Journal of Oceanology and Limnology   2023, Vol. 41 issue(1): 17-37     PDF       
http://dx.doi.org/10.1007/s00343-022-1123-4
Institute of Oceanology, Chinese Academy of Sciences
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Article Information

LI Guojing, DONG Changming, PAN Jiayi, DEVLIN Adam T., WANG Dongxiao
Influence of the upper mixed layer depth on Langmuir turbulence characteristics
Journal of Oceanology and Limnology, 41(1): 17-37
http://dx.doi.org/10.1007/s00343-022-1123-4

Article History

Received Apr. 21, 2021
accepted in principle Jul. 18, 2021
accepted for publication Jan. 12, 2022
Influence of the upper mixed layer depth on Langmuir turbulence characteristics
Guojing LI1,4, Changming DONG2, Jiayi PAN3, Adam T. DEVLIN3, Dongxiao WANG1     
1 State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou 510301, China;
2 Nanjing University of Information Science and Technology, Nanjing 210044, China;
3 Institute of Space and Earth Information Science, the Chinese University of Hong Kong, Hong Kong 999077, China;
4 Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou), Guangzhou 510301, China
Abstract: The upper mixed layer depth (h) has a significant seasonal variation in the real ocean and the low-order statistics of Langmuir turbulence are dramatically influenced by the upper mixed layer depth. To explore the influence of the upper mixed layer depth on Langmuir turbulence under the condition of the wind and wave equilibrium, the changes of Langmuir turbulence characteristics with the idealized variation of the upper mixed layer depth from very shallow (h=5 m) to deep enough (h=40 m) are studied using a non-hydrostatic large eddy simulation model. The simulation results show that there is a direct entrainment depth induced by Langmuir turbulence (hLT) within the thermocline. The normalized depthaveraged vertical velocity variance is smaller and larger than the downwind velocity variance for the ratio of the upper mixed layer to a direct entrainment depth induced by Langmuir turbulence h/hLT < 1 and h/hLT> 1, respectively, indicating that turbulence characteristics have the essential change (i.e., depth-averaged vertical velocity variance (DAVV) < depth-averaged downwind velocity variance (DADV) for shear turbulence and DAVV>DADV for Langmuir turbulence) between h/hLT < 1 and h/hLT>1. The rate of change of the normalized depth-averaged low-order statistics for h/hLT < 1 is much larger than that for h/hLT>1. The reason is that the downward pressure perturbation induced by Langmuir cells is strongly inhibited by the upward reactive force of the strong stratified thermocline for h/hLT < 1 and the effect of upward reactive force on the downward pressure perturbation becomes weak for h/hLT>1. Hence, the upper mixed layer depth has significant influences on Langmuir turbulence characteristics.
Keywords: the upper mixed layer depth    Langmuir turbulence    turbulent characteristics    large eddy simulation    
1 INTRODUCTION

Langmuir turbulence, which is induced by wave-current interactions, forms strong coherent structures and downwelling jets. It is a turbulent process which plays a key role in the turbulent mixing of the upper mixed layer (Langmuir, 1938; Skyllingstad and Denbo, 1995; Li and Garrett, 1997; McWilliams et al., 1997; Min and Noh, 2004; Polton and Belcher, 2007; Polton et al., 2008; Kukulka et al., 2013; D'Asaro, 2014; Gargett and Grosch, 2014; Yang et al., 2014; Hoecker-Martínez et al., 2016; Smyth et al., 2017). To date, the dynamic mechanism of Langmuir turbulence effects on the upper mixed layer evolution is still unclear (McWilliams et al., 2012; Basovich, 2014; Furuichi and Hibiya, 2015) and thus is not well represented in most ocean general circulation models (OGCMs) (Thorpe, 2004; Sullivan et al., 2007, 2012). The lack of influence of Langmuir turbulence on the upper mixed layer variation is one reason why the upper mixed layer depth and sea surface temperature simulated by OGCMs are generally shallower (90%–100%) and warmer (3–4 ℃), than observations in the Southern Ocean (De Boyer Montégut et al., 2004; Belcher et al., 2012). Hence, Belcher et al. (2012) suggested that Langmuir turbulence requires independent parameterization in terms of its formation mechanism, that is, wave-current interactions.

Previous studies mainly focus on Langmuir turbulence characteristics for the different wind and wave fields with a deep upper mixed layer by large eddy simulation model. The related results reveal several important characteristics consistent with observations, such as the vertical velocity structure, turbulent kinetic energy, temporal scale of Langmuir turbulence development, cold water spots in the upper mixed layer, and the deepening of the upper mixed layer (Li et al., 2009; Kukulka et al., 2010; Van Roekel et al., 2012; Gargett and Grosch, 2014). In addition, Langmuir turbulence parameterization under a deep upper mixed layer assumption was previously proposed (McWilliams and Sullivan, 2000; Smyth et al., 2002; Noh et al., 2011, 2016; Harcourt, 2013, 2015; Sutherland et al., 2014; Furuichi and Hibiya, 2015). However, when these parameterizations are incorporated into the OGCMs, the simulated upper mixed layer deepening by the OGCMs is still different from observations (Fan and Griffies, 2014; Noh et al., 2016). Observations also indicate that there is a lack of evidence to support the fact that Langmuir turbulence can directly cause the deepening of the upper mixed layer when its depth is 40–60 m (Weller and Price, 1988; Skyllingstad, 2000). Hence, these studies suggest that Langmuir turbulence parameterizations are incomplete, if they are only based on that the assumption of the upper mixed layer is deep.

Variation of the upper mixed layer depth in the real ocean is significant with the season (Belcher et al., 2012) and the low-order statics of Langmuir turbulence are a function of the upper mixed layer depth (Wijesekera et al., 2013). The effect of Langmuir turbulence on the shallow upper mixed layer has been also brought into focus (Kukulka et al., 2010; Noh et al., 2011). Kukulka et al. (2010) and Noh et al. (2011) explored the shallow upper mixed layer (i.e., h=20 m and 5 m, respectively) deepening caused by Langmuir turbulence. They analyzed the dynamic mechanism from the perspective of the turbulence kinetic energy budget, vertical velocity fields, Kelvin-Helmholtz shear instability, velocity shear, and eddy viscosity. However, they did not investigate how the variation in Langmuir turbulence characteristics changes as the upper mixed layer depth varies from shallow to deep. Through the use of large eddy simulation model, this paper presents in-depth investigations on the effects of the upper mixed layer depth on Langmuir turbulence characteristics. Our results can provide crucial information to improve Langmuir turbulence parameterization in the future modelling efforts.

Here, we focus on the influence of the upper mixed layer depth on Langmuir turbulence characteristics under the ideal conditions. We found that a direct entrainment depth of Langmuir turbulence in the interior of the thermocline (i.e., the depth of the zero of entrainment flux in the interior of thermocline) exists based on the wind and wave fields used in this study. In addition, the entrainment depth of Langmuir turbulence is proportional to the Ekman depth scale (u*/f, where u* is the wind-induced ocean surface friction velocity and f is the Coriolis force) when the thermocline appears below the upper mixed layer as suggested by Noh and Choi (2018) and the penetrateddepth of Langmuir turbulence with the different wind and wave fields (or turbulence Langmuir number) can reach the Ekman depth scale in an unstratified mixed layer (Polton and Belcher, 2007). But, a variation of the direct entrainment depth in the thermocline with changes in wind and wave fields and Coriolis force is not the topic of this paper. In our simulation cases, the stratification of the thermocline is set at a rate of dT/dz=10 K/m based on our many numerical experiments. For dT/dz=10 K/m in the thermocline, on one hand, the evident entrainment flux directly induced by Langmuir turbulence can appear in the interior of the thermocline and the upper mixed layer depth is unchanged during our integration time of 56.7 h, when the upper mixed layer depth is shallower than the direct entrainment depth of Langmuir turbulence. However, on the other hand, the entrainment flux from the thermocline is insignificant and the upper mixed layer depth is unchanged, when the upper mixed layer is deeper than the direct entrainment depth of Langmuir turbulence. Hence, the upper mixed layer is unchanged for every independent simulation case in our studies. Two scenarios of the upper mixed layer depth are considered in this study: first, the upper mixed layer depth is shallower than the direct entrainment depth of Langmuir turbulence and second, the upper mixed layer depth is deeper than the direct entrainment depth of Langmuir turbulence. We explore how the upper mixed layer depth variation affects the vertical velocity, horizontal velocity, vertical velocity skewness, vertical momentum flux, eddy viscosity, velocity shear, and turbulent kinetic energy budget generated by Langmuir turbulence. In addition, to clarify how these terms vary with changes in the upper mixed layer depth, we analyze the variations of the depth-averaged velocity variances, horizontal velocity, vertical velocity skewness, vertical momentum flux, eddy viscosity, velocity shear, and turbulent kinetic energy budget as suggested by Skyllingstad (2000), Li et al. (2005), Harcourt and D'Asaro (2008), and Wijesekera et al. (2013).

In Section 2, the large eddy simulation model and idealized experimental configuration are introduced. Section 3 analyzes the response of Langmuir turbulence to the upper mixed layer depth in terms of the horizontal velocity, vertical velocity skewness, vertical momentum flux, eddy viscosity, velocity shear, and turbulent kinetic energy budget. Section 4 presents discussion, and Section 5 gives the conclusions.

2 LARGE EDDY SIMULATION MODEL AND IDEAL EXPERIMENT SETUP

The large eddy simulation model used in this study is developed by McWilliams et al. (1997), McWilliams and Sullivan (2000), Sullivan et al. (2007), and Sullivan and McWilliams (2019). The Langmuir turbulence is modeled by the vortex force (Craik, 1977; Leibovich, 1977) (fig. 15 of Sullivan et al., 2007). Our model satisfies the following set of equations:

    (1)
    (2)
    (3)
    (4)
Fig.1 Snapshot of horizontal-cross-section of the normalized vertical velocity (w) fields at z=-0.5 m of h=5 m (a) and h=15 m (b) at t=31.2 h
Fig.2 Snapshot of distribution of the normalized three-dimensional vertical velocity (w) fields of h=5 m (a) and h=15 m (b) for t=31.2 h The black lines indicate the upper mixed layer depth.
Fig.3 Vertical profiles of mean temperature (〈T〉) normalized by the reference temperature Tr (=273 K) as suggested by Sullivan et al. (2007) (a) and mean turbulent entrainment flux (〈wʹTʹ〉) normalized by the positive surface value Q* (=1.2×10-6 (K·m)/s) (b) The horizontal black solid line denotes the depth of the significant temperature variation in the interior of the thermocline and turbulent entrainment flux in the thermocline.
Fig.4 Vertical profiles of the normalized mean velocity shear (〈S2〉) (a) and vertical velocity variance (〈2〉) (b) The inset in (b) is vertical velocity variance (〈2〉) in the internal region of -15 m < z < -13 m. The black horizontal solid line denotes the depth of the ignificant vertical velocity variance in the thermocline.
Fig.5 Comparison of the normalized depth-averaged velocity variance in three directions with a variation of h/hLT w'2 < u'2 for h/hLT < 1 and w'2>u'2 for h/hLT>1. The black orizontal solid line denotes h/hLT=1.
Fig.6 Vertical profiles of the normalized mean downwind velocity (〈u〉+us) (a) and mean crosswind velocity (〈v〉) (b) The vertical solid line denotes a zero value.
Fig.7 Variation in the normalized depth-averaged downwind velocity (u〉+us) (a) and crosswind velocity (v) (b) with a change of h/hLT The black horizontal solid line denotes h/hLT=1.
Fig.8 Vertical profiles of the mean vertical velocity skewness (〈w3〉/〈w23/2) (a) and variation in the depth-averaged vertical velocity skewness (w3/w23/2) with a change of h/hLT (b) The black horizontal solid line denotes the direct entrainment depth of Langmuir turbulence h/hLT=1.
Fig.9 Vertical profiles of the normalized mean downwind vertical momentum flux (〈uʹwʹ〉) (a) and mean crosswind vertical momentum flux (〈vʹwʹ〉) (b) The inset in (a) is vertical profiles of the normalized mean downwind vertical momentum flux (〈uʹwʹ〉) plotted by z/h.
Fig.10 Variation in the normalized depth-averaged vertical momentum fluxes in the downwind direction (uʹwʹ) (a) and crosswind direction (vʹwʹ) (b) as the h/hLT changes from h/hLT < 1 to h/hLT>1; variation in uʹwʹ/vʹwʹ as a function of the h/hLT (c) The black horizontal solid line denotes h/hLT=1.
Fig.11 Vertical profiles of the normalized mean eddy viscosity (〈Km〉) (a) and mean velocity shear (〈S2〉) (b)
Fig.12 Variation of the normalized depth-averaged mean eddy viscosity (Km) with a variation of h/hLT (a); variation of the normalized depth-averaged velocity shear (S2) with a variation of h/hLT (b) The black horizontal solid line denotes h/hLT=1.
Fig.13 Vertical profiles of the normalized total turbulent kinetic energy (〈E〉) (a), transport production (〈F〉) (b), shear production (〈PS〉) (c), Langmuir production (〈PL〉) (d), dissipation rate of turbulent kinetic energy (〈ε〉) (e), and buoyancy production (〈Pb〉) (f) The inset in (f) is the value of the normalized 〈Pb〉 plotted by z/h in the range of -3 < z/h < 0 for h=5 m.
Fig.14 Variation in the normalized depth-averaged total turbulent kinetic energy (E) (a), transport production (F) (b), shear production (PS) (c), Langmuir production (PL) (d), dissipation rate of turbulent kinetic energy (ε) (e), and buoyancy production (Pb) (f) as the h/hLT changes from h/hLT < 1 to h/hLT>1 The black horizontal solid line denotes h/hLT= 1.
Fig.15 Sketch illustrating the influence of the upper mixed layer depth (h) on Langmuir turbulence The δS is Stokes depth scale and the f is Coriolis force. (a), (b) and (c) represent the evolution of Langmuir turbulence as the upper mixed layer depth changes from shallower to deeper than the direct entrainment depth of Langmuir turbulence (i.e., the suppression effect of the upper mixed layer depth on the downwelling jets or the upward reactive force of the strong stratified thermocline to the downward pressure perturbation gradually disappears as the upper mixed layer depth increases).

where xi (i=1, 2, 3) are the Cartesian coordinates, ui (i=1, 2, 3) are the resolved velocity components in the xi directions, t is time, fk is the Coriolis parameter, ωk is the vorticity component, usj is the Stokes drift velocity, P=p/ρ0+2e/3+1/2[(ui+usi)2uiui] is the modified pressure, p is the pressure, e is the turbulent kinetic energy of the sub-grid-scale, τij=-2νt Sij is the momentum flux of the sub-grid-scale, νt is the turbulent eddy viscosity, is the strain tensor of the resolved velocities, ξijk is the standard antisymmetric tensor, δi3 is the Kronecker delta, ρ0 is the reference density, ρ=ρ0(1–αT), α is the thermal expansion coefficient, g is the acceleration due to gravity, SSGS=-τijSij is the shear production of the sub-grid-scale, is the buoyancy production of the sub-grid-scale, T0 is the reference temperature, ε=0.93e3/2/Δ (, where Δx, Δy, Δz, are the grid spacings) is the dissipation rate, is the diffusion production of the sub-grid-scale, T is the resolved temperature, is the heat flux of the sub-grid-scale, and νT is the turbulent eddy diffusivity. The νt=0.1le1/2 and terms are as suggested by Moeng (1984) and Sullivan et al. (1994), where l=Δ within the upper mixed layer and in the thermocline.

The ocean surface friction velocity (u*) is calculated based on the steady wind velocity (Liu et al., 1979) given by

    (5)
    (6)

where τa is the wind stress at the sea surface, ρo is the ocean water density, ρa is the air density, Ua is the wind velocity (z=10 m) and Cd=1.3×10-3 is the drag coefficient for Ua≤10 m/s.

The Stokes drift velocity (us) based on a full wave spectrum for a steady wind (Kenyon, 1969; McWilliams and Restrepo, 1999; Sullivan et al., 2007; Li and Fox-Kemper, 2017) is computed by

    (7)

where , , , fo=0.275 and νo=1.40 (Kenyon, 1969; Sullivan et al., 2007), n=2 (McWilliams and Restrepo, 1999).

Similar to previous studies (Sullivan et al., 2007; Noh et al., 2009; McWilliams et al., 2012), the wave fields are aligned with the wind direction. Following Grant and Belcher (2009) and Noh et al. (2011), the Coriolis parameter (f), the ocean surface friction velocity (u*), the ocean surface Stokes drift velocity (uos) and the Stokes depth scale (δs) are consistent for every simulation cases (Table 1). In order to normalize the entrainment flux, an invariable, small and positive kinematic heat flux (Q*) is imposed at the ocean surface (the positive heat flux means a heat flux out ocean) (Table 1). The turbulent Langmuir number (McWilliams et al., 1997) is in the Langmuir turbulence regime (Li et al., 2005; Sutherland et al., 2014). In addition, the surface layer (SL) Langmuir number (uSL is the average Stokes drift velocity in a surface layer 0>z>-0.2h) (Harcourt and D'Asaro, 2008) is used to understand the evolution of Langmuir turbulence characteristics with variation in the upper mixed layer depth, which is also analyzed in this paper.

Table 1 The parameters used in the eight simulation cases

The simulation experiments are not crafted as a case study of the evolution of Langmuir turbulence characteristics with the upper mixed layer depth deepening, but rather as the idealization designed to explore the changes of Langmuir turbulence characteristics at the different upper mixed layer depth. In order to demonstrate the influence of the upper mixed layer depth (h) on Langmuir turbulence characteristics, the numerical experiments are carried out with the different initial upper mixed layer depth as shown in Table 1. Sullivan et al. (2012) suggested that the wave effects will impact entrainment when δs/h>0.75, but the δs/h≤0.6 for all our simulation cases indicates that the wave effects do not directly impact entrainment.

Neutral stratification is assumed from the sea surface to the base of the upper mixed layer, and the initial temperature in the upper mixed layer is Tr=273 K (Sullivan et al., 2007). The thermocline is stably stratified at a rate of dT/dz=10 K/m, which can strongly inhibit the upper mixed layer deepening created by shear instability (McWilliams et al., 2014) and make the evident entrainment flux created by Langmuir turbulence appear in the interior of the thermocline when the upper mixed layer depth is shallower than the direct entrainment depth of Langmuir turbulence. Hence, for each independent experiment, the upper mixed layer depth is almost unchanged during our simulation time of 56.7 h, and the change of the entrainment flux in the thermocline is clearly distinguished with a variation of the upper mixed layer depth. Thus, these experiments can clearly demonstrate how Langmuir turbulence characteristics respond to the upper mixed layer depth and the direct entrainment depth of Langmuir turbulence through a fundamental difference of the entrainment flux and vertical velocity variance in the interior of the thermocline.

The simulation domain is 256 m×256 m in the horizontal direction. The number of grid points is 512×512, and the corresponding grid spacing is 0.5 m in the horizontal direction. The depth of the simulation domain is 32 m for h=5, 10, 15, 20, and 25 m and it is 64 m for h=30, 35, and 40 m, respectively. The grid spacing is 0.25 m in the vertical direction. The flux and energy carrying large-scale eddies in the upper mixed layer can be adequately resolved by the defined spatial resolution (McWilliams et al., 1997; Noh et al., 2004). A periodic condition is applied at lateral boundaries, radiation and free-slip conditions are implemented at the bottom boundary, and free-slip condition is used at the top boundary. The pseudo-spectral spatial discretization is implemented in the horizontal direction and a second-order finite differences in the vertical direction. The time integration at every step uses a third-order Runge-Kutta scheme (Moeng and Wyngaard, 1988; McWilliams et al., 1997, 2014; Sullivan and McWilliams, 2018). The details of the numerical algorithm can be found in Moeng (1984), Moeng and Wyngaard (1988), Sullivan et al. (1996), and Sullivan and Patton (2011).The ocean surface friction velocity (u*) is used to normalize the simulated physical quantities, and to allow comparison between the related quantities (Min and Noh, 2004; Sullivan et al., 2007; McWilliams et al., 2012). In addition, one might argue that the unnormalized physical quantities can also be used to study Langmuir turbulence as in previous studies (Grant and Belcher, 2009; Li et al., 2009; Kukulka et al., 2010, 2013; Noh et al., 2011; Van Roekel et al., 2012). However, the unnormalized physical quantities in these studies are mainly used to analyze the observations (Li et al., 2009; Kukulka et al., 2010, 2013; Noh et al., 2011), the variations in the turbulent Richardson number (Noh et al., 2011) and Langmuir cells (Van Roekel et al., 2012).

The initial fields are static in the all simulation cases. The lower-order statistics are generated using the spatial average in the x-y plane for every integral time step, except for the vertical velocity field. The solution reaches a statistical equilibrium state after one inertial period Tip=2π/f≈14 h (McWilliams et al., 2012). In order to eliminate the inertial oscillation, the lower order statistics are averaged over a range from t=28 h to 42 h, which corresponds to the third inertial period. The horizontal and temporal averaged value of the lower-order statistics is denoted by the angle bracket (〈•〉, the • represents the arbitrary physical quantity). The lower order statistics of Langmuir turbulence have diverse scales and are associated with the different upper mixed layer depths (Wijesekera et al., 2013). So, the lower-order statistics are averaged within the upper mixed layer as suggested by Skyllingstad (2000), Li et al. (2005), Harcourt and D'Asaro (2008), and Wijesekera et al. (2013). The depth-averaged value of the lower-order statistics (method shown in Appendix) is denoted by an overbar (·). Variation in the lower-order statistics is not evident when they are averaged during the fourth period.

3 RESULT 3.1 The entrainment depth in the thermocline

Figure 1 shows the snapshot of horizontal-cross-section of the vertical velocity (w) fields of Langmuir turbulence at z=-0.5 m for h=5 m and 15 m. The vertical velocity fields are characterized by the clear coherent structures of Langmuir cells and many Y-joints, which are consistent with the observations from Thorpe (2004) and Yang et al. (2014) and the simulations from Xuan et al.(2019, 2020). In addition, the direction of Langmuir cells spirals towards a more diagonal orientation for h=5 m as compared to that for h=15 m. This result indicates that the influence of Coriolis force on Langmuir turbulence is also affected by the upper mixed layer depth. The distance of the coherent structures induced by Langmuir turbulence for h=5 m is wider than that for h=15 m, suggesting that the intensity of Langmuir turbulence for the former is clearly weaker than that for the latter (Noh et al., 2004).

The snapshot of three-dimensional vertical velocity (w) fields of Langmuir turbulence for h=5 and 15 m are shown in Fig. 2a & b, respectively. For h=5 m, the upwelling and downwelling velocities appear below the upper mixed layer (Fig. 2a), where the depth of the upwelling and downwelling velocities is 14 m in the interior of the thermocline. For h=15 m, the upwelling and downwelling velocities do not appear below the upper mixed layer (Fig. 2b). This result implies that the vertical entrainment flux directly induced by Langmuir turbulence in the thermocline is insignificant for h=15 m based on the wind and wave fields used here.

Figure 3a displays the variation in the upper mixed layer depth and thermocline induced by Langmuir turbulence. Figure 3b shows the variation in the mean turbulence entrainment flux (〈wʹTʹ〉). Variations in the mean velocity shear (〈S2〉=(∂〈u〉/∂z)2+(∂〈v〉/∂z)2, 〈u〉 is the downwind velocity and 〈v〉 is the crosswind velocity) and the mean vertical velocity variance (〈2〉) reflects vertical mixing intensity (Li et al., 2005; Harcourt and D'Asaro, 2008; Sutherland et al., 2014) for the different upper mixed layer depth (h) are shown in Fig. 4. The upper mixed layer depth is unchanged for h=5 m and 10 m (Fig. 3a), though the more efficient entrainment from the thermocline (Fig. 3b). The upper mixing layer depth (Sutherland et al., 2014) "deepens" for h=5 and 10 m, because the mean turbulence entrainment flux (Fig. 3b) and mean vertical velocity variance (Fig. 4b) extend to z=-14 m (Sutherland et al., 2014). The upper mixed layer depth is unchanged for h≥15 m (Fig. 3a), owing to the very weak entrainment from the thermocline (Fig. 3b). It is also found that, for h=5 and 10 m, variation in the thermocline structure is evident from the upper mixed layer base to z=-14 m (Fig. 3a), since the enhanced mean turbulence entrainment can extend to z=-14 m (Fig. 3b). The insignificant mean turbulence entrainment flux only appears at the upper mixed layer base for h≥15 m (Fig. 3b), which induces that the thermocline structure remains unchanged for h≥15 m. Furthermore, the mean velocity shear (Fig. 4a) is very small below the base of the upper mixed layer, suggesting that shear instability cannot induce the significant mean turbulence entrainment flux in the interior of the thermocline for h=5 and 10 m. The mean vertical velocity variance extends to z=-14 m for h=5 m and 10 m (Fig. 4b), implying that the significant mean turbulence entrainment flux in the thermocline is directly caused by the downwelling and upwelling velocities of Langmuir turbulence (Kukulka et al., 2010). Notably, the large mean vertical velocity variance cannot extend below the upper mixed layer base for h≥15 m, showing that a variation of the upper mixed layer depth for h≥15 m cannot be directly affected by Langmuir turbulence (Sullivan et al., 2007; Kukulka et al., 2010). The above results show that, for the cases of h=5 and 10 m, the entrainment flux over the range of -hz≥-14 m (Fig. 3b) is directly induced by Langmuir turbulence (Fig. 4b). However, when h≥15 m, the entrainment flux does not appear at a depth deeper than the upper mixed layer depth (Fig. 3b). Thus, there exists a depth of the evident entrainment flux in the interior of thermocline directly associated with Langmuir turbulence in this study, which may imply that the mixing mechanism at the base of the upper mixed layer (i.e., the mixing is induced by the downwelling and upwelling velocities or the shear instability), and we define the depth as the direct entrainment depth of Langmuir turbulence (i.e., the zero of entrainment flux in the thermocline), hLT. Based on the wind and wave fields used here, the hLT for the present study is found to be 14 m. To highlight the variation in the vertical profiles of the mean temperature, turbulence entrainment flux, velocity shear and vertical velocity variance for h=5 and 10 m, z is not normalized by the upper mixed layer depth (hLT). In addition, variation in their profiles for h≥25 m is also not shown.

For Langmuir turbulence (Lat≤0.3), the normalized depth-averaged vertical velocity variance (w'2) is larger than the normalized depth-averaged downwind velocity variance (u'2), while w'2 is smaller than u'2 for shear turbulence (Lat>0.3) (Li et al., 2005; Sutherland et al., 2014). Hence, to determine whether there is a fundamental difference of turbulence characteristics between h/hLT < 1 and h/hLT>1, variations in u'2 and w'2 with the h/hLT are used to indicate the influence of the upper mixed layer depth on the turbulence characteristics (Fig. 5). For h/hLT < 1, w'2 < u'2 representing shear turbulence characteristics (Lat>0.3) is inconsistent with the results of Li et al. (2005) and Sutherland et al. (2014), that is, w'2>u'2 for Lat≤0.3. For h/hLT>1, w'2>u'2 agrees with the results of Li et al. (2005) and Sutherland et al. (2014). This result suggests that there is a fundamental difference in the turbulence characteristics between h/hLT < 1 and h/hLT>1. In addition, w'2 is the largest for h/hLT≈1 (h=15 m), meaning that in this study, Langmuir turbulence is the strongest for h/hLT≈1 (Li et al., 2005; Harcourt and D'Asaro, 2008).

According to the above results, the direct entrainment depth of Langmuir turbulence in the interior of the thermocline (hLT=14 m) can be identified. In the following sections, the different characteristics and evolution of Langmuir turbulence parameters between h/hLT < 1 and h/hLT>1 are analyzed in relation to the different upper mixed layer depth. The reason for the fundamental difference of turbulence characteristics between h/hLT < 1 and h/hLT>1 is explained in Section 4.

A large eddy simulation model with the current grid size cannot resolve dynamic parameters, such as horizontal velocity, vertical momentum flux and turbulent kinetic energy budget, in a stratified thermocline (Wang et al., 1998; Skyllingstad et al., 1999; Furuichi and Hibiya, 2015). However, the evident entrainment flux for h=5 and 10 m appears in the interior of the thermocline (Fig. 3b). This is attributed to the fact that turbulent eddy scales for turbulent entrainment flux are much larger than those for the dynamic parameters (Wang et al., 1998; Skyllingstad et al., 1999; Skyllingstad, 2000; Grant and Belcher, 2009; Furuichi and Hibiya, 2015). Hence, the depth-averaged low-order statistics of Langmuir turbulence over the upper mixed layer should not include the dynamic quantities below the upper mixed layer.

3.2 Horizontal velocity

Vertical profiles of the normalized mean horizontal velocity (〈u〉+us and 〈v〉) (Fig. 6) show the influence of variations in the upper mixed layer depth on the magnitude of mean horizontal velocity. When the upper mixed layer changes from 5 to 15 m, a decrease in mean horizontal velocity magnitude is evident. This indicates that the influence of the upper mixed layer depth on the magnitude of mean horizontal velocity has a significant change as the upper mixed layer depth changes from shallower to slightly deeper than hLT=14 m. From 15 to 40 m, the mean horizontal velocity is slowly reduced, implying that the upper mixed layer depth variation has little effect on the mean horizontal velocity when the upper mixed layer depth is deeper than hLT=14 m. The downwind velocity has a negative value (Fig. 6a), consistent with previous studies (McWilliams et al., 1997; Sullivan et al., 2007).

To differentiate between the mean horizontal velocity variation when the h/hLT are smaller and larger than h/hLT=1, the normalized depth-averaged mean horizontal velocity (u〉+us and v) are calculated within the upper mixed layer depth (h) (Fig. 7) as suggested by Skyllingstad (2000), Li et al. (2005), Harcourt and D'Asaro (2008), and Wijesekera et al. (2013). The rate of change of u〉+us (Fig. 7a) and v (Fig. 7b) for h/hLT < 1 is much larger than that for h/hLT>1, showing that h/hLT=1 is a critical value at which the rate of change of u〉+us and v with the h/hLT variation is significantly different. In addition, u〉+us is smaller as compared to v, which is consistent with previous studies (Li et al., 2005; Sutherland et al., 2014) based on a deep upper mixed layer assumption.

3.3 Vertical velocity skewness

The asymmetry of the downward and upward velocities, and coherent structures produced by Langmuir turbulence are measured by the mean vertical velocity skewness, given by 〈w3〉/〈w23/2 (McWilliams et al., 1997; Sullivan et al., 2007; Li et al., 2009). Figure 8a illustrates that the maximum 〈w3〉/〈w23/2 appears near the upper mixed layer base, consisting with the result of McWilliams et al. (1997). The 〈w3〉/〈w23/2 strengthens with an increase of the upper mixed layer depth owing to that the inhibiting effect of the very strong stratification in the thermocline on the 〈w3〉/〈w23/2 becomes weak with an increase of the upper mixed layer depth. The 〈w3〉/〈w23/2 < 1 for h≤35 m that is inconsistent with the 〈w3〉/〈w23/2>1 (McWilliams et al., 1997), while the 〈w3〉/〈w23/2>1 for h=40 m that agrees with the result of McWilliams et al. (1997). This reason is that the very strong stratification in the thermocline significantly inhibits 〈w3〉/〈w23/2 for h≤35 m and the impact of the thermocline on 〈w3〉/〈w23/2 for h=40 m is very weak.

Figure 8b shows the depth-averaged vertical velocity skewness (w3/w23/2). When the h/hLT varies from h/hLT < 1 to h/hLT≈1 (h=15 m), the depth-averaged vertical velocity skewness is enhanced significantly. In comparison, a small change is observed when h/hLT>1. The result shows that the rate of change of the depth-averaged vertical velocity skewness for h/hLT < 1 is much large than that for h/hLT>1.

3.4 Vertical momentum flux, eddy viscosity, and velocity shear

When wind and wave fields are aligned, the horizontal flow is homogeneous and the flow is static as z→∞, hence the momentum budget (McWilliams et al., 1997; Sullivan et al., 2007) is given by

    (8)
    (9)

It is the surface turbulence stress when the depthintegration is curried out. Vertical profiles of the normalized mean vertical momentum flux are shown in Fig. 9. With a change in the upper mixed layer depth, the variation in the mean downwind vertical momentum flux (〈uʹwʹ〉) is significantly different from the mean crosswind vertical momentum flux (〈vʹwʹ〉). The mean downwind vertical momentum flux increases when the upper mixed layer depth changes from 5 to 15 m, whereas it decreases when the upper mixed layer depth changes from 15 to 40 m (Fig. 9a). This result suggests that the variation of the vertical momentum transport in the downwind direction is fundamentally different for h < hLT and h>hLT. The increase of the mean crosswind vertical momentum flux is clear when the upper mixed layer depth changes from 5 to 40 m (Fig. 9b), due to that the mean crosswind vertical momentum increases with an increase of the upper mixed layer depth (Grant and Belcher, 2009). Furthermore, the mean downwind vertical momentum flux is always larger than the mean crosswind vertical momentum flux. The shape of the mean vertical momentum flux vertical profiles is similar to those from previous studies that are based on a deep upper mixed layer assumption (McWilliams et al., 1997; Sullivan et al., 2007), except for h=5 and 10 m where the mean vertical momentum flux is not obvious near the upper mixed layer base.

It is worth noting that for h=5 and 10 m, the relatively significant mean vertical momentum flux in the crosswind direction appears below the upper mixed layer base (Fig. 9b), which shows that internal gravity waves are generated by the downward pressure perturbation induced by Langmuir turbulence impinging on the top of the pycnocline (Polton et al., 2008).

Variations in the normalized depth-averaged vertical momentum flux in the downwind (uʹwʹ) and crosswind (vʹwʹ) directions are shown in Fig. 10a and b, respectively. The rate of increase of uʹwʹ when h/hLT < 1 is larger as compared to the rate of decrease when the h/hLT>1 (Fig. 10a). This result suggests that the rate of change for uʹwʹ is completely different between h/hLT < 1 and h/hLT>1. The vʹwʹ varies insignificantly when h/hLT < 1 (Fig. 10b), due to the intensity of Langmuir turbulence is still suppressed by the upper mixed layer depth. This result indicates that the upper mixed layer depth variation has a very weak influence on a variation of the vʹwʹ, which is inconsistent with previous study of Grant and Belcher (2009), that is, the vʹwʹ increases with an increase of the upper mixed layer under a deep upper mixed layer condition. Furthermore, it increases linearly for h/hLT≥0.71 (h≥10 m). This observation implies that vʹwʹ increases quickly when the h/hLT changes from h/hLT=0.71 (h=10 m) to h/hLT≈1 (h=15 m) in this study. Figure 10c shows the ratio between uʹwʹ and vʹwʹ, that is uʹwʹ/vʹwʹ. Very small variations of uʹwʹ/vʹwʹ are observed when h/hLT changes from h/hLT < 1 to h/hLT≈1, suggesting that both uʹwʹ and vʹwʹ increase almost proportionally due to that the suppress effect of the upper mixed layer depth on Langmuir turbulence intensity becomes weak. For h/hLT>1, a linear decrease of uʹwʹ/vʹwʹ is noted, due to the decreased uʹwʹ (Fig. 10a) and the increased vʹwʹ (Fig. 10b). This result reveals that the relative magnitude between uʹwʹ and vʹwʹ is unchanged for h/hLT≤1 and enlarges significantly for h/hLT≥1 (Fig. 10c).

The mean vertical eddy viscosity (〈Km〉) is a key parameter in Langmuir turbulence parameterization (McWilliams and Sullivan, 2000; Sullivan et al., 2007; Harcourt, 2015). The 〈Km〉 is given by:

    (10)

Figure 11a shows the vertical profiles of the normalized mean 〈Km〉. The 〈Km〉 weakens when the upper mixed layer depth changes from 5 to 40 m. The reason is that the energy input by the wind and wave fields spreads over a larger space by Langmuir turbulence with an increase of the upper mixed layer depth. As the upper mixed layer depth changes from 15 to 40 m, the values of 〈Km〉 are gradually close to one another with the increasing depth within the upper mixed layer, consistent with the results of Skyllingstad (2000) and Thorpe (2004), which are also based on a deep upper mixed layer assumption.

The vertical profiles of the normalized mean velocity shear (〈S2〉=(∂〈u〉/∂z)2+(∂〈v〉/∂z)2) are shown in Fig. 11b. Large values of 〈S2〉 are located near the surface layer and the base of the upper mixed layer. The effective upper mixed layer depth is reduced by vertical entrainment relative to the initial upper mixed layer for h=5 and 10 m (Fig. 3a). Hence, the depth of the large values of 〈S2〉 near the upper mixed layer base for h=5 and 10 m has a small decrease. Moreover, except near the surface layer, the relatively large 〈S2〉 for h=5 and 10 m appears over a larger range relative to that for h≥15 m (Fig. 11b).

Figure 12a shows the variation in the normalized depth-averaged eddy viscosity (Km). The Km is the largest for h/hL≈1 (Fig. 12a), showing that mixing is the most efficient when the upper mixed layer depth is close to the direct entrainment depth of Langmuir turbulence hLT=14 m. The Km enhances and weakens when the h/hLT≤1 and h/hLT≥1, respectively, revealing that the rate of change of the Km with a change in the h/hLT is opposite between h/hLT < 1 and h/hLT>1.

Figure 12b shows the normalized depth-averaged velocity shear (S2) as a function of the h/hLT. There is a clear decrease in S2 when the h/hLT≤1, while it decreases relatively less when the h/hLT≥1 (Fig. 12b). This result demonstrates that the rate of decrease of S2 with the h/hLT is different when the h/hLT≤1 and the h/hLT≥1.

3.5 Turbulent kinetic energy budget

We now analyze the change in turbulent kinetic energy budget as the upper mixed layer depth varies in order to investigate the dynamic mechanisms that drive the upper mixed layer deepening.

The mean turbulent kinetic energy budget equation (McWilliams et al., 1997; Skyllingstad, 2000; Polton and Belcher, 2007; Noh et al., 2009) is given by:

    (11)

where 〈E〉 (=〈uiʹuiʹ〉/2) is the total turbulent kinetic energy, 〈F〉(=-∂(〈wʹPʹ〉+〈wʹEʹ〉)/∂z), where is the modified pressure perturbation and is the total turbulent kinetic energy perturbation) is the transport production, 〈PS〉(=-〈uʹwʹ〉∂〈u〉/∂z-〈vʹwʹ〉∂〈v〉/∂z) is the shear production, 〈PL〉 (=-〈uʹwʹ〉∂us/∂z) is the Langmuir production, 〈ε〉 is the dissipation rate of turbulent kinetic energy, and 〈Pb〉 (=〈αgwʹTʹ〉, 〈wʹTʹ〉 is the entrainment flux) is the buoyancy production.

The vertical profiles of the normalized mean turbulent kinetic energy budget terms for h≥15 m are close each other, thus the vertical profiles of the meanturbulent kinetic energy budgets for h=5 and 15 m are only shown to briefly illustrate the difference between the upper mixed layer shallower and deeper than the direct entrainment depth (hLT=14 m). The vertical profiles of the mean turbulent kinetic energy budget terms for h=5 and h=15 m are shown in Fig. 13. The values of 〈E〉 (Fig. 13a) and 〈Pb〉 (Fig. 13f) for h=5 m are much larger as compared to those for h=15 m. The 〈PL〉 for h=5 m is much smaller than that for h=15 m (Fig. 13d), indicating that intensity of Langmuir turbulence for former is weaker than that for the later. The 〈ε〉 for h=5 m are much smaller than that for h=15 m. These results suggest that 〈Pb〉 significantly contributes to the turbulent kinetic energy budget when the upper mixed layer depth is shallower than the direct entrainment depth of Langmuir turbulence (hLT=14 m). Furthermore, the value of 〈E〉 below the upper mixed layer base has a very small contribution to turbulent kinetic energy budget, which is consistent with previous studies (Sullivan et al., 2007; McWilliams et al., 2014; Pearson et al., 2015).

The 〈F〉 for h=5 m is smaller than that for h=15 m (Fig. 13b), implying that the downward transport of the energy input by the wind and wave fields for h=5 m is greatly suppressed relative to that for h=15 m. 〈PS〉 (Fig. 13c) and 〈Pb〉 (Fig. 13f) have relatively large values in the interval -0.4 h>z>-0.9 h for h=5 m as compared to that for h=15 m, which shows that the suppressed effects for h=5 m is stronger than that for h=15 m. 〈Pb〉 has a very small contribution to the turbulent kinetic energy budget terms for h=15 m (Fig. 13), suggesting that Langmuir turbulence cannot directly cause the upper mixed layer deepening when the upper mixed layer depth is deeper than hLT=14 m, which is in agreement with previous observations of Weller and Price (1988) and Skyllingstad (2000).

Comparison between 〈PS〉 (Fig. 13c) and 〈PL〉 (Fig. 13d) shows that, for h=5 m, 〈PS〉 is much larger as compared to 〈PL〉 below z=-0.4 h, suggesting that shear instability dominates the mixing. PS is much smaller than 〈PL〉 for h=15 m, which is consistent with the results from Kukulka et al. (2010). These results imply that the dynamic mechanism that influences the turbulent kinetic energy budget below z=-0.4 h is completely different between h=5 m and h=15 m.

The variations in the normalized depth-averaged total turbulent kinetic energy (E), transport production (F), shear production (PS), Langmuir production (PL), dissipation rate (ε), and buoyancy production (Pb) are shown in Fig. 14. The rate of decrease of E (Fig. 14a) is high when the h/hLT≤1 and is slow when the h/hLT≥1. The PS (Fig. 14c) and Pb (Fig. 14f) terms decay significantly for h/hLT≤1, while variation in Pb is very small and PS enhances for h/hLT≥1, indicating that variation in these terms is significant between h/hLT≤1 and h/hLT≥1. The rate of decrease of the PL (Fig. 14d) and ε (Fig. 14e) terms do not vary significantly between h/hLT≤1 and h/hLT≥1.

In addition, the F (Fig. 14b) term is much smaller than the other terms, which implies that transport production contributes very little to the turbulent kinetic energy budget whether the h/hLT≤1 or the h/hLT≥1, in agreement with previous study of McWilliams et al. (1997) based on a deep upper mixed layer assumption. The Pb (Fig. 14f) term is very large for h/hLT < 1 and very small for h/hLT>1 (Fig. 14f). This result shows that the buoyancy production contributes differently to the turbulent kinetic energy budget when the h/hLT < 1 and the h/hLT>1. In addition, PS decreases for h/hLT < 1 and increases for h/hLT>1 (Fig. 14c), which is significantly different from the other terms of the turbulent kinetic energy budget.

4 DISCUSSION

Turbulence characteristics are fundamentally different between h/hLT < 1 and h/hLT>1 (h is the upper mixed layer depth, hLT=14 m is the direct entrainment depth of Langmuir turbulence (hLT) in the thermocline (Fig. 5) based on the wind and wave fields used here). The normalized depth-averaged velocity variance in the vertical direction (w'2) is smaller than that in the downwind direction (u'2) for h/hLT < 1 and w'2 is greater than u'2 for h/hLT>1. An interpretation of this fundamental difference of turbulence characteristics between h/hLT < 1 and h/hLT>1 is presented in Fig. 15. The Langmuir cells induced by wave-current interactions are over the range of the Stokes depth scale (Leibovich, 1980; Skyllingstad, 2000; Polton and Belcher, 2007) and the impact depth of the strong downwelling jets is much larger than Stokes depth scale (Skyllingstad, 2000; Polton and Belcher, 2007) due to the downward pressure perturbation (Suzuki and Fox-Kemper, 2016). Hence, when the upper mixed layer depth is much less than the impact depth of the strong downwelling jets (h/hLT < 1), the strong downward pressure perturbation can be strongly inhibited by the upward reactive force of the strong stratified thermocline (Fig. 15), which causes w'2 to be smaller than u'2 for h/hLT < 1 (Fig. 5). As the h/hLT increases for h/hLT < 1, the inhibition effect of the upward reactive force of the strong stratified thermocline on the strong downward pressure perturbation weakens (Fig. 15), so that the downwelling jets further develop, that is, w'2 enhances and u'2 weakens (Fig. 5). When the upper mixed layer depth does not inhibit the strong downwelling jets (Fig. 15), Langmuir turbulence can develop fully, reflected by the fact that w'2 is larger than u'2 for h/hLT>1 (Fig. 5). This result suggests that the intrinsic characteristic of Langmuir turbulence, that is, w'2>u'2, is not suitable when the upper mixed layer depth is much less than the direct entrainment depth of Langmuir turbulence in the interior of the thermocline. This is also the reason why the characteristics and variations in the other low-order statistics of Langmuir turbulence analyzed in this study have a significant or complete difference between h/hLT < 1 and h/hLT>1.

In our simulations, the wind and wave fields are kept constant and the upper mixed layer depth is varied from 5 to 40 m. Hence, another possible choice of a nondimensional parameter related to the variation of the upper mixed layer depth is the surface layer Langmuir number (u* is the surface friction velocity, uSL is the average Stokes drift velocity in a surface layer 0>z>-0.2 h), which can be used to quantify the intensity of the Langmuir turbulence (Harcourt and D'Asaro, 2008). Figure 16a shows that LaSL increases as the upper mixed layer depth changes from 5 to 40 m, which implies that the intensity of Langmuir turbulence decays (Harcourt and D'Asaro, 2008; Li and Fox-Kemper, 2017).

Fig.16 Variation in the surface layer Langmuir number (LaSL) as a function of the upper mixed layer depth (h) (a); variation in the LaSL as a function of the h/hLT (b); variation in the normalized depth-averaged vertical velocity variance (w'2) with a change of the LaSL (c); variation in the normalized (w'2) with a change of the h/hLT (d) The black horizontal solid line indicates: in (a): the direct entrainment depth of Langmuir turbulence hLT=14 m; in (b): h/hLT=1; in (c): the LaSL corresponding to h/hLT=1; in (d): h/hLT=1.

The normalized depth-averaged vertical velocity variance (w'2) is also a metric used to quantify the intensity of Langmuir turbulence (Li et al., 2005; Harcourt and D'Asaro, 2008; Li and Fox-Kemper, 2017) and vertical mixing (Tseng and D'Asaro, 2004; Van Roekel et al., 2012; Pearson et al., 2015). A comparison of Fig. 16bc shows that when the h/hLT changes from h/hLT < 1 to h/hLT≈1, enhancement in the normalized w'2 is significant when LaSL increases (Fig. 16c), which is contrary to observations from previous studies (Harcourt and D'Asaro, 2008; Li and Fox-Kemper, 2017), that is, the normalized w'2 should weaken as LaSL increases. This inconsistency is attributed to the fact that the intensity of Langmuir turbulence measured by LaSL is based on the assumption that the upper mixed layer is deep enough to allow for the full development of Langmuir turbulence (Harcourt and D'Asaro, 2008; Van Roekel et al., 2012; Li and Fox-Kemper, 2017). The intensity of Langmuir turbulence is suppressed when h/hLT is much less than h/hLT= 1 in this paper and is enhanced evidently as h/hLT varies from h/hLT < 1 to h/hLT≈1 (Fig. 16c). However, when the h/hLT varies from h/hLT≈1 to h/hLT> 1, the normalized w'2 declines as LaSL increases, which is generally consistent with previous studies from Harcourt and D'Asaro (2008) and Li and Fox-Kemper (2017). Furthermore, as shown in Fig. 16d, the normalized w'2 increases when the h/hLT changes from h/hLT < 1 to h/hLT≈1 and decreases from h/hLT≈1 to h/hLT>1. This result suggests that a variation of Langmuir turbulence intensity is completely different for h/hLT < 1 and h/hLT>1 (Fig. 16d), which is agreement with that the normalized w'2 first strengthens and then weakens with an increase of LaSL (Fig. 16c). In addition, the normalized w'2 is largest for h/hLT≈1.

The curvature of the profiles of the normalized downwind vertical momentum flux (〈uʹwʹ〉) strengthens with an increase of the ratio between the upper mixed layer depth (h) and the Ekman depth scale (u*/f), that is, fh/u* (Grant and Belcher, 2009). However, the curvature of the 〈uʹwʹ〉 first weakens and then strengthens, when the fh/u* changes from fh/u* < fhLT/u* to fh/u*>fhLT/u* (fhLT/u*=0.271) as shown in Fig. 17. This result shows that the change in the curvature of the 〈uʹwʹ〉 with an increase of the fh/u* is different between fh/u* < fhLT/u* and fh/u*>fhLT/u*.

Fig.17 Variation in the vertical profiles of the normalized mean downwind vertical momentum flux (〈u'w'〉) plotted by z/h with a change of fh/u* The fhLT/u*=0.271.

The shear production of turbulent kinetic energy (〈PS〉=-〈uʹwʹ〉∂〈u〉/∂z–〈vʹwʹ〉∂〈v〉/∂z) in the interval below z=-0.45 h for h=5 m is much larger than that for h=15 m (Fig. 13c). This is attributed mainly to the fact that the vertical gradient of the mean horizontal velocity below z=-0.45 h for h=5 m is significantly larger than that for h=15 m (Fig. 18), because the difference in the downwind vertical momentum flux between h=5 m and h=15 m is very small (Fig. 9a), and the magnitude of the crosswind vertical momentum flux for h=5 m is smaller than that for h=15 m (Fig. 9b). In addition, when the h/hLT≤1, a decrease in depth-averaged shear production PS is observed (Fig. 14c). This can be attributed to the significant decay in the gradient of the mean horizontal velocity below z=-0.45 h (Fig. 18), since the magnitude of the vertical momentum flux in both downwind and crosswind directions are almost similar (Fig. 9). However, when the h/hLT>1, enhancement in PS is small (Fig. 14c). The reason is that the increase in the vertical gradient of the mean horizontal velocity for h/hLT>1 appears primarily in the downwind direction near the surface layer (not shown). Moreover, although the crosswind vertical momentum flux is evidently enhanced (Fig. 9b) and the downwind vertical momentum flux is slightly weakened (Fig. 9a), the crosswind vertical momentum flux is still smaller than the downwind vertical momentum. Hence, an increase in the crosswind vertical momentum flux has a small effect on the enhancement of PS for h/hLT>1.

Fig.18 Vertical profiles of the normalized gradient of mean downwind velocity (∂〈u〉/∂〈z〉) (a) and mean crosswind velocity (∂v/∂z) (b)

The depth-averaged buoyancy production Pb for h/hLT < 1 (Fig. 14f) has nearly the same order of magnitude as the depth-averaged total turbulent kinetic energy EE (Fig. 14a), shear production PS (Fig. 14c) and dissipation rate ε (Fig. 14e), suggesting that Pb contributes significantly to the turbulent kinetic energy budget. This result does not agree with previous results (McWilliams et al., 1997; Noh et al., 2009), that is, Pb has a very small contribution to the turbulent kinetic energy budget. The reason is due to that Langmuir turbulence characteristics are not affected by the deep upper mixed layer depth (McWilliams et al., 1997; Noh et al., 2009). However, when h/hLT>1, the effect of Pb on the turbulent kinetic energy budget is very weak (Fig. 14f), which is consistent with previous studies based on a deep upper mixed layer assumption (Skyllingstad, 2000; Polton and Belcher, 2007). These results demonstrate that when the upper mixed layer depth (h=5 and 10 m) is shallower than the direct entrainment depth of Langmuir turbulence (hLT=14 m) (h/hLT < 1) in this study, the influence of Pb on the turbulent kinetic energy budget should not be neglected, owing to intense entrainment flux (〈wʹTʹ〉) from the thermocline (Fig. 2b), that is, buoyancy production Pb =〈αgwʹTʹ〉 (Fig. 13f), where α is the thermal expansion coefficient and g is the acceleration due to gravity.

Furthermore, the relationship between the Stokes depth scale (δs) and hLT may indicate the hLT of the broad applicability. The rate of change of the δs/h for h/hLT < 1 is much larger than that for h/hLT>1 with a change of the h/hL (Fig. 19), which is similar to a variation of the depth-averaged horizontal velocity (Fig. 7), velocity shear (Fig. 12b), Langmuir production (Fig. 14d), dissipation rate (Fig. 14e) and buoyancy production (Fig. 14f). This result implies that the rate of change of the related depth-averaged physical quantiles has the internal connection with the rate of change of the δs/h when the h/hL changes from h/hL < 1 to h/hL>1. The internal connection is owing to that Langmuir cells induced by the wave-current interactions are in the range of Stokes depth scale (δs) (Polton and Belcher, 2007) and the downward pressure perturbation can penetrate very deeply (Suzuki and Fox-Kemper, 2016), the downward pressure perturbation is strongly inhibited by the thermocline for h/hL < 1 (or δs/h>δs/hLT) and the inhibited effect of the thermocline on the downward pressure perturbation quickly weakens for h/hL>1 (or δs/h < δs/hLT) (Fig. 15) with an increase of the h/hL (or a decrease of the δs/h).

Fig.19 Variation of the ratio between the Stokes depth scale (δs) and the upper mixed layer depth (h), i.e., δs/h, with a variation of h/hLT The black horizontal solid line indicates the h/hLT=1 and the black vertical solid line indicates the δs/hLT.

The above discussion indicates that when the upper mixed layer depth (h=5 and 10 m) is shallower than direct entrainment depth of Langmuir turbulence (hLT=14 m) (h/hLT < 1) based on the wind and wave fields used here, Langmuir turbulence is strongly inhibited by a shallow upper mixed layer and is strongly enhanced when h/hLT < 1 (Fig. 15). Hence, many related low-order statistics analyzed in this study change significantly with the variation in the h/hLT. Nevertheless, Langmuir turbulence develops fully when the h/hLT>1. Therefore, variation in the related low-order statistics is small as the upper mixed layer depth varies. Moreover, the relative magnitudes of the related low-order statistics between h/hLT < 1 and h/hLT>1 often have a fundamental difference.

5 CONCLUSION

This study employs a large eddy simulation model to investigate the influence of the idealized upper mixed layer depth variation on Langmuir turbulence characteristics under ideal conditions. There is a direct entrainment depth induced by Langmuir turbulence (hLT=14 m). The results of a varying upper mixed layer depth, h (=5, 10, 15, 20, 25, 30, 35, and 40 m), can be classified into two categories, in which the upper mixed layer depth is either shallower or deeper than the direct entrainment depth of Langmuir turbulence (hLT=14 m) (h/hLT < 1 or h/hLT>1) in the thermocline based on the wind and wave fields used here. The normalized depth-averaged vertical velocity variance (w'2) is smaller than the normalized depth-averaged downwind velocity variance (u'2) for h/hLT < 1, demonstrating that an intrinsic characteristic of Langmuir turbulence, that is, w'2>u'2, is not suitable for h/hLT < 1. For h/hLT>1, w'2>u'2 represents a fully developed Langmuir turbulence. Furthermore, eddy viscosity increases (decreases) for h/hLT < 1 (h/hLT>1). In addition, the strongest eddy viscosity is observed for h/hLT≈1. The reason is that the the downward pressure perturbation induced by Langmuir cells is strongly inhibited by the upward reactive force of the thermocline for h/hLT < 1 and the effect of upward reactive force on the downward pressure perturbation becomes weak for h/hLT>1.

Whenh/hLT < 1, the magnitudes of the most low-order statistics of Langmuir turbulence quickly enhances as the h/hLT increases. When h/hLT>1, both the vertical momentum flux in the crosswind direction and the shear production of the turbulent kinetic energy are enhanced as the h/hLT increases, while changes in the other low-order statistics are very small.

Finally, the turbulence characteristics of Langmuir turbulence between hhLT and hhLT are completely different, suggesting that improvements of Langmuir turbulence parameterizations should consider that the variation in Langmuir turbulence characteristics between hhLT and hhLT has a fundamental difference.

6 DATA AVAILABILITY STATEMENT

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

7 ACKNOWLEDGMENT

The large eddy simulation model is provided by the National Center for Atmospheric Research. All numerical calculations were carried out at the High Performance Computing Center (HPCC) of the South China Sea Institute of Oceanology, Chinese Academy of Sciences.

8 APPENDIX

Calculation method of the depth-averaged value

The depth-averaged arbitrary physical quantity (ψ) is computed as (the overbar denotes the depth averaged operation), where the N is the number of vertical grid cells from the sea surface to the upper mixed layer base.

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