all of which will go to the ocean.
If wind input and dissipation in the wave field were in equilibrium, the air-side stress would be equal to the total water-side stress. By water-side stress is meant the stress as seen by the Eulerian ocean, i.e., the momentum flux from the waves. However, most of the time waves are not in equilibrium (Janssen, 2012; Janssen et al., 2013), giving differences in air-side and water-side stress of the order of 5–10%, with occasional departures much larger in cases where the wind suddenly slackens. Likewise, in cases with sudden onset of strong winds the input from the wind field will be much larger than the dissipation to the ocean, lowering the water-side stress to values well below 70% of its normal ratio to the air-side stress. The water-side stress thus equals the total atmospheric stress minus the momentum flux absorbed by the wave field (positive) minus the momentum injected from breaking waves to the ocean (negative), . Here, the dissipation source term is assumed to include all relevant dissipative processes. This can be written (ECMWF, 2013)
The stress from waves is archived as a normalized quantity and is applied as a factor to the air-side stress in our implementation in NEMO.
2.4 Mixing parameterizations
The TKE equation with Reynolds averages can be written
Here, is the TKE per unit mass (with the turbulent velocity) and the dissipation rate (see e.g. Stull (1988), p 152). NEMO has the option of modelling the evolution of TKE with local closure (a prognostic equation in only, see Stull 1988 pp 203–208 and Pope 2000 pp 369–373). Assuming that the advective terms are small in comparison and making the gradient transport approximation where turbulent coefficients are proportional to the gradients in the mean quantities, we arrive at
This is the standard one-equation formulation for NEMO (see the reference manual for NEMO v3.4, Madec and the NEMO team 2012, pp 176–177). Here is the mixing length. The buoyancy term is assumed proportional to the local Brunt-Väisälä frequency,
and the shear production is related to the shear of the mean flow,
Two non-standard mixing processes present in NEMO’s TKE scheme warrant our attention. The first is an artificial boost to the TKE known as the ETAU parameterization which is pegged to the surface TKE with an exponential vertical decay (see Madec and the NEMO team (2012), Sec 10.1),
The depth scale can vary with longitude from 0.5 m at the Equator to 30 m poleward of or be fixed at 10 m. The coefficient, here , can also be varied. The second mixing process of interest to us is a parameterization of Langmuir turbulence according to Axell (2002) which has been implemented in NEMO. The vertical velocity of the Langmuir cells is assumed to peak at , half the maximum depth to which Langmuir cells penetrate,
Here is the surface Stokes drift speed and is a coefficient. Axell (2002), by making an analogy with the characteristic convective velocity scale (D’Alessio et al., 1998) further assumed the Langmuir production term in the TKE equation (10) could be written
This production term will attain a maximum value in the interior of the mixed layer.
Craig and Banner (1994), CB94 hereafter, demonstrated that as waves break they will considerably modify the vertical dissipation profile from the traditional law-of-the-wall where dissipation (Stull, 1988). With wave breaking CB94 found dissipation . Terray et al. (1996) and Drennan et al. (1996) later demonstrated that the observed dissipation rates under breaking waves are indeed much higher than anticipated by the law of the wall. CB94’s model has since been extended to a two-equation turbulence model by Burchard (2001), who demonstrated that the injection of turbulent kinetic energy from breaking waves was sufficient to successfully model the evolution of the mixed layer representative of North Sea conditions. CB94 suggested that the flux of turbulence kinetic energy (TKE) should be related to the water friction velocity as
CB94 assumed that was a constant , but noted that its range would probably be between 50 and 150, depending on the sea state (see also Mellor and Blumberg 2004). The TKE flux from breaking waves is related (see e.g. Janssen et al. 2004; Rascle et al. 2006; Janssen 2012; Janssen et al. 2013) to the dissipation source function of a spectral wave model as
For consistency we have written the energy flux from the waves (thus always negative), , normalized by the air friction velocity . In NEMO the energy flux from breaking waves is introduced as a Dirichlet boundary condition on TKE, following Mellor and Blumberg (2004). It is assumed that in the wave-affected layer the mixing length can be set to a constant where the surface roughness length relates to the significant wave height as , and that in this near-surface region diffusion balances dissipation. In this case the TKE equation takes a simple exponential solution [see Eq (10) by Mellor and Blumberg 2004],
Here the inverse length scale is
with and given by Mellor and Yamada (1982). This is how the flux from breaking waves is implemented in NEMO v3.4. This allows the following simple boundary condition
However, the inverse depth scale (20) is sea state dependent, and for a wave height of, say, 2.5 m, which is close to the global mean, . Thus, varies rapidly with depth, and we have modified the boundary condition (21) by weighting the surface value by the thickness of the topmost level to attain an average value more representative of the turbulence near the surface of the model,
It is clear that as the vertical resolution increases the difference between and becomes smaller, and in the limit the two coincide. The weighting (23) is thus less important with higher vertical resolution. It is worth noting that the exponential profile (19) assumed by Mellor and Blumberg (2004) is only valid very near the surface, and in fact CB94 had already found the solution to the more general case where the mixing length is allowed to vary with depth. We have not implemented this operationally, but preliminary tests suggest that the effect is to roughly double the depth over which the TKE from breaking waves is distributed. The derivation is presented in the appendix.
2.5 The Stokes-Coriolis forcing
Waves set up a Lagrangian displacement in the down-wave direction known as the Stokes drift velocity (Stokes, 1847). Although it decays rapidly with depth, it can be substantial near the surface (). In combination with the Earth’s rotation it adds an additional veering to the upper-ocean currents known as the Stokes-Coriolis force (Hasselmann, 1970),
Here is the Coriolis frequency, is the upward unit vector, is the pressure and is the stress. The full two-dimensional spectrum is in principle required to compute the Stokes drift velocity profile (Janssen et al., 2004; Janssen, 2012),
This is computationally demanding and full two-dimensional wave spectra from a numerical wave model (see e.g. ECMWF 2013) may not always available. It is therefore customary to introduce a simplified, monochromatic Stokes drift profile (see e.g., Carniel et al. (2005); Polton et al. (2005); Saetra et al. (2007); Tamura et al. (2012)). However, it was shown by Breivik et al. (2014) that this profile is a poor match to the full profile and that the following parameterization gives a considerable improvement,
Here the subscript “e” distinguishes the approximate profile from the full Stokes drift velocity profile (25). The surface Stokes drift velocity vector is computed by ECWAM and is available both in ERA-Interim (Dee et al., 2011) and from the operational ECMWF forecasts (ECMWF, 2013).
To compute the profile (26) we must find the inverse depth scale . This is related to the transport through the exponential integral (Abramowitz and Stegun, 1972) and can be solved analytically (Breivik et al., 2014) to yield
Rearranging we get the following expression for the inverse depth scale,
Here , thus
The -th order spectral moment is defined as
The mean frequency is defined as (World Meteorological Organization, 1998; Holthuijsen, 2007) and the significant wave height . We can derive the first moment from the integrated parameters of a wave model or from wave observations and find an estimate for the Stokes transport,
Here is the unit vector in the direction of the Stokes transport. We approximate the Stokes transport direction by the surface Stokes drift (see Breivik et al. (2014)).
3 Ocean-only Forced Model Experiments
The NEMO model is run on a tripolar ORCA 1 grid configuration with 42 vertical levels. The uppermost level is 10 m thick. The model is coupled to LIM2, a two-level thermodynamic-dynamic sea ice model (Fichefet and Maqueda, 1997; Bouillon et al., 2009) and is relaxed weakly towards a climatology in temperature (3-yr e-folding time). No sea surface temperature (SST) relaxation is performed. The ORCA grid is such that the resolution is increased towards the Equator (roughly ) to better resolve tropical waves (see Madec and the NEMO team (2012) for details on the ORCA grid).
The atmospheric and wave forcing fields have been computed from the ERA-Interim reanalysis (Simmons et al., 2007; Dee et al., 2011). ERA-Interim is a continuously updated atmospheric and wave field reanalysis starting in 1979. The resolution of the wave model is on the Equator but the resolution is kept approximately constant globally through the use of a quasi-regular latitude-longitude grid where grid points are progressively removed toward the poles (Janssen, 2004). Similarly, the atmospheric model fields are archived on a reduced Gaussian grid of approximately resolution at the Equator. Some care has to be taken when interpolating between these grids, in particular where wave parameters are interpolated from the ECWAM grid to the ORCA grid. NEMO requires fluxes to be defined in all ocean points. However, there are discrepancies between the ice coverage and the land-sea mask of the wave grid and the ocean grid. This is solved by reverting to the ECMWF drag law (Janssen, 2008; Edson et al., 2013) where ECWAM has ice or land,
The coefficients are , , and . Here is the 10-m wind speed from ERA-Interim. Where ECWAM and NEMO agree on open water, the stress is computed from the drag coefficient of ECWAM,
Here, is the neutral 10-m wind speed, available on the ECWAM grid. The conversion to water-side stress is implemented as
where is the ratio of water-side to air-side stress (see also Eq 34). It is this parameter which is archived by ERA-Interim.
A standard integration period covering the ERA-Interim period from 1979 up until the end of 2009 has been used in the following. A summary of the settings for the model runs can be found in Tables 1 and 2. Four experiments with the new wave-related effects are presented, all compared against a control experiment, CTRL, where standard settings are used for NEMO. The CTRL experiment includes a parameterization of the TKE flux from breaking waves (CB94) without explicit sea state information (see Eq (17) and also Madec and the NEMO team (2012), Sec 10.1) but has no averaging over the topmost model level (22). The stress in CTRL is computed using the ECMWF drag law (32) with air-side stress (33). In all runs the NEMOVAR observation operator relating model state to observation space () is applied. This allows a comparison of model integrations (in our case without assimilation) against the large number of quality-controlled temperature and temperature-salinity profiles compiled in the EN3 data set (Ingleby and Huddleston, 2007). Seasonal averages (December to January, DJF, and June to August, JJA) over the period 1989-2008 are used to compare SST fields in the following.
The first wave experiment, TAUOC, uses the water-side stress described in Sec 2.3 together with the ECWAM drag coefficient (see Sec 2.2). The effect is confined mostly to areas with rapidly developing weather systems in the extra-tropics (Fig 1), where the sea state will be quite far from equilibrium. There is a slight weakening of the wind stress along the west coast of South America and along the coast of south-west Africa, leading to decreased upwelling. This is mainly a consequence of differences between the ECWAM drag coefficient and the drag law (32) caused by limited fetch near the coast. The overall effect is a slight reduction of the bias in the tropics compared to EN3 near-surface temperature measurements (0.1 K, not shown). In the tropics, differences are most likely due to differences in the drag over swell compared to the drag law (32) used for the CTRL experiment.
The second experiment, WTKE, introduces the TKE flux (18) from ECWAM (described in Sec 2.4). The differences found in the extra-tropics amount to more than 2 K (Fig 2). The large difference suggests that the standard settings of NEMO overdoes the mixing due to waves, especially with coarse vertical resolution (cf Eq 22).
The STCOR experiment introduces Stokes-Coriolis forcing from ECWAM as described in Sec 2.5. The largest impact (Fig 3) is found in areas with extra-tropical cyclones in combination with strong temperature gradients, such as across the Gulf Stream and the Kuro-Shio.
Another experiment called LOW where a Law-of-the-wall boundary condition is applied, i.e. no TKE flux from breaking waves, was also performed. The motivation is to establish a lower bound on the mixing. The difference between the WTKE and LOW runs is much smaller (not shown) than the difference between CTRL and WTKE.
We now combine the three wave experiments TAUOC, WTKE and STCOR in one experiment referred to as the WAVE run. Fig 4 shows the standard deviation (upper curves) and the biases of the CTRL run (blue) (blue) (blue) (blue) (blue) (blue) (blue) (blue) and a wave run including all three wave effects (green). The most striking feature is the large-amplitude annual cycle observed for the bias of the CTRL run. This amplitude is much weaker for the run with wave effects, and the seasonality is not at all so clear. Due to the huge differences to the mixing in the runs with and without wave effects, the heat uptake of the ocean also differs significantly. Fig 5 shows the global total heat content anomaly relative to 1979 (the start of the period) for the experiment with wave effects (green) v the CTRL experiment (blue). We also show the ocean reanalysis ORAS4 (red) which is based on an earlier version of NEMO (v3.0), see Balmaseda et al. (2013), with observations assimilated using the NEMOVAR 3D-VAR assimilation system (Mogensen et al., 2012a). The reanalysis covers the period 1957 to present, with forcing provided by ERA-40 (Uppala et al., 2005) and later ERA-Interim from 1979 and onwards. The similarity with ORAS4 is clear, and the trends almost identical. It is also clear that the impact of the wave mixing establishes itself within the first two years of the integration. The pronounced annual cycle (dashed lines) is due to the difference in oceanic volume in the southern and northern hemispheres.
To further assess the impact on the surface temperature we now compare with OI_v2, an SST analysis by Reynolds et al. (2002). The comparison with the CTRL run in Fig 6a reveals large biases, especially in the summer hemisphere. These biases are reduced when wave effects are included (Fig 6b), especially in the northern extra-tropics. This is mainly due to a more correct level of mixing. The long-term SST fields from ORAS4 are very similar to OI_v2 since a strong relaxation to the OI_v2 gridded SST was applied as a flux correction between December 1981 and December 2009 (Balmaseda et al., 2013) and are not shown here.
Finally, to test the relative impact of the ETAU TKE boost (14) and the Langmuir turbulence parameterization (16), we switched off the Langmuir circulation (NOLC) and ETAU (NOETAU) in two additional experiments. As is evident from Fig 7, the Langmuir turbulence has only modest impact on the temperature in the mixed layer in the extra-tropics (50 m depth in the northern extra-tropics is shown in Fig 7). Switching off ETAU has a much larger impact on the mixing, both in terms of bias and random error.
4 The Coupled Atmosphere-Wave-Ocean Model
Coupling between the wave model and the ocean model was first implemented operationally in the ensemble suite of the IFS in Cycle 40R1 in November 2013, but was limited to the mixing (WTKE) and Stokes-Coriolis forcing (STCOR). The seasonal integrations described here include in addition the modified stress (TAUOC) and the ice model LIM2 but are run at lower spatial resolution than the operational ensemble forecasts. Fields are exchanged every three hours (the coupling timestep) between IFS/ECWAM and NEMO. The atmospheric model is run with a spectral truncation of T255 (corresponding to roughly 78 km) with 91 vertical levels to 1 Pa. ECWAM is run at 1.5 resolution while NEMO is run on the ORCA 1 grid described in Sec 3. Fig 8 shows a flow chart outlining the sequence of execution of the various components of the IFS-ECWAM-NEMO single executable over two coupling timesteps, which can be summarized as follows.
The atmospheric component (IFS) is integrated (internal time step 2,700 s) one coupling time step (10,800 s), yielding wind fields for ECWAM as well as radiation and evaporation minus precipitation fields for NEMO
ECWAM (internal time step 900 s) is tightly coupled to IFS and returns sea surface roughness to the atmospheric boundary layer at every atmospheric time step. After one NEMO coupling time step, ECWAM also gives Stokes drift velocity, turbulent kinetic energy and waterside stress to NEMO
NEMO is integrated (internal time step 3,600 s) one coupling time step, yielding SST and surface currents to IFS
All model components have now been integrated one coupling time step and the sequence begins anew
Surface currents and SST are communicated back to the atmospheric model and will affect the stress and the temperature of the boundary layer. This in turn affects the oceanic wave field, but there is presently no direct feedback from NEMO to ECWAM. The coupling between the different components is described in more detail by Mogensen et al. (2012b). Here we compare a setup for seasonal integrations to seven months, with three ensemble members starting from 1 May and 1 November for the period 1993-2012. The CTRL experiment is run with a standard CB94 type wave mixing which corresponds to (Eq 21), similar to the ocean-only CTRL experiment presented in Sec 3. The wind stress is computed using the ECMWF drag law (32) and 10-m wind vectors from IFS. The wave experiment includes the three processes TAUOC, WTKE and STCOR described in Sec 2 and Sec 3. Fig 9 reveals large differences in the bias in the northern extra-tropics relative to ERA-Interim for the boreal summer (JJA) at a lead time of one to three months from 1 May. Panel (a) shows the bias of the CTRL run, with large cold biases in the northern extra-tropics. These biases are broadly similar to what was found from the ocean-only (forced) runs presented in Sec 3 (cf Fig 2). Panel (b) reveals the biases in the run with wave effects to be generally much smaller, although there is a certain deterioration of the upwelling area along the coast of Baja California. Note that the cold bias in the eastern equatorial Pacific (the “cold tongue”) is reduced slightly. The results are similar for the southern hemisphere summer (DJF), although the bias reduction is not as strong as for the northern hemisphere. The seasonal variation of the bias found for the forced (ocean only) runs in Fig 4 is also present in the coupled integrations, see Fig 10. Again, the bias is greatly reduced in the wave run.
We have introduced three wave effects in NEMO, namely the sea-state dependent water-side stress, the energy flux from breaking waves and the Stokes-Coriolis force. Using ocean-only integrations and experiments with a coupled system consisting of the atmospheric model IFS, the wave model ECWAM and NEMO, we demonstrated that the impact of the wave effects is particularly noticeable in the extra-tropics. Of the three processes, the modification of the mixing (WTKE) has the largest impact (Fig 2), but as we discuss below, this is also related to the additional mixing found in NEMO. The impact of the modified stress (TAUOC) and Stokes-Coriolis (STCOR) is also significant (on the order of 0.5 K locally, see Figs 1 and 3). It is also important to note that compared to the law-of-the-wall experiment (LOW), the WTKE differs by only 0.5 K. This again suggests that the CTRL experiment has too vigorous mixing. In ocean-only integrations we see a reduction of the temperature bias in the mixed layer, particularly in the extra-tropical summer (Fig 4). This manifests itself in a more realistic oceanic heat uptake (Fig 5). The coupled seasonal integrations show a similar reduction in bias (compared to ERA-Interim, Fig 9) as the ocean-only wave run (see Fig 10). The mixing is strongly influenced by the ETAU parameterization (14). It is clear that the TKE scheme (11) has too shallow mixing without this parameterization, and it is also clear that the present parameterization of Langmuir turbulence (16) does very little due to its vertical structure which has the distinguishing feature that it will put most if not all of the enhanced turbulence deep into the mixed layer and nothing near the surface. It is thus unable to transport down enough heat to make a substantial difference. This explains why its impact on the temperature in the OSBL (Fig 7) is so much smaller than that from the ETAU term. Its vertical profile (15) is very different from Langmuir parameterizations involving the shear of the Stokes drift velocity profile (McWilliams et al., 1997; Polton and Belcher, 2007; Grant and Belcher, 2009),
Here, is the horizontal velocity fluctuations and the vertical. Due to the strong shear of the Stokes drift profile this would add a larger contribution near the surface. The ETAU profile (14) is quite similar to (35) and it appears to act as a parameterization for Langmuir turbulence with its characteristic exponential decay with depth (35), or similarly mixing by non-breaking waves (Qiao et al., 2004; Babanin, 2006; Huang et al., 2011). It facilitates deeper penetration of mixing from surface processes than what is normally assumed from breaking waves (Grant and Belcher, 2009; Belcher et al., 2012).
6 Concluding remarks and further work
The ocean-only integrations and coupled seasonal integrations all suggest that the right level of mixing is very important for reducing the temperature bias in the upper part of the ocean and also for the oceanic heat uptake. An important result is that introducing wave-enhanced mixing must be done in such a way that the thickness of the uppermost layer is accounted for. This is done with the present implementation by weighting with the thickness of the layer and is essential with model configurations with a thick uppermost level, e.g. ORCA1L42 as discussed here. This would not be necessary if a flux boundary condition had been used for the TKE from breaking waves, but this is not the case in NEMO, which uses the surface boundary condition first proposed by Mellor and Blumberg (2004). The impact of mixing on SST is clearly shown in Fig 6 where runs with and without wave effects are compared with the OI_v2 SST analysis. That the seasonal cycle of the CTRL run is distorted by too vigorous mixing is clear, and Fig 4 shows that the annual cycle in biases extends well below the surface. The conclusions from the ocean-only experiments that temperature biases are reduced by introduction of wave-induced mixing are borne out by the seasonal coupled integrations which essentially show the same bias reduction (Fig 9). It is important to note here, though, that the additional ad hoc deep mixing in NEMO interacts with the surface processes and that without this additional mixing the model fails to mix deeply enough. We speculate that this mechanism is really masking Langmuir turbulence or mixing from non-breaking waves. More work is clearly needed with ocean circulation models and coupled models to fully answer the question of which mixing processes are dominant in the OSBL, but it is clear that getting the mixing right is a balancing act between the right deep mixing and the right mixing near the surface, and these processes are probably all wave-related.
These results are relevant for assessing the impact surface waves have on climate projections (Babanin et al., 2009; Fan and Griffies, 2014), and a natural next step would be to investigate the impact of waves on long, decadal to century-wide integrations (see also the Co-ordinated Ocean-Wave Climate Projections (COWCLIP) initiative, Hemer et al. (2012)). One candidate for forcing ocean-only integrations would be the recently completed ERA 20th century reanalysis (ERA-20C, see Poli et al. 2013, Hersbach et al. 2013, de Boisséson et al. 2014, and Dee et al. 2014). This opens up the possibility of running century-long NEMO integrations with wave effects from a state-of-the-art version of ECWAM (Bidlot, 2012) since all relevant parameters have been archived in the new reanalysis. For coupled climate projections, the nearest candidate would be EC-Earth (Hazeleger et al., 2010, 2012) which operates a modified version of an earlier cycle of IFS. Such experiments would help determining the importance of waves in the climate system, rather than just the impact of climate change on the wave climate.
Appendix A The dissipation profile
The exponential profile (19) for the balance of
assumed by Mellor and Blumberg (2004) is only valid very near the surface where the mixing length can be assumed constant = . CB94 presented the solution to the more general case where the mixing length is allowed to vary with depth. This equation has a power-law solution [cf CB94, Eq (23)],
This leads to a slightly more complicated expression for the vertical average (22),
This work has been supported by the European Union project MyWave
(grant FP7-SPACE-2011-284455). This paper is in partial fulfilment of MyWave
deliverables D1.3 and D1.4. All datasets and model integrations presented in
this study are archived in ECMWF’s MARS and ECFS databases.
For more information on how to access MARS and ECFS, see
We would like to thank the two anonymous reviewers for thorough reviews with suggestions which helped us make the article more succinct and to the point.
- Abramowitz and Stegun (1972) Abramowitz, M., and I. A. Stegun (Eds.) (1972), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, 1064 pp., Dover, New York.
- Ardhuin and Jenkins (2006) Ardhuin, F., and A. Jenkins (2006), On the Interaction of Surface Waves and Upper Ocean Turbulence, J Phys Oceanogr, 36, 551–557, doi:10.1175/2009JPO2862.1.
- Axell (2002) Axell, L. B. (2002), Wind-driven internal waves and Langmuir circulations in a numerical ocean model of the southern Baltic Sea, J Geophys Res, 107(C11), 20, doi:10.1029/2001JC000922.
- Babanin (2006) Babanin, A. V. (2006), On a wave-induced turbulence and a wave-mixed upper ocean layer, Geophys Res Lett, 33(20), 6, doi:10.1029/2006GL027308.
- Babanin and Haus (2009) Babanin, A. V., and B. K. Haus (2009), On the existence of water turbulence induced by nonbreaking surface waves, J Phys Oceanogr, 39(10), 2675–2679, doi:10.1175/2009JPO4202.1.
- Babanin et al. (2009) Babanin, A. V., A. Ganopolski, and W. R. Phillips (2009), Wave-induced upper-ocean mixing in a climate model of intermediate complexity, Ocean Model, 29(3), 189–197, doi:10.1016/j.ocemod.2009.04.003.
- Balmaseda et al. (2013) Balmaseda, M. A., K. Mogensen, and A. T. Weaver (2013), Evaluation of the ECMWF ocean reanalysis system ORAS4, Q J R Meteorol Soc, 139(674), 1132–1161, doi:10.1002/qj.2063.
- Belcher et al. (2012) Belcher, S. E., A. L. M. Grant, K. E. Hanley, B. Fox-Kemper, L. Van Roekel, P. P. Sullivan, W. G. Large, A. Brown, A. Hines, D. Calvert, A. Rutgersson, H. Pettersson, J.-R. Bidlot, P. A. E. M. Janssen, and J. A. Polton (2012), A global perspective on Langmuir turbulence in the ocean surface boundary layer, Geophys Res Lett, 39(18), 9, doi:10.1029/2012GL052932.
- Bidlot (2012) Bidlot, J.-R. (2012), Present status of wave forecasting at ECMWF, in Workshop on Ocean Waves, 25-27 June 2012. Available online at http://www.ecmwf.int/publications/, p. 15.
- Blanke and Delecluse (1993) Blanke, B., and P. Delecluse (1993), Variability of the Tropical Atlantic Ocean Simulated by a General Circulation Model with Two Different Mixed Layer Physics, J Phys Oceanogr, 23, 1363–1388, doi:10/c3hndq.
- Blumberg and Mellor (1987) Blumberg, A. F., and G. L. Mellor (1987), A description of a three-dimensional coastal ocean circulation model, in Three-Dimensional Coastal Ocean Models, edited by N. S. Heaps, AGU Coastal and Estuarine Series 4, American Geophysical Union, Washington D C.
- Bouillon et al. (2009) Bouillon, S., M. Á. Morales Maqueda, V. Legat, and T. Fichefet (2009), An elastic-viscous-plastic sea ice model formulated on Arakawa B and C grids, Ocean Model, 27(3), 174–184, doi:10.1016/j.ocemod.2009.01.004.
- Breivik and Allen (2008) Breivik, Ø., and A. A. Allen (2008), An operational search and rescue model for the Norwegian Sea and the North Sea, J Marine Syst, 69(1–2), 99–113, arXiv:1111.1102, doi:10.1016/j.jmarsys.2007.02.010.
- Breivik et al. (2013) Breivik, Ø., A. Allen, C. Maisondieu, and M. Olagnon (2013), Advances in Search and Rescue at Sea, Ocean Dynam, 63(1), 83–88, doi:10/jtx.
- Breivik et al. (2014) Breivik, Ø., P. Janssen, and J. Bidlot (2014), Approximate Stokes Drift Profiles in Deep Water, J Phys Oceanogr, 44(9), 2433–2445, arXiv:1406.5039, doi:10.1175/JPO-D-14-0020.1.
- Broström et al. (2014) Broström, G., K. H. Christensen, M. Drivdal, and J. E. Weber (2014), Note on Coriolis-Stokes force and energy, Ocean Dynam, 64(7), 1039–1045, doi:10.1007/s10236-014-0723-8.
- Burchard (2001) Burchard, H. (2001), Simulating the wave-enhanced layer under breaking surface waves with two-equation turbulence models, J Phys Oceanogr, 31(11), 3133–3145, doi:10/fvc4n5.
- Carniel et al. (2005) Carniel, S., M. Sclavo, L. H. Kantha, and C. A. Clayson (2005), Langmuir cells and mixing in the upper ocean, Il Nuovo Cimento C Geophysics Space Physics C, 28C, 33–54, doi:10.1393/ncc/i2005-10022-8.
- Cavaleri et al. (2007) Cavaleri, L., J. Alves, F. Ardhuin, A. Babanin, M. Banner, K. Belibassakis, M. Benoit, M. Donelan, J. Groeneweg, T. Herbers, P. Hwang, P. Janssen, T. Janssen, I. Lavrenov, R. Magne, J. Monbaliu, M. Onorato, V. Polnikov, D. Resio, W. Rogers, A. Sheremet, J. M. Smith, H. Tolman, G. van Vledder, J. Wolf, and I. Young (2007), Wave modelling–The state of the art, Prog Oceanogr, 75(4), 603–674, doi:10.1016/j.pocean.2007.05.005.
- Cavaleri et al. (2012) Cavaleri, L., B. Fox-Kemper, and M. Hemer (2012), Wind waves in the coupled climate system, Bull Am Meteor Soc, 93(11), 1651–1661, doi:10.1175/BAMS-D-11-00170.1.
- Charnock (1955) Charnock, H. (1955), Wind stress on a water surface, Q J R Meteorol Soc, 81(350), 639–640, doi:10.1002/qj.49708135027.
- Craig (1996) Craig, P. D. (1996), Velocity profiles and surface roughness under breaking waves, J Geophys Res, 101(C1), 1265–1277, doi:10.1029/95JC03220.
- Craig and Banner (1994) Craig, P. D., and M. L. Banner (1994), Modeling wave-enhanced turbulence in the ocean surface layer, J Phys Oceanogr, 24(12), 2546–2559, doi:10/df29g2.
- D’Alessio et al. (1998) D’Alessio, S., K. Abdella, and N. McFarlane (1998), A new second-order turbulence closure scheme for modeling the oceanic mixed layer, J Phys Oceanogr, 28(8), 1624–1641, doi:10/bwq3zv.
- Davidson et al. (2009) Davidson, F. J. M., A. Allen, G. B. Brassington, Ø. Breivik, P. Daniel, M. Kamachi, S. Sato, B. King, F. Lefevre, M. Sutton, and H. Kaneko (2009), Applications of GODAE ocean current forecasts to search and rescue and ship routing, Oceanography, 22(3), 176–181, doi:10.5670/oceanog.2009.76.
- de Boisséson et al. (2014) de Boisséson, E., M. A. Balmaseda, S. Abdalla, E. Källén, and P. A. E. M. Janssen (2014), How robust is the recent strengthening of the Tropical Pacific trade winds?, Geophys Res Lett, pp. 4398–4405, doi:10.1002/2014GL060257.
- Dee et al. (2011) Dee, D., S. Uppala, A. Simmons, P. Berrisford, P. Poli, S. Kobayashi, U. Andrae, M. Balmaseda, G. Balsamo, P. Bauer, B. P, A. Beljaars, L. van de Berg, J. Bidlot, N. Bormann, et al. (2011), The ERA-Interim reanalysis: Configuration and performance of the data assimilation system, Q J R Meteorol Soc, 137(656), 553–597, doi:10.1002/qj.828.
- Dee et al. (2014) Dee, D., M. Balmaseda, G. Balsamo, R. Engelen, A. Simmons, and J.-N. Thépaut (2014), Toward a consistent reanalysis of the climate system, Bull Am Meteor Soc, 95, 1235–1248, doi:10.1175/BAMS-D-13-00043.1.
- Drennan et al. (1996) Drennan, W., M. Donelan, E. Terray, and K. Katsaros (1996), Oceanic turbulence dissipation measurements in SWADE, J Phys Oceanogr, 26(5), 808–815, doi:10/dftwnb.
- ECMWF (2013) ECMWF (2013), IFS Documentation CY40r1, Part VII: ECMWF Wave Model, ECMWF Model Documentation, European Centre for Medium-Range Weather Forecasts.
- Edson et al. (2013) Edson, J., V. Jampana, R. Weller, S. Bigorre, A. Plueddemann, C. Fairall, S. Miller, L. Mahrt, D. Vickers, and H. Hersbach (2013), On the Exchange of Momentum over the Open Ocean, J Phys Oceanogr, 43, 1589–1610, doi:10.1175/JPO-D-12-0173.1.
- Fan and Griffies (2014) Fan, Y., and S. M. Griffies (2014), Impacts of parameterized Langmuir turbulence and non-breaking wave mixing in global climate simulations, J Climate, doi:10.1175/JCLI-D-13-00583.1.
- Fan et al. (2009) Fan, Y., I. Ginis, and T. Hara (2009), The effect of wind-wave-current interaction on air-sea momentum fluxes and ocean response in tropical cyclones, J Phys Oceanogr, 39(4), 1019–1034, doi:10.1175/2008JPO4066.1.
- Fichefet and Maqueda (1997) Fichefet, T., and M. A. M. Maqueda (1997), Sensitivity of a global sea ice model to the treatment of ice thermodynamics and dynamics, J Geophys Res, 102(C6), 12,609–12,646, doi:10.1029/97JC00480.
- Gaspar et al. (1990) Gaspar, P., Y. Grégoris, and J.-M. Lefevre (1990), A simple eddy kinetic energy model for simulations of the oceanic vertical mixing: Tests at Station Papa and long-term upper ocean study site, J Geophys Res, 95(C9), 16,179–16,193, doi:10.1029/JC095iC09p16179.
- Grant and Belcher (2009) Grant, A. L., and S. E. Belcher (2009), Characteristics of Langmuir turbulence in the ocean mixed layer, J Phys Oceanogr, 39(8), 1871–1887, doi:10.1175/2009JPO4119.1.
- Hackett et al. (2006) Hackett, B., Ø. Breivik, and C. Wettre (2006), Forecasting the drift of objects and substances in the oceans, in Ocean Weather Forecasting: An Integrated View of Oceanography, edited by E. P. Chassignet and J. Verron, pp. 507–523, Springer, doi:10.1007/1-4020-4028-8_23.
- Hasselmann (1970) Hasselmann, K. (1970), Wave-driven inertial oscillations, Geophys Astrophys Fluid Dyn, 1(3-4), 463–502, doi:10.1080/03091927009365783.
- Hasselmann et al. (1988) Hasselmann, S., K. Hasselmann, E. Bauer, P. A. E. M. Janssen, G. J. Komen, L. Bertotti, P. Lionello, A. Guillaume, V. C. Cardone, J. A. Greenwood, M. Reistad, L. Zambresky, and J. A. Ewing (1988), The WAM model—a third generation ocean wave prediction model, J Phys Oceanogr, 18, 1775–1810, doi:10/bhs3rr.
- Hazeleger et al. (2010) Hazeleger, W., C. Severijns, T. Semmler, S. Stefanescu, S. Yang, X. Wang, K. Wyser, E. Dutra, J. M. Baldasano, R. Bintanja, et al. (2010), EC-earth: a seamless earth-system prediction approach in action, Bull Am Meteor Soc, 91(10), 1357–1363, doi:10.1175/2010BAMS2877.1.
- Hazeleger et al. (2012) Hazeleger, W., X. Wang, C. Severijns, S. Ştefănescu, R. Bintanja, A. Sterl, K. Wyser, T. Semmler, S. Yang, B. Van den Hurk, , T. van Noije, E. van der Linden, and K. van der Wiel (2012), EC-Earth V2.2: description and validation of a new seamless earth system prediction model, Climate Dynamics, 39(11), 2611–2629, doi:10.1007/s00382-011-1228-5.
- Hemer et al. (2012) Hemer, M. A., X. L. Wang, R. Weisse, and V. R. Swail (2012), Advancing wind-waves climate science: The COWCLIP project, Bull Am Meteor Soc, 93(6), 791–796, doi:10.1175/BAMS-D-11-00184.1.
- Hersbach et al. (2013) Hersbach, H., C. Peubey, A. Simmons, P. Poli, D. Dee, and P. Berrisford (2013), ERA-20CM: a twentieth century atmospheric model ensemble, ERA Report Series 16, European Centre for Medium-Range Weather Forecasts, http://www.ecmwf.int/publications.
- Holthuijsen (2007) Holthuijsen, L. (2007), Waves in Oceanic and Coastal Waters, 387 pp., Cambridge University Press.
- Huang et al. (2011) Huang, C. J., F. Qiao, Z. Song, and T. Ezer (2011), Improving simulations of the upper ocean by inclusion of surface waves in the Mellor-Yamada turbulence scheme, J Geophys Res, 116(C1), doi:10.1029/2010JC006320.
- Ingleby and Huddleston (2007) Ingleby, B., and M. Huddleston (2007), Quality control of ocean temperature and salinity profiles Historical and real-time data, J Marine Syst, 65, 158–175, doi:10.1016/j.jmarsys.2005.11.019.
- Janssen (1989) Janssen, P. (1989), Wave-induced stress and the drag of air flow over sea waves, J Phys Oceanogr, 19(6), 745–754, doi:10/fsz7vd.
- Janssen (1991) Janssen, P. (1991), Quasi-linear theory of wind-wave generation applied to wave forecasting, J Phys Oceanogr, 21(11), 1631–1642, doi:10/b77wvv.
- Janssen (2004) Janssen, P. (2004), The interaction of ocean waves and wind, 300 pp., Cambridge University Press, Cambridge, UK.
- Janssen (2008) Janssen, P. (2008), Air-sea interaction through waves, in ECMWF Workshop on Ocean-Atmosphere Interactions, 10-12 November 2008. Available online at http://www.ecmwf.int/publications/, pp. 47–60.
- Janssen (2012) Janssen, P. (2012), Ocean Wave Effects on the Daily Cycle in SST, J Geophys Res, 117, 24, doi:10/mth.
- Janssen et al. (2004) Janssen, P., O. Saetra, C. Wettre, H. Hersbach, and J. Bidlot (2004), Impact of the sea state on the atmosphere and ocean, in Annales hydrographiques, vol. 3-772, pp. 3.1–3.23, Service hydrographique et océanographique de la marine.
- Janssen et al. (2013) Janssen, P., Ø. Breivik, K. Mogensen, F. Vitart, M. Balmaseda, J. Bidlot, S. Keeley, M. Leutbecher, L. Magnusson, and F. Molteni (2013), Air-Sea Interaction and Surface Waves, ECMWF Technical Memorandum 712, European Centre for Medium-Range Weather Forecasts.
- Jenkins (1987) Jenkins, A. D. (1987), Wind and wave induced currents in a rotating sea with depth-varying eddy viscosity, J Phys Oceanogr, 17, 938–951, doi:10/fdwvq2.
- Kantha and Clayson (2004) Kantha, L. H., and C. A. Clayson (2004), On the effect of surface gravity waves on mixing in the oceanic mixed layer, Ocean Model, 6(2), 101–124, doi:10.1016/S1463-5003(02)00062-8.
- Komen et al. (1994) Komen, G. J., L. Cavaleri, M. Donelan, K. Hasselmann, S. Hasselmann, and P. A. E. M. Janssen (1994), Dynamics and Modelling of Ocean Waves, 532 pp., Cambridge University Press, Cambridge.
- Madec and the NEMO team (2012) Madec, G., and the NEMO team (2012), Nemo ocean engine v3.4, Note du Pole de modélisation, available from http://www.nemo-ocean.eu/ 27, Institut Pierre Simon Laplace.
- McWilliams et al. (1997) McWilliams, J., P. Sullivan, and C.-H. Moeng (1997), Langmuir turbulence in the ocean, J Fluid Mech, 334(1), 1–30, doi:10.1017/S0022112096004375.
- McWilliams and Restrepo (1999) McWilliams, J. C., and J. M. Restrepo (1999), The Wave-driven Ocean Circulation, J Phys Oceanogr, 29(10), 2523–2540, doi:10/dwj9tj.
- McWilliams and Sullivan (2000) McWilliams, J. C., and P. P. Sullivan (2000), Vertical mixing by Langmuir circulations, Spill Science and Technology Bulletin, 6(3), 225–237, doi:10.1016/S1353-2561(01)00041-X.
- Mellor and Blumberg (2004) Mellor, G., and A. Blumberg (2004), Wave breaking and ocean surface layer thermal response, J Phys Oceanogr, 34(3), 693–698, doi:10/bftfk9.
- Mellor and Yamada (1982) Mellor, G. L., and T. Yamada (1982), Development of a turbulent closure model for geophysical fluid problems, Rev Geophys Space Phys, 20, 851–875, doi:10.1029/RG020i004p00851.
- Mogensen et al. (2012a) Mogensen, K., M. A. Balmaseda, and A. Weaver (2012a), The NEMOVAR ocean data assimilation system as implemented in the ECMWF ocean analysis for System 4, ECMWF Technical Memorandum 668, European Centre for Medium-Range Weather Forecasts.
- Mogensen et al. (2012b) Mogensen, K., S. Keeley, and P. Towers (2012b), Coupling of the NEMO and IFS models in a single executable, ECMWF Technical Memorandum 673, European Centre for Medium-Range Weather Forecasts.
- Monahan (1971) Monahan, E. (1971), Oceanic whitecaps, J Phys Oceanogr, 1, 139–144, doi:10/d7z784.
- Poli et al. (2013) Poli, P., H. Hersbach, D. Tan, D. Dee, J. Thépaut, A. Simmons, C. Peubey, P. Laloyaux, T. Komori, P. Berrisford, et al. (2013), The data assimilation system and initial performance evaluation of the ECMWF pilot reanalysis of the 20th-century assimilating surface observations only (ERA-20C), ERA Report Series 14, European Centre for Medium-Range Weather Forecasts, http://www.ecmwf.int/publications.
- Polton (2009) Polton, J. A. (2009), A wave averaged energy equation: Comment on “global estimates of wind energy input to subinertial motions in the Ekman-Stokes layer” by Bin Liu, Kejian Wu and Changlong Guan, Journal of Oceanography, 65(5), 665–668, doi:10.1007/s10872-009-0057-1.
- Polton and Belcher (2007) Polton, J. A., and S. E. Belcher (2007), Langmuir turbulence and deeply penetrating jets in an unstratified mixed layer, J Geophys Res, 112(C9), 11, doi:10.1029/2007JC004205.
- Polton et al. (2005) Polton, J. A., D. M. Lewis, and S. E. Belcher (2005), The role of wave-induced Coriolis-Stokes forcing on the wind-driven mixed layer, J Phys Oceanogr, 35(4), 444–457, doi:10.1175/JPO2701.1.
- Pope (2000) Pope, S. B. (2000), Turbulent flows, 776 pp., Cambridge university press.
- Qiao et al. (2004) Qiao, F., Y. Yuan, Y. Yang, Q. Zheng, C. Xia, and J. Ma (2004), Wave-induced mixing in the upper ocean: Distribution and application to a global ocean circulation model, Geophys Res Lett, 31(11), 4, doi:10.1029/2004GL019824.
- Rascle et al. (2006) Rascle, N., F. Ardhuin, and E. Terray (2006), Drift and mixing under the ocean surface: A coherent one-dimensional description with application to unstratified conditions, J Geophys Res, 111(C3), 16, doi:10.1029/2005JC003004.
- Reynolds et al. (2002) Reynolds, R. W., N. A. Rayner, T. M. Smith, D. C. Stokes, and W. Wang (2002), An improved in situ and satellite SST analysis for climate, J Climate, 15(13), 1609–1625, doi:10/cdpxf6.
- Ris et al. (1999) Ris, R. C., L. H. Holthuijsen, and N. Booij (1999), A third-generation wave model for coastal regions 2. Verification, J Geophys Res, 104(C4), 7667–7681, doi:10.1029/1998JC900123.
- Röhrs et al. (2014) Röhrs, J., K. H. Christensen, F. Vikebø, S. Sundby, Ø. Saetra, and G. Broström (2014), Wave-induced transport and vertical mixing of pelagic eggs and larvae, Limnol Oceanogr, 59, 1213–1227, doi:10.4319/lo.2014.59.4.1213.
- Saetra et al. (2007) Saetra, Ø., J. Albretsen, and P. Janssen (2007), Sea-State-Dependent Momentum Fluxes for Ocean Modeling, J Phys Oceanogr, 37(11), 2714–2725, doi:10.1175/2007JPO3582.1.
- Simmons et al. (2007) Simmons, A., S. Uppala, D. Dee, and S. Kobayashi (2007), ERA-Interim: New ECMWF reanalysis products from 1989 onwards, ECMWF newsletter, 110, 25–35.
- Skyllingstad and Denbo (1995) Skyllingstad, E. D., and D. W. Denbo (1995), An ocean large-eddy simulation of Langmuir circulations and convection in the surface mixed layer, J Geophys Res, 100(C5), 8501–8522, doi:10.1029/94JC03202.
- Smyth et al. (2002) Smyth, W. D., E. D. Skyllingstad, G. B. Crawford, and H. Wijesekera (2002), Nonlocal fluxes and Stokes drift effects in the K-profile parameterization, Ocean Dynam, 52(3), 104–115, doi:10.1007/s10236-002-0012-9.
- Stokes (1847) Stokes, G. G. (1847), On the theory of oscillatory waves, Trans Cambridge Philos Soc, 8, 441–455.
- Stull (1988) Stull, R. B. (1988), An introduction to boundary layer meteorology, 666 pp., Kluwer, New York.
- Sullivan et al. (2004) Sullivan, P. P., J. C. McWilliams, and W. K. Melville (2004), The oceanic boundary layer driven by wave breaking with stochastic variability. Part 1. Direct numerical simulations, J Fluid Mech, 507, 143–174, doi:10.1017/S0022112004008882.
- Tamura et al. (2012) Tamura, H., Y. Miyazawa, and L.-Y. Oey (2012), The Stokes drift and wave induced-mass flux in the North Pacific, J Geophys Res, 117(C8), 14, doi:10.1029/2012JC008113.
- Teixeira and Belcher (2002) Teixeira, M., and S. Belcher (2002), On the distortion of turbulence by a progressive surface wave, J Fluid Mech, 458, 229–267, doi:10.1017/S0022112002007838.
- Terray et al. (1996) Terray, E. A., M. A. Donelan, Y. C. Agrawal, W. M. Drennan, K. K. Kahma, A. J. W. III, P. A. Hwang, and S. A. Kitaigorodskii (1996), Estimates of kinetic energy dissipation under breaking waves, J Phys Oceanogr, 26(5), 792–807, doi:10/b5zg6t.
- Tolman (1991) Tolman, H. L. (1991), A Third-Generation Model for Wind Waves on Slowly Varying, Unsteady, and Inhomogeneous Depths and Currents, J Phys Oceanogr, 21(6), 782–797, doi:10/ddxwxn.
- Tolman et al. (2002) Tolman, H. L., B. Balasubramaniyan, L. D. Burroughs, D. V. Chalikov, Y. Y. Chao, H. S. Chen, and V. M. Gerald (2002), Development and Implementation of Wind-Generated Ocean Surface Wave Models at NCEP, Wea Forecasting, 17(2), 311–333, doi:10/d74ttq.
- Uppala et al. (2005) Uppala, S., P. Kållberg, A. Simmons, et al. (2005), The ERA-40 Re-analysis, Q J R Meteorol Soc, 131, 2961–3012, doi:10.1256/qj.04.176.
- Weber (1983) Weber, J. E. (1983), Steady Wind- and Wave-Induced Currents in the Open Ocean, J Phys Oceanogr, 13, 524–530, doi:10/djz6md.
- World Meteorological Organization (1998) World Meteorological Organization (1998), Guide to wave analysis and forecasting, World Meteorological Organization, Geneva, Switzerland, 2 ed.
- Wu (1979) Wu, J. (1979), Oceanic whitecaps and sea state, J Phys Oceanogr, 9(5), 1064–1068, doi:10/bsjxrx.
|Vertical resolution||42 levs (10 m top lev)|
|Time step||3600 s|
|3D damping to clim||ON (3 yr Newtonian relaxation)|
|Experiment description||Stress||TKE flux||Stokes-Coriolis||Langmuir||ETAU|
|CTRL:||Drag law||CB94 TKE flux||Off||On||On|
|Control experiment||Eq (32)||Eq (17)|
|TAUOC: Water-side||ECWAM||as CTRL||Off||On||On|
|WTKE: Sea-state||as CTRL||ECWAM||Off||On||On|
|dependent TKE flux||TKE flux (18)|
|STCOR: Stokes-Coriolis||as CTRL||as CTRL||ECWAM||On||On|
|WAVE:||as TAUOC||as WTKE||as STCOR||On||On|
|All three wave effects|
|LOW: Law-of-the-wall||as CTRL||Off||Off||On||On|
|NOLC: Langmuir off||as CTRL||as CTRL||Off||Off||On|
|NOETAU: ETAU off||as CTRL||as CTRL||Off||On||Off|