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    Sketch of three-dimensional LES domain featuring wind-driven flow over a no-slip bottom and either a surface cooling or heating flux.

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    Downwind- and time-averaged downwind vorticity fluctuations over crosswind and vertical directions. Vorticity is scaled by uτ/δ.

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    Mean downwind velocity profile (a) throughout the entire water column and (b) in the bottom half of the water column. In the log-law profile in (b), κ = 0.41 (von Kármán’s constant) and C = 5.5.

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    RMS of downwind, crosswind, and vertical velocity fluctuations.

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    Downwind- and time-averaged crosswind velocity fluctuations for flows with Langmuir forcing with Raτ = 0, 211, and 5000.

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    As in Fig. 5, but for vertical velocity fluctuations .

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    As in Fig. 5, but for downwind velocity fluctuations .

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    As in Fig. 5, but for temperature fluctuations .

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    (a) Mean downwind velocity profile as a function of depth; (b) mean velocity shear; (c) mean downwind velocity near surface, plotted as , where is at the surface; and (d) mean downwind velocity near bottom in flows with Langmuir forcing. Note that in (a) no significant difference is observable between the cases with Raτ = 0 and 211. In the log-law profile, in (c), κ = 0.41 and C = 1. In (d), κ = 0.41 and C = 5.5.

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    RMS of (a) downwind, (b) crosswind, and (c) vertical velocity and (d) effective Reynolds shear stress in flows with Langmuir forcing for varying values of the Rayleigh number.

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    Contribution from supercell structures to (a) crosswind and (b) vertical velocity variance for varying values of the Rayleigh number in simulations with Langmuir forcing.

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    Selected terms in the budgets of turbulent kinetic energy (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 0, 211, and 5000; denotes distance from surface in wall units.

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    Selected budgets of turbulent kinetic energy (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 500, 1000, and 1500; denotes distance from surface in wall units.

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    Selected budgets of Reynolds shear stress (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 0, 211, and 5000; denotes distance from surface in wall units.

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    (a) Mean downwind velocity profile as a function of depth, (b) mean velocity shear, (c) mean velocity near the surface, and (d) mean velocity near wall in flows with Langmuir forcing and varying Richardson numbers. The wind-only case corresponds to wind-driven flow without LC and no surface heat flux.

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    (a) Mean temperature and (b) temperature gradient throughout full water column depth in flows with Langmuir forcing and varying Richardson numbers. Note that .

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    Downwind- and time-averaged vertical velocity fluctuations for flows with Langmuir forcing with Riτ = 0, 100, and 500.

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    RMS of velocity and effective Reynolds shear stress in flows with Langmuir forcing and varying Richardson numbers.

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    Selected budgets of turbulent kinetic energy (scaled by ) in bulk of water column in flows with Langmuir forcing with Riτ = 0, 100, and 500; denotes distance from bottom in wall units.

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Large-Eddy Simulation of a Coastal Ocean under the Combined Effects of Surface Heat Fluxes and Full-Depth Langmuir Circulation

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  • 1 Civil and Environmental Engineering, University of South Florida, Tampa, Florida
  • | 2 Center for Coastal Physical Oceanography, and Department of Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, Virginia
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Abstract

Results are presented from the large-eddy simulations (LES) of a wind-driven flow representative of the shallow coastal ocean under the influences of Langmuir forcing and surface heating and cooling fluxes. Langmuir (wind and surface gravity wave) forcing leads to the generation of Langmuir turbulence consisting of a wide range of Langmuir circulations (LCs) or parallel, counterrotating vortices that are aligned roughly in the direction of the wind. In unstratified, shallow coastal regions, the largest of the LCs reach the bottom of the water column. Full-depth LCs are investigated under surface waves with a significant wave height of 1.2 m and a dominant wavelength of 90 m and wave period of 8 s, for a wind speed of 7.8 m s−1 in a 15-m-deep coastal shelf region. Both unstable and stable stratification are imposed by constant surface heat fluxes and an adiabatic bottom wall. Simulations are characterized by Rayleigh and Richardson numbers representative of surface buoyancy forcing relative to wind forcing. For the particular combination of Langmuir forcing parameters studied, although surface cooling is able to augment the strength of LC, a significantly high cooling flux of 560 W m−2 (such that the Rayleigh number is Raτ = 1000) is required in order for turbulence kinetic energy generation by convection to exceed Langmuir production. Such a transition is expected at a lower heat flux for weaker wind and wave conditions and thus weaker LCs than those studied. Furthermore, a surface heating flux of approximately 281 W m−2 (such that the Richardson number is Riτ = 500) is able to inhibit vertical mixing of LC, particularly in the bottom half of the water column, allowing stable stratification to develop.

Corresponding author address: Andrés E. Tejada-Martínez, Civil and Environmental Engineering, University of South Florida, 4202 E Fowler Ave., Tampa, FL 33612. E-mail: aetejada@eng.usf.edu

Abstract

Results are presented from the large-eddy simulations (LES) of a wind-driven flow representative of the shallow coastal ocean under the influences of Langmuir forcing and surface heating and cooling fluxes. Langmuir (wind and surface gravity wave) forcing leads to the generation of Langmuir turbulence consisting of a wide range of Langmuir circulations (LCs) or parallel, counterrotating vortices that are aligned roughly in the direction of the wind. In unstratified, shallow coastal regions, the largest of the LCs reach the bottom of the water column. Full-depth LCs are investigated under surface waves with a significant wave height of 1.2 m and a dominant wavelength of 90 m and wave period of 8 s, for a wind speed of 7.8 m s−1 in a 15-m-deep coastal shelf region. Both unstable and stable stratification are imposed by constant surface heat fluxes and an adiabatic bottom wall. Simulations are characterized by Rayleigh and Richardson numbers representative of surface buoyancy forcing relative to wind forcing. For the particular combination of Langmuir forcing parameters studied, although surface cooling is able to augment the strength of LC, a significantly high cooling flux of 560 W m−2 (such that the Rayleigh number is Raτ = 1000) is required in order for turbulence kinetic energy generation by convection to exceed Langmuir production. Such a transition is expected at a lower heat flux for weaker wind and wave conditions and thus weaker LCs than those studied. Furthermore, a surface heating flux of approximately 281 W m−2 (such that the Richardson number is Riτ = 500) is able to inhibit vertical mixing of LC, particularly in the bottom half of the water column, allowing stable stratification to develop.

Corresponding author address: Andrés E. Tejada-Martínez, Civil and Environmental Engineering, University of South Florida, 4202 E Fowler Ave., Tampa, FL 33612. E-mail: aetejada@eng.usf.edu

1. Introduction

Turbulence in the coastal ocean can be generated by a variety of forcing mechanisms including tidally driven shear, wave breaking, and convection-driven buoyancy, among others. The vertical mixing induced by such phenomena has been a prominent area of research for many years due to the significant impact it can have on nutrient transport, surface gas exchange, contaminant dispersion, and other products of the vertical transport of mass, momentum, and heat.

Since its original characterization by Langmuir in 1938, Langmuir turbulence has become widely acknowledged as one of the key contributors to vertical mixing in the upper ocean. This turbulence is commonly observed in the form of Langmuir circulation (LC), which consists of pairs of parallel, counterrotating vortices approximately aligned in the direction of the wind and is generated by the interaction of the wind-driven shear current and the surface gravity wave–driven Stokes drift. This interaction is capable of tilting vertical vorticity into the direction of wave propagation (Craik and Leibovich 1976).

A description of laboratory and oceanic observations of LC up to the early 2000s has been provided by Thorpe (2005). The oceanic observations were made in the upper-ocean mixed layer (UOML), where LC is typically observed and these confirmed Langmuir’s deduction of the main features of LC. LC has more recently also been observed in shallow shelf coastal regions in which turbulence-induced vertical mixing can lead to homogenization of most of the water column. For example, Gargett et al. (2004) and Gargett and Wells (2007) described observations of the largest of the LC spanning the full depth of an unstratified water column using a five-beam acoustic Doppler current profiler with a 40-cm spatial resolution in shallow (15 m) water on the shelf some 6–7 km off the New Jersey coast. Much more detail on these full-depth LCs or “Langmuir supercells” was presented by Gargett and Wells (2007), including comparison with large-eddy simulations (LES) simulations of LC of Tejada-Martínez and Grosch (2007) performed with the well-known Craik–Leibovich (C–L) vortex force [representing wave–current interaction generating Langmuir turbulence (Craik and Leibovich 1976)]. After removing the mean flow, the main features of the remaining flow include those of the Langmuir supercells: 1) downwelling limbs of the cells that are narrower than upwelling limbs with the downward velocities larger than the upward velocities, 2) downwind jets under the downwelling regions, and 3) correct phasing of the crosswind velocity relative to the vertical velocity. In the two-part sequence of Gargett and Wells (2007) and Tejada-Martínez and Grosch (2007), the comparison of field observations of full-depth LC with the LES showed remarkable agreement in many aspects, including 1) the overall similarity of the observed LC structure to that from the LES; 2) in both the observations and the LES, downwind variance has a near-bottom maximum, and crosswind variance is less than downwind variance near the bottom boundary but increases to exceed it in the upper water column; 3) in both the observations and the LES results, profiles of the downwind vertical component of the Reynolds stress tensor are significantly different from zero, but profiles of the crosswind vertical component are approximately zero; and 4) the structure of the Lumley invariant maps (Lumley 1979; Simonsen and Krogstad 2005) for the observations and LES agree in the lower half of the water column, where they are dramatically different from the map of an LES without the Craik–Leibovich vortex force showing that the anisotropy of the energy-containing eddies depends strongly on the action of the C–L vortex force.

More recently, Kukulka et al. (2011, 2012) and Li et al. (2013) have conducted LES of Langmuir turbulence in the coastal ocean. In particular Kukulka et al. (2011) investigated the effect of a crosswind tidal current on the structure of full-depth Langmuir cells, showing that tidal forcing can lead to the merging and weakening of the cells. Guided by the field measurements of Gerbi et al. (2009) and others, Li et al. (2013) extended the earlier simulations of Kukulka et al. (2012) to include the effects of wave breaking on the structure of the turbulence. Li et al. (2013) concluded that the wave breaking is the dominant source for turbulence generation near the surface with Langmuir turbulence prevailing deeper in the water column. Note that the wind and wave forcing in the LES of Kukulka et al. (2011, 2012) and Li et al. (2013) is such that the Langmuir turbulence is not as intense in the lower half of the water column as that in the simulations of Tejada-Martínez and Grosch (2007).

Gargett and Grosch (2014) presented the analysis of 170 field-measured records of turbulent processes in shallow water with combined forcing by wind stress, the Langmuir vortex force, and surface cooling. The analysis is based on a new scaling using the surface stress velocity uτ and a time scale tw characteristic of the growth of Langmuir circulation. This scaling leads to characterizing the data with two dimensionless numbers: a Langmuir number La [not the well-known turbulent Langmuir number Lat defined by McWilliams et al. (1997)] and a Rayleigh number Ra. The Langmuir number is inversely proportional to wind forcing–relative wave forcing, and the Rayleigh number is indicative of the strength of surface buoyancy relative to shear. The properties of two turbulent archetypes, one full-depth LC and the other full-depth convection, were described in detail. It was shown that these two archetypes lie in distinct regions of the log(La, Ra) plane. Situations in which neither process dominates lie in a band in the log(La, Ra) plane between the archetypes with the relative dominance of full-depth LC or full-depth convection given by proximity to either archetype. It was also shown that, if LC dominated, the surface Stokes velocity uso was linearly related to uτ so that it was impossible to differentiate between different scalings involving both uso and uτ. Finally, it was found that there were no cases among the 170 records for which direct wind stress forcing dominated.

Gargett et al. (2014) gave a very detailed description of a complete Langmuir supercell event, including the structure of LC in the surface layer of the stratified water column that existed before the onset of the Langmuir supercell event, the evolution of the LC, and its cessation. The start of a Langmuir supercell event occurs when there are surface waves of the intermediate type that “feel bottom” and the three conditions for full-depth LC are satisfied: 1) an unstratified water column, 2) La < 0.3, and 3) |Ra| < 105. The end of the LC event occurs if one or both of the two latter conditions is not met or by the reappearance of stratification. In the case where there is stratification and LC is confined to the well-mixed surface layer, first-mode internal waves with the frequency of the stratified layer are generated. The active surface layer LC was not a cause of mixed layer deepening; rather, the deepening occurred when the Richardson number was lowered by increased shear. Because of its quasi-organized structure and enhanced vertical velocity, LC can increase momentum transfer to the surface layer, thus increasing flow acceleration and thereby contributing to the shear instability that deepens the surface layer.

Observations of LC in stratified, shallow-water columns with a surface mixed layer have recently been reported by Scully et al. (2015). These observations were made from a tower on a broad, western shoal (14-m water depth) in the central portion of Chesapeake Bay. The instrumentation included six vector acoustic Doppler velocimeters (ADVs) having 2-m vertical separation with the uppermost ADV 1.5 m below the water surface. The effects of both LC and convection were apparent in these observations. In these observations the salinity stratification in the bay was almost always strong enough that full-depth LC was not observed. It was found that, when the turbulent Langmuir number Lat was less than 0.5, the surface mixed layer had 1) increased vertical turbulent kinetic energy, 2) decreased anisotropy, and 3) negative vertical velocity skewness, indicating that the downwelling was both strong in narrow regions and the upwelling was weak in broad regions. These effects were attributed to LC driven by the vortex force, but it was also noted that convection appears to have also contributed. The observations of Scully et al. (2015) never or almost never showed the structure of either of the two archetypes, pure LC and pure convection, that were described by Gargett and Grosch (2014). Rather these observations in Chesapeake Bay appear to show that its vertical mixing is driven by the dynamics of a combination of LC and convection for most of their observations.

LES can be used to complement field observations because of its ability to isolate individual contributors to, or inhibitors of, turbulence and vertical mixing. Alongside LES of wind-driven flows with the C–L vortex forcing generating full-depth LC in a shallow-water column (e.g., Tejada-Martínez and Grosch 2007), similar full-depth structures can also be found in pressure gradient-driven flows with zero surface shear and thus no wave forcing in which a surface cooling flux is applied (Walker et al. 2014). Cooling-induced buoyancy was found to lead to the development of full-depth, counterrotating cell pairs, denoted “convection supercells” because of their similarity with Langmuir supercells. The initiation of vertical mixing and development of convective cellular structures in the presence of an unstable temperature gradient has been established experimentally [Tilgner et al. (1993) is one of many such examples], and extensive studies consider deep convection in the global ocean on scales significantly larger than that of the present work (Paluszkiewicz and Garwood 1994; Marshall and Schott 1999). However, such full-depth, convection-induced structures in a simulation representative of a tidal boundary layer by Walker et al. (2014) were unique.

Although typically wind and wave forcing would be also present in the field, with sufficient heat loss from the ocean surface, it is possible for cooling-induced buoyant motion to dominate surface turbulence (Shay and Gregg 1986). However, for example, in Gargett et al. (2004), despite diurnal surface cooling fluxes up to approximately 200 W m−2, the full-depth Langmuir cells remained nearly unchanged. This suggests Langmuir forcing remained the dominant cause of the turbulent structures even in the presence of other forcing mechanisms.

Quantification of the point at which a transition from Langmuir to convective turbulence occurs has been attempted by means of the Hoenikker number Ho, defined as the ratio of the buoyant forcing that drives thermal convection to the Craik–Leibovich vortex force that drives LC. It was suggested by McWilliams et al. (1997) that Ho > 1 leads to the transition to convection-driven turbulence. Following this, Li et al. (2005) observed that the value of Ho required for this transition to occur is around Ho = O(1) and additionally that “typical” heat fluxes and wind speeds generate only Ho = O(0.01). More recently, Gargett and Grosch (2014) have also demonstrated that it is possible to determine the point at which convection is dominant by identifying the region in which log(Ra) > 7.5 and log(La) > 0.

Several questions remain unanswered: How do we determine which of these influences (C–L vortex forcing or surface cooling) is the primary driver for generating full-depth cells observed in shallow coastal regions? Are we able to quantify the respective contributions of each mechanism? Can we identify the point at which the transition between the dominance of these two respective mechanisms could occur in the field? The first series of tests to be considered in the present work involve the application of turbulence-enhancing surface cooling fluxes to a flow with full-depth LC in an attempt to answer these questions.

A secondary aim of the present work is to assess the impact of a surface heating flux on the full-depth LC. In this opposite scenario, stable density stratification provides resistance to mixing, inhibiting vertical motion and reducing the vertical transfer of momentum. Taylor et al. (2005) considered the case of an open channel with stable stratification imposed at the free surface by a constant heating flux. An adiabatic solid lower wall was used and a stably density stratified pycnocline was seen to develop throughout the channel, inhibiting the mixing induced by the turbulence generated by flow interaction with the solid wall at the lower boundary. Similar results have been found with a fixed density difference across a channel in which the Dirichlet boundary conditions impose a lower density at the surface than the bottom (Armenio and Sarkar 2002). Prior LES studies of Langmuir cells under surface heating have been made for cells in the UOML capped below by a pycnocline (Min and Noh 2004; Kukulka et al. 2013). In particular, in the UOML simulations of Min and Noh (2004), the penetration depth of the Langmuir cells decreases with increasing surface heat flux and, as such, the crosswind width of the cells becomes smaller.

In the present work, LES is used to assess the impact of a combination of both stable and unstable surface heat fluxes and Langmuir forcing on the turbulence statistics of wind-driven flows in a domain representative of a shallow coastal shelf region. Several values of a Rayleigh number representative of the surface buoyancy relative to wind shear forcing will be tested, which correspond to varying strengths of the surface cooling flux. Following this, several values of the analogous Richardson number will be tested, corresponding to varying strengths of the surface heating flux. By analyzing mean velocity profiles, root-mean-square (RMS) and partially averaged velocity fluctuations, and budgets of turbulent kinetic energy and Reynolds shear stress, we intend to gain a clearer insight into the relative contribution of each forcing mechanism in the generation and inhibition of full-depth cell structures.

2. Governing equations

In nondimensional form, the incompressible Navier–Stokes equations (under the Boussinesq approximation), continuity equation, and advection–diffusion equation for temperature are, respectively,
e1
e2
e3
These equations correspond to the LES equations in which application of a homogeneous low-pass spatial filter is denoted by an overbar. The filtered or resolved temperature, pressure, velocity, and vorticity are, respectively, , , , and . Furthermore, x1, x2, and x3 represent the downwind, crosswind, and vertical directions. In (1), , where θb is the instantaneous averaged over the x1 and x2 directions [the interested reader can find further information about this buoyancy term in Tejada-Martínez et al. (2009) and Armenio and Sarkar (2002)].
Equations (1)(3) above have been made dimensionless using wind stress friction velocity uτ = (τw/ρ0)1/2 (where τw is the constant wind stress at the surface, and ρ0 is the density) and water column half depth δ = H/2 (where H is the water column depth), giving rise to the constant Reynolds number, turbulent Langmuir number, and Prandtl number:
e4
where ν is molecular kinematic viscosity, and α is thermal diffusivity. The characteristic Stokes drift velocity in (4) is defined by us = ωKa2, where ω is the dominant frequency, K is the dominant wavenumber (defined by K = 2π/λ with λ the wavelength), and a is the amplitude of the surface gravity waves generating Langmuir circulation.
Equation (3) has been nondimensionalized with , a characteristic temperature based on the temperature gradient imposed at the surface [, where Q ≥ 0 is the constant heat flux at the upper surface, and k is the thermal conductivity]. This gives rise to the constant Rayleigh number Raτ, in the case of surface cooling, or the constant Richardson number Riτ, in the case of surface heating:
e5
where β is the coefficient of thermal expansion, and Cp is the heat capacity. Note that in (1), in the case of surface heating, the buoyancy term (the fifth term on the right-hand side of this equation) appears with a negative sign in front.
In (1) and (3), the LES subgrid-scale (SGS) stress and SGS buoyancy flux τij and λj are, respectively, modeled using a dynamic mixed model (Morinishi and Vasilyev 2001) and a dynamic Smagorinsky model (Smagorinsky 1963):
e6
with as the spatial filter width; Cs and are Smagorinsky coefficients, is the filtered strain-rate tensor, and is its norm. The model coefficients Cs, CL, and are computed dynamically using the well-known Germano identity (Lilly 1992; Morinishi and Vasilyev 2001). Note that in the SGS stress model above, the stress is split into an isotropic component and a deviatoric (or trace free) component. The isotropic component of the stress is lumped into the pressure in the momentum equation, and the deviatoric component τij is modeled as given in (6). The terms on the right-hand side of τij in (6) are both deviatoric: is deviatoric due to the incompressibility condition in (2), and the term is deviatoric by definition, given as , where δij is Kronecker’s delta. Details of the implementation of the dynamic mixed and dynamic Smagorinsky models above, including the filters used, is given in Tejada-Martínez and Grosch (2007).
The fourth term on the right-hand side of the momentum equation in (1) represents the Craik–Leibovich vortex forcing or Langmuir forcing, which models the generation of LC through the interaction of the Stokes drift, driven by surface waves, and the vertical shear of the current. As can be seen, this term features the vector cross product between the Stokes drift velocity and the flow vorticity; its components are . The dimensionless, wind-aligned Stokes drift velocity is defined following Phillips (1977) and LeBlond and Mysak (1978) as
e7
where H is the water depth, K is the dominant wavenumber of surface gravity waves, and x3 is the depth in the vertical direction varying from 0 at the surface to −H at the bottom of the water column.

Note that the Navier–Stokes equations featuring this Craik–Leibovich vortex force term are commonly known as the Craik–Leibovich equations (McWilliams et al. 1997).

The last term on the right-hand side of (3) represents a source term used to make the temperature statistically steady (Leighton et al. 2003). Without this term, the temperature would decay continuously throughout the domain in the case of the surface cooling boundary condition and vice versa in the case of the surface heating boundary condition. The source term is s = 1/(2PrReτ) for surface cooling and s = −1/(2PrReτ) for surface heating.

3. Flow configuration

We model a shallow, coastal ocean region 2δ or H in depth by a three-dimensional domain in which the flow is driven by a wind stress with zero normal flow imposed at the surface and no slip imposed at the bottom wall (see Fig. 1). Periodicity is enforced in both downwind x1 and crosswind x2 directions, implying spatial homogeneity in these horizontal directions, characteristic of a flow unaffected by lateral boundaries. In this flow configuration, in the mean, the wall friction velocity is equal to the constant wind stress friction velocity uτ.

Fig. 1.
Fig. 1.

Sketch of three-dimensional LES domain featuring wind-driven flow over a no-slip bottom and either a surface cooling or heating flux.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

The wind stress is applied such that the Reynolds number in (4) is 395. This Reynolds number is much lower than in the field and has been chosen in order to resolve the surface and bottom viscous sublayers expected in such shear stress–driven flows. This avoids the need to use a wall model (Piomelli and Ballaras 2002) that assumes the presence of a well-developed bottom log layer. As shown by Tejada-Martínez et al. (2012), full-depth Langmuir cells can disrupt the bottom log layer, thereby invalidating the assumption of the wall model. Note that in the coastal ocean Langmuir turbulence simulations of Kukulka et al. (2011), Kukulka et al. (2012), and Li et al. (2013), a wall model was used because the Langmuir cells in those cases were deemed insufficiently strong to disrupt the bottom log layer. This is not the case for the Langmuir cells simulated here, as will be shown farther below.

Although the present simulations are at a lower Reynolds number than in the field, this Reynolds number is sufficiently high that the resolved Langmuir supercells simulated possess similar structure as that of observed supercells in the field, while scaling up favorably in terms of the range of fluctuating velocity components. This will be described in greater detail in the results section.

The Prandtl number in (4) has been set to 1 also in order to obtain a computationally feasible simulation. This setting is not expected to alter the trends observed in the LES results associated with stratification, as justified by Enstad et al. (2006).

Following the field measurements of Gargett et al. (2004) and Gargett and Wells (2007) of full-depth Langmuir cells, Lat in (4) is set to 0.7, and the wavelength of the surface waves is λ = 6H [with K = 2π/λ = 2π/6H in (7)]. These wind and wave forcing parameters correspond to surface waves with a significant wave height of 1.2 m, wavelength of 90 m, wave period of 8 s, and wind stress of 0.1 N m−2 for a wind speed of 7.8 m s−1 in a 15-m-deep coastal shelf region (Gargett et al. 2004).

A constant temperature gradient (respectively negative for surface cooling and positive for surface heating) is applied at the surface, and the bottom wall is set to adiabatic (see Fig. 1). Various Rayleigh numbers (for surface cooling) and Richardson numbers (for surface heating) are investigated. For example, a case with Raτ = 211 corresponds to a surface cooling flux of approximately Q = 120 W m−2, and a case with Riτ = 100 corresponds to a heating flux of approximately Q = 56 W m−2; both heat fluxes are observed during the field measurements of full-depth LC of Gargett et al. (2004) and Gargett and Wells (2007). Results from simulations with more intense heat fluxes will also be presented. Note that the values of Q reported above were obtained from the definition of Raτ and Riτ in (5) with seawater properties (heat capacity, density, and thermal expansion coefficient) at 20°C obtained from Kaye and Laby (1995) and uτ = 0.01 m s−1 from the field measurements of Gargett et al. (2004) and Gargett and Wells (2007). The viscosity and diffusivity were obtained from the definitions of the Reynolds and Prandtl numbers in (4) with uτ = 0.01 m s−1 and δ = 7.5 m, following Gargett et al. (2004) and Gargett and Wells (2007), and Reτ = 395 and Pr = 1 as discussed earlier. The resulting viscosity and diffusivity for these Reynolds and Prandtl numbers may be interpreted as an eddy viscosity and eddy diffusivity.

The computational domain is of dimensions 4πδ × (8/3)πδ × 2δ in the respective downwind x1, crosswind x2, and vertical x3 directions, which, given a half depth δ = 7.5 m [following Gargett et al. (2004)], corresponds to 94.2 m × 60 m × 15 m in dimensional form. Gargett et al. (2004) observed the approximate crosswind length of one Langmuir supercell to be between 3 and 6 times the water column depth, and Tejada-Martínez and Grosch (2007) confirmed this result computationally. As such, the current computational domain was chosen in order to be able to contain one Langmuir supercell.

The computational domain was discretized with 32 × 64 × 97 points in the x1, x2, and x3 directions, respectively. These points were equally spaced in the former two directions and stretched in the latter using a hyperbolic function symmetric about the middepth in order to resolve the molecular sublayers, or large velocity and temperature gradients, at the top and bottom boundaries. For all flows, excluding a case with extreme surface cooling such that Raτ = 5000 (Q = 2810 W m−2), the first grid point above the bottom wall is at distance from the wall, where the plus superscript represents viscous wall scaling defined by and , where . Similarly, the first grid point below the top of the domain (i.e., the surface) is at distance from the surface. The case with Raτ = 5000 has thinner surface boundary layers than the other flows, and thus the vertical grid spacing is adapted such that the first grid point below the surface is at distance from the surface. Similarly, for this case the first grid point above the wall is at distance from the wall.

Scaling up the LES to the field measurements of Gargett et al. (2004), the LES mesh possesses 2.9-m resolution in the downwind direction and 1-m resolution in the crosswind direction. For all flows, except for the case with Raτ = 5000, vertical mesh resolution varies between 0.02 m near the wall and surface and 0.35 m in the middle of the water column. For the Raτ = 5000 case, the vertical mesh resolution varies between 0.002 m near the wall and surface and 0.56 m in the middle of the water column.

The computational solver is based on the fractional step scheme described by Tejada-Martínez and Grosch (2007). Horizontal directions (x1 and x2) are treated spectrally using fast Fourier transforms, and the vertical (x3) direction is treated with fifth- or sixth-order, compact, finite-difference stencils. The interested reader is directed to Tejada-Martínez and Grosch (2007) for further information on the numerical method employed.

Turbulent quantities are ensemble averaged over x1, x2 and time t and obtained after flows had reached statistical equilibrium. The variables presented in the following results are dimensional and are nondimensionalized by a reference or characteristic scale given in the corresponding figures.

4. Results

a. Surface cooling

1) Preliminary simulations

As discussed in the introduction, in boundary layer flows cooling-induced buoyancy has been found to lead to the development of full-depth, counterrotating cell pairs, denoted convection supercells, which are similar in structure to full-depth Langmuir cells (i.e., Langmuir supercells; Walker et al. 2014). Initially, we demonstrate this similarity by a comparison of three flows with configuration described earlier in section 3: a wind-driven flow in the absence of surface cooling (and thus no surface buoyancy forcing) and absence of C–L vortex (Langmuir) forcing, a wind-driven flow with surface cooling only (with Raτ = 5000), and a wind-driven flow with Langmuir forcing only. From this, we are able to demonstrate the shared features of full-depth cells initiated by each respective mechanism in this idealized problem.

To visualize the structures present in each case, we consider downwind vorticity fluctuations across the crosswind and vertical extent of the domain. We average the downwind vorticity fluctuations over the downwind direction x1 and time in order to capture the well-maintained, full-depth structures of the flows. Figure 2 shows the averaged vorticity fluctuation for the wind-only, wind and Langmuir forcing (wind and LC), and wind and surface cooling cases.

Fig. 2.
Fig. 2.

Downwind- and time-averaged downwind vorticity fluctuations over crosswind and vertical directions. Vorticity is scaled by uτ/δ.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Notice in the case with wind only (Fig. 2a), the constant surface wind stress leads to two pairs of full-depth counterrotating vortices or two cells. These vortices are elongated in the direction of the wind and alternate in sign in the crosswind direction. These structures are known as Couette cells, which have been previously observed both experimentally and computationally in the flow between two parallel plates moving in opposite directions (Couette flow), for example, by Lee and Kim (1991) and Papavassiliou and Hanratty (1997).

When Langmuir (wind and wave) forcing is applied to the wind-only flow and thus the Couette cells in Fig. 2a, the Couette cells merge, strengthen, and become more coherent, giving rise to the Langmuir supercell in Fig. 2b. Similarly, when surface cooling is applied to the wind-only flow and thus the Couette cells in Fig. 2a, the cells merge, strengthen, and become more coherent, giving rise to the convective supercell in Fig. 2c.

In the field, the events leading to a Langmuir supercell are much different than in our computations since winds and waves occur simultaneously. As discussed in the introduction, Gargett et al. (2014) give a detailed description of a complete Langmuir supercell event in the field including the structure of LC in the surface layer of the stratified water column that existed before the onset of the Langmuir supercell event, the evolution of the LC, and its cessation.

The present simulations are not able to capture the events leading to a Langmuir supercell in the field; however, the Langmuir supercell resolved numerically and under statistical equilibrium is in good agreement with the Langmuir supercells measured in the field, as described in Gargett et al. (2004), Gargett and Wells (2007), and Tejada-Martínez and Grosch (2007). Furthermore in Tejada-Martínez et al. (2009), instantaneous velocity fluctuations in the present flow with LC with zero surface heat flux have been scaled up or, better yet, redimensionalized using the wind stress friction velocity uτ measured in the field by Gargett et al. (2004) and Gargett and Wells (2007) in the presence of full-depth Langmuir cells. The redimensionalized LES velocity fluctuations agree well with those measured in the field. For example, downwind and crosswind velocity fluctuations range from −8 to +8 cm s−1, while vertical velocity fluctuations range from −4 to +4 cm s−1. In summary, LES with Langmuir (wind and wave) forcing parameters consistent with the full-depth LC field measurements of Gargett et al. (2004) and Gargett and Wells (2007) yield Langmuir turbulence in agreement with the field measurements in terms of structure and intensity, despite the lower Reynolds number of the LES compared to the Reynolds number in the field. This can be attributed to Reτ = 395 being sufficiently large to allow for scale invariance between the simulated (computed) and the field turbulence in the core region of the flow away from the surface and bottom viscous sublayers where viscous effects are important. Thus, results obtained here within the core flow region are expected to be applicable to the real ocean.

In Fig. 2c, the application of surface cooling with Raτ = 5000 to the wind-driven flow without Langmuir forcing is seen to generate a comparable structure to a full-depth Langmuir cell; one large, full-depth, and coherent cell is observed but with considerable lower intensity and coherency compared to the case with Langmuir forcing in Fig. 2b. This is consistent with the work of Gargett and Grosch (2014), characterizing archetypes of full-depth Langmuir circulation and full-depth convection in an unstratified, shallow coastal ocean. The full-depth cell generated by surface cooling is also similar to that obtained by Walker et al. (2014) in a pressure gradient–driven flow with zero wind stress and the same surface cooling as the wind-driven flow shown here. The cell (in Fig. 2c) will be referred to as a convective supercell following Walker et al. (2014).

To further assess similarities and differences of the two forms of supercell identified in the vertical plane color maps, we can consider the mean downwind or streamwise velocity profiles throughout the entire water column (Fig. 3a) and in the lower half (Fig. 3b), the latter emphasizing the bottom boundary layer [i.e., the viscous sublayer (), the buffer layer (), and the log layer ()]. In Fig. 3a, we can primarily see that in a case without either LC or cooling present, the velocity demonstrates an S-shaped profile featuring a positive gradient throughout the water column, as is to be expected in a simple wind-driven flow (Papavassiliou and Hanratty 1997). This profile is altered in both cases with an additional mechanism applied. In both the case with LC and the case with surface cooling, the increased vertical mixing induced by the presence of a full-depth supercell leads to greater homogenization of mean downwind velocity (momentum) throughout the water column and a significant thinning of the surface boundary layer. In Fig. 3b, where the wind-driven flow is approximately characterized by an expected log-law profile (indicated in the figure by the light solid line) that represents the behavior of a typical turbulent shear stress–driven flow over a no-slip wall (Papavassiliou and Hanratty 1997), in both the case with LC and surface cooling, this log-law profile is disrupted by the presence of a supercell. The log-law disruption and overall disruption of log-layer dynamics in the case with LC are analyzed in depth by Tejada-Martínez et al. (2012) and Sinha et al. (2015).

Fig. 3.
Fig. 3.

Mean downwind velocity profile (a) throughout the entire water column and (b) in the bottom half of the water column. In the log-law profile in (b), κ = 0.41 (von Kármán’s constant) and C = 5.5.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Additionally, in order to distinguish between the mean flow and the turbulent fluctuations, we employ the classical Reynolds decomposition in which the LES velocity is decomposed as , where brackets denote the averaging over time and the streamwise and cross-stream directions (Reynolds averaging), and the prime indicates the velocity fluctuation component. The RMS of the velocity components is computed as , , and , respectively.

In Fig. 4, it can be seen that for the case with wind only, the velocity RMS components are approximately homogeneous in the bulk of the water column. In contrast, in the cases featuring Langmuir and convection supercells, higher magnitudes of crosswind velocity RMS can be observed at the near surface and near bottom of the water column when compared to the middepth, visible in Fig. 4b. In Fig. 4c, observe that the vertical velocity RMS is significantly greater throughout the bulk of the water column in the flows with Langmuir and convection supercells, compared to the flow with wind only. Notice, however, that the RMS of downwind velocity in the case with convection (surface cooling) is closer to that of the wind-only flow than the flow with LC.

Fig. 4.
Fig. 4.

RMS of downwind, crosswind, and vertical velocity fluctuations.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

2) Simulations with LC and surface cooling

Based on the previous preliminary understanding of the similarities in cell structures observed in flows under the influence of a surface cooling flux and Langmuir forcing, respectively, we can compare the turbulence statistics for the flows in which both mechanisms are present. In this section, the major differences in equilibrium dynamics are examined between three flows: 1) with LC and zero surface buoyancy forcing (thus Raτ = 0), 2) with LC and a moderate surface cooling flux (such that Raτ = 211), and 3) with LC and a strong surface cooling flux (such that Raτ = 5000). In the case with Raτ = 0, the fluid is cooled at the surface, but the temperature is treated as a passive scalar and does not affect the flow dynamics; hence, this flow has zero surface buoyancy forcing. As noted earlier, the Rayleigh number of 211 was calculated based on field measurements of LC in a coastal region by Gargett et al. (2004) and Gargett and Wells (2007) where the observed wind stress was approximately 0.1 N m−2 and the surface cooling flux was approximately 120 W m−2. As heat fluxes of this magnitude are observed frequently in coastal environments, we refer to this as a moderate surface cooling flux. A Rayleigh number of 5000 represents our strong cooling flux, as was applied in the idealized wind-driven flows described earlier. For the wind speed (and resulting wind stress) in the field measurements of Gargett et al. (2004) and Gargett and Wells (2007), this Rayleigh number corresponds to an unrealistically high surface cooling flux of 2810 W m−2. Unlike the value of Raτ = 211 previously described, this was not taken from field measurements and instead was estimated as an approximate upper limit to a strong cooling flux that could be modeled in the present LES within the time and accuracy restraints of the current domain and discretization.

As seen earlier, the full-depth coherent turbulent structures present in our flows with Langmuir circulation with and without surface cooling can be visualized in terms of downwind vorticity. However, instead, in Figs. 5, 6, and 7 the cells are visualized in terms of velocity fluctuations averaged over the downwind direction x1 and time in order to better reveal differences in cell structure when surface cooling is applied. The regions of surface convergence and bottom divergence of the supercells can be observed in terms of crosswind velocity fluctuations in Fig. 5. The surface convergence of each cell generates a full-depth region of downwelling, that is, a full-depth region of negative vertical velocity fluctuations, as observed in Fig. 6. Meanwhile, the bottom convergence of each cell generates a full-depth region of upwelling or, better yet, positive vertical velocity fluctuations. Note that the downwelling limb of each cell, in Fig. 6, coincides with a full-depth region of positive, downwind, velocity fluctuations, observed in Fig. 7. In summary, the regions of surface and bottom convergence and divergence of crosswind velocity fluctuations and full-depth positive and negative vertical velocity fluctuations representing regions of upwelling and downwelling constitute a full-depth supercell resolved in the LES, as was seen earlier in terms of downwind vorticity.

Fig. 5.
Fig. 5.

Downwind- and time-averaged crosswind velocity fluctuations for flows with Langmuir forcing with Raτ = 0, 211, and 5000.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for vertical velocity fluctuations .

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for downwind velocity fluctuations .

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

In Figs. 57, all three cases (Raτ = 0, 211, 5000) appear similar, indicating that the application of cooling is not significantly altering the shape of the cells; for example, their cross-stream width and height is maintained. However, an increased level of symmetry and an increase in the intensity of the upwelling and downwelling regions or limbs is visible at Raτ = 5000. Furthermore, in the Raτ = 0 and 211 cases the downwind velocity fluctuations are intensified near the bottom of the water column, consistent with the field measurements of Gargett et al. (2004) and Gargett and Wells (2007). Meanwhile, such intensification is absent in the flow with Raτ = 5000.

Downwind- and time-averaged temperature fluctuations are observed in Fig. 8. Comparing Figs. 6 and 8, it is seen that in flows without surface buoyancy forcing and with moderate cooling (Raτ = 0, 211) the downwelling limbs of the cells serve to bring colder fluid from the surface to the bottom, and the upwelling limbs serve to bring warmer fluid from the bottom to the surface. This temperature structure is less discernible in the flow with strong cooling (Raτ = 5000) due to the greater vertical mixing of temperature in this idealized case.

Fig. 8.
Fig. 8.

As in Fig. 5, but for temperature fluctuations .

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

(i) Mean velocity profiles

Mean downwind velocity profiles and associated vertical shear as functions of depth are shown in Fig. 9. Results are shown for wind-driven flows with LC with Raτ = 0, 211, and 5000. Profiles from wind-driven flow without both Langmuir circulation and surface buoyancy forcing (again denoted wind only) are included for comparison.

Fig. 9.
Fig. 9.

(a) Mean downwind velocity profile as a function of depth; (b) mean velocity shear; (c) mean downwind velocity near surface, plotted as , where is at the surface; and (d) mean downwind velocity near bottom in flows with Langmuir forcing. Note that in (a) no significant difference is observable between the cases with Raτ = 0 and 211. In the log-law profile, in (c), κ = 0.41 and C = 1. In (d), κ = 0.41 and C = 5.5.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Considering first the whole column depth (Fig. 9a), in the Raτ = 0 case with LC present, the vertical mixing induced by the up- and downwelling limbs of the Langmuir cell structure (described earlier) causes flow velocity homogenization throughout the bulk of the water column, leading to a near-uniform profile over the bulk of the flow. When surface cooling is applied with Raτ = 5000, this homogenization reaches farther up toward the surface, making the surface boundary layer thinner and suggesting an increase in the strength of the vertical mixing near the surface. Note that the velocity profile with a Rayleigh number of 211 is not noticeably different from the LC case with Raτ = 0. In all cases with LC, the mean velocity profiles exhibit slight negative vertical shear, especially in the upper half of the water column, as seen in Fig. 9b.

The mean velocity in the upper half of the water column is plotted in Fig. 9c in the form of the difference between the surface mean velocity and the mean velocity throughout the water column. The log-law profile in Fig. 9c represents the velocity behavior for a typical turbulent flow at a shear stress–driven surface (Papavassiliou and Hanratty 1997) and is indicated by the plus symbols; it can be seen that the wind-driven case without both LC and surface buoyancy forcing follows this expected profile closely. In all flows with LC present we see a deviation from the classical log-law velocity profile. When cooling with Raτ = 5000 is applied, this deviation is greater than in the LC cases with Raτ = 0 and 211.

In Fig. 9d the flow velocity is in the lower wall boundary layer region. Notice the similarity between the log-layer behavior of the flow with wind only and the expected theoretical log-layer profile. For all values of the Rayleigh number, the inclusion of LC again leads to the disruption of this expected profile. However, notice that as the Rayleigh number reaches 5000, the values of the velocity are closer to those of the theoretical log-law compared to the other LC cases. In the Raτ = 0 case, the downwelling limb of the cells bring high speed fluid closer to the bottom of the water column, thereby causing the deviation from the log law (Tejada-Martínez et al. 2012). In the LC flow with Raτ = 5000, the upwelling limb is closer in intensity and width to the downwelling limb than in the flows with Raτ = 0, 211 (see Fig. 6). Thus, the upwelling limb in the Raτ = 5000 case is able to provide a greater offset of the log-law deviation caused by the downwelling limb, resulting in an overall lesser deviation from the log law near the bottom.

The main observations here are that (i) the case with Raτ = 211 demonstrates little, if any, difference from the case with Raτ = 0, further suggesting that the moderate surface cooling is too weak to cause significant alterations to the preexisting Langmuir cells structure; and (ii) at a higher cooling flux of Raτ = 5000, the strengthened turbulent structure is able to cause greater vertical mixing closer to the surface, causing significant deviation of the mean downwind velocity from the log law near the surface.

Differences between the Raτ = 5000 case and the Raτ = 0 and Raτ = 211 cases can be further quantified in terms of the surface and bottom boundary layer thickness listed in Table 1. The thickness is arbitrarily defined as the distance from the surface or the wall to the point where the mean downwind velocity is 99% of the mean downwind velocity at the middle of the water column. As can be seen in Table 1, Langmuir forcing with zero surface buoyancy forcing (Raτ = 0) significantly decreases the thickness of the surface and bottom boundary layers relative to the flow driven by wind only (without Langmuir forcing and without surface buoyancy). The application of surface cooling to the flow with LC with increasing Raτ further decreases the thickness of the surface boundary layer. However, the application of surface cooling with varying values of Raτ to the flow with LC does not significantly alter the bottom boundary layer thickness.

Table 1.

Surface and bottom boundary layer (BL) thickness in flows with Langmuir forcing and varying Rayleigh numbers.

Table 1.

(ii) RMS profiles

A comparison of velocity RMS can be conducted to further assess the impact of cooling for the LC cases with a varying Rayleigh number. First, observe the presence of the cell structure in the RMS profiles in Fig. 10. For all cases with LC, the regions of surface and bottom convergence and divergence (observed earlier in Fig. 5) are identifiable by the higher magnitudes of crosswind velocity RMS at the near surface and near bottom of the water column when compared to the middepth, visible in Fig. 10b. In Fig. 10c, the vertical velocity RMS is at a maximum in the center of the water column, representative of the up- and downwelling limbs of the cells that are also intensified at middepth (Fig. 6).

Fig. 10.
Fig. 10.

RMS of (a) downwind, (b) crosswind, and (c) vertical velocity and (d) effective Reynolds shear stress in flows with Langmuir forcing for varying values of the Rayleigh number.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

In Fig. 10a, the downwind velocity RMS in the case with Raτ = 5000 demonstrates an increased velocity fluctuation in the center of the water column with a less distinct gradient than the cases with lower values of the Rayleigh number. This is due to the increased level of symmetry with increasing Raτ in the structure of the full-depth cells mentioned earlier. The symmetry can be observed, for example, in terms of the downwind velocity fluctuation that is significantly intensified near the bottom for the Raτ = 0 and Raτ = 211 cases (Figs. 7a,b) but not so for the Raτ = 5000 case (Fig. 7c).

In Figs. 10b and 10c, we can see that an increase in Raτ gives rise to increases in the vertical and crosswind velocity RMS throughout the entire water column. Slight increases are seen with the moderate cooling flux, but the case with Raτ = 5000 shows the most distinct increases relative to the case with Raτ = 0.

Figure 10d shows profiles of the effective Reynolds shear stress, consisting of the resolved dominant Reynolds shear stress and the corresponding mean subgrid-scale stress . Because of the homogenization induced by the full-depth cells, the resolved viscous shear stress in the interior of the water column is small, leaving the effective Reynolds shear stress to transfer the wind stress down through this region. Hence, the effective dimensional Reynolds shear stress scaled by (plotted in Fig. 10d) is nearly 1 throughout the interior of the water column for all cases. Note that for the case with Raτ = 5000, the enhanced near-surface mixing induced by the surface cooling leads to higher values of the effective Reynolds shear stress, reaching closer to the surface compared to the Raτ = 0, 211 cases.

(iii) Cell strength
To further quantify cell strengthening as the Rayleigh number increases, we can assess the contribution of the full-depth cells to the turbulent crosswind and vertical velocity fluctuations, indicating the relative strength of the surface convergence and bottom divergence zones of the cells and associated upwelling and downwelling limbs. Using a triple decomposition defined by Tejada-Martínez and Grosch (2007), the contribution due to downwind coherent structures (CS) to the components of the Reynolds stress tensor is defined as
e8
where denotes the averaging in the downwind direction x1 and in time, and denotes averaging in the downwind and crosswind directions, x1 and x2, and in time.

Although from the results shown above the Raτ = 0 and 211 cases are exhibiting nearly the same turbulence intensities, in Fig. 11 it is visible from the depth profiles of and that LC with Raτ = 211 does in fact lead to an increase in cell strength when compared to the LC case without surface buoyancy (Raτ = 0; Fig. 11). At a Rayleigh number of 5000, this increase in cell strength is even more substantial.

Fig. 11.
Fig. 11.

Contribution from supercell structures to (a) crosswind and (b) vertical velocity variance for varying values of the Rayleigh number in simulations with Langmuir forcing.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

(iv) Budgets of turbulent kinetic energy
Next we consider the sinks and sources of turbulent kinetic energy (TKE). Following the classical Reynolds decomposition, we consider the transport equation for resolved turbulent kinetic energy as follows:
e9
where is the resolved TKE, and
e10
These terms in respective order represent the pressure transport rate A, the viscous diffusion rate D, the viscous dissipation rate ε, the SGS transport rate TSGS, the SGS dissipation rate εSGS, the mean (Eulerian) shear production rate P, the turbulent transport rate T, rate of TKE generation or destruction by surface buoyancy forcing B, and the Stokes drift shear (Langmuir) production rate Q in flows with LC. Note that the budget terms sum to zero in all cases when the flows are in statistical equilibrium.

To better highlight key contrasting points between the different cases with Langmuir forcing and surface buoyancy, we focus the discussion on mean (Eulerian) shear production, Stokes drift shear production rate, and the rate of TKE generation by buoyancy. In the cases with Raτ = 0 and 211, in Fig. 12, it can be seen that in the near-surface region, mean shear and Stokes drift shear dominate TKE production over buoyancy. In the interior or core of the water column, mixing induced by the turbulence and associated Langmuir supercell diminishes mean shear and the latter becomes negative and thus a TKE sink. When surface cooling is applied with Raτ = 5000, the enhanced vertical mixing reaches closer to the surface and the mean shear diminishes, becoming negative at a faster rate with the distance from the surface than in the cases with Raτ = 0 and 211. This is consistent with Fig. 9b, showing mean shear for these cases. Meanwhile, TKE generation by buoyancy clearly dominates over production by the Stokes drift shear and mean shear, suggesting the supercell structure visible in the case with Raτ = 5000 is primarily driven by the cooling-induced buoyant motion.

Fig. 12.
Fig. 12.

Selected terms in the budgets of turbulent kinetic energy (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 0, 211, and 5000; denotes distance from surface in wall units.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Based on the implication that at some point between the Rayleigh numbers of 211 and 5000 a transition had occurred from Langmuir-driven to convective-driven turbulence, we can determine the point at which this transition occurred. This involved finding the value of the Rayleigh number for which buoyancy would overtake the Stokes drift shear in terms of being the greater source of turbulent kinetic energy. From Fig. 13 it can be seen that at a Rayleigh number of 1000, these sources are approximately equal in their contribution throughout the bulk of the water column, excluding the near-surface region (not shown). For any Rayleigh number above Raτ = 1000, convection will be the greater source of turbulent kinetic energy in the bulk portion of the water column.

Fig. 13.
Fig. 13.

Selected budgets of turbulent kinetic energy (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 500, 1000, and 1500; denotes distance from surface in wall units.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Interestingly, note that production by Stokes drift shear remains nearly unchanged for all cases in Fig. 13. The reason for this is that both components of this production term, and , remain nearly constant. The Stokes drift shear remains constant because the Stokes drift does not change as wind and wave forcing remains constant across all cases. Further, as described earlier, the homogenization of momentum in the core of the water column (away from the viscous surface and bottom sublayers) by the turbulence results in a near-zero viscous shear stress in the interior of the water column, leaving the Reynolds shear stress as the primary carrier of the wind stress and thus (in dimensional form) in the interior for all cases. In Fig. 13, as Raτ increases, buoyancy contribution to TKE generation increases, eventually surpassing the relative fixed threshold of the mean Stokes drift shear production. The previous observations suggest that a transition to convection-dominated turbulence could be expected at Raτ < 1000 for flows with lesser Stokes drift shear than in the present study and thus weaker Langmuir cells. Note that Raτ = 1000 corresponds to a surface cooling flux of Q = 560 W m−2, which is much higher than in typical ocean conditions. Thus, the transition to convection-dominated supercells would be expected to occur at more typical values of Q for weaker Langmuir cells than those studied here.

We consider these findings now in terms of the Hoenikker number (Li et al. 2005; Gargett and Grosch 2014), which is the ratio of the buoyant forcing that drives thermal convection to the vortex force that drives LC and is defined by
e11
where again β is the thermal expansion coefficient, g is the gravitational acceleration, Q > 0 is the surface heat flux, ρ0 is the density, Cp is the specific heat, and K is the wavenumber of the dominant surface waves. Our primary Rayleigh numbers of 211 and 5000 correspond to Hoenikker numbers of approximately 1 and 25, respectively. A Rayleigh number of 1000 generates Ho = 4.9, suggesting that when Ho > 4.9, the transition to convection-driven turbulence has occurred. This value lies comfortably within the approximate ranges identified by Li et al. (2005) and McWilliams et al. (1997) of Ho = O(1) and Ho > 1, respectively, for convection-dominated turbulence.
More recently, using field measurements in a shallow shelf region, Gargett and Grosch (2014) have investigated full-depth turbulence in shallow coastal regions under various combinations of wind forcing, Langmuir forcing, and surface buoyancy forcing due to surface cooling for varying Langmuir numbers La and Rayleigh numbers Ra. The Ra is a Rayleigh number defined in a different manner to the one used in the present study but still describing the importance of destabilizing buoyancy forces relative to shear forces, and La is a Langmuir number defined as ; the definition of Lat was provided earlier. The definition of the Rayleigh number by Gargett and Grosch (2014) is
e12
where again β is the coefficient of thermal expansion, k is the thermal conductivity, and Q is the heat flux. The time scale is defined as
e13
where dUs/dx3 and are the dimensional vertical gradients of Stokes drift and the mean streamwise velocity at the surface. For the dataset analyzed, Gargett and Grosch (2014) demonstrated that an archetype of full-depth convection without the effects of Langmuir forcing occurs for log(Ra) ≈ 8 and log(La) ≈ 5, while an archetype of full-depth Langmuir cells purely driven by Langmuir forcing occurs when log(Ra) ≈ 4.2 and log(La) ≈ −1.1. Using the definitions for Ra and La, we can calculate that the flow with LC with Raτ = 211 is characterized by log(Ra) = 5.58 and log(La) = −0.71, while the flow with LC with Raτ = 1000 possesses log(Ra) = 7.1 along with log(La) = −0.71. Within the full-depth Langmuir turbulence and convective turbulence archetypes established by Gargett and Grosch (2014), the LC case with Raτ = 1000 is a hybrid case driven partially by convection and partially by Langmuir forcing, whereas the case with Raτ = 211 is closer to the flow driven purely by Langmuir forcing. Finally, the case with LC with Raτ = 5000 is characterized by log(Ra) = 8.7 along with log(La) = −0.71. This case falls outside of the Ra and La range exhibited by the field measurements of Gargett and Grosch (2014) as expected, given that the surface heat flux for this case is unrealistically high (Q = 2810 W m−2).
(v) Budgets of resolved Reynolds shear stress
Next we can consider budgets of resolved Reynolds stress. The transport equations for the resolved Reynolds shear stress is
e14
where
e15
These terms represent the pressure transport rate A, the viscous diffusion rate D, the viscous dissipation rate ε, the SGS transport rate TSGS, the SGS dissipation rate εSGS, the pressure–strain redistribution (PSR), the mean (Langmuir) shear production rate P, the turbulent transport rate T, the buoyancy term B, and the Stokes drift shear (Langmuir) production Q.

We consider the Reynolds shear stress budgets in the upper half of the water column within the region between and from the surface to assess turbulent transport associated with the presence of the full-depth cells. It has been found that coherent cells such as Langmuir supercells can act as a nonlocal source for Reynolds shear stress (Kukulka et al. 2012; Sinha et al. 2015); hence, we can compare the contributing sources to the Reynolds shear stress and particularly turbulent (nonlocal) transport in cases with and without surface buoyancy forcing. In the Reynolds shear stress budgets seen in Fig. 14, increasing the Rayleigh number up to Raτ = 5000 significantly increases turbulent transport as a source serving to partially balance the increasing sink of Reynolds shear stress caused by negative mean shear. The latter is important to recognize because it implies that a hybrid Langmuir–convection turbulence closure for Reynolds shear stress in shallow water (suitable for coarse-scale general circulation models) would need to incorporate a parameterization of turbulent (nonlocal) transport. Note that even for the case with Raτ = 0, we have shown in Sinha et al. (2015) that in shallow-water Langmuir turbulence, the role of turbulent transport of the Reynolds shear stress is significant compared to its role in turbulence driven purely by mean shear, where the turbulent transport is negligible. And furthermore, in Sinha et al. (2015), we have shown that in order to account for this increased role of turbulent transport associated with shallow-water Langmuir turbulence, a closure of the Reynolds shear stress should contain a nonlocal transport term in addition to the usual local term based on transport down local gradients. The present Langmuir turbulence results with cooling at Raτ = 5000 show that the importance of turbulent transport and thus the nonlocal term in a Reynolds shear stress closure can become even greater with the addition of surface cooling. The implication is that a more general closure [than that in Sinha et al. (2015)] should be developed that takes into account the combined effects of Langmuir turbulence and convective turbulence on nonlocal (turbulent) transport.

Fig. 14.
Fig. 14.

Selected budgets of Reynolds shear stress (scaled by ) in the upper half of the water column in flows with Langmuir forcing with Raτ = 0, 211, and 5000; denotes distance from surface in wall units.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

b. Surface heating

Next, the results of wind-driven flows with LC and surface heating flux are presented. Recall that the strength of the stabilizing buoyancy force associated with the surface heating flux with respect to the wind-driven shear is measured through the Richardson number specified in (5), where Q > 0 is now the surface heating flux. Note that in the Riτ = 0 case, the fluid is warmed at the surface but the temperature acts as a passive scalar and thus the flow is calculated without surface buoyancy forcing.

1) Mean velocity profiles

The mean downwind velocity profile is roughly uniform throughout the full depth of the water column in the cases with Riτ = 0 and 100 because of the high level of vertical mixing induced by the LC (Fig. 15a). When the Richardson number increases to 500, notice that the majority of the upper half of the water column still shows a near-homogeneous velocity profile similar to that of the Riτ = 0, 100 cases despite stronger heating intensity. However, in the lower half of the water column, observe that this velocity profile attains a positive gradient similar to that of wind-driven flow without LC and without surface buoyancy forcing, suggesting vertical mixing and associated velocity homogenization is no longer reaching full depth. This behavior in velocity can also be seen in terms of velocity shear in Fig. 15b. In the upper half of the water column, the mean velocity shear in the Riτ = 500 case trends toward the Riτ = 0, 100 cases, although not reaching negative values as in the latter two. Meanwhile in the lower half of the water column, the mean velocity shear in the Riτ = 500 case trends toward the wind-driven case without LC and no surface buoyancy forcing (i.e., the wind-only case).

Fig. 15.
Fig. 15.

(a) Mean downwind velocity profile as a function of depth, (b) mean velocity shear, (c) mean velocity near the surface, and (d) mean velocity near wall in flows with Langmuir forcing and varying Richardson numbers. The wind-only case corresponds to wind-driven flow without LC and no surface heat flux.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

Figure 15c shows the mean velocity near the surface. The log-law profile is indicated in the figure by the thin black line and represents the behavior of a typical turbulent flow underneath a wind-driven surface; it is clear to see that a wind-driven case without LC or surface buoyancy forcing closely follows this expected profile. In all flows with LC we see the disruption of the classical log-law velocity profile. When heating with Riτ = 500 is applied, this disruption is least among the various cases analyzed, yet note for future discussion this disruption is still present and significant.

Figure 15d shows the mean velocity in terms of wall units at the lower wall. The log-law profile is again indicated in the upper half of the figure by the narrow black line and represents the behavior of a typical turbulent flow over a no-slip boundary. The flows in which LC is present with Riτ = 0, 25, and 100 demonstrate a disruption of the classical log-law profile due to the increased vertical mixing within the water column caused by the LC. When the stronger heating is applied (at Riτ = 500), the flow velocity profile is again characterized by the well-developed, log-law profile, indicative of the weakening of the vertical mixing causing the disruption. This suggests that at a Richardson number of 500, the full-depth mixing caused by the Langmuir cells is inhibited to a sufficient extent that the flow behavior in the near-wall log layer is similar to a case without LC present.

The interesting thing to note here is that although there is some inhibition of the strength of vertical mixing in the upper half of the water column (indicated by the decrease in turbulence-induced disruption of the boundary layer), this inhibition is limited and the Langmuir-induced mixing is still present. On the contrary, in the bottom half of the water column, these mean velocity profiles would suggest that the Langmuir turbulence has been almost completely diminished at Riτ = 500, as the flow behavior in the bottom log region is similar to that in a wind-driven flow without Langmuir and surface buoyancy forcing (i.e., the wind-only case). The implication here is that the application of a stable density gradient through surface heating is able to limit the depth to which mixing from Langmuir supercells is able to penetrate within the water column.

2) Temperature profiles

We next consider the mean temperature profile throughout the water column and also the temperature gradients for each of the values of the Richardson number simulated (see Fig. 16). For the flow with LC with Riτ = 0, the temperature is a passive scalar, and the buoyancy force in the momentum equations is zero. Thus, the temperature distribution in the Riτ = 0 case is controlled by LC, and it has the shape of the mean velocity profile seen in Fig. 9a for the case with Raτ = 0 (equivalent to Riτ = 0). With Riτ = 100, there is a surface buoyancy force but Fig. 16a demonstrates that it is a minor contributor to the force balance, as the temperature profiles for Riτ = 100 and Riτ = 0 are close. As the surface heat flux increases (e.g., up to Riτ = 500) significant changes can be observed. For example, the thickness of the surface temperature boundary layer correspondingly increases (Fig. 16a), with the region of the higher temperature gradient extending to a greater depth from the surface (Fig. 16b). Considering the case with greatest heating (Riτ = 500), in Fig. 16a we can see a significant positive gradient is present throughout the lower half of the column. Thus, a stable temperature stratification is prevalent here, as confirmed in Fig. 16b, unlike in the other cases with lower Riτ.

Fig. 16.
Fig. 16.

(a) Mean temperature and (b) temperature gradient throughout full water column depth in flows with Langmuir forcing and varying Richardson numbers. Note that .

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

3) Full-depth turbulent structures

We consider vertical velocity fluctuations averaged in time and over the downwind direction in order to see more clearly the impact of an increasing Richardson number on the upwelling and downwelling limbs of full-depth Langmuir cells by plotting these averaged fluctuations over the cross-stream and vertical directions (Fig. 17). As previously, the flow with Riτ = 0 (no surface buoyancy forcing) demonstrates the presence of full-depth regions of negative and positive vertical velocity fluctuations representing the downwelling and upwelling limbs of the resolved Langmuir supercell. In the case with Riτ = 100 (Fig. 17), little difference is seen in the case with Riτ = 0, but as the Richardson number increases to Riτ = 500, we can clearly see that the intensity of the upwelling and downwelling limbs is diminished especially in the lower half of the water column. Thus, the turbulence has been significantly inhibited near the lower wall. This supports the implication made through previous analysis of the mean velocity and temperature profiles in the upper and lower boundary layers that the application of stable stratification by surface heating is able to limit the depth to which full-depth LC is able to generate strong mixing.

Fig. 17.
Fig. 17.

Downwind- and time-averaged vertical velocity fluctuations for flows with Langmuir forcing with Riτ = 0, 100, and 500.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

4) RMS profiles

Returning to profiles of the RMS of velocity fluctuations (Figs. 18a,b,c), in the horizontal directions (downwind and crosswind), the influence of the surface heat flux at a Richardson number of 500 is substantial. The downwind velocity RMS increases with the Richardson number, consistent with the increase of mean shear caused by the surface heating. In the lower wall region, at Riτ = 500, the stable stratification is seen to suppress the crosswind RMS, consistent with the weakening of the crosswind velocity fluctuations near the lower wall, which identify the bottom divergence zone of the LC as seen through the partially averaged crosswind velocity fluctuations in Fig. 5. However, in the case with Riτ = 500, at the surface the characteristic patterns of persistent Langmuir cell structures are still present. Most noticeably the increase in the crosswind RMS relative to the middle of the water column corresponding to the presence of strong surface convergence and divergence zones of the LC is still present, albeit at a lower magnitude. In Fig. 18c, at a Richardson number of 500, we can see a substantial decrease in the vertical velocity RMS. Again, as the surface heat flux inhibits the strength and coherence of the Langmuir cells, the full-depth upwelling and downwelling limbs of the cells become weaker, especially in the middle of the water column, leading to the decrease in vertical velocity RMS observed here. Finally, as seen in Fig. 18d, the effective Reynolds shear stress is nearly the same for all cases. A weakened Langmuir turbulence by surface heating is offset by shear turbulence becoming more dominant, and thus the effective Reynolds shear stress remains nearly the same across all cases.

Fig. 18.
Fig. 18.

RMS of velocity and effective Reynolds shear stress in flows with Langmuir forcing and varying Richardson numbers.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

5) Budgets of turbulent kinetic energy

Figure 19 shows selected terms of the budgets of turbulent kinetic energy for the region in the bulk of the water column (−1.8 < x3/δ < −0.2, corresponding to approximately 100 < < 690 from the lower wall). Here, the most significant differences between the flows with and without the application of a surface heating flux can be observed. The most noteworthy observation to be made is the difference in distribution of turbulent transport relative to other source terms. In the upper half of the water column, turbulent transport is a significant source attaining values close to those of production by Stokes drift shear in the Riτ = 0 and 100 cases and even surpassing the values of production by Stokes drift shear in the Riτ = 500 case. In the lower half of the water column, turbulent transport remains a significant source for the Riτ = 0 and 100 cases but not for the Riτ = 500 case. In the latter case, in the lower half of the water column, turbulent transport becomes much less than production by mean shear, consistent with the earlier finding that an increasing surface heat flux inhibits the depth to which mixing by LC reaches.

Fig. 19.
Fig. 19.

Selected budgets of turbulent kinetic energy (scaled by ) in bulk of water column in flows with Langmuir forcing with Riτ = 0, 100, and 500; denotes distance from bottom in wall units.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0168.1

5. Discussion and conclusions

We have analyzed diagnostics of flows with full-depth Langmuir circulation and both cooling and heating surface heat fluxes to assess their corresponding contributions to either the strength or inhibition of full-depth mixing.

It was found that although the overall cell features remain intact, increasing surface cooling and thus increasing the Rayleigh number when applied to a fully mixed water column in the presence of Langmuir forcing can lead to strengthening of full-depth structures. For the wind and wave (Langmuir) forcing conditions studied [i.e., those of Gargett et al. (2004) and Gargett and Wells (2007)] at a Rayleigh number of 5000, the increase in cell strength with respect to the case with zero surface buoyancy forcing (Raτ = 0) is sufficient to dramatically alter the turbulence dynamics, with buoyancy becoming the dominant source of TKE. However, at a Rayleigh number of 211 representing a moderate surface cooling flux of approximately 120 W m−2, despite the tendency of convection to create a similar structure to full-depth LC, it is unable to overtake Langmuir forcing (i.e., production by Stokes drift shear) as the dominant mechanism in turbulence generation, and Langmuir forcing remains the primary cause of the observed full-depth cell structures.

The present simulations have been motivated by the field measurements of Gargett et al. (2004) and Gargett and Wells (2007) of full-depth Langmuir cells in a shallow-water region under surface heat fluxes ranging from approximately Q = 150 W m−2 for ocean heat loss (surface cooling) to Q = 200 W m−2 for ocean heat gain (surface heating). Gargett et al. (2004) and Gargett and Wells (2007) reported no significant changes in the structure of the full-depth Langmuir cells under these heating and cooling surface heat fluxes. The present LES of full-depth Langmuir cells [with wind and wave forcing obtained from the field measurements of Gargett et al. (2004) and Gargett and Wells (2007)] showed no significant differences between the Raτ = 0 case and the cases with Raτ = 211 (surface cooling at Q = 120 W m−2) and Riτ = 100 (surface heating at Q = 56 W m−2) and thus have further confirmed their field measurements. The LES results with Raτ = 0, 211, and 1000 (surface cooling at Q = 560 W m−2) are also consistent with the more recent shallow-water field measurements of Scully et al. (2015) and Scully (2014) of Langmuir cells with Laτ = 0.38 under surface cooling up through Q = 409 W m−2. For example, Scully (2014) concludes that the surface cooling is able to enhance the Langmuir cells but that “LC is the dominant mechanism driving coherent low frequent motions.”

In the present study, simulations with higher heat fluxes, such that Raτ = 5000 (Q = 2810 W m−2) and Riτ = 500 (Q = 281 W m−2), have been performed. Although the case Raτ = 5000 is characterized by an unrealistic high heat flux for the particular flow problem investigated [i.e., with the Langmuir (wind and wave) forcing conditions of Gargett et al. (2004) and Gargett and Wells (2007)], this case highlights how the destabilizing surface heat flux and associated convective forcing may significantly alter the structure of existing Langmuir cells while becoming the dominant source of turbulent kinetic energy (TKE) generation over Stokes drift shear (Langmuir) production and mean (Eulerian) shear production. Additional simulations showed that for Raτ greater than approximately 1000 (Q = 560 W m−2), buoyancy is able to overtake production by Stokes drift shear in the production of TKE, signifying a transition from Langmuir-dominated turbulence to convection-dominated turbulence. Note that Raτ may increase not solely by increasing Q, but also by decreasing wind speed and associated wind stress friction velocity uτ [see (5)]. Thus, milder wind conditions than those of Gargett et al. (2004) and Gargett and Wells (2007) could yield a regime in which forcing by convection becomes greater than Langmuir forcing at lower values of Q (i.e., lower than Q = 560 W m−2). This situation would arise in the presence of weaker Langmuir cells than those analyzed here due to the lower wind speed. This along with the significant contribution of buoyant forces to the turbulence dynamics and cell structure in some of the cases investigated here implies there is potential for cells found in the field to be hybrids, with the turbulence observed being generated by a combination of Langmuir and convective forcing as also described by Gargett and Grosch (2014).

For the surface heating case, such that Riτ = 5000, the surface heating significantly weakens the cells in the lower portion of the water column. Consequently, the cells are not able to generate as much mixing in the near-wall region as the cases with lower Riτ, thereby allowing a pycnocline to develop in the lower half of the water column. We hypothesize that in these simulations, a sufficiently high surface heating flux (such that Riτ > 500) could impede the penetration of the Langmuir cells (rather than simply weakening the cells in the lower portion of the water column) and in the process lead to cells of smaller crosswind width, as reported by Min and Noh (2004), for the upper-ocean mixed layer.

Finally, analysis of TKE and Reynolds shear stress budgets showed that turbulent (nonlocal) transport becomes a significant contributor during instances of active full-depth supercells generated by Langmuir forcing and strengthened by convection (surface cooling). Thus, turbulent transport closures for TKE and Reynolds shear stress transport equations need to be developed that account for the combined effects of Langmuir turbulence and convective turbulence. In the presence of full-depth supercells, mean shear diminishes and can become negative, causing mean shear to act as a sink in the budgets of TKE and Reynolds shear stress production. Thus, classical closures of the Reynolds shear stress in terms of the vertical gradient of mean velocity (i.e., mean shear) need to be supplemented with a nonlocal transport term accounting for the nonlocal transport induced by the supercells driven under the combination of Langmuir forcing and surface cooling.

Acknowledgments

The authors were partially supported through National Science Foundation Awards 0846510, 0927724, and 0927054. This research was also made possible by a grant from The Gulf of Mexico Research Initiative. Data are publicly available through the Gulf of Mexico Research Initiative Information & Data Cooperative (GRIIDC) [https://data.gulfresearchinitiative.org (doi:10.7266/N7736NXV)].

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