Boundary Layer Energetics of Rapid Wind and Wave Forced Mixing Events

Eric D. Skyllingstad aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Roger M. Samelson aCollege of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

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Harper Simmons bApplied Physics Laboratory, University of Washington, Seattle, Washington

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Lou S. Laurent bApplied Physics Laboratory, University of Washington, Seattle, Washington

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Sophia Merrifield cScripps Institution of Oceanography, La Jolla, California

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Thilo Klenz dUniversity of Alaska Fairbanks, Fairbanks, Alaska

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Luca Centurioni cScripps Institution of Oceanography, La Jolla, California

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Abstract

The observed development of deep mixed layers and the dependence of intense, deep-mixing events on wind and wave conditions are studied using an ocean LES model with and without an imposed Stokes-drift wave forcing. Model results are compared to glider measurements of the ocean vertical temperature, salinity, and turbulence kinetic energy (TKE) dissipation rate structure collected in the Icelandic Basin. Observed wind stress reached 0.8 N m−2 with significant wave height of 4–6 m, while boundary layer depths reached 180 m. We find that wave forcing, via the commonly used Stokes drift vortex force parameterization, is crucial for accurate prediction of boundary layer depth as characterized by measured and predicted TKE dissipation rate profiles. Analysis of the boundary layer kinetic energy (KE) budget using a modified total Lagrangian-mean energy equation, derived for the wave-averaged Boussinesq equations by requiring that the rotational inertial terms vanish identically as in the standard energy budget without Stokes forcing, suggests that wind work should be calculated using both the surface current and surface Stokes drift. A large percentage of total wind energy is transferred to model TKE via regular and Stokes drift shear production and dissipated. However, resonance by clockwise rotation of the winds can greatly enhance the generation of inertial current mean KE (MKE). Without resonance, TKE production is about 5 times greater than MKE generation, whereas with resonance this ratio decreases to roughly 2. The results have implications for the problem of estimating the global kinetic energy budget of the ocean.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Skyllingstad, eric.skyllingstad@oregonstate.edu

Abstract

The observed development of deep mixed layers and the dependence of intense, deep-mixing events on wind and wave conditions are studied using an ocean LES model with and without an imposed Stokes-drift wave forcing. Model results are compared to glider measurements of the ocean vertical temperature, salinity, and turbulence kinetic energy (TKE) dissipation rate structure collected in the Icelandic Basin. Observed wind stress reached 0.8 N m−2 with significant wave height of 4–6 m, while boundary layer depths reached 180 m. We find that wave forcing, via the commonly used Stokes drift vortex force parameterization, is crucial for accurate prediction of boundary layer depth as characterized by measured and predicted TKE dissipation rate profiles. Analysis of the boundary layer kinetic energy (KE) budget using a modified total Lagrangian-mean energy equation, derived for the wave-averaged Boussinesq equations by requiring that the rotational inertial terms vanish identically as in the standard energy budget without Stokes forcing, suggests that wind work should be calculated using both the surface current and surface Stokes drift. A large percentage of total wind energy is transferred to model TKE via regular and Stokes drift shear production and dissipated. However, resonance by clockwise rotation of the winds can greatly enhance the generation of inertial current mean KE (MKE). Without resonance, TKE production is about 5 times greater than MKE generation, whereas with resonance this ratio decreases to roughly 2. The results have implications for the problem of estimating the global kinetic energy budget of the ocean.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Eric Skyllingstad, eric.skyllingstad@oregonstate.edu

1. Introduction

Boundary layer interaction between the atmosphere and ocean mediates the exchange of properties between the two fluid systems and is critical for coupled processes important for Earth’s climate system. In the ocean, surface mixing is generated by turbulence produced through wind-forced shear instability, surface wave processes such as breaking and Langmuir turbulence, and buoyancy instability created by cooling and increased salinity. Typical ocean mixed layer depths range from less than 10 m in regions with strong solar heating and low winds to deep mixed layers extending up to 1000 m in polar regions that experience extreme surface cooling (Marshall and Schott 1999; de Boyer Montégut et al. 2004). Outside of polar convective regimes, mixed layer depths greater than ∼80 m are mostly limited to regions with strong wind forcing from intense, wintertime atmospheric cyclones.

Storm-produced deep mixing events are important because they cause irreversible changes in temperature and salinity resulting from entrainment of relatively cold, saline water from below the mixed layer. Storm winds can also force strong inertial currents and long-lived near-inertial gravity waves that propagate through the water column. Accurate calculation of near-inertial gravity wave energy flux from the ocean boundary layer (OBL) depends on estimates of the work done by the wind on inertial currents and the loss of OBL kinetic energy through conversion to potential energy via entrainment at the boundary layer base and turbulence kinetic energy dissipation. Each of these energy budget terms represents a significant challenge for estimating the global power contained in near-inertial waves that are forced in the OBL (e.g., see Alford 2020).

Obtaining direct measurements of deep mixing events is difficult given the particularly harsh sea-state conditions and infrequent event occurrence. As a result, there are few direct observations of the processes responsible for driving turbulence to large depths. Two primary mechanisms are thought to be responsible for storm generation of rapid mixing events and enhanced shear at the OBL base. First, strong wind-driven OBL currents are needed, which requires optimal flux of wind momentum into the ocean. Wind forced currents are initially unbalanced and rotate at the local inertial period (∼12 h at high latitude). If winds are constant, then after 1/2 of an inertial cycle, the wind will oppose the current and reduce the momentum flux. Optimal, resonant transfer of wind momentum is produced either by winds that rotate anticyclonically with inertial frequency, thereby staying in phase with the wind-forced current, or by winds that have constant direction but cycle in strength with an inertial period, thereby having the strongest velocity when the wind and current have a matching direction (e.g., Skyllingstad et al. 2000). These interactions are responsible for enhanced mixing that has been observed in both midlatitude storms (Large and Crawford 1995) and propagating hurricanes (Sanford et al. 2011), both of which generate local effective anticyclonic rotation through changes in the wind direction. At high latitudes, resonant forcing of surface waves by fast-moving polar lows (e.g., Orimolade et al. 2016) may also enhance near-surface wave-driven mixing under wave-forcing conditions that would otherwise be fetch limited.

The second mechanism necessary for driving rapid OBL growth is vertical momentum transport by Langmuir circulation (LC) and related OBL coherent roll structures. Langmuir circulation arises as an instability of near-surface vertical shear that is generally associated with wave-driven momentum or vorticity fluxes. Coherent roll structures are produced that efficiently transport water properties from the surface to the OBL interior. LES studies of observed OBLs show that, without the near-surface shear that supports Langmuir circulation, model mixed layers deepen too slowly (see Wang et al. 2018). Langmuir circulation effectively mixes momentum in the OBL, generating a velocity structure more in line with a slab model, with maximum shear at the boundary layer base and enhanced entrainment mixing. In recent years, the global importance of Langmuir circulation in defining the ocean boundary layer thickness has been recognized (Belcher et al. 2012). Improved representations of mixing including the effects of Langmuir circulation have been developed (e.g., Smyth et al. 2002; McWilliams et al. 2012) and are being adapted in global ocean applications (e.g., Noh et al. 2016). The accuracy of these methods is a current area of research (e.g., Li et al. 2019) and most have not been thoroughly tested. Further studies are needed with comparisons between OBL measurements, mixing parameterizations, and LES cases.

As the OBL depth increases, the effects of surface forcing are diluted by the larger mass of water, which smooths out the vertical shear of horizontal momentum. Langmuir turbulence increases the efficiency of OBL deepening by providing a more direct transport of momentum from the surface to the OBL base. However, the effectiveness of this transport should also decrease as the mixed layer depth increases. In the northern Atlantic and Pacific oceans, OBL deepening is constrained by the underlying seasonal pycnocline where entrainment of cold, saline water rapidly converts mixed layer kinetic energy into potential energy (Sanford et al. 2011). The limiting effect of the pycnocline causes a relatively slow growth of the OBL over winter, with depth reaching a maximum in late spring. As solar heating increases in the spring, the OBL gradually restratifies. Late spring storms can rapidly mix these weakly stratified layers and provide a unique laboratory for studying the relationship between surface wind/wave forcing and mixed layer growth over much shorter time periods than typical seasonal OBL deepening.

It is generally accepted that surface wave forcing is an important component in upper ocean mixing and numerous LES studies have examined how waves can influence OBL turbulence (e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997; Noh et al. 2004). However, only a few modeling experiments have directly compared observed turbulence variables with LES. Most notable are studies using neutrally buoyant floats by D’Asaro et al. (2014) and measurements using acoustic Doppler current velocimetry, for example, by Gargett and Wells (2007). Another commonly observed ocean turbulence variable is turbulence dissipation rate, which is calculated from shear measured over millimeter distances associated with the Kolmogorov scale (Lueck et al. 2002). Skyllingstad et al. (1999) compared measured turbulent and temperature variance dissipation rates with LES values to demonstrate feasibility of modeling ocean turbulence associated with the central Pacific equatorial undercurrent. These experiments included the Craik–Leibovich (CL) (Craik and Leibovich 1976) wave parameterization, however, surface waves were not a significant factor in the turbulence kinetic energy budget for the modeled conditions. Most studies using turbulence dissipation rate to understand wave effects have focused on the upper 10 m of the ocean where both LC and wave breaking are dominant sources of turbulence (e.g., Gerbi et al. 2009).

Deep mixing of momentum by LC also has a direct influence on the wind energy flux or wind work at the ocean surface. Estimates of wind work use the product of the wind stress and upper ocean currents, so both wind and near surface current direction are critical for accurate wind work calculation. However, increased vertical transport of momentum away from the surface can limit the strength of this work term and change the kinetic energy budget for upper ocean inertial currents. Both the dissipation of kinetic energy by turbulence and the uncertainty in the wind work term have hampered studies that attempt to estimate the global energy flux into near inertial gravity waves (e.g., D’Asaro 1985; Alford 2020).

Numerous LES modeling studies of the OBL with and without Stokes wave forcing have reported on the significant effects of Langmuir circulation on the upper ocean turbulence energy budget (e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997; Polton and Belcher 2007). However, very little research has focused on the total pathway that wind energy takes after entering the ocean, and in particular, the role of Stokes wave forcing in the total wind work. Wind forced currents are a key component of the global ocean circulation, hence defining the partitioning of input wind energy into turbulence and mean currents is critical for accurate current prediction.

Our goal in this paper is to use measurements of the OBL in the North Atlantic along with large-eddy simulation (LES) to both demonstrate how LC affects deep mixing, examine how kinetic energy is partitioned between mean currents and turbulence, and explain how this partition changes because of variations in the direction of wind forcing. A description of the glider observations and the LES model are presented in section 2 along with the model experimental design. Three observed wind events are examined over an 8-day period, focusing on the pathway that kinetic energy follows from the wind and wave surface boundary conditions. Waves are parameterized using the CL equations (Craik and Leibovich 1976), with the kinetic energy budgets following the Lagrangian form of the CL equations (see Suzuki and Fox-Kemper 2016). Analysis of the simulation results are presented in section 3 and compared with glider measurements. Simulations without surface wave CL forcing are conducted to demonstrate the importance of Langmuir turbulence in setting the depth of the ocean mixed layer. Analysis of the kinetic energy budget is presented in section 4. For each event, we relate the wind work to kinetic energy associated with inertial current formation, shear production, and Stokes production of turbulence. Section 5 presents a discussion and conclusions from our study.

2. Methods

a. Observations

Much of the motivation for this work stems from measurements taken in May 2018 from the R/V Armstrong during a research cruise in the Icelandic Basin as part of the ONR-funded Near-Inertial Shear and Kinetic Energy in the North Atlantic Experiment (NISKINe). A series of mixing events were observed using an ocean glider outfitted with CTD and microstructure sensors capable of making measurements that allow for estimates of the turbulent dissipation rate ε. The sampling system was a Teledyne Webb Slocum Glider with a Rockland Scientific Microrider, as described by St. Laurent and Merrifield (2017). Details of how the system operates, and how the various sensors are used with the measured motion of the glider to estimate ε are given in that reference and also in Fer et al. (2014) who discuss an identical system. The turbulence dissipation rate is defined as
ε=ν(uixj+ujxi)2,
where the quantity in parentheses is an element of the rate of strain tensor; {i, j} = {1, 2, 3} are indices for the zonal, meridional, and vertical coordinate and velocity components; ν is the kinematic viscosity of water; and a time average and sum over repeated indices is assumed. In practice, our estimate is achieved through the measurement of sub-centimeter-scale vertical shear (u^/z) using airfoil probes (Lueck 2003), where u^ is the measured horizontal velocity. We then invoke isotropy relations (Hinze 1975) to express the dissipation rate in terms of a single microscale shear component, such that ε=(15/2)ν(u^/z)2¯. Using cospectral analysis, we remove the platform induced vibrations from the probe records using the method of Goodman et al. (2006). Typically, this correction applies to frequencies between about 10 and 60 Hz. In the resulting “clean” shear spectra, coherent vibrational energy between the profiler platform and the shear records has been removed, allowing for resolved dissipation rates as low as about 5 × 10−11 W kg−1. When used for measurements near the sea surface, dissipation rate estimates from the glider have been demonstrated to match those from mooring based methodologies (Zippel et al. 2021).

During the NISKINe cruise, Slocum glider Husker conducted a 10-day mission from 18 to 29 May profiling to 200 m. A series of storms were sampled, with peak wind speeds reaching 20 m s−1 and wave heights reaching 6 m. The glider was deployed on the cyclonic, cool side of a mesoscale dipole, and was advected along and across the front separating the two eddies, crossing to the warm side on 20 May. Lagrangian drifters were also deployed across the front to measure winds and wave parameters.

The combination of synoptic glider, ship, and drifter data enabled the construction of a meteorological dataset built from wind and wave buoys and shipboard measurements by selecting appropriate data from the platform nearest to the glider at a given time. This amounted to distances < 40 km for wind speed and direction, humidity, air temperature, and radiative forcing and <90 km for wave measurements. The directional wave spectra (DWS; Centurioni 2018) wave buoy drifters measured directional spectral of surface gravity waves 0.03–0.50 Hz. The MiniMet drifters (Centurioni 2018; Klenz et al. 2022) use a high-quality sonic anemometer and an internal compass to measure the wind velocity. Wind stress, τh, was derived using the COARE 3.5 algorithm (Edson et al. 2013) with significant wave height and phase speed of the dominant waves from the DWS wave buoys. Accounting for sea state in the COARE 3.5 algorithm results in a reduction of computed wind stress by 10% for large winds (>20 m s−1) with little change for weaker winds. Ship meteorological parameters such as downwelling short and longwave radiation plus glider measured sea surface temperatures allow for estimation of the buoyancy fluxes at the glider using the COARE 3.5 algorithms.

Three main wind-forced mixing events are notable in these observed turbulence dissipation rate time series (Fig. 1). Here, we define the OBL depth in terms of the dissipation rate with values dropping below approximately 10−8 m2 s−3 indicating the maximum depth of mixing. The first wind event, starting late on 21 May, generates the deepest mixed layer, extending to about 180 m. Subsequent wind pulses on 26 and 27 May are not as strong, with mixing extending to about 100-m depth. For each event, the deepest extent of large turbulence dissipation rates descends rapidly over a 6–12-h period, with the maximum depth penetration occurring at the time of peak wind stress. Wind direction shows some counterclockwise rotation (backing) in the first event, very little rotation in the second event, and strong clockwise (veering) rotation in the third event. The veering in the final wind event corresponds to rotation following the inertial current and favors increased resonance, consistent with the rapid deepening of the OBL during this event, even though the wind event was shorter than the previous storms.

Fig. 1.
Fig. 1.

(a) Estimated wind stress and (b) glider measured ocean turbulence dissipation rates over the study period (time in UTC) along with (c) temperature and (d) salinity.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

Temperature and salinity profiles from the glider (Figs. 1c,d) show that the glider was moving through various water masses associated with mesoscale eddy circulations, especially between 20 and 21 May. However, this background structure is, for the most part, weakly stratified with minor subsurface fronts and weak geostrophic shear that have little effect on surface-forced mixing during the time period of our study. At any given point, the vertical structure over the depth of the glider path is relatively uniform, with only small variations in density (this is why we use turbulence dissipation rate to indicate the OBL depth). We hypothesize that the vertical extent of wind and wave forced mixing is mostly a function of the input energy, and the strength of shear at the base of the mixed layer as it moves downward. Unfortunately, we do not have ocean velocity measurements that are concurrent with the glider to directly verify this hypothesis. Surface buoyancy flux forcing (not shown) is relatively small in comparison with the wind forcing as shown by the relatively small changes in temperature. We note a significant reduction in salinity on 27 May, suggestive of rainfall. This freshwater pulse is quickly mixed downward by the third wind event and may play a role in limiting the deepening of the OBL.

Forcing for Langmuir turbulence in the model requires estimates of the depth-dependent Stokes drift profile, which appears in the Craik–Leibovich vortex force wave parameterization (Craik and Leibovich 1976) and through a Coriolis term. The surface wave environment near the glider path (Fig. 2) is constructed using wave buoy measurements noted above. Stokes drift is computed using Kenyon’s (1969) directional spectrum F estimate,
Us=2F(k)gke2kzkdk,
where k = |k| and z is the depth, simplified in terms of the measured omnidirectional spectrum ϕ(k) as
Us(z)=2k1k2ϕ(k)gke2kzkdk.
Stokes drift velocity components are calculated by assuming they follow the wind direction.
Fig. 2.
Fig. 2.

(a) Surface significant wave height, (b) surface Stokes drift velocity, (c) friction velocity, and (d) turbulent Langmuir number.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

The relative importance of wind forcing versus wave forcing of turbulence can be estimated using the turbulent Langmuir number,
Lat=(u*So)1/2,
where the friction velocity u*=τh/ρ and S0 = Us(z = 0). Values of Lat range from about 0.2 to 0.4 as shown in Fig. 2d, which are conditions that favor wave-dominated turbulence (Li et al. 2005).

b. Model description and experimental design

Simulations are conducted using the LES model described in Skyllingstad et al. (1999, 2000). Briefly, the model is based on the Deardorff (1980) equation set with subgrid-scale turbulence parameterization from Ducros et al. (1996). Surface wave forcing is implemented via the CL vortex force and Coriolis terms mentioned above. All simulations are performed using uniform grid spacing of 1.5 m for a periodic horizontal domain of 1080 m × 1080 m with a depth of 225 m. Both top and bottom boundaries are rigid lids with Rayleigh damping of velocities performed over the bottom 10 grid points as a “sponge” condition to suppress reflection of downward propagating internal waves. Two 72-h experiments are performed using initial profiles taken from the glider as shown in Fig. 1 from 0000 UTC 21 May 2018 to 0000 UTC 25 May 2018 (shown in Figs. 1c,d). Simulations are started at rest with surface wind stress and Stokes drift forcing as shown in Figs. 1 and 2. Surface fluxes of momentum, sensible, and latent heat are applied as top boundary conditions for velocity and temperature subgrid fluxes. Surface equivalent salinity flux is also set to include effects of rainfall. Penetrative solar radiation is modeled using ocean transparency data as in Skyllingstad et al. (1999) and net infrared radiation is applied at the surface using the measured radiation balance. We consider simulations with identical wind forcing, but with CL wave forcing included in one case but not the other (designated as W-CL and W, respectively). These two scenarios examine the role of surface waves in setting both the strength of turbulence and the OBL depth.

3. Simulated boundary layer evolution

We first compare the simulated turbulence dissipation rate from cases W and W-CL with measured values to gain some perspective on the accuracy of the model and importance of surface wave forcing. In both cases, adding the CL Stokes forcing term produces boundary layer depths and turbulence intensities that are much closer to the glider measured fields than the pure wind forced case. For example, maximum boundary layer depth at hour 36 for the 21 May case (Fig. 3) is about 80 m for case W and 120 m for case W-CL. Boundary layer depths for the second wind event (Fig. 4, hour 20) are roughly 60 m without wave forcing and 100 m with wave forcing suggesting a similarly robust response.

Fig. 3.
Fig. 3.

Vertical profiles of log10 turbulence dissipation rate from (a) the glider, (b) simulation case W, and (c) simulation case W-CL. Contours of simulated log10 dissipation rate equal to −8 for case W (red) and W-CL (black) are overlaid on each panel. Simulation results are horizontal averages. The initial time is 0000 UTC 21 May 2018.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for an initial time of 0000 UTC 25 May 2018.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

In general, the instantaneous turbulence dissipation rate follows the wind forcing, with a lag accounting for vertical transport of momentum from the surface. We note that elevated turbulence dissipation rates below 100-m depth persist after the main wind event at hour 36 on 22 May for about 24 h, suggesting ongoing turbulence production from elevated shear at depth (Fig. 3). In contrast, elevated dissipation rate in the simulations at 30–60-m depth in the 25 May case persists much longer than the measurements (Fig. 4). For example, turbulence dissipation rate at 40-m depth decreases rapidly at approximately hour 36 in the observations to values of about log10(ε) = −10 versus −8 in case W-CL. As we note in the introduction, overprediction of turbulence in the LES could arise from excess inertial shear that in turn is caused by artificial confinement of radiating NIW energy within the computational domain. We explore this possibility further in section 5. Excess simulated inertial velocities could also arise from initializing the Eulerian model velocity at rest when there may have been currents initially opposing the wind stress and inferred Stokes drift, which would have reduced wind energy input.

Plots of the simulated temperature and salinity (Figs. 5 and 6) closely follow the evolution of the turbulence dissipation rate in the upper 40 m and provide an explanation for shoaling of the OBL depth. Much of this structure is hidden in the measured temperature and salinity (Fig. 1) by horizontal water mass variability (note the difference in color bar scales). For the 21 May case, solar warming has the dominant influence on stratification, limiting OBL depth particularly in case W. Rainwater flux on 25 May adds to the upper ocean stratification, especially around hour 48 when approximately 3 cm of rain accumulates in the surface layer. Wind deepening of the OBL is limited by stratification from solar heating and by this surface freshwater flux from rain, as shown by the slow deepening of turbulence dissipation rate between hours 48 and 60 (Fig. 4). The rapid increase in stratification in response to the rain effectively traps wind forced momentum above 20-m depth as shown by current speed plots in Fig. 7. Rotation of the wind stress vector on 25 May between hours 48 and 60 is evident in the large acceleration near the surface for both case W and W-CL. Near resonance after hour 60 produces strong shear and rapid destruction of the warm, fresh layer resulting from the rain event at hour 48. Overall, increased mixing of momentum when wave forcing is applied results in shear at greater depths and rapid deepening of the boundary layer.

Fig. 5.
Fig. 5.

Horizontally averaged potential temperature from (a) case W and (b) case W-CL along with horizontally averaged salinity from (c) case W and (d) case W-CL for the simulation of 21 May 2018.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for the simulation of 25 May 2018.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

Fig. 7.
Fig. 7.

Magnitude of the horizontally averaged currents for the 21 May 2018 simulation for case (a) W and (b) W-CL and for the 25 May 2018 simulation for case (c) W and (d) W-CL.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

4. Kinetic energy analysis

a. Kinetic energy budget equations

Wind and wave energy input are the main drivers for the deep mixing events that we examine in this paper. To aid in understanding the energy pathways, we analyze the LES model equations in a form that is equivalent to a version of the wave-averaged Boussinesq (WAB) equations described in Suzuki and Fox-Kemper (2016), modified by addition of the LES subgrid terms. Using the notation of Suzuki and Fox-Kemper (2016) our budget is derived from the WAB equations cast in terms of the wave filtered Lagrangian vector velocity, uL = u + uS,
tuL+(uL)uL+fk^×uL+p=b+(uiuj˜)uL×(×uS)+tuS.
Here u is the standard Eulerian current for a fixed location; uS is the Stokes velocity of the surface waves and is assumed to have only depth dependence; f is the Coriolis parameter; b=ρg/ρok^ is the buoyancy force with gravitational acceleration g, background density ρo, and the density perturbation ρ′ = ρρo; p is pressure divided by ρ; and uiuj˜, is the subgrid-scale momentum flux defined below. The main assumption in using (2) is that surface waves are of small wave slope and velocities are time averages that filter out the rapid orbital velocities associated with surface gravity waves. This application is similar to Wagner et al. (2021) who used (2) to examine the effects of swell generated currents on turbulent mixing with LES. The LES model equations are formulated numerically in terms of the standard momentum equations for u supplemented by the CL vortex-force and Stokes–Coriolis terms but, as noted by Suzuki and Fox-Kemper [2016, their Eqs. (1) and (4)], the two forms are equivalent.
From (2) we can derive a budget equation for the mean kinetic energy MKE and turbulent kinetic energy TKE defined as
MKE=12uiL2,
TKE=12ui2,
where ui are the velocity components; i, j, k represent directions x, y, z; angle brackets denote the horizontal mean; and ui denotes turbulence velocities defined using ui=uiLuiL=uiui. This relationship holds because Uis is assumed to be independent of x and y. Kinetic energy budget equations are derived by replacing uiL in (2) with ui+uiL, multiplying the resulting equation by uiL or ui, and applying the averaging operator to yield MKE and TKE, respectively. Subgrid momentum flux is parameterized as uiuj˜=Km[(ui/xj)+(uj/xi)], where Km is the subgrid eddy viscosity.
For the LES periodic domain, horizontal averaging eliminates horizontal flux terms and the average vertical velocity is identically zero, yielding the following budget equations
MKEt=uiu3Uix3+uiu3Uisx3(Uiuiu3˜)x3(Uisuu3˜)x3+uiu3˜(Ui+Uis)x3+(Ui+Uis)Uist,
TKEt=uiu3Uix3uiu3Uisx3u3TKE+uiuiu3˜+u3px3u3gρρo+uiuj˜uixj,
where uiu3˜ is bounded at the surface by the input wind stress τh, Ui = 〈ui〉, i = {1, 2}, are the average horizontal velocity components, and UiS are the Stokes drift components, which are independent of x, y directions. Terms in the MKE budget are, in order, time rate of change of MKE or MKE storage, exchange of energy with the TKE budget by shear production from mean shear and by shear production from the Stokes drift, two vertical flux terms from the subgrid-scale diffusion, two terms that typically dissipate MKE via subgrid eddy viscosity, and forcing terms imposed by slow variations in the wave field. Terms in the TKE budget are, time rate of change of TKE or TKE storage, shear production from the mean shear and from the Stokes drift shear (which offset the loss terms in the MKE budget), vertical transport by turbulence, subgrid-scale mixing, and pressure, followed by buoyancy production, and dissipation.

It is important to note that the Stokes–Coriolis work term that appears in the standard mean energy budget (e.g., Suzuki and Fox-Kemper 2016) does not appear in the form (3), because the mean velocity for the MKE in (3) combines the Eulerian mean velocity and the Stokes drift. Indeed, a primary motivation for using (2) as the basis of the energy analysis is that multiplying (2) by uL annihilates all the cross-product inertial terms [ fk^×uL and uL×(×uS)], so that these terms vanish identically from the energy equation in the same way that the analogous rotational terms vanish from the standard energy equation without added wave forcing (Stokes drift). From this point of view, the use of the “Lagrangian” MKE based on uL is effectively demanded by the form of the cross-product terms in (2). It should also be noted that this form of the energy budget includes a nonstandard term (Ui+Uis)(Uis/t) in (3); however, this additional nonstandard term is proportional to the rate of change of the Stokes-drift velocity and so vanishes or is small, respectively, under steady or quasi-steady forcing conditions.

From (3) and (4), equations for the depth integrated energy budgets are
MKEtdz=uhwUhzdz+uhwUhszdzUsfcτhUsfcsτh,
TKEt=uhwUhzdzuhwUhszdzwgρρodz+uiwj˜uixjdz,
where Usfc = Ui(z = 0) and Usfcs=Uis(z=0) are the mean velocity and Stokes drift at the surface and we have neglected the MKE subgrid term in (3), which is much smaller than the shear loss terms. The nonstandard term proportional to the rate of change of Stokes drift in (3) is also small and has been neglected. Transport terms in the LES domain integrate to zero except for momentum, which is bounded at the model top by the surface stress, (uiu3˜)top=τh and set to zero for the rigid bottom. The equal and opposite shear production terms in (5) and in (6) represent the exchange of energy between the mean flow and turbulence. Typically, energy moves from the mean flow to turbulence through this term and is removed through the subgrid-scale TKE dissipation term or by increasing the mean potential energy through the buoyancy production term.

We employ a sponge layer at the model bottom that absorbs internal wave energy contained in the pressure transport. In addition, imposing a uniform surface wind stress prevents convergence and divergence of the surface velocity field that would lead to inertial gravity wave propagation of energy and momentum from the surface boundary layer into the ocean interior. Because of this missing term, the LES mixed layer energy budget using (5) and (6) is likely an overestimate of turbulence. Wind energy lost to near-inertial internal waves would decrease the mean velocity response and consequently reduce the shear production of turbulence. Nevertheless, (5) and (6) are a correct representation of the LES energy budget and as such should provide a useful framework to help understand how much energy is gained by the mean flow versus lost to dissipation, particularly for short-time-scale wind events that would not have spun up a strong inertial wave response.

Two terms in the MKE budget (5) are of special note
Ew=Usfcτh,
Es=Usfcsτh.
The first term is commonly referred to as the “wind work” and represents the flux of kinetic energy from the wind into the mean flow. The second, “Stokes wind-work” term is similar and represents an effective flux of energy from wind working on the imposed wave Stokes drift. In general, estimates of the flux of wind energy into the ocean will depend critically on the measured surface current velocity, which may differ substantially for observations made by fixed or drifting instruments or at slightly different near-surface depths. For the budget analysis, we evaluate Usfc and Usfcs at z = −0.5Δz, where the horizontal components of velocity are calculated on the staggered grid.

Our analysis can be interpreted as the kinetic energy of the Lagrangian mean flow in the context of generalized Lagrangian mean theory (e.g., Gjaja and Holm 1996; Holm 1996). By analogy with the standard shear production term, the Stokes shear production term in the TKE balance (3) now appears with opposite sign in the corresponding MKE balance. Similarly, the standard wind-work term in (2) is supplemented in the corresponding total mean kinetic energy balance by the analogous, effective Stokes wind-work term Es. Under steady conditions, the total energy budget obtained by summing (3) and (4) is then a closed balance between surface work and dissipation, without the effective body-force work terms that arise from the Stokes–Coriolis force (e.g., Suzuki and Fox-Kemper 2016) in the standard formulation.

b. Budget analysis

Examination of the boundary layer evolution shown in Figs. 37 indicates a complex interaction between energy input through wind stress and the overall growth and decay of ocean boundary layer currents. Stratification and wave forced turbulence are both important for determining the surface current, which is used to calculate the wind work term in the mean kinetic energy budget. We first focus on the MKE and the wind-work terms, Ew and Es, as an overview of the energy budget.

Figure 8 shows that MKE is generally larger without Stokes drift forcing during strong wind events, particularly for the 21 May case. This is also true for the 25 May case when the wind direction is relatively constant at the beginning of the simulation. After hour 36, we note that the larger MKE alternates between the Stokes and no Stokes cases. Wind direction and speed are fluctuating more rapidly at this time, suggesting that energy is at times being removed from the mean flow by opposing winds. In general, the additional KE from the Stokes drift velocity is a small fraction of MKE because Stokes drift decays rapidly with depth compared to the Eulerian velocity.

Fig. 8.
Fig. 8.

(a) MKE and (b) wind-work terms for (left) 21 May 2018 and (right) 25 May 2018. All integrated variables are scaled by the model depth.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

Wind work is maximized when the wind and current direction are positively correlated, which is more likely for winds rotating at the inertial period in a clockwise direction, for example at hour 60 in the 25 May case. A large wind work term is also likely for short, strong wind events, for which there is not enough time for the currents to rotate significantly, for example, for the wind event at hour 30 in the 21 May case. Sustained winds in one direction as from hour 12 to 36 in the 25 May case, generate progressively less wind work as the surface velocity rotates out of alignment with the wind stress, and can lead to rapid decrease in MKE if the wind opposes the surface current. Both of the wind events noted above are followed by steep decreases in MKE that are at least partially a result of inertial current rotation and reduction of effective wind work.

In general, adding the Stokes drift generates a significant decrease in the traditional wind work Ew that is compensated by Es. Estimates of wind work are often the basis for determining the strength of near-inertial currents and the resulting vertically propagating near-inertial gravity waves (e.g., Alford 2020). The relation between the wind-work processes represented in the wave-averaged LES model and the much more complicated, wave-phase-dependent energy flux pathways that occur at the wavy air–sea interface is not obvious. One possible interpretation or inference, however, is that surface current measurements or estimates that do not include a wave-driven Stokes drift (or its wind-driven equivalent; see, e.g., Samelson 2022) may in many cases underpredict the flux of wind energy. With surface waves, near surface Eulerian currents in the model are decreased because of enhanced vertical momentum transport from Langmuir circulation. However, this decrease is offset by the addition of the Stokes drift current, providing an additional energy source for the MKE. Inspection of Fig. 8 suggests that sum of Ew and Es is nearly equal to Ew without Stokes forcing. This compensation, and the effect on TKE production, are explored further below.

The pathway that energy follows when going from wind stress to near-inertial currents involves the generation of MKE and TKE, with TKE typically representing a local loss term through dissipation and frictional heating of the fluid. Examination of the MKE and TKE storage terms shown in Fig. 9 for our experiments shows that the change in MKE is a small fraction of the total wind work term and TKE storage is insignificant. One exception for MKE storage is the second wind event in the 25 May case, where MKE storage is similar in magnitude to the total wind work for both cases after hour 60.

Fig. 9.
Fig. 9.

Storage term for MKE and TKE along with wind work for (a) case W and (b) W-CL for (left) 21 May and (right) 25 May.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

MKE storage usually has a pattern of positive values through most of the wind event followed by negative values as the wind event decays, for example as shown for 21 May in Fig. 9a between hours 24 and 48. This behavior is consistent with rotating inertial currents that initially are aligned with the wind but eventually turn to an opposing direction that acts to remove energy from the flow. Rotation of the currents also will cause greater internal mixed layer shear as the surface acceleration continues in the direction of the wind, while deeper water motion rotates away from the wind direction. The unique behavior noted for the second wind event in the 25 May case is an example of winds that have rotation in a clockwise direction, thereby enhancing the inertial currents and, specifically, the surface current. Inspection of the wind stress for this case (Fig. 1a) on 27 and 28 May, shows wind direction from the south that shifts to winds from the west, indicating a clockwise rotation of the wind vector that results in wind–inertial current resonance (e.g., Large and Crawford 1995). Wind events earlier in the record typically have near constant direction from the south without much current resonance.

c. Energy pathways and Stokes-forced compensation

Over most of both simulation periods, both the MKE and the standard wind work Ew are systematically smaller for the Stokes-forced W-CL cases than for the W cases, which have no Stokes forcing (Fig. 8). Reduced MKE in case W-CL is reflected in the vertically integrated TKE budget as shown in Fig. 10. For both cases, the TKE budget described by (6) is dominated by shear production and dissipation with buoyancy production (not shown) having a much smaller role. When wave effects are included with the Stokes terms, Langmuir circulation shifts TKE produced from Eulerian shear production to Stokes shear production. Enhanced vertical momentum transport by Langmuir turbulence in case W-CL causes a significant decrease in the vertical shear of the horizontal currents (e.g., Fig. 7) and the resulting standard shear production of turbulence. Much of this reduction in standard shear production is compensated by the Stokes shear production and the greater depth of the mixed layer, thereby generating an integrated dissipation rate that is only modestly reduced in case W-CL in comparison with case W. The reduction in integrated dissipation for case W-CL suggests that wave effects represented by addition of the Stokes drift terms do not directly introduce a new source of wave KE into the boundary layer. Instead, it is still the effective wind-work that provides the KE source, some of which is represented in the W-CL case as Stokes wind-work.

Fig. 10.
Fig. 10.

Terms from the TKE budget equation (a) without Stokes and (b) with Stokes for (left) 21 May and (right) 25 May. Terms are shear production (red), dissipation (black), Stokes shear production (blue), wind work Ew (red dash), and Stokes wind work Es (blue dash).

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

The TKE budget shows that in the W cases the standard shear production essentially balances dissipation, while in the W-CL cases, it is the Stokes shear production that balances most of the dissipation (Fig. 10). Without wave Stokes terms, much of the input wind work goes toward the shear production term as shown in Fig. 10a. With Stokes terms (Fig. 10b), a similar balance between the Stokes wind work Es and Stokes shear production term is noted, with normal shear production explaining less than half of the energy transfer from MKE to TKE. Budget terms associated with the Stokes drift are the main drivers of energy transfer in the wave forced system. Again, the second 25 May wind event presents an exception to this rule. Partial resonance during this event increases the strength of the surface current and wind work term for both cases. However, shear production does not grow with increasing MKE as the direction of the mean current is aligned with the wind stress. In this case, reduced shear production and corresponding dissipation leads to stronger mean currents and greater storage of MKE.

The strong similarity between the dissipation rate from case W and W-CL suggests that the sum of the work terms, Ew and Es, for case W-CL should be similar to Ew for case W, as was noted above. This comparison is shown in Fig. 11 and indicates that the inclusion of Stokes terms, while causing significant changes in the boundary layer depth and overall vertical structure, does not have a large effect on the overall KE budget. Because the wind forcing is identical in the two pairs of simulations, the near equality of the total wind-work indicates a near compensation of the imposed Stokes surface velocity by the LES surface velocity. The main effect of the Stokes drift vortex term in (2) is greater boundary layer mixing by Langmuir turbulence, but without a large change in the overall energy budget. In this sense, the Stokes drift term acts as a catalyst, altering the transport of kinetic energy but not the input energy flux or dissipation.

Fig. 11.
Fig. 11.

Wind work for case W and wind work plus Stokes wind work for case W-CL from (a) 21 May and (b) 25 May.

Citation: Journal of Physical Oceanography 53, 8; 10.1175/JPO-D-22-0150.1

5. Discussion

Past studies of Langmuir circulation using LES have treated Stokes drift terms as an added energy source representing conversion of energy from waves into the Eulerian turbulence velocity [see, e.g., Eq. (18) from Suzuki and Fox-Kemper (2016)]. Using this analysis approach leads to a mean energy balance that is missing the surface work term from the Stokes current and underestimates the flow of energy from the wind into the boundary layer through the wind work term. Here we use an energy budget based on the WAB equations cast in terms of a Lagrangian velocity that combines the Eulerian velocity with the Stokes velocity. This approach can be motivated by requiring that the rotational inertial terms vanish from the energy budget in the usual way. Analysis of this system yields a closed budget under steady or quasi-steady conditions, with wind work computed for the total Lagrangian current balanced by the dissipation of turbulence energy, and shear production computed for the total Lagrangian current providing the conversion between mean and turbulent kinetic energies.

A perhaps surprising but revealing model result is that the addition of the Stokes terms yields a system where the shear production terms compensate so that the total turbulence production is nearly constant regardless of the strength of the wave forcing. In addition, the production terms are nearly equal to the indicated wind work values, Ew and Es. Perhaps the key to this equivalence is the idea of total wind work conservation that is tied to the wind stress and does not depend on the addition of a Stokes drift. Both wind work terms come from the MKE budget and are part of the subgrid-scale term that is analogous to the vertical flux term responsible for the exchange of kinetic energy from MKE to TKE via shear production. Consequently, if wind work is primarily a function of surface momentum exchange (wind stress), then the production terms must balance the wind work terms.

For most of the three observed wind events from the glider record, the wind direction was relatively constant and the duration of the events was near the inertial period. For the first two storms, TKE dissipation was typically about 5 times larger than MKE production, limiting the strength of boundary layer currents. The third event was forced by a system with wind rotation near the inertial period, resulting in enhanced, resonant ocean currents. For this case, MKE production was much greater with values about half the TKE dissipation rate. This result suggests that resonant wind production of inertial currents could be a major source of energy for deep propagating inertial gravity waves that would be lost to dissipation without resonance.

Resonance between wind direction and upper ocean currents is known to cause enhanced boundary layer growth as demonstrated in Large and Crawford (1995) and modeled using LES in Skyllingstad et al. (2000). Our results show that resonance in general causes a reduction in bulk mixed layer turbulence for a given wind work magnitude, with increased MKE coming at the expense of decreased TKE. This is not inconsistent with enhanced entrainment at the mixed layer base, which is produced locally by higher shear from the elevated mixed layer MKE but does not dominate the boundary layer average TKE budget. The addition of the Stokes drift vortex force and Langmuir turbulence, while altering the TKE production terms and overall MKE, does not change this basic scenario for resonant wind forcing.

Our results without Stokes drift forcing are broadly consistent with the analysis of near-inertial wind work conducted by Alford (2020), who estimated that between 37% and 19% of global wind work computed using the Price–Weller–Pinkel (PWP; Price et al. 1986) and K-profile (Large et al. 1994) mixed layer models is available for near-inertial current generation. For boundary layers shallower than u*/(Nf)1/2, where N is the stratification immediately below the boundary layer, the available fraction of wind work was much smaller, consistent with the nonresonant simulated wind events on 22 and 25 May, which have most of the wind work going toward generation of TKE by shear production. In agreement with the larger available fractions found by Alford (2020), the resonant event on 27 May has roughly half of the input wind work going toward TKE production. One-dimensional mixed layer models such as PWP have been calibrated to yield accurate wind-forced boundary layer evolution using surface stress without explicitly considering waves. As we show here, addition of the Stokes drift, while affecting the vertical boundary layer structure, does not have a significant effect on the total KE budget as long as the Stokes drift wind work term is included in the total energy input.

6. Summary

Estimation of the global ocean kinetic energy budget requires not only accurate wind and wave energy flux, but also the partition of ocean kinetic energy into average currents, turbulence, and internal waves. Here we apply an ocean LES model with and without an imposed Stokes-drift wave forcing to examine the dependence of intense, deep-mixing events on wind and wave conditions and the pathway that energy follows in the development of deep boundary layers. Comparison of simulated turbulence dissipation rates to glider measurements shows that wave forcing, via the commonly used Stokes drift vortex force parameterization, is crucial for accurate prediction of boundary layer depth and turbulence dissipation rate magnitude. We derive a total energy budget—closed under steady or quasi-steady forcing conditions—for a Lagrangian form of the WAB equations by demanding that the rotational inertial terms vanish from the energy balance, and show how wind energy enters the system through both surface and Stokes drift currents in the resulting wind work formulation. Our analysis shows that the overall KE budget changes only slightly with the addition of Stokes wave forcing, with increased turbulence from Langmuir circulation and Stokes shear production offset by decreased standard shear production. Predicted integrated TKE dissipation rates are similar with and without Stokes forcing. Variable surface wave conditions, which would generate a local rate of change of the Stokes drift and alter the MKE budget (3), are neglected in this study and should be evaluated in future work.

Changes in the partition of MKE and TKE production by wind and wave work can have important consequences for estimating the generation of near inertial gravity waves and overall vertical mixing in the ocean boundary layer. For resonant winds, less wind and wave energy goes toward turbulence and dissipation, resulting in stronger OBL currents. Slab mixed layer models often used for global wind work estimation are not necessarily configured to represent these processes, which alters the strength of the surface current used to calculate the wind work term. Higher-order turbulence closures (e.g., Harcourt 2013), may yield more accurate results by resolving the mixed layer shear and thereby increasing the surface current and wind work term. Recent developments in ocean measurement systems have provided unprecedented datasets for studying the ocean mixed layer during intense storms. The hope is that as these observations become more commonplace, we will gain more physical insight on processes that control wind and wave energy input into the ocean and resulting near-inertial energy generation.

Acknowledgments.

This research was supported by the Office of Naval Research under Award N00014-18-1-2083 (E. Skyllingstad and R. Samelson) and Award N00014-18-1-2386 (H. Simmons and T. Klenz).

Data availability statement.

All shipboard and drifter data collected as part of the U.S. Office of Naval Research Near-Inertial Shear and Kinetic Energy in the North Atlantic experiment (NISKINe) is still in the process of being organized and archived. All data will be fully released to the public when the program formally concludes. NISKINe data presented in this study are available from the corresponding author upon reasonable request. Model output will be included with this archived data.

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  • Fig. 1.

    (a) Estimated wind stress and (b) glider measured ocean turbulence dissipation rates over the study period (time in UTC) along with (c) temperature and (d) salinity.

  • Fig. 2.

    (a) Surface significant wave height, (b) surface Stokes drift velocity, (c) friction velocity, and (d) turbulent Langmuir number.

  • Fig. 3.

    Vertical profiles of log10 turbulence dissipation rate from (a) the glider, (b) simulation case W, and (c) simulation case W-CL. Contours of simulated log10 dissipation rate equal to −8 for case W (red) and W-CL (black) are overlaid on each panel. Simulation results are horizontal averages. The initial time is 0000 UTC 21 May 2018.

  • Fig. 4.

    As in Fig. 3, but for an initial time of 0000 UTC 25 May 2018.

  • Fig. 5.

    Horizontally averaged potential temperature from (a) case W and (b) case W-CL along with horizontally averaged salinity from (c) case W and (d) case W-CL for the simulation of 21 May 2018.

  • Fig. 6.

    As in Fig. 5, but for the simulation of 25 May 2018.

  • Fig. 7.

    Magnitude of the horizontally averaged currents for the 21 May 2018 simulation for case (a) W and (b) W-CL and for the 25 May 2018 simulation for case (c) W and (d) W-CL.

  • Fig. 8.

    (a) MKE and (b) wind-work terms for (left) 21 May 2018 and (right) 25 May 2018. All integrated variables are scaled by the model depth.

  • Fig. 9.

    Storage term for MKE and TKE along with wind work for (a) case W and (b) W-CL for (left) 21 May and (right) 25 May.

  • Fig. 10.

    Terms from the TKE budget equation (a) without Stokes and (b) with Stokes for (left) 21 May and (right) 25 May. Terms are shear production (red), dissipation (black), Stokes shear production (blue), wind work Ew (red dash), and Stokes wind work Es (blue dash).

  • Fig. 11.

    Wind work for case W and wind work plus Stokes wind work for case W-CL from (a) 21 May and (b) 25 May.

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