a. Wind deepening

The first wind-driven case considered is that conducted by Martin (1985, 1986) where comparisons have been made with the Mellor–Yamada level 2 (1974), level 2.5 (1982), and Therry-Lacarrère (1983) turbulence closure schemes, as well as with the Niiler (1975), Garwood (1977), and Price–Weller–Pinkel (1986) bulk models. For this experiment the initial stratification was 0.05° C m⁻¹, the salinity was constant at 35 psu, the surface heat and salinity fluxes were zero, and the inertial period was 24 h. Displayed in Table 2 are boundary and mixed layer depths after 5 days of forcing for wind stress values of 1, 4, and 16 dyn cm⁻² (1 dyn cm⁻² = 0.1 N m⁻²) acting on an initially shearless and motionless fluid having U(z, t = 0) = V(z, t = 0) = 0. As expected, the final depths are proportional to the square root of the wind stress. Perhaps surprisingly, under this forcing the present model appears to behave more like the Garwood and Niiler bulk models than like the Mellor–Yamada models. The reason for this lies in the stability functions utilized in those schemes. For example, in the level 2.5 scheme S_m and S_h, the eddy viscosity coefficients for momentum and heat respectively, can be determined algebraically through (Mellor 1989):

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where

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and (A₁, A₂, B₁, B₂, C₁) = (0.92, 0.74, 16.6, 10.1, 0.08). These expressions result from the closure approximations that they have made and do not apply to our formulation. As a curiosity, we imposed these constraints on S_m and S_h and found that our results came in close agreement with theirs.

The next case considered corresponds to a steady wind stress of 0.3 N m⁻², constant temperature gradient of 0.1° C m⁻¹ and initial velocity profiles U(z, t = 0) = V(z, t = 0) = 0 as used in the work by Pollard et al. (1973). Plotted in Fig. 1 is the mixed layer depth according to the PRT model and our boundary layer depth versus time. Time has been made dimensionless by normalizing it with half the inertial period, T = π/f. The diagram shows that for early times the two models agree well and behave like h ∼ u∗(t/N). This behavior was also confirmed in the work of Kundu (1981) by assuming a self-similar structure in the flow for early times. After half an inertial period, deepening in the PRT model is arrested by rotation and the mixed layer depth remains constant, while the present model shows a transition occurring in the growth rate. The small oscillations observed are from fluctuations in shear at the base due to mixing, while the larger oscillation is due to inertial motions.

b. Heating

For the heating experiment the model was initialized as in Martin (1985, 1986) with a uniform temperature from the surface down to a depth of 100 m and a linear stable stratification of 0.05° C m⁻¹ below and zero initial velocity profiles. Also, it was forced with a constant wind stress of 1 dyn cm⁻² and heat fluxes of 150, 600, 2400 ly day⁻¹ (1 ly day⁻¹ = 0.484 W m⁻²) applied at the surface (with no solar radiation). This case represents the opposing effects of mixing due to wind generated turbulence and the stabilizing effect of surface heating. As in the wind-deepening case, all the turbulent fluxes collapse to downgradient expressions. In Table 3 the boundary layer depth of the present model is contrasted with the mixed layer depths of several models after 2 days of forcing. Our model behaves closest to the models of Mellor–Yamada for this stable case. For stronger heating all the differential models behave similarly. The Niiler and Garwood bulk models produce shallow mixed layer depths for the heating experiment. It was observed that our model produced significant stratification near the surface for high surface heat fluxes indicating that mixing was not strong enough to keep the mixed layer temperature uniform. This is to be expected though since the source of heat is at the surface with no penetration into the fluid column, as would be the case if solar radiation were present. As commented in Martin (1986), this also occurred with the other turbulence closure schemes; the bulk models, on the other hand, respond quickly and hence display little stratification. The heating case produces an example of the difficulties that arose in using the Ri_B and ΔT criteria for determining h. Because of the large stratification that can occur near the surface, these two criteria predicted an unrealistically shallow depth for high heating rates, whereas the TKE criterion was not so affected by the presence of such a stratification. One must bear in mind though that this heating experiment is unrealistic since high heating rates are expected to occur as a result of solar radiation and hence allowed to decay beneath the surface. This was not the case in this experiment.

c. Cooling

We begin examination of the effects of convection by simulating the case considered by Martin (1985, 1986). Here, the initial conditions were identical to those in the wind-deepening case. Forcing is due to a constant wind stress of 1 dyn cm⁻² and surface fluxes of −100, −200, and −300 ly day⁻¹. Table 4 compares the boundary and mixed layer depths of several models after a forcing duration of 120 days. The long forcing period simulates the cooling typical of the winter season and also magnifies the differences in the various models. The shallowest mixed layer depths result from the Mellor–Yamada, Price–Weller–Pinkel, and Therry–Lacarrère models, while the deepest from the Niiler model. The further deepening beyond the shallowest depths in the present model can be attributed to the nonlocal and countergradient contributions to the turbulent fluxes, which are activated in convective situations. Plotted in Fig. 2 is the normalized heat flux profile at the end of the 120 day run for the cooling rate of −300 ly day⁻¹. The diagram reveals that the magnitude of the flux generated at the base of the mixed layer is about 15% of the surface value, which is consistent with observational findings on convective boundary layers (Stull 1976).

The next comparisons are for the important case of free convection. Listed in Table 5 are the boundary layer depths for the experiment proposed by Large et al. using their similarity-based KPP model. The surface is subject to a steady cooling of −100 W m⁻² and has an initial linear stable stratification of 0.1° C m⁻¹. As with all these idealized forcing examples, the salinity was kept constant. The depth shown is after 3 days of cooling and are in good agreement. Also presented in Table 6 are boundary layer depths at various times for a case conducted with the third-order model of André–Lacarrère. The agreement for this experiment is within 11%. Here, the surface is cooled at −200 W m⁻² and the initial profiles as well as the profiles at later times are illustrated in Figs. 3a and 3b. Also shown in these diagrams are the corresponding results of André–Lacarrère, which are represented as dots. The agreement demonstrates that our second-order model does reasonably well in reproducing their velocity and temperature structures. In Fig. 3a we see that convectively generated turbulence is capable of efficiently mixing momentum throughout the mixed layer in contrast to wind generated turbulence, which typically produces an almost constant shear throughout the mixed layer. Also, the temperature profiles in Fig. 3b remain slightly unstable near the surface due to the cooling.

d. Ocean weather station datasets

In addition to the above simple forcing experiments, the model was further tested against observational data taken from Ocean Weather Stations (OWS) November (N) and Papa (P) located in the North Pacific at 30°N, 140°W and 50°N, 145°W respectively. For these simulations the model was run with a uniform 5-m grid spacing that extended down to a depth of 200 m and a 15-min time step. The sensitivity of the results to the grid spacing will be briefly discussed later in this section. Since previous studies such as Martin (1985, 1986), Large et al. (1994), and Kantha and Clayson (1994) have focused on the year 1961, we have decided to follow suit and begin our simulations on 1 January, which we denote as day 0. The datasets consist of 3-h meteorological and BT observations that were used to initialize the temperature profiles. Unfortunately, no velocity, salinity, or TKE measurements are available. This, of course, raises the concern of how to initialize these profiles as well as the sensitivity to these unknown profiles. These sensitivity issues will be addressed later when we present the results for OWS P. From the meteorological data (which included SST, air, wet-bulb, and dewpoint temperatures, wind speed and direction, cloud cover, and surface air pressure) surface fluxes of heat, momentum, evaporation, and radiation can be computed. To do this we utilized the bulk formulas that are fully described in Martin (1985). Fluxes required by the model between the 3-h measurements were obtained by interpolation. As previously mentioned, the boundary layer depth can be determined from the observed BT using a temperature jump criterion. It was defined as the depth at which the temperature drops by 0.1°C from the sea surface value. The effects of advection were not taken into account in our simulation, and the consequences of this will be commented on later. The extinction of solar radiation in seawater as classified by Jerlov (1976) according to turbidity was used in the model with optical type I at OWS N and type II at OWS P. Last, we have assumed a 6% surface albedo at both stations.

At OWS N salinity was held constant during the yearlong simulation. The justification for ignoring salinity effects is supported by the 20 years of analysis of meteorological and oceanographic data performed by Dorman et al. (1974). Also, the velocity and TKE profiles were initially set to zero. The clear-sky insolation was corrected for cloud cover using the correction and solar transmission coefficient set forth by Reed (1977). The results of the simulation are shown in Figs. 4a–c. Obvious features of the simulated SST are spikes that occur during the spring and summer. Similar spikes are also present in the models used by Martin (1985, 1986) and that of Kantha and Clayson (1994) and are the result of the combination of strong heating and weak winds acting on a shallow mixed layer. Figures 4a and 4b contrast the observed and simulated SST and boundary layer depth respectively. Large differences in the SST are apparent in Fig. 4c. However, as explained in Martin (1985), the observed SST at OWS N does show some spurious features where the temperature jumps abruptly and should be taken into account when evaluating the comparison.

For simulations carried out at OWS P the clear-sky insolation was corrected for cloud cover using the correction and solar transmission coefficient of Tabata (1964). Salinity must be treated more carefully since the works of Beatty (1977) and Tabata (1965) both report that a strong halocline exists below a depth of ∼125 m. The climatological data of Beatty was used to initialize the salinity profile. To investigate the sensitivity to salinity effects we ran several simulations. First, we ran a trial simulation having a constant initial salinity profile (instead of the climatological data of Beatty), which of course ignored the halocline. In the second simulation we set the salinity flux to zero during the entire run, as did Martin, and used the climatological data for the initial salinity profile. For the third simulation we followed the suggestion of Gaspar (1988), which is to keep the initial salinity profile constant throughout the entire year (again using the profile of Beatty). Last, a fourth simulation was performed similar to the second with the exception that the salinity flux was not set to zero but estimated based upon evaporation that was computed and precipitation that was specified according to the observed mean annual cycle of Tabata (1965). For the three simulations, excluding the first case, the resulting SST profiles were only slightly different from one another and thus justifies using the climatological data as a reasonable initial salinity profile. As to be expected, ignoring the halocline produced a significantly deeper mixed layer during the wintertime.

To investigate the sensitivity to the initial velocity profiles we again ran several simulations whereby different initial profiles were selected ranging from the trivial choice U(z, t = 0) = V(z, t = 0) = 0 to various linear profiles. The results showed negligible dependence on the initial velocity profiles. The velocity profiles after a short time, on the order of days, were virtually independent of the initial profiles. Last, the sensitivity to the initial TKE profile was also studied. Here we considered two initial profiles: the trivial choice e(z, t = 0) = 0 and expression (A8) derived in the appendix. Once again the results revealed negligible dependence on the initial TKE profile. Presented and contrasted in Figs. 5a–c are observed and simulated SST and boundary layer depth profiles for the case where evaporation and precipitation were incorporated. The agreement between the observed and simulated SST, as shown in Fig. 5a, is good as is the agreement in h portrayed in Fig. 5b. Figure 5c reveals that noticeable discrepancies occur in the final and summertime SST where in both instances the model overestimates the observed value. The final SST is overestimated by about 0.9°C. A similar overestimation was also obtained by Kantha and Clayson and as they mention could be partly attributed to the neglect of advection effects. Large et al. (1994) point out that horizontal advection is important during the period running from September to February. Another apparent difference is an underpredicted mixed layer depth during the winter due to the strong halocline. Since the observed depth is based solely on temperature, it mimics the thermocline depth and does not take into account the halocline. Plotted in Fig. 6 are simulated and observed temperature profiles at various times during the year-long simulation. This illustrates that the model is able to reasonably capture the seasonal temperature structure in addition to the seasonal SST. We also carried out experiments where the grid spacing was varied. Illustrated in Fig. 7 are the SST profiles corresponding to the three resolutions: Δz = 2.5 m, Δz = 5 m, and Δz = 10 m. The three profiles are in good agreement and show the largest difference occurring during the summertime. This is to be expected though since the mixed layer can get quite shallow and sensitive to the minimum depth allowed, which was set equal to the grid spacing Δz. As a final note we wish to comment that computing h using the different criteria mentioned (i.e., using Ri_B and ΔT) produced similar profiles for the boundary layer depth as that presented. Since the agreement with the observed depth is reasonable, this supports our approach in using the TKE extinction method.

a. Langmuir circulation

Langmuir circulations can be described as organized convective motions in the surface layer of the ocean that play a prominent role in upper-layer mixing. Apart from surface and internal gravity waves, Langmuir cells are considered to be the most important coherent structures in the ocean boundary layer according to Weller and Price (1988). While numerous theories have been proposed to explain these circulations (see Pollard 1977), the prevailing theory is that due to Craik and Leibovich (1976). This theory claims that Langmuir circulations are an instability arising from the nonlinear interaction between the Stokes drift and the frictional wind drift current that is initiated by a vortex force, which enters into the momentum equations. This model is known as the CL2 instability mechanism. Visible manifestations of Langmuir circulations are streaks (or windrows) formed nearly parallel to the wind caused by a series of subsurface counterrotating vortices that closely resemble convection rolls [although they form regardless of surface heating or cooling, the effects of heating or cooling was numerically investigated by Li and Garrett (1995)]. These circulations normally do not exist in winds less than approximately 3 m s⁻¹. Also, they realign themselves with shifts in the wind direction. The wind drift current has larger velocities in the convergence zones. Langmuir (1938) believed that this is because the water there had been on the surface since it rose from the divergence zone and therefore had been accelerated by the wind for the longest period of time. Langmuir also claimed that the vortices are largely responsible for the formation of the thermocline and the maintenance of mixed layers. These circulations have been successfully modeled by Leibovich (1977), Leibovich and Radhakrishnan (1977), and Li and Garrett (1993) using two-dimensional models. Recently, three-dimensional large-eddy simulations of Langmuir circulations have been performed by Skyllingstad and Denbo (1995) and McWilliams et al. (1997), while a three-dimensional model was advanced by Tandon and Leibovich (1995).

To date, ocean models have not yet included the effects of Langmuir circulation. However, using the Price–Weller–Pinkel bulk model Li et al. (1995) and Li and Garrett (1997) have recently proposed introducing another criterion, called the Langmuir circulation index LC_n, to determine if deepening should occur. This condition is used in addition to the bulk Richardson number criterion and generally produces a deeper mixed layer. The physics behind this criterion is based upon the Froude number, Fr = w_d/(Nh), where w_d is the maximum downwelling velocity produced by Langmuir circulations in a homogeneous fluid. Their numerical simulations show that this quantity is constant. In the present scheme we have accounted for the effects of Langmuir circulation through the following modifications to the model when the winds exceed 3 m s⁻¹:

1. Adding to the convective velocity, w∗, a mechanically driven contribution due to Langmuir circulation. Here we exploit the nonlocal versatility of our parameterizations for third moments; and

2. including in the equations the contributions arising from the vortex force responsible for producing Langmuir circulations. As we will see this modifies the TKE and momentum equations.

An obvious choice for the mechanical contribution to the convective velocity w∗_LC is to use the maximum downwelling velocity w_d. With |w_d| ≈ (0.0025 to 0.0085)(U²_w + V²_w (Leibovich 1983) we can write

w_LCc(U²_w + V²_w)

where c lies in the range 0.0025 < c < 0.0085. The vortex force that must be included in the momentum equations is given by V_s × ω (Leibovich 1977), where V_s is the Stokes drift velocity and ω is the absolute vorticity. The Stokes drift can be parameterized as in Li and Garrett (1993) for a fully developed sea:

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with

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Owing to the assumption of horizontal homogeneity, adding the vortex force to the momentum equations only affects the Coriolis terms. This has been pointed out in the very recent work of McWilliams et al. (1997). The Coriolis terms in Eqs. (1) and (2) become f(V + V_s) and −f(U + U_s), respectively, which now include the Stokes drift. Another consequence of adding the vortex force to the momentum equations is a modification to the TKE equation, which now takes the form

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A full derivation of the above is given in the work by Skyllingstad and Denbo (1995). The result is that the following new terms appear on the right-hand side of the TKE equation:

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which can be rewritten as

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Written in this way we see that the shear production and transport terms have been modified by the inclusion of the vortex force. The shear production term (SP) now reads

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To investigate the outcome of these modifications we ran the yearlong simulation at OWS P with Langmuir circulation since the winds are stronger there and the dataset is of better quality. The results of that simulation with c = 0.006 are illustrated in Figs. 8a,b. In Fig. 8a we plot the boundary layer depth, which shows episodes of dramatic deepening at various instances; illustrated in Fig. 8b is a running 5-day average of the depth difference with and without Langmuir circulation. It is clear from these diagrams that the effects of Langmuir circulation, as implemented here, can lead to further deepening. This deepening is especially prevalent during the fall and winter when convection and winds are stronger. The only noticeable change in the sea surface temperature worth mentioning is a drop in the final SST of about 0.2°C compared to the case with no Langmuir circulation. The inclusion of Langmuir circulation does improve the agreement with observation in that the rms value of the difference between the observed and simulated h is reduced slightly and also the overestimation in SST near the end of the simulation is lowered. As a final note, the modified shear production term suggests redefining the bulk Richardson number Ri_B as follows:

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where

V_s²U_sz_sU_sh²V_sz_sV_sh²

Using the prescribed expressions for the Stokes drift with h ≫ L and z_s → 0, we find that

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It is interesting that the modified expression for Ri_B comes in close agreement with one proposed by Vogelezang and Holtslag (1998), for atmospheric use under stable conditions. Also, Large et al. (1994) suggest a form for Ri_B that is similar to that above whereby they replace |ΔV_s|² with a turbulent velocity. We point out that the above expression for Ri_B was not used in our case since the TKE extinction method was implemented to compute h and we offer the above only as an alternative.

b. Breaking waves

The recent measurements of Drennan et al. (1996), Terray et al. (1996), Anis and Moum (1992), Agrawal et al. (1992), and Gargett (1989), to mention a few, support the idea of enhanced turbulence in the oceanic surface layer as a result of wind wave action. In previous studies, for example Kundu (1980) and Klein and Coantic (1981), wave breaking is taken into account by prescribing a nonzero TKE flux at the surface z = 0. Following this approach, as in the recent work of Craig and Banner (1994), we proceed to investigate the consequences of this on the present scheme. This involves examining the limiting behavior of the TKE equation (5) in its steady-state form near the surface. In the absence of a surface buoyancy flux we expect that

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The TKE equation then assumes the form of a nonlinear ordinary differential equation given by

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where ζ = |z|/z₀, n² = 3/(Bk²S_mα_e) and λ = Bu⁴_*/S_m. This equation represents a three-way balance between shear, diffusion, and dissipation. In the absence of diffusion we get the familiar shear–dissipation balance given by q = λ^1/4. Craig and Banner also considered the balance between diffusion and dissipation to arrive at the dominant solution

qCζ^−n/3

where C′ is an arbitrary constant. Physically, this solution represents the decay in TKE from the surface where it is presumably generated. Our objective here is to present a more general solution to (18) that reduces to the above result when the right-hand side of (18) tends to zero. We begin by making the substitution (1 + ζ) = exp(ξ) and cast the equation in dimensionless form by defining q̂ = q/λ^1/4 and rescaling ξ as ξ̂ = nξ. Then, in terms of q̂², Eq. (18) can be rewritten as

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In this form the shear–dissipation balance becomes simply q̂ = 1. Next we set q̂² = 1 + F, where F denotes an enhancement factor above the shear–dissipation value due to wave breaking, and define R = dF/dξ̂ so that dR/dξ̂ = RdR/dF. With these in place and after some algebra we obtain

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In arriving at this expression we have invoked the condition that R = 0 when F = 0 and have chosen the negative root to give the desired asymptotic behavior, which is that q̂² → 1 sufficiently far away from the surface. Before presenting the exact solution to (20) we first illustrate two limiting cases:

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which recovers the solution of Craig and Banner, and

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which corresponds to the shear–dissipation limit. This demonstrates that, if wave breaking significantly enhances the turbulence, then the balance is established between dissipation and diffusion with little contribution coming from shear. The general solution to (20) is given by

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where C is an arbitrary constant and

ηF(3 + F)(1 + F)

It can be shown that the solution given by (21) collapses to the limiting forms above when the appropriate limits are taken.

In order to investigate the role of wave breaking we consider the limiting case of large F since in such circumstances wave breaking is expected to be the main source of turbulence. In this case we have F → C(1 + ζ)^−2n/3 and q₂ → λF in terms of the original variables. If we impose a TKE flux condition as in Kundu (1980) and Klein and Coantic (1981)

ew′₀−mu³_*

where m is a parameter ranging from about 10 to as high as 100, this enables us to determine the constant C and yields C = 4m^2/3S_m/B. Since the near surface structure of the TKE is known, this information can be used to convert the above flux condition into a Dirichlet condition and is equivalent to adding a wave breaking component to the coefficient α₁ in the TKE boundary condition giving

α₁m^2/3α₁

This limiting case considered becomes more and more valid as m increases. As well, we redefine z₀ with a Charnock-type formula

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as discussed in Craig and Banner. As noted in the recent review by Melville (1996), the specification of z₀ is an unresolved issue plaguing turbulence closure modeling. Craig and Banner found that in order to yield reasonable agreement with dissipation measurements much larger values of z₀ in the range 0.1–8 m had to be used. Simulations were run with these modifications and although these changes enhanced the turbulence near the surface, it showed a negligible effect in further deepening the mixed layer. This is also consistent with the findings of Kundu (1980) and Klein and Coantic (1981). The explanation for this, which is consistent with the above analysis, is that deeper beneath the surface the TKE becomes controlled by the mean flow through shear production and not by its surface conditions. Since the effects of wave breaking do not directly enter into the equations for the mean, little change in the TKE occurs at larger depths.

Considerable interest has been directed toward the vertical structure of ε and the prediction of its decay rate since the dissipation is a quantity that can be measured with reasonable confidence. Recent measurements (cited earlier) verified the existence of a wave zone characterized by enhanced dissipation due to wave breaking well in excess of

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as suggested by the classical logarithmic boundary layer over a rigid surface. The dissipation was first estimated to decay as z^−a with a in the range 3.0–4.6. However, as discussed in Melville (1996) the recent papers by Terray et al. (1996) and Drennan et al. (1996) both point to a slower z⁻² decay. The decay rate predicted by the Craig and Banner model is given by z^−3.4. According to the analysis presented above, our dissipation is found to decay with a power law behavior having an index of −2.2 and thus comes in close agreement with the currently accepted inverse square law.