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    Time series of (top) MLD (h), (middle) the buoyancy flux at the MLD [bw(z = h)], and (bottom) Δbw(h)/bw0(h). Here, Δbw(h) = bw(h) − bw0(h), and bw0(h) is the buoyancy flux in the absence of LC (La = ∞). Blue is L0 (La = ∞); green is L1 (La = 0.64); red is L2 (La = 0.32); and black is the theoretical prediction by Pollard et al. (1973), which is valid only up to t = π/f: (a) N1 (N2 = 10−5 s−2) and (b) N3 (N2 = 2 × 10−4 s−2).

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    Variation of Δbw(h)/bw0(h) with hΔB/υL2. Here data were obtained by calculating the value of bw(z = h) averaged over the period of 20 m < h < 30 m from each experiment. Horizontal bars represent the corresponding range of hΔB/υL2.

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    Profiles of velocity shear S2 and stratification N2 [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

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    (top) Profiles of Ri and (bottom) time series of Ri (z = h) [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

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    Profiles of eddy viscosity Km and eddy diffusivity Kh [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

  • View in gallery

    Profiles of the mixing length scale l and the turbulent velocity scale q. Here, l was calculated from Kh = Shql with Sh = 0.49 [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

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    Scatterplots l/l0 vs (left) Ri and (right) Rt (N3). Data were obtained every hour at each grid depth for z < h from Kh: (a) L0 and (b) L2.

  • View in gallery

    Scatterplots in the logarithmic scale for l/l0 vs Rt. Data were obtained every hour at each grid depth for z < h [blue (N1, L0), light blue (N3, L0), green (N1, L1), light green (N3, L1), red (N1, L2), and light red (N3, L2)]. The black solid and dashed lines represent the predictions from l/l0 = (1 + αRt)−1/2 and l/l0 = γ(1 + αγ2Rt)−1/2, where α = 50 and γ = 10: (a) Km and (b) Kh.

  • View in gallery

    Time series of (top) MLD (h) and (bottom) the buoyancy flux at the MLD [bw(z = h)] for the case L2 (solid line is λ = 40 m and dashed line is λ = 80 m): (a) N1 and (b) N3.

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Influence of Langmuir Circulation on the Deepening of the Wind-Mixed Layer

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  • 1 Department of Atmospheric Sciences, and Global Environmental Laboratory, Yonsei University, Seoul, South Korea
  • | 2 Institute of Meteorology and Climatology, University of Hannover, Hannover, Germany
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Abstract

Analysis of large eddy simulation data reveals that Langmuir circulation (LC) induces a significant enhancement of the mixed layer deepening, only if the mixed layer depth (MLD) h is shallow and the buoyancy jump across it ΔB is small, when simulations are initiated by applying the wind stress to a motionless mixed layer with stratification. The difference in the entrainment rate between the cases with and without LC decreases with hΔBL2, where υL is the velocity scale of LC. The ratio of the mixing length scale l between the cases with and without LC is close to 1 for larger Rt [=(Nl0/q)2; Rt > ∼1], but it increases to above 10 with the decrease of Rt, where N is the Brunt–Väisälä frequency and q and l0 are the velocity and length scales of turbulence in the homogeneous layer. It is also found that, in the presence of LC, the effect of stratification on vertical mixing should be parameterized in terms of Rt instead of Ri (=(N/S)2), because velocity shear S is no longer a dominant source of turbulence. The parameterization is provided by l/l0 = (1 + αRt)−1/2 with α ∼ 50, regardless of the presence of LC. However, LC makes l0 much larger than conventionally used for the boundary layer.

Corresponding author address: Yign Noh, Department of Atmospheric Sciences/Global Environmental Laboratory, Yonsei University, 262 Seongsanno, Seodaemun-gu, Seoul 120-749, South Korea. Email: noh@yonsei.ac.kr

Abstract

Analysis of large eddy simulation data reveals that Langmuir circulation (LC) induces a significant enhancement of the mixed layer deepening, only if the mixed layer depth (MLD) h is shallow and the buoyancy jump across it ΔB is small, when simulations are initiated by applying the wind stress to a motionless mixed layer with stratification. The difference in the entrainment rate between the cases with and without LC decreases with hΔBL2, where υL is the velocity scale of LC. The ratio of the mixing length scale l between the cases with and without LC is close to 1 for larger Rt [=(Nl0/q)2; Rt > ∼1], but it increases to above 10 with the decrease of Rt, where N is the Brunt–Väisälä frequency and q and l0 are the velocity and length scales of turbulence in the homogeneous layer. It is also found that, in the presence of LC, the effect of stratification on vertical mixing should be parameterized in terms of Rt instead of Ri (=(N/S)2), because velocity shear S is no longer a dominant source of turbulence. The parameterization is provided by l/l0 = (1 + αRt)−1/2 with α ∼ 50, regardless of the presence of LC. However, LC makes l0 much larger than conventionally used for the boundary layer.

Corresponding author address: Yign Noh, Department of Atmospheric Sciences/Global Environmental Laboratory, Yonsei University, 262 Seongsanno, Seodaemun-gu, Seoul 120-749, South Korea. Email: noh@yonsei.ac.kr

1. Introduction

Langmuir circulation (LC), which appears in the form of an array of alternating horizontal roll vortices with axes aligned roughly with the wind, represents one of the most important characteristics of the ocean mixed layer (see, e.g., Leibovich 1983; Smith 2001; Thorpe 2004). The prevailing theory of LC is that of Craik and Leibovich (1976), which describes the formation of LC in terms of instability brought on by the interaction of the Stokes drift with the wind-driven surface shear current. The instability is initiated by an additional “vortex force” term in the momentum equation as us × ω, where us is the Stokes drift velocity and ω is vorticity.

Various features of LC have been reported from field observations (Weller and Price 1988; Plueddemann et al. 1996; Smith 1992, 1998; D’Asaro and Dairiki 1997; Thorpe et al. 2003; Gargett and Wells 2007). Meanwhile, recent progress in large eddy simulation (LES) has enabled us to investigate the dynamical process of LC directly (Skyllingstad and Denbo 1995; McWilliams et al. 1997; Noh et al. 2004; Min and Noh 2004; Polton and Belcher 2007; Li et al. 2005; Tejada-Martínez and Grosch 2007; Grant and Belcher 2009; Sullivan et al. 2007; Harcourt and D’Asaro 2008; Gerbi et al. 2009).

The significance of LC in the ocean mixed layer can be represented by the turbulent Langmuir number La [=(u*/Us)1/2], where u* is the frictional velocity and Us is the Stokes drift velocity at the surface (McWilliams et al. 1997). In particular, Li et al. (2005) and Grant and Belcher (2009) found that the transition between Langmuir turbulence and shear turbulence occurs at La ∼ 0.5–2. The velocity of LC υL is shown to scale as υL ∼ (Usu*2)1/3 (Min and Noh 2004; Skyllingstad 2001; Harcourt and D’Asaro 2008; Grant and Belcher 2009), although there are observational evidences suggesting different scaling (Plueddemann et al. 1996; Smith 1998). In the presence of LC, vertical turbulent kinetic energy (TKE) is enhanced, and TKE production is dominated by the divergence of TKE flux, rather than shear production, in the mixed layer (McWilliams et al. 1997; Noh et al. 2004, 2009; Li et al. 2005; Polton and Belcher 2007; Gerbi et al. 2009).

It has been well established that, within the mixed layer, LC enhances vertical mixing greatly, resulting in the uniform profiles of temperature and velocity (McWilliams et al. 1997; Noh et al. 2004; Li et al. 2005; Skyllingstad 2005). Nonetheless, the question of how the mixed layer deepening is affected by the presence of LC still remains to be resolved. Evidence of enhancements to mixed layer deepening or entrainment at the mixed layer depth (MLD) in the presence of LC has been reported (Li et al. 1995; Sullivan et al. 2007; Grant and Belcher 2009; Kukulka et al. 2009). On the other hand, Weller and Price (1988) and Thorpe et al. (2003) observed no evidence that LC has a direct role in the mixed layer deepening. Skyllingstad et al. (2000) suggested from the LES results that the effects of LC are mostly confined to the initial stages of mixed layer growth.

Note that, in the presence of LC, the transport of heat and momentum can occur by means of large-scale eddies, contrary to the wall boundary layer dominated by small-scale shear-driven eddies near the wall. From this perspective, large-scale eddies associated with LC can be regarded as behaving similarly to convective eddies. However, unlike convective eddies, which are driven continuously by buoyancy force, eddies generated by LC are essentially unforced and behave inertially, because the vortex force is confined to near the surface by the decay scale of the Stokes drift. The intensity of these eddies thus decreases rapidly with depth, unlike convective eddies.

In spite of uncertainty as to the role of LC in the mixed layer deepening, there have been several attempts to incorporate its effects into the mixed layer model (Li et al. 1995; Li and Garrett 1997; D’Alessio et al. 1998; Smith 1998; McWilliams and Sullivan 2000; Smyth et al. 2002; Kantha and Clayson 2004). They considered the modification of the criterion at the MLD controlling the mixed layer deepening (Li and Garrett 1997; Smith 1998; D’Alessio et al. 1998), the inclusion of nonlocal mixing by LC (McWilliams and Sullivan 2000; Smyth et al. 2002), or the increase of TKE (D’Alessio et al. 1998; Kantha and Clayson 2004).

To evaluate and improve the mixed layer models, however, it is essential to understand properly the physical process by which LC influences the mixed layer deepening. Therefore, in the present work, we carried out LES experiments of the wind-mixed layer deepening under various conditions and analyzed the results with an aim to resolve the question on the role of LC in the mixed layer deepening and to provide information for the parameterization of its effect.

2. Simulation

The LES model used in the present simulation is similar to those used in Noh et al. (2004, 2006, 2009, 2010), Min and Noh (2004), and Noh and Nakada (2010), which have been developed based on the Parallelized LES Model (PALM) (Raasch and Schröter 2001). Langmuir circulation is realized by the Craik–Leibovich vortex force (Craik and Leibovich 1976), and wave breaking is represented by stochastic forcing. For simplicity, we assumed that both the wind stress and wave fields are in the x direction and further assumed that the wave field is steady and monochromatic. The associated Stokes velocity is then given by us = Us exp(−4πz/λ) with Us = (2πa/λ)2(/2π)1/2, where a is the wave height, λ is the wavelength, and g is the gravitational acceleration.

Integration is initiated by applying the wind stress to a motionless fluid. The wind stress at the surface is given by u* = 0.02 m s−1, and no heat flux is applied at the surface. The initial MLD is set to be h0 = 5 m, and the density is linearly stratified below the MLD with N2 (=∂B/∂z) = 10−5, 5 × 10−5, and 2 × 10−4 s−2, which will be referred to as the case N1, N2, and N3, respectively. The buoyancy jump across the MLD ΔB is given by N2h0/2 initially, so it can be calculated at the subsequent time automatically by ΔB = N2h/2. Three cases of different intensity of LC are considered with a = 0, 0.5, and 1 m, which correspond to La = ∞, 0.64, and 0.32, respectively, and will be referred to as the cases L0, L1, and L2. The wavelength is fixed as λ = 40 m, but the case with λ = 80 m is also simulated for comparison. The model domain is always 300 m in the horizontal direction but 120 m (N1 and N2) or 80 m (N3) in the vertical direction. The grid size is 1.25 m in all directions. A free-slip boundary condition is applied at the bottom. The Coriolis parameter is given by f = 10−4 s−1. Quasi-equilibrium state of turbulent boundary layer is reached at about 10h0/u* (∼1 h) after the onset of the wind stress. Simulations are performed for 16 h.

3. Results

a. Sensitivity of the mixed layer deepening to LC

Figure 1 compares the time series of MLD h, the buoyancy flux at the MLD bw(z = h), and Δbw(h)/bw0(h), which are obtained from the simulations with different intensity of LC (L0, L1, and L2) and different stratification below the MLD (N1 and N3). Here Δbw(h) represents the difference of bw(z = h) between the cases with and without LC, that is, Δbw(h) = bw(h) − bw0(h), where bw0(h) is the value of bw(h) in the absence of LC (La = ∞). Here bw(z = h) is calculated from the variation of potential energy, following Ayotte et al. (1996). The MLD is defined as the depth of the maximum N2, as in Li and Garrett (1997). Large fluctuation of bw(z = h) is due to the discontinuous increase of h over the grid size. In particular, the fluctuation of Δbw(h)/bw0(h) becomes very large in the later stage of the case N1, as bw0(h) approaches zero.

As illustrated by Pollard et al. (1973), Niiler (1975), and Phillips (1977), the mixed layer deepening passes through two distinct stages after the onset of wind, as a result of inertial oscillation: a rapid formation and deepening of the mixed layer until half the inertial period π/f (∼8.7 h) and the slow erosion thereafter. The theoretical prediction by Pollard et al. (1973) such as h = u*[4(1 − cosft)/f2N2]1/4, based on the assumption Ri (=N2/S2) = 1 at the MLD, is also shown in Fig. 1a, where S {=[(∂U/∂z)2 + (∂V/∂z)2]1/2} is velocity shear. Note that the Pollard et al.’s prediction is valid only up to t = π/f. It is interesting that a better agreement is found between LES results and the Pollard’s prediction, when LC is stronger (L2). It may be related to the fact that more uniform temperature and velocity profiles are generated under stronger LC, which is consistent with the hypothesis used in Pollard’s model.

Langmuir circulation is found to enhance the mixed layer deepening in general. However, the most remarkable feature in Fig. 1 is that the effect of LC, represented by Δbw(h)/bw0(h), is stronger when h is shallower in the initial stage and when stratification is weaker, as in the case N1. On the other hand, the effect of LC is negligible when stratification is strong and MLD is deep, as in the later stage of N3.

b. Variation of entrainment with LC

The factors to determine the entrainment rate we (=dh/dt) in the presence of LC can be represented as
i1520-0485-41-3-472-e1
Dimensional analysis leads to
i1520-0485-41-3-472-e2
where υL [=(Usu*2)1/3] and hΔB/υL2 [=(hΔB/u*2)(u*/υL)2] are used instead of Us and υL/u*. The momentum difference across MLD [(ΔU)2 + (ΔV)2]1/2 cannot be an independent parameter, because it is determined by u*, h, N, and f, as long as momentum is generated by wind stress.
If h/λ and h/(u*/f ) remain invariant, we assume the functional form of (2) as we/u* = f1(hΔB/u*2)[1 + f2(hΔB/υL2)], where f2 approaches zero with the increase of hΔB/υL2. In this case, the difference of we between the cases with and without LC Δwe or equivalently Δbw(h) (=ΔweΔB) can be expressed as
i1520-0485-41-3-472-e3

Figure 2 clearly identifies that Δbw(h)/bw0(h) decreases with hΔB/υL2, which means that the effect of LC on entrainment are significant only for small h and ΔB, as observed in Fig. 1. Here data are obtained by calculating the value of bw(z = h) averaged over the period corresponding 20 m < h < 30 m from each experiment to eliminate the dependence on h/λ and h/(u*/f ), so the values in Fig. 2 actually represent the averaged ones over a certain range of hΔB/υL2, shown by horizontal bars. Note that Δbw(h) in Fig. 2 represents the difference of bw(h) at the same h, whereas Δbw(h) in Fig. 1 represents the difference of bw(h) at the same t. Figure 2 also implies that, if h and ΔB are the same, the entrainment rate increases with La−1 for smaller La−1, but becomes insensitive to La−1 for larger La−1, which is consistent with Fig. 16b in Grant and Belcher (2009).

To incorporate the effect of LC, Li et al. (1995) and Smith (1998) modified the mixed layer model by Price et al. (1986), or the PWP model, which assumes that the mixed layer deepening occurs by shear-driven turbulence when the condition
i1520-0485-41-3-472-e4
is satisfied at the MLD, where ΔU is the velocity jump across the MLD. In the presence of LC, they suggested that the deepening can also occur, not only when the condition (4) is satisfied but also when hΔB/υL2 is smaller than a critical value, or
i1520-0485-41-3-472-e5
In particular, Smith (1998) suggested a critical value as C = 9.8 with υL = (Usu*2)1/3. It implies that an entrained parcel must acquire kinetic energy to overcome the increase in potential energy corresponding to its being mixed over the MLD (Smith 1998).

The present result, shown in Fig. 2, is conceptually similar to the mixed layer model by Li et al. (1995) or Smith (1998), if their models are interpreted as that the mixed layer deepening by shear-driven turbulence continues until the condition in (4) is satisfied, and the additional deepening by LC occurs if the condition in (5) is satisfied. The present results imply, however, that the additional entrainment decreases gradually with hΔB/υL2 rather than disappears abruptly at a critical value of hΔB/υL2. The criterion hΔB/υL2 < 9.8, suggested by Smith (1998), roughly represents the regime of the significant effect of LC in Fig. 2.

The present result is also consistent with the previous reports that the effect of LC is mostly confined to the initial stages of mixed layer growth, and there is no evidence that LC has a direct role in mixing near the base of a 40–60-m-deep mixed layer (Weller and Price 1988; Skyllingstad et al. 2000; Thorpe 2004). The LES results (Sullivan et al. 2007; Grant and Belcher 2009; Kukulka et al. 2009), in which LC induces a noticeable increase of entrainment, were obtained under the condition in which hΔB/υL2 is less than 40. Note that these simulations started with the initial profiles with ΔT = 0°C, h0 larger than 30 m, and weak stratification below (e.g., ∂T/∂z = 0.01 K m−1).

c. Modification of profiles in the mixed layer under LC

Here we investigate how the profiles of various physical quantities in the mixed layer, such as velocity shear, stratification, eddy viscosity and diffusivity, and the velocity and length scales of turbulence, are modified under the influence of LC, which helps to clarify the mechanism for the enhanced entrainment under LC. Profiles are made at t = 2 h for the case N1 and at t = 15 h for the case N3, which represent the typical cases of strong and weak effects, respectively, of LC on the mixed layer deepening, as mentioned in previous sections.

The values of S2 are drastically decreased in the presence of LC in both cases of N1 and N3, as already reported in previous reports (McWilliams et al. 1997; Noh et al. 2004; Li et al. 2005; Skyllingstad 2005) (Fig. 3). Especially in the case of L2, S2 almost disappears at a certain depth (z/h ∼ 0.2), suggesting that momentum transport may occur by means of nonlocal mixing (McWilliams and Sullivan 2000; Smyth et al. 2002). Furthermore, Fig. 3 shows that S2 decreases at the MLD as well as within the mixed layer. It should be mentioned, however, that the magnitude of shear production at the MLD is nearly the same, regardless of LC (not shown). The values of N2 also decrease in the presence of LC, especially in the case of N1, but they are not so sensitive to the presence of LC as S2. Generally speaking, LC always enhances vertical mixing greatly within the mixed layer, regardless of whether the contribution of LC to the mixed layer deepening is significant.

As expected from the fact that LC is more effective to reduce S2 than N2, Ri (=N2/S2) becomes larger under LC within the mixed layer (Fig. 4). It is also found that the decrease of Ri is less sensitive near the MLD than within the mixed layer. The time series of Ri (z = h) shows that Ri maintains a constant value at the MLD up to tπ/f after the initial stage: that is, Ri (z = h) ∼ 0.8, which is consistent with the assumption made in Pollard et al. (1973) [Ri (z = h) = 1]. After the inertial period (t > π/f ), however, it increases exponentially with time, as the mean velocity within the mixed layer decreases a result of inertial oscillation, as shown by Noh et al. (2010).

It is possible to consider that the increased entrainment under LC is attributed to the enhanced shear at the MLD resulting from the stronger mixing of momentum by LC within the mixed layer, as mentioned in Thorpe (2004). The decrease of S2 (z = h) in the presence of LC suggests, however, that it is more likely to be attributed to the direct effect of engulfment by large-scale eddies of LC impinging on the MLD, similarly to the case of convective eddies.

In accordance with the enhanced vertical mixing within the mixed layer, much larger eddy viscosity and diffusivity, Km and Kh, are found in the presence of LC, in both N1 and N3 (Fig. 5). Here Km and Kh are calculated by −uwU/∂zυwV/∂z = Km[(∂U/∂z)2 + (∂V/∂z)2] and −bw = KhB/∂z. Note that negative Km appears near the surface at L2, as observed by Sullivan et al. (2007), reflecting the nonlocal nature of vertical mixing by LC.

The profiles of velocity and length scales of turbulence, q [=(uiui)1/2] and l, indicate that the increase of Km and Kh under LC is attributed predominantly to the increase of l, although both q and l increase under LC (Fig. 6). Here, l is calculated from Kh = Shql with Sh = 0.49 (Mellor and Yamada 1982; Noh and Kim 1999). It indicates that the increase of TKE itself is not sufficient to explain the stronger vertical mixing under LC. It is also interesting to find that the ratio of l between the cases with and without LC is much larger than 1 when N2 is small, such as at z/h ∼ 0.2, but it is close to 1 when N2 becomes large, such as at z/h ∼ 0.8. This will be discussed further in the next section.

d. Parameterization of vertical mixing in the presence of LC

The suppression of vertical mixing under stratification is often parameterized in terms of Ri (=N2/S2) in mixed layer models (Pacanowski and Philander 1981; Mellor and Yamada 1982; Kantha and Clayson 1994; Canuto et al. 2001). It implies that, as in the case of the atmospheric boundary layer, entrainment is dominantly contributed by turbulent eddies generated by velocity shear. This type of parameterization is usually evaluated based on the assumption of local equilibrium Ps + Pb = ε with the neglect of TKE flux, where Ps is shear production, Pb is buoyancy production/decay, and ε is dissipation rate (Mellor and Yamada 1982; Kantha and Clayson 1994; Canuto et al. 2001). It has been found, however, that the divergence of TKE flux becomes a dominant turbulence source in the presence of LC (Noh et al. 2004; Sullivan et al. 2007; Polton and Belcher 2007; Grant and Belcher 2009).

Noh and Kim (1999) suggested that the parameterization of stratification on vertical mixing in the ocean mixed layer should be parameterized in terms of the Richardson number based on TKE itself {i.e., Rt [=(Nl0/q)2], rather than Ri}, because shear production is no longer a dominant source of TKE. Here, l0 is the length scale in the homogeneous mixed layer. In particular, they suggested a formula such as
i1520-0485-41-3-472-e6
with an empirical constant α, based on the assumption that l approaches the buoyancy length scale lb (=q/N) with increasing stratification. Here, the length scale l0 is prescribed by an empirical formula as
i1520-0485-41-3-472-e7
where κ is the von Kármán constant (=0.4) and the length scale at the sea surface is given by z0 = 1 m (Noh and Kim 1999). The mixed layer model based on this parameterization is shown to reproduce well the realistic upper-ocean structure (Noh and Kim 1999; Noh et al. 2002, 2005, 2007; Hasumi and Emori 2004; Rascle and Ardhuin 2009).

Figure 7 examines how l/l0 varies with Ri and Rt for the case of N3 with and without LC (L2-N3 and L0-N3). The data were obtained every hour at each grid depth from the surface to MLD. The data l/l0 versus Ri show a good collapse in the absence of LC, but they show a large scatter without apparent correlation in the presence of LC. It indicates that Ri is an appropriate parameter only in the case where shear production is a dominant source of TKE as in the atmospheric boundary layer. On the other hand, the data l/l0 versus Rt show a good collapse in both cases with and without LC. Moreover, the functional forms of correlation are similar to each other, suggesting the universality of the parameterization in terms of Rt. Figure 7 convinces us that vertical mixing should be parameterized in terms of Rt instead of Ri in the ocean mixed layer.

To clarify further the relationship between l/l0 and Rt, we plot the data from six experiments (L0-N1, L1-N1, L2-N1, L0-N3, L1-N3, and L2-N3) in the logarithmic scale for both Km and Kh (Fig. 8). To calculate l/l0 for Km, Km = Smql is used with Sm = 0.39 (Mellor and Yamada 1982; Noh and Kim 1999), and the data with negative l/l0, originating from the negative Km near the surface in the case L2 (Fig. 4), are not included. The general pattern is similar for Km and Kh, although data are slightly more scattered for Km.

It is remarkable to find that the data l/l0 versus Rt more or less collapse at larger Rt (Rt > ∼1), regardless of the presence of LC, as the similarity in the relationship between l/l0 and Rt in Fig. 7 suggests. In particular, the relationship between l/l0 and Rt is found to follow (6) with α ∼ 50, as shown by a solid line. The slope corresponding to l/l0 ∝ Rt−1/2 indicates that l approaches α−1/2lb with increasing Rt.

Meanwhile, Fig. 8 also shows that, at smaller Rt (Rt < ∼1), l/l0 becomes larger in the case with LC than without LC, and the ratio of l/l0 between the cases with and without LC continues to increase to above 10 with the decrease of Rt. The ratio is larger for stronger LC. Note also that l increases by more than 10 times in the presence of LC, when N2 almost disappears at z ∼ 0.2h, but it is less affected by LC, when N2 becomes larger at z ∼ 0.8h in Fig. 6. This confirms again the fact that the influence of LC on the mixed layer deepening is important only for weaker stratification and shallower depth, as already found in Figs. 1 and 2.

The general pattern for l/l0 versus Rt, shown in Fig. 8, can be explained in terms of the increase of l0 under LC. Equation (6) predicts that l approaches l0, as Rt approaches 0, whereas it approaches α−1/2lb independent of l0, as Rt becomes large. Because the length scale in the presence of LC is much larger than the one given by (7), l/l0 becomes much larger than 1 in Fig. 8 as Rt approaches 0. Consequently, we expect that the parameterization (6) remains valid, regardless of the presence of LC, but a much larger value of l0 should be used in the presence of LC. For example, if l0 is replaced by γl0, (6) is modified to
i1520-0485-41-3-472-e8
The prediction from (8) with γ = 10 is shown to predict well the relationship between l/l0 and Rt for the case L2 (Fig. 7). The length scale in the presence of LC may depend on λ and h as well as La, however.

e. Sensitivity to the e-folding depth of the Stokes drift

It has been reported from the recent LES works that the vertical TKE can be affected by the e-folding depth of the Stokes drift, or λ, when λ/h is large (Harcourt and D’Asaro 2008; Grant and Belcher 2009). To investigate the sensitivity of λ to the mixed layer deepening, we compared the results with λ = 40 and 80 m for the case L2. Figure 9 shows that the effect of λ is insignificant in both N1 and N3, although the mixed layer deepening is slightly faster for larger λ in the initial stage.

4. Conclusions

Analysis of LES data reveals the consistent pattern with regard to the influence of LC on the mixed layer deepening in the present paper. That is, LC induces a significant enhancement of the mixed layer deepening only if h and ΔB are small. This property is attested by the facts that the difference of the entrainment at the MLD between the cases with and without LC decreases with hΔB/υL2 (Fig. 2) and that the ratio of the mixing length scale l between the cases with and without LC is close to 1 for larger Rt (Rt > ∼1) but continues to increase to above 10 with the decrease of Rt (Fig. 8). The present result is in agreement with previous reports that the effect of LC is mostly confined to the initial stage of mixed layer growth (Weller and Price 1988; Skyllingstad et al. 2000; Thorpe 2004). It is also conceptually similar to the model suggested by Li et al. (1995) and Smith (2001) that the additional deepening of the mixed layer occurs in the presence of LC, when hΔB/υL2 becomes less than a critical value.

The magnitude of velocity shear at the MLD tends to be slightly decreased in the presence of LC (Fig. 3). It suggests that the engulfment by large-scale eddies of LC impinging on the density interface at the MLD may contribute to the increased entrainment directly, similarly to convective eddies, rather than the increased shear at the MLD. However, unlike convective eddies, which is driven continuously by buoyancy force, eddies generated by LC are essentially unforced and behave inertially, because the vortex force is confined to near the surface. Therefore, it is expected that eddies generated by LC are efficient for the vertical mixing under weak stratification, such as within the mixed layer or across the MLD with small ΔB, but their contribution to the vertical mixing becomes negligible under strong stratification, such as across the MLD with large ΔB. Moreover, the effect of LC decreases with depth, because the intensity of eddies generated by LC decreases rapidly with depth, unlike convective eddies.

Furthermore, it is clearly illustrated that the effect of stratification on vertical mixing in the presence of LC should be parameterized in terms of Rt [=(Nl0/q)2] instead of Ri [=(N/S)2], because shear production is not a dominant source of TKE any more. In particular, the mixing length scale is parameterized by l/l0 = (1 + αRt)−1/2 with α ∼ 50, as suggested by Noh and Kim (1999), regardless of the presence of LC. However, the presence of LC makes l0 much larger than conventionally used for the boundary layer. The inclusion of nonlocal mixing may be necessary for the more elaborate parameterization of vertical mixing, especially to account for negative Km near the surface at small La (Fig. 5), but the present result suggests that the effect of LC can be largely represented by the increase of the mixing length scale. It is also found that the increase of Km and Kh under LC is attributed predominantly to the increase of l, although both q and l increase under LC.

Finally, it is important to mention that the present simulation is initiated by applying the wind stress to a motionless mixed layer with stratification, which is more relevant to storm events. However, most analyses are performed after quasi-equilibrium state is reached at about 10h0/u* (∼1 h) after the onset of the wind stress, which implies that the present result can be applied more generally.

Acknowledgments

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (MEST) (NRF-2009-C1AAA001-0093068), by the Korean Ocean Research and Development Institute (KORDI) research program PE98512, and by the project “Research for the Meteorological Observation Technology and Its Application” at the National Institute of Meteorological Research.

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Fig. 1.
Fig. 1.

Time series of (top) MLD (h), (middle) the buoyancy flux at the MLD [bw(z = h)], and (bottom) Δbw(h)/bw0(h). Here, Δbw(h) = bw(h) − bw0(h), and bw0(h) is the buoyancy flux in the absence of LC (La = ∞). Blue is L0 (La = ∞); green is L1 (La = 0.64); red is L2 (La = 0.32); and black is the theoretical prediction by Pollard et al. (1973), which is valid only up to t = π/f: (a) N1 (N2 = 10−5 s−2) and (b) N3 (N2 = 2 × 10−4 s−2).

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 2.
Fig. 2.

Variation of Δbw(h)/bw0(h) with hΔB/υL2. Here data were obtained by calculating the value of bw(z = h) averaged over the period of 20 m < h < 30 m from each experiment. Horizontal bars represent the corresponding range of hΔB/υL2.

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 3.
Fig. 3.

Profiles of velocity shear S2 and stratification N2 [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 4.
Fig. 4.

(top) Profiles of Ri and (bottom) time series of Ri (z = h) [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 5.
Fig. 5.

Profiles of eddy viscosity Km and eddy diffusivity Kh [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 6.
Fig. 6.

Profiles of the mixing length scale l and the turbulent velocity scale q. Here, l was calculated from Kh = Shql with Sh = 0.49 [dotted line is L0 (La = ∞), dashed line is L1 (La = 0.64), and solid line is L2 (La = 0.32)]: (a) N1 (t = 2 h) and (b) N3 (t = 15 h).

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 7.
Fig. 7.

Scatterplots l/l0 vs (left) Ri and (right) Rt (N3). Data were obtained every hour at each grid depth for z < h from Kh: (a) L0 and (b) L2.

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 8.
Fig. 8.

Scatterplots in the logarithmic scale for l/l0 vs Rt. Data were obtained every hour at each grid depth for z < h [blue (N1, L0), light blue (N3, L0), green (N1, L1), light green (N3, L1), red (N1, L2), and light red (N3, L2)]. The black solid and dashed lines represent the predictions from l/l0 = (1 + αRt)−1/2 and l/l0 = γ(1 + αγ2Rt)−1/2, where α = 50 and γ = 10: (a) Km and (b) Kh.

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

Fig. 9.
Fig. 9.

Time series of (top) MLD (h) and (bottom) the buoyancy flux at the MLD [bw(z = h)] for the case L2 (solid line is λ = 40 m and dashed line is λ = 80 m): (a) N1 and (b) N3.

Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4494.1

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