## 1. Introduction

Studies of the upper ocean advance the understanding necessary to develop rules relating the turbulence to the forcing and the ocean state. These mixing rules allow turbulent fluxes to be estimated from information provided by observations, numerical models, or a combination of both through data assimilation. Unfortunately, the technology to measure fluxes directly in the open ocean does not yet exist, so observational knowledge is limited to indirect inferences, such as those from ocean microstructure (e.g., Moum et al. 2013).

In ocean general circulation models vertical mixing schemes employ a set of established rules to parameterize the vertical fluxes, whose divergences appear as terms in the prognostic equations for the resolved ocean state. Such schemes come in two distinct flavors, that Burchard et al. (2008) term statistical turbulence models (STM) and empirical turbulence models (ETM). STMs solve the Reynolds-averaged Navier–Stokes equations with a variety of second-order closures (e.g., Mellor and Yamada 1982; Kantha and Clayson 1994; Harcourt 2015) and some higher order (e.g., Cheng et al. 2002; Canuto et al. 2009). Empirical relationships simplify ETMs, such as the mixed-layer models of Kraus and Turner (1967) lineage, and of Price et al. (1986) and the *K*-profile parameterization (KPP) of the ocean boundary layer (Large et al. 1994). In the absence of direct measurements, evaluation of all schemes, as well as guidance on further developments, have relied heavily on comparisons of modeled and observed ocean states. However, discrepancies are caused both by inadequate rules and by forcing error, with no means to discriminate and the possibility of compensating errors a further complication.

An extensive comparison by Li et al. (2019) of 11 vertical mixing schemes, including both STM and ETM, demonstrates the current lack of community consensus. The primary metric is the mixed layer depth (MLD) of de Boyer Montégut (2004), and a robust conclusion is that the inclusion of Langmuir turbulence produces deeper MLD. Although this measure would be insufficient for either showing comprehensive agreement or for judging relative merit, it does expose “limited understanding and numerical deficiencies.” In particular, no two schemes agree globally over the annual cycle, so overall they do not perform as well as might be expected based on the theory, observations, and numerical experiments applied to their development. One of the focused demonstrations of disparate behavior is the annual cycle at Ocean Weather Station Papa, but unbalanced surface fluxes preclude the quantitative evaluations against observations of both the forcing and ocean state found in Martin (1985) and Large (1996), for example.

Large-eddy simulation (LES) is fundamentally different and offers unique opportunities, because it can be used to study turbulence (Burchard et al. 2008). LES models solve the Navier–Stokes equations by resolving the vertical fluxes down to scales of order a meter, such that subgrid-scale (SGS) contributions are small and relatively well parameterized. LES is key to many recent advances in the understanding and hence modeling of upper-ocean mixing physics (e.g., McWilliams et al. 1997; Wang et al. 1998; Grant and Belcher 2009; Harcourt and D’Asaro 2008; van Roekel et al. 2012; Li and Fox-Kemper 2017). The present study of the diurnal cycle follows directly from two particular examples: Large et al. (2019a, hereafter L19a) extends Monin–Obukhov similarity theory (Monin and Obukhov 1954) to include surface wave forcing (Stokes drift), and Large et al. (2019b, hereafter L19b) relates this forcing to both the local and the nonlocal transports of buoyancy and momentum throughout the boundary layer. Hereafter, these works will collectively be referred to as L19. These results and additional analyses of the same LES data (Large et al. 2021) are herein combined to develop rules for boundary layer entrainment and detrainment and for misalignment of the shear and turbulent stress vectors when changes in the forcing may be too rapid for equilibrium assumptions.

The LES model, the surface flux and wave forcing, and the various simulations are detailed L19, so only the most relevant aspects are recapped in section 2. Germane results from L19 are summarized in section 3, where there is also a novel analysis of similarity theory in the stably forced surface layer. In section 4, an empirical rule that includes Stokes forcing is developed for entrainment into both convective nighttime and stable afternoon boundary layers. In addition, the orientation of the stress vector is related to the Eulerian and Lagrangian shear over different vertical scales. Section 5 focuses on detrainment, which is broadly defined as daytime turbulence that depends strongly on past history, such as found below a stable morning boundary layer. In section 6, the diurnal cycles of buoyancy and momentum fluxes in the upper ocean are parameterized in two distinct ways, then both are evaluated against LES fluxes to take advantage of the LES mean ocean state and turbulence statistics being consistent with the specified “error-free” forcing.

## 2. LES of the Southern Ocean boundary layer

Large-eddy simulations of the ocean boundary layer are driven by the observed forcing shown in Fig. 1 at the Southern Ocean Flux Site (SOFS) near 47°S, 140°E (Schultz et al. 2012). As detailed in L19a, there are two cases from April 2010: AprS with wave forcing from realistic Stokes drift profiles and AprN without. Two additional cases from June 2010 are similarly denoted as JunS and JunN. Also utilized here is an idealized case, D24S, where the surface waves are always perfectly aligned with a westerly wind. All cases use the SOFS inertial period of 16.4 h. This study of the diurnal cycle focuses on AprS, because it is the only Stokes case with a period of surface buoyancy gain (Fig. 1c), beginning with the transition at *t*_{US} = 32.75 h from unstable buoyancy forcing (effective cooling) to stable (heating), until the reverse at *t*_{SU} = 40.75 h. The other cases extend the range of forcing conditions, including the prior April diurnal cycle, as well as another two in June (noon at hours 6 and 30) when there is net cooling throughout.

The LES model is well documented (Moeng 1984; Sullivan et al. 1994; McWilliams et al. 1997) and widely used in ocean boundary layer studies (e.g., Sullivan et al. 2012; van Roekel et al. 2012; Kukulka et al. 2013). The modifications to account for surface wave effects are extensive and summarized in L19a, as are the initial conditions and grid choices. Initially, the inversion depth of maximum stratification *d*_{i} is about 180 m in April, 222 m in June, and 96 m for D24S. The model solves the wave-averaged, incompressible, and Boussinesq Craik–Leibovich equation set (Craik and Leibovich 1976; McWilliams et al. 1997). Its thermodynamic variable is buoyancy *g*(1 − *ρ*/*ρ*_{o}), relative to a reference density *ρ*_{o}, where *g* is gravitational acceleration and *ρ* is ocean density. It produces evolving profiles of horizontal and time mean buoyancy Θ and horizontal Eulerian flow **U**, with orthogonal components, *U* and *V*. The corresponding turbulent vertical fluxes, ⟨*wθ*⟩, ⟨*wu*⟩, and ⟨*wυ*⟩, are given by the correlations of vertical velocity *w* with fluctuations, *θ*, *u*, and *υ*, plus small SGS contributions. Similarly, the variance of velocity fluctuations plus SGS give the turbulent kinetic energy, TKE, whose dissipation *ε*_{TKE} is parameterized. Hourly statistics converge and are computed half-hourly in order to track turbulent responses to the variable forcing that are faster than the responses of stratification and shear, and hence of eddy diffusivity and viscosity. Thus, only every second calculation is independent.

### a. Meteorological and Stokes (wave) forcing

*ρ*defines the kinematic wind stress

^{−1}. The wind direction of Fig. 1b is in degrees clockwise from north and there is always an eastward (westerly) component. The surface buoyancy flux

*B*

_{0}includes both heat and freshwater forcing, and defines the equivalent heating of Fig. 1c:

*α*= 1.7 × 10

^{−4}K

^{−1}, and

*C*

_{p}is ocean heat capacity, with

*ρC*

_{p}= 4.2 MJ m

^{−3}K

^{−1}. The maximum heating of 200 W m

^{−2}is much less than reached at noon in summer or at low latitudes, when the simplification of absorbing all the solar radiation in the first model level would not be as defensible.

Surface waves are imposed as profiles of Stokes drift

*ε*= 0.1, of a boundary layer of depth

*h*. In L19a, surface wave effects are incorporated into Monin–Obukhov similarity theory by regarding the sources of TKE over this layer as a forcing. Specifically, the contributions of Eulerian shear, Stokes shear, and buoyancy are given by the integrals

*P*

_{U},

*P*

_{S}, and

*P*

_{B}, that are parameterized in L19a, with updates coming in section 6.

Figure 2 shows the AprS distributions below 1 m of TKE, its dissipation, and its three production terms of both signs. The column integrals of these three terms above the inversion depth (Fig. 2d) are displayed as time series in Fig. 1. The buoyancy integral is often small (Fig. 1f), because of significant TKE suppression (⟨*wθ*⟩ < 0), and can become negative with heating (*Q*_{0} > 0). Usually the column Eulerian production is about ^{−1}, and the Eulerian shear production rapidly increases. There is a similar response prior to hour 37 in June, followed by the MRW period of moderate winds and counterinertial rotation.

The AprS nondimensional distributions below 1 m of (a) TKE and its (b) buoyancy production, (c) Eulerian shear production, (d) Stokes shear production, and (e) dissipation. The contours are irregular at ±0.10, ±0.30, ±1.0, ±3.0, ±10.0, and 30, with negative contours gray. The time series (red) show the boundary layer depth *h*^{L19} (solid), the entrainment depth *d*_{E} (dotted line), the inversion depth *d*_{i} (long-dashed line) of maximum stratification, and *E*_{W}. The vertical dashed lines demark the stable forcing between *t*_{US} and *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The AprS nondimensional distributions below 1 m of (a) TKE and its (b) buoyancy production, (c) Eulerian shear production, (d) Stokes shear production, and (e) dissipation. The contours are irregular at ±0.10, ±0.30, ±1.0, ±3.0, ±10.0, and 30, with negative contours gray. The time series (red) show the boundary layer depth *h*^{L19} (solid), the entrainment depth *d*_{E} (dotted line), the inversion depth *d*_{i} (long-dashed line) of maximum stratification, and *E*_{W}. The vertical dashed lines demark the stable forcing between *t*_{US} and *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The AprS nondimensional distributions below 1 m of (a) TKE and its (b) buoyancy production, (c) Eulerian shear production, (d) Stokes shear production, and (e) dissipation. The contours are irregular at ±0.10, ±0.30, ±1.0, ±3.0, ±10.0, and 30, with negative contours gray. The time series (red) show the boundary layer depth *h*^{L19} (solid), the entrainment depth *d*_{E} (dotted line), the inversion depth *d*_{i} (long-dashed line) of maximum stratification, and *E*_{W}. The vertical dashed lines demark the stable forcing between *t*_{US} and *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

### b. Diagnostic depths, regimes, and layers

Surface forced turbulence, including Stokes effects, dominates at depths, *d* = −*z*, in the boundary layer where *σ* = *d*/*h* is a natural vertical coordinate. Table 1 catalogs various boundary layer depths as well as other diagnostic and empirical depths and angles. In L19, the buoyancy flux (e.g., Fig. 2b) is used to diagnose the entrainment depth (*d*_{E}, dotted red trace) where the entrainment buoyancy flux *B*_{E} is a negative minimum, as well as a deeper boundary layer depth *h*^{L19} (solid red traces) where the flux approaches a local maximum, which is usually near zero. However, when the surface forcing does not dominate, this maximum can be shallow and significantly positive, such that the boundary layer and interpretation of *h*^{L19} become ambiguous. In particular, Fig. 2b shows a very abrupt collapse of *h*^{L19} at *t*_{US}, followed by two hours of deepening then 2 h of shoaling, but this behavior is reflected in neither TKE (Fig. 2a), nor its dissipation (Fig. 2e).

Catalog of depths and of angles between horizontal vectors, including empirical relationships, brief descriptions, basic dependencies, and references.

The examples of Fig. 3 show profiles of buoyancy flux (blue traces) and smoothed stratification (red traces). The stratification is scaled with depth, so it is constant where the buoyancy profile is logarithmic. The additional scaling by *κ* = 0.4, but becomes 10^{5}/*g* in the lower, in order to capture transitions across *B*_{0} = 0, and to display the local diurnal maximum at depths, *d*_{i} is more than 1000 and far off scale. These examples demonstrate that the diurnal cycle can be partitioned into at least four regimes, namely, a stable boundary layer (SBL), a detraining boundary layer (DBL), wind/wave dominated entrainment (*E*_{W}), and the most common regime *E*_{C} (Fig. 1; solid) when convective entrainment is more important. The familiar partition into unstable and stable boundary layers does not capture the diurnal cycle, because detrainment (regime DBL) becomes evident prior to *t*_{US} as the dominance of surface forcing diminishes, and characteristics of stable afternoon entrainment (regime *E*_{W}) can persist well past *t*_{SU}.

Profiles of buoyancy flux (blue) normalized by the greater of |*B*_{0}| or |*B*_{E}|, and of smoothed stratification (red) scaled (top) by ^{5}*d*/*g* from AprS regimes: (a) *E*_{C} at hour 2, (b) DBL at hour 34, with the entrainment depth at *t*_{ED} = 30.5 h, (c) SBL at hour 37 (*B*_{E} = − *B*_{0}), and (d) *E*_{W} at hour 40. The depths are described in Table 1.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Profiles of buoyancy flux (blue) normalized by the greater of |*B*_{0}| or |*B*_{E}|, and of smoothed stratification (red) scaled (top) by ^{5}*d*/*g* from AprS regimes: (a) *E*_{C} at hour 2, (b) DBL at hour 34, with the entrainment depth at *t*_{ED} = 30.5 h, (c) SBL at hour 37 (*B*_{E} = − *B*_{0}), and (d) *E*_{W} at hour 40. The depths are described in Table 1.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Profiles of buoyancy flux (blue) normalized by the greater of |*B*_{0}| or |*B*_{E}|, and of smoothed stratification (red) scaled (top) by ^{5}*d*/*g* from AprS regimes: (a) *E*_{C} at hour 2, (b) DBL at hour 34, with the entrainment depth at *t*_{ED} = 30.5 h, (c) SBL at hour 37 (*B*_{E} = − *B*_{0}), and (d) *E*_{W} at hour 40. The depths are described in Table 1.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Both entrainment regimes are characterized by buoyancy loss (∂_{z}⟨*wθ*⟩ > 0) between the surface and *d*_{E} and by gain below as denser water is mixed into the boundary layer. Regime *E*_{C} covers convective entrainment into strong stratification near the inversion. In the typical example of Fig. 3a, stratification is unstable above 80 m and the nearly logarithmic buoyancy profile is consistent with similarity theory even beyond the surface layer. Also, ⟨*wθ*⟩ is countergradient between 75 and 100 m. Apart from the WRW period, Fig. 2 shows that TKE scales well with *h*^{L19}, Stokes shear production dominates near the surface, and scaled dissipation over the upper half of the boundary layer decreases steadily from about *κ*^{−1} = 2.5 at the surface to about 1.

The onset of solar radiation, with weak winds initiates a transition from regime *E*_{C} to DBL at time, *t*_{ED} = 30.5 h of AprS. Regime DBL has unstable forcing (*B*_{0} < 0) to *t*_{US} and lasts until a transition to regime SBL at about *t*_{DS} = 35.5 h. The profiles of Fig. 3b are representative of the entire regime, including a period of JunS (Fig. 1; dotted blue). The robust characteristics of detrainment are negligible entrainment; negative curvature of the buoyancy flux profile in a detrainment zone from the surface to near the entrainment depth at the transition [*d*_{E}(*t*_{ED}) about 155 m]; *h*^{L19} approaches the depth *d*_{MAX} where the flux is a positive maximum denoted as *B*_{MAX}; no logarithmic layer above *h*^{L19}; and extensive regions of countergradient buoyancy flux both above and below *d*_{MAX}. In Fig. 2b, *d*_{MAX} steadily deepens from the surface at *t*_{ED} to about 100 m at *t*_{DS}, while

The Fig. 3c example from regime SBL (from *t*_{DS} until the transition to regime *E*_{W} at *t*_{SE} = 38.5 h) appears to be a near equilibrium stable boundary layer, because there is no evidence of either entrainment (*d*_{E} = 0), or detrainment (*B*_{MAX} ≤ 0). Its characteristic depth is relatively shallow, with *E*_{W} when the wind–wave forcing and diurnal shear overcome the afternoon surface buoyancy gain and stop the diurnal sea surface temperature warming in early afternoon (e.g., Filipiak et al. 2012; Large and Caron 2015). This entrainment continues after *t*_{SU}, while buoyancy forcing remains negligible, and deepens *E*_{W} (Fig. 3d) is also distinguished from regime *E*_{C}, by evidence of neither a logarithmic profile, nor countergradient buoyancy flux.

## 3. The upper-ocean boundary layer

*σ*< 1) is to express the turbulent vertical fluxes as a local downgradient flux proportional to the local vertical gradient plus a nonlocal term representing everything else. For buoyancy this partition gives

*K*

_{s}is an eddy diffusivity and −

*γ*

_{θ}is the effective nonlocal buoyancy gradient (Deardorff 1972). Furthermore, L19b finds that there is usually a nonzero angle Ω of the kinematic turbulent stress

_{W}of the wind from the shear (Table 1). In the coordinate system with the

*U*current component aligned with this shear and

*V*orthogonal, ∂

_{z}

*V*is zero and the partition becomes

*K*

_{m}is an eddy viscosity, −

*γ*

_{u}is an effective nonlocal shear, and the momentum flux vector is

*m*,

*s*) to refer to momentum or scalars, a general form for boundary layer viscosity and diffusivity is

*w*

_{m,s}, whose L19a parameterizations are updated in section 6a. The similarity functions

*ϕ*

_{m,s}and

*χ*

_{m,s}are formulated in section 3a.

The nondimensional shape functions *G*_{m,s} are found to be quite similar in L19b, and the combined averages over 20 bins are shown in Fig. 4 (*G*_{C} (black trace) is constructed. It is the piecewise linear interpolation between *σ* between 0.05 and 0.525. Nearer the surface (bottom panel), *G*_{C} = *σ*(1 − 3*σ*), such that *G*_{C}/*σ* → 1 at the surface. Deeper, the behavior of *G*_{s} affects entrainment, so it is revisited in Fig. 4 (top panel). For *σ* > 0.525, the composite becomes *G*_{C} = 0.287(1 − *σ*)^{2}. This quadratic is a very good fit to the L19b bin averages of *G*_{m} (red crosses) from the Stokes cases and it falls well within a standard deviation of the *G*_{s} bin averages from all cases (blue diamonds). However, it lies systematically below *G*_{m} > *G*_{C} > *G*_{s} from AprN and JunN, but these cases are not a present focus.

Nondimensional shape functions of (top) *σ* > 0.5 and (bottom) *σ* < 0.5: the revised composite *G*_{C} (solid black traces); and the composite bin averages *G*_{m} from the Stokes forced cases of L19b (red crosses), including AprS, and *G*_{s} from all cases of L19b (blue diamonds). Vertical bars extend ±1 standard deviation from *G*_{s}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Nondimensional shape functions of (top) *σ* > 0.5 and (bottom) *σ* < 0.5: the revised composite *G*_{C} (solid black traces); and the composite bin averages *G*_{m} from the Stokes forced cases of L19b (red crosses), including AprS, and *G*_{s} from all cases of L19b (blue diamonds). Vertical bars extend ±1 standard deviation from *G*_{s}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Nondimensional shape functions of (top) *σ* > 0.5 and (bottom) *σ* < 0.5: the revised composite *G*_{C} (solid black traces); and the composite bin averages *G*_{m} from the Stokes forced cases of L19b (red crosses), including AprS, and *G*_{s} from all cases of L19b (blue diamonds). Vertical bars extend ±1 standard deviation from *G*_{s}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

_{θ}in (5) and Γ

_{υ}in (6) are similar to buoyancy in the convective atmospheric boundary layer, with Γ

_{θ}= 0 in the stable:

_{W}) from the wind and the empirical coefficients are

*R*

_{m}= 3.5 and

*R*

_{s}= 4.7.

### The surface layer

*εh*) has distinctive physics described by semiempirical Monin–Obukhov similarity theory. Following L19a, nondimensional gradients are related to

*ϕ*

_{m,s}and

*χ*

_{m,s}. These gradients can be defined to be consistent with (5) and (6):

*γ*

_{θ}=

*γ*

_{u}= 0, of L19a, with

**U**the Eulerian flow, not the Lagrangian. Furthermore, a justified wave parameter

*ξ*is the fraction of the total surface layer source of TKE

*ξ*> 0.35 are

*ξ*= 0),

*χ*

_{m,s}becomes 1, so linear interpolations are taken for

*ξ*between 0.0 and 0.35, along with constant extrapolation beyond the minimum values of quadratics at

*ξ*= 0.73 in (12) and

*ξ*= 0.89 in (13). These minima are about 0.18 and 0.25, respectively.

*ζ*=

*d*/

*L*, where

*ϕ*

_{m,s}=

*Ψ*

_{m,s}in the unstable surface layer of AprN and JunN, show that

*A*

_{m}= 14 and

*A*

_{s}= 25, are consistent with

*ϕ*

_{m,s}are extremely noisy, because of limited depth averaging. However, the LES TKE source integrals give

*ξ*from (11) and hence estimates of

*χ*

_{m,s}, as well as of

*ζ*from (14), so that

*ϕ*

_{m,s}= (Ψ

_{m,s}/

*χ*

_{m,s}) can be computed from AprS, where the Stokes forcing further reduces the noise. This exercise verifies (9) and (10) over the range 0.02 <

*σ*< 0.12. Furthermore, Fig. 5, as well as AprN despite excessive scatter, suggest the stable stability functions (dashed lines)

*B*

_{m}= 14 and

*B*

_{s}= 5, whereas both coefficients are about 5 in the atmosphere (Högström 1988). There is also an indication that Stokes forcing induces small nonlocal fluxes (negative

*γ*

_{θ}and

*γ*

_{u}), but there are other possibilities and the effects are small compared to

*B*

_{m}= 14 rather than 5. Therefore, in the absence of additional evidence, the usual assumption of

*γ*

_{θ}=

*γ*

_{u}= 0, for stable boundary layers should suffice for present purposes.

Similarity functions of the stability parameter (14) calculated from hourly statistics every half hour during the stable buoyancy forcing of AprS for (a) momentum, with *ϕ*_{m} = 1 + 14*ζ* (dashed) and (b) buoyancy, with *ϕ*_{s} = 1 + 5*ζ* (dashed).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Similarity functions of the stability parameter (14) calculated from hourly statistics every half hour during the stable buoyancy forcing of AprS for (a) momentum, with *ϕ*_{m} = 1 + 14*ζ* (dashed) and (b) buoyancy, with *ϕ*_{s} = 1 + 5*ζ* (dashed).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Similarity functions of the stability parameter (14) calculated from hourly statistics every half hour during the stable buoyancy forcing of AprS for (a) momentum, with *ϕ*_{m} = 1 + 14*ζ* (dashed) and (b) buoyancy, with *ϕ*_{s} = 1 + 5*ζ* (dashed).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

## 4. Entrainment

A fraction of the turbulent kinetic energy produced in a boundary layer is not dissipated, but converted into column potential energy when denser water is entrained at depth then mixed throughout. An empirical rule for atmospheric convection says that the ratio of entrainment to surface buoyancy flux (*B*_{E}/*B*_{0}) is constant (Ball 1960) at about 0.2 (Tennekes 1973). An interpretation of this rule is that the integrated buoyancy source (⟨*wθ*⟩ > 0) of TKE is proportional to *wθ*⟩ < 0), whose integral over the boundary layer scales with *B*_{E}*h*. Therefore, a more general rule for ocean entrainment is formulated by empirically relating a scale for the rate of column potential energy increase to all three independent drivers of boundary layer turbulence in L19, namely, the surface layer source integrals [Eqs. (2)–(4)]. The particular scale is a robust diagnostic of buoyancy flux profiles, namely, the product of entrainment flux and depth −*B*_{E}*d*_{E}.

Since entrainment (*B*_{E} < 0) requires stable stratification near the entrainment depth, there should be a dependence on a function *F*(*δρ*) of a bulk density difference, *δρ* > 0. For present purposes *F*(*δρ*) is unity, until a much greater range of ocean conditions shows how it approaches zero for an unstratified water column, and possibly increases for entrainment into the much stronger and shallower stratification of a seasonal pycnocline, for example.

*B*

_{E}

*d*

_{E}< 3 cm

^{3}s

^{−3}falls to 0.41. Alternatively, a trivariate regression against the three TKE source integrals gives

*F*(

*δρ*) = 1, and neglecting a small intercept. This relationship is shown in Fig. 6b, where the behavior near the origin is much improved and the 1:1 line (dotted) is the best fit by construction. The correlation coefficient is greater than 0.96, so (17) becomes a potentially useful entrainment rule, when combined with the updated parameterizations of

*P*

_{U},

*P*

_{S}, and

*P*

_{B}from section 6a.

The entrainment rule from independent hourly statistics showing the product −*B*_{E}*d*_{E} vs (a) net surface layer production of TKE

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The entrainment rule from independent hourly statistics showing the product −*B*_{E}*d*_{E} vs (a) net surface layer production of TKE

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The entrainment rule from independent hourly statistics showing the product −*B*_{E}*d*_{E} vs (a) net surface layer production of TKE

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

_{SL}is a surface layer Langmuir number, with

*P*

_{U}from (2) and

*P*

_{S}from (3), whereas all Langmuir effects are captured by the single

*E*

_{C}. With

*h*=

*h*

^{L19}, they usually differ by less than 30%, and the mean bias (ratio of means) of 5% low from (18) is within the uncertainty of

*h*, since it becomes 3% high with

*h*=

*d*

_{E}. There are only marginal changes with the Reichl and Li (2019) dependency on

*B*

_{E}

*d*

_{E}, whereas that of (18) is only 2%. The Grant and Belcher (2009) scaling, −

*B*

_{E}

*d*

_{E}= (

*d*

_{E}/

*h*

^{L19})0.045La

^{−2}

*Q*

_{0}< 0;

A similar procedure can relate any measure of entrainment to the source integrals over different depth ranges. For example, *M*_{E}, the integral of (4) over the entrainment zone of ⟨*wθ*⟩ < 0, gives the rate of conversion of TKE to potential energy across the zone. It is the basis for determining the all-important boundary layer depth in the ePBL scheme of Reichl and Hallberg (2018) and its extension to Langmuir turbulence by Reichl and Li (2019). Its correlation coefficient with *M*_{E} is an excellent predictor of *B*_{E} in regime *E*_{C} (0.98 correlation; 3% low bias). However, in regime *E*_{W} there is more than a factor of 2 high mean bias, despite high values of (−*B*_{E}/|*B*_{0}| > 1), because −*M*_{E} integrals are relatively even higher as a consequence of the entrainment zone extending from the surface to more than twice the entrainment depth in Fig. 3d, for example. Also contributing are the highly nonlinear buoyancy flux profiles that appear to be a consequence of prior conditions. Nonetheless, this approach appears to be viable at least for regime *E*_{C} where the profiles are more linear and the ePBL assumption that the depth of neutral stratification bounds the entrainment zone could be modified to account for the 20 m difference in Fig. 3a, for example. In regime *E*_{W} the relatively high values of −*B*_{E} are largely compensated by smaller entrainment depths in the −*B*_{E}*d*_{E} product of Fig. 6 and (17).

Another application of the procedure relates −*B*_{E}*d*_{E} to the source integrals above −*h*^{L19}, −*d*_{E}, or −*d*_{i}. With any of these limits, the resulting regression coefficients corresponding to those of (17) become 0.020, 0.035, and 0.27. Respectively, these values are the fractions of TKE produced by Eulerian shear, Stokes shear, and buoyancy that go to increasing potential energy rather than dissipating. Buoyancy is by far the most efficient with over 25% going to entrainment, and so contributes about 10% of the regime *E*_{C} driving of AprS before the winds weaken at hour 15, and as much as 90% afterward, but is negligible in regime *E*_{W}. Although much less efficient, the Stokes source usually contributes more than 70% of the entrainment driving, except for the WRW period. The largest Eulerian shear contribution of AprS is only about 20% in regime *E*_{W}.

### a. Entraining boundary layer depth

*h*

^{BL}, where the entrainment flux,

*B*

_{E}< 0, equals the buoyancy flux from (5), given the diffusivity of (7), and Γ

_{θ}from (8). It is the first depth below

*d*

_{E}, where

*σ*

_{E}= (

*d*

_{E}/

*h*

^{BL}) > 0.225, where

*G*

_{C}is maximum (Fig. 4). Figure 7b compares the boundary layer depths,

*h*

^{BL}and

*h*

^{L19}from all the entrainment regimes of cases with Stokes forcing (red) and without (blue). Depths around 220 m are from June, those from D24S are near 100 m, and the others are from April. Each cluster displays more spread in

*h*

^{L19}, which Fig. 2 indicates is due to hourly variability not reflected in either TKE, or ⟨

*wθ*⟩. Therefore, there has been no attempt to parameterize

*h*

^{L19}. However, the entrainment rule (17) allows

*h*

^{BL}to be parameterized as

*h*

^{ER}in section 6, so it is the preferred choice, and compared to other depths (Table 1) in Fig. 7. It is less than 10 m deeper than

*d*

_{E}(Fig. 7a), except in regime

*E*

_{W}of AprS as it increases from 50 to 100 m, and also when

*d*

_{E}is less than about 15 m following the transition from regime SBL.

Comparison of the boundary layer depth *h*^{BL} to other Table 1 depths diagnosed from 224 independent hourly statistics throughout the entrainment regimes of AprS, JunS, and D24S with Stokes forcing (red) and from AprN and JunN (blue): (a) entrainment depth *d*_{E}, (b) *h*^{L19}, (c) the KPP boundary layer depth of Danabasoglu et al. (2006), and (d) parameterized *h*^{ER} from the entrainment rule (section 6).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of the boundary layer depth *h*^{BL} to other Table 1 depths diagnosed from 224 independent hourly statistics throughout the entrainment regimes of AprS, JunS, and D24S with Stokes forcing (red) and from AprN and JunN (blue): (a) entrainment depth *d*_{E}, (b) *h*^{L19}, (c) the KPP boundary layer depth of Danabasoglu et al. (2006), and (d) parameterized *h*^{ER} from the entrainment rule (section 6).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of the boundary layer depth *h*^{BL} to other Table 1 depths diagnosed from 224 independent hourly statistics throughout the entrainment regimes of AprS, JunS, and D24S with Stokes forcing (red) and from AprN and JunN (blue): (a) entrainment depth *d*_{E}, (b) *h*^{L19}, (c) the KPP boundary layer depth of Danabasoglu et al. (2006), and (d) parameterized *h*^{ER} from the entrainment rule (section 6).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Figure 7c compares the KPP boundary layer depth calculation, *h*^{KPP}, as updated in Danabasoglu et al. (2006). The major differences are during regime *E*_{W} when *h*^{KPP} becomes greater than *h*^{L19} > *h*^{BL}, because it is less affected by the weak diurnal stratification near *h*^{KPP} would increase with the Li and Fox-Kemper (2017) modifications to incorporate the effects of Langmuir turbulence. Also in regime *E*_{W}, *h*^{KPP} is shallower in AprS than AprN, but not if the calculation uses Lagrangian shear instead of just the Eulerian, which is much reduced by Stokes forcing.

### b. Flux profiles during entrainment

The defining characteristic of the entrainment regimes is buoyancy loss (∂_{z}⟨*wθ*⟩ > 0) between the surface and the entrainment depth, with gain below to beyond the boundary layer depth. Buoyancy loss through the surface (*B*_{0} < 0) is usual (Fig. 3a), but not necessary (Fig. 3d). Furthermore, the buoyancy flux varies smoothly with depth, as also seen in Fig. 8 (upper panels; solid profiles). Positive curvature across the boundary layer is a singular feature of regime *E*_{W}, when the diurnal stratification weakens, and its local maximum deepens steadily from *G*_{C}, of Fig. 4, rather than *h*^{BL}, but even so it tends to be shallower than *h*^{L19}. Therefore, the buoyancy flux at *h*^{BL} can be significantly negative (e.g., Fig. 8).

Vertical profiles during the entrainment regimes *E*_{C} (hours 9 and 26), and *E*_{W} (hours 39 and 44) from AprS: (top) buoyancy flux normalized by the greater of |*B*_{0}| or |*B*_{E}| (solid blue), and the parameterizations from section 6, ⟨*wθ*⟩^{FP} (dashed) and ⟨*wθ*⟩^{KP} (dotted); (middle) stress magnitude normalized by *τ*^{FP} (dashed) and *τ*^{KP} (dotted); and (bottom) vector orientations from the wind of the shear −Ω_{W} (red profiles) and of the stress Ω_{τ} (solid blue) and its parameterizations *h*^{AM} from (21), and from section 6, *h*^{ER} are consistent with (17).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Vertical profiles during the entrainment regimes *E*_{C} (hours 9 and 26), and *E*_{W} (hours 39 and 44) from AprS: (top) buoyancy flux normalized by the greater of |*B*_{0}| or |*B*_{E}| (solid blue), and the parameterizations from section 6, ⟨*wθ*⟩^{FP} (dashed) and ⟨*wθ*⟩^{KP} (dotted); (middle) stress magnitude normalized by *τ*^{FP} (dashed) and *τ*^{KP} (dotted); and (bottom) vector orientations from the wind of the shear −Ω_{W} (red profiles) and of the stress Ω_{τ} (solid blue) and its parameterizations *h*^{AM} from (21), and from section 6, *h*^{ER} are consistent with (17).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Vertical profiles during the entrainment regimes *E*_{C} (hours 9 and 26), and *E*_{W} (hours 39 and 44) from AprS: (top) buoyancy flux normalized by the greater of |*B*_{0}| or |*B*_{E}| (solid blue), and the parameterizations from section 6, ⟨*wθ*⟩^{FP} (dashed) and ⟨*wθ*⟩^{KP} (dotted); (middle) stress magnitude normalized by *τ*^{FP} (dashed) and *τ*^{KP} (dotted); and (bottom) vector orientations from the wind of the shear −Ω_{W} (red profiles) and of the stress Ω_{τ} (solid blue) and its parameterizations *h*^{AM} from (21), and from section 6, *h*^{ER} are consistent with (17).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Figure 8 also shows the magnitude *τ* of the turbulent stress and momentum flux (middle panels; solid blue profiles) decreasing monotonically from *h*^{BL}, except over the WRW period. At hour 26, for example, the profile remains smooth, but its negative curvature suggests a depth-dependent delay in the response to the varying wind. A similar response is evident in the June MRW period. Thus, these two periods of counterinertial wind rotation are not typical of regime *E*_{C}, but will be designated as such until it can be determined how relevant parameters, such as the depth of maximum *τ* (≈70 m at hour 26), depend on the wind’s time history.

The Eulerian shear vector _{W} (Table 1) can have a complicated vertical structure. The free slip boundary condition aligns the stress and shear vectors with the wind, Ω = Ω_{W} = 0, only at the surface. As the shear veers with depth, the stress stays more aligned with the wind, so its angle relative to the wind vector, Ω_{τ} = Ω − Ω_{W}, diverges from the angle of the shear from the wind, −Ω_{W}. The angle Ω_{τ} is almost always in an inertial sense (positive in the Southern Hemisphere) and increases steadily with depth, as shown in Fig. 8 (bottom panels; solid dark blue). This angle should be related to orientations of the shear relative to the wind at multiple scales. A simple, yet effective, representation of the large scale is the angle ^{−2}*π*/4 and 3*π*/4. Angles outside this range are linearly downweighted and not used if less than −*π*/2, or greater than 5*π*/4.

_{τ}in the boundary layer, namely,

_{τ}(solid blue) across regimes, makes formulating the momentum flux as a magnitude and direction relative to the wind an attractive option. However, the Fig. 8 differences between −Ω

_{W}(solid red) and Ω

_{τ}demonstrate that the common assumption of stress and shear alignment is not generally valid. The dramatic swings of −Ω

_{W}over the upper 20 m appear to be due to Stokes effects, because they are also evident in D24S and to a lesser extent in JunS, but in neither AprN, nor JunN. In the WRW example, the stress vector is oriented more than 180° in an inertial sense from the shear vector between about 5 and 70 m, yet

## 5. The stable boundary layer and detrainment

Figure 9 shows time histories of various depths (Table 1) that bound the zones and layers of AprS (red) and AprN (blue) during the single day of stable buoyancy forcing. These depths are shown along with the profiles of ⟨*wθ*⟩, *τ* and Ω_{τ} in the four examples of Fig. 10. The boundary layer depth of choice, *h*, is shown in Fig. 9a (solid traces). It follows *h*^{BL} when the buoyancy forcing is unstable, and the depth, *h*^{AM} (Fig. 9c), of a growing morning boundary layer from the unstable to stable transition at *t*_{US} in regime DBL until the reverse transition at *t*_{SU} in regime *E*_{W}.

Half-hourly time variability of diagnostic and time-integrated depths (Table 1) from hourly statistics of AprS (red) and AprN (blue) during regimes DBL, SBL, and *E*_{W}: (a) *h*^{BL} and the boundary layer depth of choice, *h* (solid traces); (b) *h*^{L19} and *d*_{MKE} and the depth of the morning boundary layer, *h*^{AM} (solid traces) from (21); and (d) depth of the maximum positive buoyancy flux *d*_{MAX}, both as diagnosed (symbols) and as integrated from (22), and the afternoon entrainment depth *d*_{E}. The vertical dashed lines mark the regime transitions, near times *t*_{ED} (*E*_{C} to DBL), *t*_{DS} (DBL to SBL), and *t*_{SE} (SBL to *E*_{W}). The transition from unstable to stable buoyancy forcing is shown at *t*_{US} and from stable to unstable at *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Half-hourly time variability of diagnostic and time-integrated depths (Table 1) from hourly statistics of AprS (red) and AprN (blue) during regimes DBL, SBL, and *E*_{W}: (a) *h*^{BL} and the boundary layer depth of choice, *h* (solid traces); (b) *h*^{L19} and *d*_{MKE} and the depth of the morning boundary layer, *h*^{AM} (solid traces) from (21); and (d) depth of the maximum positive buoyancy flux *d*_{MAX}, both as diagnosed (symbols) and as integrated from (22), and the afternoon entrainment depth *d*_{E}. The vertical dashed lines mark the regime transitions, near times *t*_{ED} (*E*_{C} to DBL), *t*_{DS} (DBL to SBL), and *t*_{SE} (SBL to *E*_{W}). The transition from unstable to stable buoyancy forcing is shown at *t*_{US} and from stable to unstable at *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Half-hourly time variability of diagnostic and time-integrated depths (Table 1) from hourly statistics of AprS (red) and AprN (blue) during regimes DBL, SBL, and *E*_{W}: (a) *h*^{BL} and the boundary layer depth of choice, *h* (solid traces); (b) *h*^{L19} and *d*_{MKE} and the depth of the morning boundary layer, *h*^{AM} (solid traces) from (21); and (d) depth of the maximum positive buoyancy flux *d*_{MAX}, both as diagnosed (symbols) and as integrated from (22), and the afternoon entrainment depth *d*_{E}. The vertical dashed lines mark the regime transitions, near times *t*_{ED} (*E*_{C} to DBL), *t*_{DS} (DBL to SBL), and *t*_{SE} (SBL to *E*_{W}). The transition from unstable to stable buoyancy forcing is shown at *t*_{US} and from stable to unstable at *t*_{SU}.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 8, but for the detrainment regime DBL (hours 32, 33, and 34), with *d*_{MKE}. Also, the buoyancy flux is normalized with the greater of |*B*_{0}| or |*B*_{max}| and angles are shown only for *τ* > 0.05

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 8, but for the detrainment regime DBL (hours 32, 33, and 34), with *d*_{MKE}. Also, the buoyancy flux is normalized with the greater of |*B*_{0}| or |*B*_{max}| and angles are shown only for *τ* > 0.05

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 8, but for the detrainment regime DBL (hours 32, 33, and 34), with *d*_{MKE}. Also, the buoyancy flux is normalized with the greater of |*B*_{0}| or |*B*_{max}| and angles are shown only for *τ* > 0.05

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The boundary layer depth *h*^{L19} of Fig. 9b is always available (Fig. 2). However, a depth where the integrated Lagrangian shear production of TKE balances the buoyant suppression (gray triangles) can be found only between hours 34 and 37. In stable atmospheric boundary layers, this depth has been associated with the Monin–Obukhov depth (Wyngaard 2010), which appears to be the case near hour 35 of AprS for

*d*

_{MKE}, where the flow speed, and hence the mean kinetic energy, are at minima. Figure 9c shows this depth increasing throughout the day and into the night. It provides a bound on

*h*, because dominant surface forcing would maintain

*h*

^{L19}(Fig. 9b) does not remain within the

*d*

_{MKE}bound and, therefore, is not a viable boundary layer depth. However, assuming internal morning boundary layers grow at a rate proportional to the wind and wave forcing, tempered by buoyancy, leads to

*h*

^{AM}in Fig. 9c (solid traces) remains less than

*d*

_{MKE}(symbols) and matches both

*h*

^{BL}and

*h*

^{L19}during regime

*E*

_{W}, for both AprS and AprN. These criteria are also satisfied throughout regime SBL by

*E*

_{W}allows

*h*

^{BL}and

*h*

^{ER}to be calculated.

### a. Detrainment

Following section 2, Fig. 9d shows the depth *d*_{MAX} of maximum buoyancy flux, *B*_{MAX} > 0, during detrainment. Its appearance marks the transition to regime DBL at *t*_{ED} = 30.75 h. Such a maximum remains clearly discernible at hour 34 in Fig. 10, but not in regime SBL. In the detrainment examples of Fig. 10 (top panels), *d*_{MAX} separates buoyancy gain from the loss down to near *d*_{E}(*t*_{ED}), the entrainment depth at *t*_{ED}. In contrast, the corresponding stress profiles (middle panels) do not appear to reflect *d*_{MAX}, but decrease monotonically from *d*_{MKE} (hour 32), or *d*_{E}(*t*_{ED}) at hours 33 and 34, or *h*_{AM} (regime SBL).

*d*

_{MAX}deepens more rapidly during the stable hours of regime DBL than the unstable, and with Stokes forcing than without. Following (21), this behavior is captured (solid traces) by integrating

*t*

_{ED}, with empirical coefficients

*a*

_{1}=

*a*

_{2}= 0.002 until

*t*

_{US}, and

*a*

_{1}= 0.20,

*a*

_{2}= 0.05 afterward, for both AprS and AprN. If turbulence remained steady after

*t*

_{ED}, then

*B*

_{MAX}would be the buoyancy flux at

*d*

_{MAX}at time

*t*

_{ED}. However, turbulent transport and dissipation are two processes of change. The vertical structure of Fig. 2b, suggests that transport enhances the near surface buoyancy flux, and assuming that it does not change the integral (4) above the entrainment zone, and that the profile at

*t*

_{ED}is linear from −

*B*

_{0}at the surface to

*f*

_{B}

*B*

_{0}at

*d*

_{E}, gives a detrainment rule, akin to the entrainment rule (17), but involving prior forcing and

*d*

_{MAX}from (22):

*f*

_{E}= 1.15 is the effective near surface enhancement. A simple means of accounting for turbulent dissipation and its reduction of

*B*

_{MAX}to zero is with

*f*

_{B}= 0.5; more than −

*B*

_{E}/

*B*

_{0}at

*t*

_{ED}, but less than at hour 9 (Fig. 8), for example. The comparison of Fig. 11a shows that (23) then represents the early decrease of

*B*

_{MAX}from near

*B*

_{0}(

*t*

_{ED}) and its later approach to zero, during regime DBL of both AprS (red) and AprN (blue). This general behavior is captured also during regime DBL of JunS (black), but is sensitive to when the integration of (22) begins at

*t*

_{ED}.

Evaluation of the rules for regime DBL detrainment and for regime *E*_{W} afternoon and nighttime entrainment from AprS (red) and AprN (blue) and from JunN and JunS (black): (a) *B*_{MAX}, normalized by *B*_{0} < 0 at the time *t* = *t*_{ED} of transition to regime DBL and (b) *B*_{E}, normalized by *B*_{0} > 0 at time *t* = *t*_{SE} of transition to afternoon entrainment. Values are calculated half-hourly from hourly statistics.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Evaluation of the rules for regime DBL detrainment and for regime *E*_{W} afternoon and nighttime entrainment from AprS (red) and AprN (blue) and from JunN and JunS (black): (a) *B*_{MAX}, normalized by *B*_{0} < 0 at the time *t* = *t*_{ED} of transition to regime DBL and (b) *B*_{E}, normalized by *B*_{0} > 0 at time *t* = *t*_{SE} of transition to afternoon entrainment. Values are calculated half-hourly from hourly statistics.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Evaluation of the rules for regime DBL detrainment and for regime *E*_{W} afternoon and nighttime entrainment from AprS (red) and AprN (blue) and from JunN and JunS (black): (a) *B*_{MAX}, normalized by *B*_{0} < 0 at the time *t* = *t*_{ED} of transition to regime DBL and (b) *B*_{E}, normalized by *B*_{0} > 0 at time *t* = *t*_{SE} of transition to afternoon entrainment. Values are calculated half-hourly from hourly statistics.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

_{τ}(solid blue) is tracked by

*h*=

*h*

^{AM}during regime SBL. However, a consequence of

*d*

_{MKE}is a nearby change in the sign of Ω, which is reflected in a detrainment zone parameterization of Ω

_{τ},

_{τ}across all depths below 2 m. Thus,

*d*

_{MKE}. Below

*d*

_{MKE}in both regimes DBL and SBL, the stress orientation is better represented by

*τ*falls below about 0.05

*h*

^{AM}and

*h*

^{AM}and

*d*

_{MKE}.

In Fig. 2a, the turbulent kinetic energy in the detrainment zone between *h* = *h*^{AM} and *d*_{E}(*t*_{ED}) decreases smoothly with both depth and time and by *t*_{DS} it is nearly uniform at about *d*_{MAX}, nor *d*_{MKE}.

## 6. Empirical turbulence models of the diurnal cycle

The vertical fluxes of buoyancy and momentum are parameterized in terms of the ocean state, surface fluxes and Stokes drift in two distinct ways. The first ETM (*K*-profile) is based on diffusivity and viscosity in the boundary layer. In the detrainment zone, for example, the rules from section 5a are formulated as analytic profiles of buoyancy flux and stress magnitude, along with *F*-profile) extends this concept to all regimes by utilizing _{υ} (6) directly shows the error associated with the common assumption of downgradient viscosity, but its distribution is not shown here, because further interpretation is hampered by a coordinate that rotates with depth.

### a. K-profile (KP) boundary layer fluxes

_{υ}from (8) and the boundary layer depth differs (Fig. 7c). In addition, the composite shape function (Fig. 4) and the nondimensional gradients (9) and (10) simplify the KP fluxes in the surface layer (

*σ*<

*ε*) to

^{KP}= sin

^{−1}(−Γ

_{υ}/

*τ*

^{KP}), so that as (

*G*

_{C}/

*σ*) converges to 1 at the surface the boundary conditions (⟨

*wθ*⟩ = −

*B*

_{0};

*ε*<

*σ*< 1),

*G*

_{C}as defined by (7) relates the vertical fluxes and gradients of (5) and (6):

_{θ}and Γ

_{υ}are given by (8), and

*γ*

_{u}= 0 (section 3). Furthermore, nonzero fluxes at a parameterized boundary layer depth

*h*

^{PAR}are themselves parameterized as downgradient fluxes by specifying local eddy transfer coefficients

*υ*

_{m,s}:

*H*(

*d*), which for present convenience decreases linearly from unity at

*h*

^{PAR}to zero at

*d*

_{E}and at 1.2

*h*

^{PAR}.

*ζ*from (14) and

*ξ*from (11) and hence

*εh*

^{L19}to the center of the topmost LES grid cell. Therefore, there are slight modifications here for the different integration limits (−

*εh*to 0) and for the exclusion of sinks of TKE in (2), (3), and (4), namely, the coefficients of (8), (12), and (13) as well as

*P*

_{U}and

*P*

_{S}, and hence of the turbulent velocity scales. For example, assuming a linear decrease of

*τ*from the surface wind stress to zero at

*h*

^{PAR}, as seen in Fig. 8, allows the integrand of (2) to be approximated, such that TKE sources (cosΩ

^{PAR}> 0) give

These two new parameterizations are evaluated in Fig. 12. Despite differences between *h*^{BL} and *h*^{ER} (Fig. 7d), the 1:1 lines (dotted) are good fits to cases with Stokes drift (red) and without (blue) over the entire range, though there is somewhat more spread of the black symbols from the WRW period in April and the MRW in June. Nevertheless, all the points of Fig. 12 give correlations of 0.99 for both *P*_{U} values from shallow stable boundary layers, both with and without Stokes drift. The more consistent behavior when TKE sinks are excluded suggests that the associated energy loss is largely compensated by reduced dissipation, rather than by weaker entrainment.

Evaluation of the parameterizations of the surface layer TKE production integrals, ^{−2}

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Evaluation of the parameterizations of the surface layer TKE production integrals, ^{−2}

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Evaluation of the parameterizations of the surface layer TKE production integrals, ^{−2}

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The *K*-profile fluxes also depend on a depth scale such as a boundary layer, or entrainment depth. A very simple approach would be to use the MLD as defined by de Boyer Montégut (2004) as the depth where the density (buoyancy) difference from the surface first exceeds a 0.03 kg m^{−3} threshold. However, this depth remains nearly constant at about 172 m throughout all regimes of AprS. A much smaller threshold is needed to detect the diurnal cycle. For example, with 0.006 kg m^{−3}, the modified

Comparison of (a) nondimensional buoyancy flux distribution from AprS to the parameterizations of (b) ⟨*wθ*⟩^{FP} (Table 2) and (c) ⟨*wθ*⟩^{KP} from (26). The contours are irregular at 0.0, ±0.05, ±0.10, ±0.30, and 0.60, with negative contours gray. The time series are boundary layer depth *h* (solid red) and *d*_{E} (dotted red) in (a); *h*^{PAR} (solid) and *t*_{ED} value over regimes DBL and SBL in (b) and (c). Also shown in (c) is a mixed layer depth ^{−3} (dashed black). Vertical dashed lines demark the unstable to stable transition at *t*_{US} = 32.75 h and the return to unstable at *t*_{SU} = 40.75 h.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of (a) nondimensional buoyancy flux distribution from AprS to the parameterizations of (b) ⟨*wθ*⟩^{FP} (Table 2) and (c) ⟨*wθ*⟩^{KP} from (26). The contours are irregular at 0.0, ±0.05, ±0.10, ±0.30, and 0.60, with negative contours gray. The time series are boundary layer depth *h* (solid red) and *d*_{E} (dotted red) in (a); *h*^{PAR} (solid) and *t*_{ED} value over regimes DBL and SBL in (b) and (c). Also shown in (c) is a mixed layer depth ^{−3} (dashed black). Vertical dashed lines demark the unstable to stable transition at *t*_{US} = 32.75 h and the return to unstable at *t*_{SU} = 40.75 h.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of (a) nondimensional buoyancy flux distribution from AprS to the parameterizations of (b) ⟨*wθ*⟩^{FP} (Table 2) and (c) ⟨*wθ*⟩^{KP} from (26). The contours are irregular at 0.0, ±0.05, ±0.10, ±0.30, and 0.60, with negative contours gray. The time series are boundary layer depth *h* (solid red) and *d*_{E} (dotted red) in (a); *h*^{PAR} (solid) and *t*_{ED} value over regimes DBL and SBL in (b) and (c). Also shown in (c) is a mixed layer depth ^{−3} (dashed black). Vertical dashed lines demark the unstable to stable transition at *t*_{US} = 32.75 h and the return to unstable at *t*_{SU} = 40.75 h.

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

The alternative, *h*^{PAR}, is adapted from sections 4a and 5. Whenever the buoyancy forcing is unstable, the shallowest depth *h*^{ER} is found where a shallower depth *h*^{BL} and *h*^{ER} over the unstable entrainment regimes, though *h*^{BL} has about twice the spread. With stable forcing *h*^{PAR} equals *h*^{AM} shown in (Fig. 9) from integrating (21). Thus, *h*^{PAR} varies more smoothly in time (Figs. 13b,c) than either *h*^{BL} (Fig. 13a) or *h*^{L19} (Fig. 2).

Figures 8 and 10 also show *K*-profile fluxes (dotted blue) throughout the boundary layer, with *υ*_{m,s} = 0 in (29) convenient for the purpose of revealing major problems above the entrainment depth. Perhaps the most prevalent (bottom panels) is that during entrainment Ω^{KP} = tan^{−1}(⟨*wυ*⟩^{KP}/⟨*wu*⟩^{KP}) tracks the near surface veering of the shear, such that _{W}) (red) and deviates radically from Ω_{τ} (solid blue), though there is often better agreement at depth. Also, the entrainment is too vigorous throughout regime *E*_{W} (e.g., by about 40% at hour 39). In addition, the positive bias in *τ*^{KP} between 20- and 60-m depth over the WRW period and into regime DBL, suggests that the shear, stress and variable forcing are far from equilibrium. Finally, *K*-profile fluxes in the detrainment examples of Fig. 10 are not faithful representations in the boundary layer and are missing below, even though they display considerable fidelity below 40 m at hour 32. Perhaps less serious issues include discontinuities at *d* = *εh*^{PAR}, as well as the fluxes at *h*^{PAR}. A possible approach to the former would be to blend the surface layer fluxes from (25) over a range of depths, and the latter could be overcome by tuning the function *H*(*d*) and the values of *υ*_{m,s} in (29). However, such solutions may not be universally applicable.

### b. F-profile (FP) parameterizations

To utilize *h*^{AM} and to address detrainment and afternoon entrainment, the appropriate regimes (section 2b) need to be identified without knowledge of the buoyancy flux, especially DBL and *E*_{W} that are most problematic for *K*-profile. For this purpose the key factors are the meteorological forcing (Fig. 1); the Langmuir number, La; the Monin–Obukhov depth *L*; the local maximum stratification at *Q*_{0}; and the entrainment forcing, *L*| > 2000 m). When it is more stable with solar radiation, regime SBL covers steady midday forcing (|Δ*Q*_{0}| < 0.4*Q*_{0}), regime DBL has a more rapid morning increase, while regime *E*_{W} begins with a more rapid afternoon decrease and continues to cover weakly forced (<5 cm^{3} s^{−3}), shallow (*E*_{C}, including for the present the WRW and MRW periods. The exceptions are times of unstable detrainment (Fig. 1), when there is a distinct *B*_{MAX} > 0 at *d*_{MAX} > 0. Although the circumstances may not be unique, these brief DBL periods are identified by a particular combination; namely, solar radiation, Stokes forcing (La^{−2} > 0), very weak entrainment forcing (less than 1 cm^{3} s^{−3}), and cooling that is neither too great (−*Q*_{0} < 70 W m^{−2}), nor increasing too much (Δ*Q*_{0} > 0.1 *Q*_{0}). Thus, regime *E*_{C} includes all of JunN (*L* < 0; La^{−2} = 0), as well as AprS and JunS days when entrainment forcing is sufficiently strong.

*wθ*⟩

^{FP}and

*τ*

^{FP}, from the surface to beyond the boundary layer, including the detrainment zone. The downwind and crosswind momentum flux components are calculated as

*wθ*⟩ and

*τ*profiles are nearly linear from the surface to

*h*

^{AM}, as indicated in Table 2.

Summary of the section 6b construction of *F*-profile fluxes as polynomial segments and angles from the wind, for each diurnal cycle regimes. A single “+” superscript denotes where a segment’s vertical gradient is zero (e.g., *z*_{M}, where the linear and quadratic regime *E*_{C} segments match with a continuous gradient, and −*z*_{B}, where the fluxes and their gradients become zero below *h*. Table 1 provides references for the key depths, including the regime DBL time integral d_{MAX}(*t*) and the extent of the detrainment zone *z*_{D} at a prior entrainment depth.

There also appears to be a high degree of equilibrium, including the entrainment, in regime *E*_{C}, so it is also relatively straightforward, with nearly linear flux profiles throughout most of the boundary layer. For buoyancy, however, the entrainment point *h*^{PAR} = *h*^{ER}. The unique quadratic from (*h*^{PAR}; _{M} where it meets the linear segment from the surface and the gradient is continuous, as in Table 2. Although Fig. 8 (26 h) shows that the stress magnitude is not yet well represented in the WRW period, _{τ} throughout the boundary layer, despite the 180° deviations from Ω. This example is also representative of JunS’s MRW period.

*τ*

^{FP}is also linear to

*h*

^{PAR}=

*h*

^{ER}in regime

*E*

_{W}(Table 2). However, entrainment into weak diurnal stratification, first through the afternoon then into the night is an

*F*-profile challenge too. The overly strong

*K*-profile entrainment suggests that the fluxes and gradients are not in sufficient equilibrium with the weak entrainment forcing (<5 cm

^{3}s

^{−3}) for

*K*-profile constructs. In particular, the examples from Fig. 8 suggest that afternoon entrainment has a contribution from prior regime SBL generated turbulence, which following section 5a, is related to the buoyancy flux at the time

*t*

_{SE}of transition from regime SBL to

*E*

_{W}. As in regime

*E*

_{C}, the contribution from current forcing is characterized by

*E*

_{W}is, therefore, parameterized by

*t*

_{SE}and depth

*h*

^{AM}. This relation is evaluated in Fig. 11b over all of regime

*E*

_{W}for both AprS (red) and AprN (blue). The 0.70 coefficient is chosen to give good estimates of the strongest entrainment from early in the regime, and to keep the AprS agreement within a factor of 2. A quadratic from the surface to zero gradient at the entrainment point

*wθ*⟩

^{FP}in the top segment of regime

*E*

_{W}, and a smooth connection to another quadratic from

*h*

^{ER}, as indicated in Table 2.

In regimes *E*_{C}, *E*_{W}, and SBL, there is a meaningful boundary layer depth, *h*^{PAR}, where (29) can be used to give estimates of nonzero fluxes, *h*^{PAR} and goes to zero with zero gradient at a computed depth −*z*_{B}. However, in the *F* profiles (dashed blue) of Figs. 8 and 10, *z* = −*h*^{PAR}.

The shortcomings of *K*-profile fluxes indicate that regime DBL is far from equilibrium, such that fluxes are nonlocal in time as well as depth. Hence, they do not reflect a boundary layer depth. Nonetheless, complete *F* profiles are constructed, as shown in Table 2. First, (22) is integrated from *t*_{ED} to give a deepening *d*_{MAX} (Fig. 9d), then the detrainment rule (23), as verified in Fig. 11, gives *B*_{MAX} diminishing with time. Also, an earlier entrainment depth and buoyancy flux at time *t*_{ED} remain a constant endpoint throughout DBL. Curvature in flux profiles may also reflect dependencies on time history. In particular, regime DBL profiles of ⟨*wθ*⟩^{FP} are cubic functions of depth over three segments that connect with continuous gradients at both *d*_{MAX} and *d*_{E}(*t*_{ED}). The deepest segment is cosmetic rather than physical, but smoothly goes to zero at the inversion. Also, *τ*^{FP} is a cubic from

### c. Evaluation

For evaluation purposes, *υ*_{s} = 0.1 cm^{2} s^{−1} in (29), in rough accord with regime *E*_{C}. Although primarily for cosmetic purposes, this diffusivity is consistent with purposeful tracer releases in the pycnocline (Ledwell et al. 1993), but is only about one-fifth the regime *E*_{W} values. In general *υ*_{m} = 0. At *H*(*d*) in (26) is zero, so in regime *E*_{C} ⟨*wθ*⟩^{FP} and ⟨*wθ*⟩^{KP} are equal by construction, even though the latter may not be the local negative extremum (e.g., Fig. 8).

Figure 13 shows that a number of features seen in the diurnal cycle of buoyancy flux in the boundary layer are represented by both schemes. Examples include the collapse of the boundary layer at *t*_{US}, variations in the vertical extent of the entrainment zone (negative gray contours), as well as the depth and strength of entrainment. One of the more obvious successes is the response to the daytime solar heating between hours 9 and 17, while the buoyancy forcing remains unstable. At all depths in Fig. 13c the primary response is that of Γ_{s} from (8) to variations in *B*_{0} as the sun rises then sets. This response only affects Fig. 13b directly at *d*_{E} (Fig. 13a). The discontinuity at the surface layer limit of (25) is readily apparent throughout Fig. 13c.

The statistics compiled in Table 3 provide more quantitative evaluations of different regimes and layers, with row numbers provided for reference. They show the *F*-profile fluxes to have higher fidelity; sometimes significantly. For example (row A.1), in the surface layer of all regimes the ⟨*wθ*⟩^{FP} regression gives a correlation coefficient greater than 0.99, a slope very near unity and intercept near zero, there is no mean bias, *R*_{M} = 1.00, and a small root-mean-square difference, *δ*_{rms} = 0.024. Even though surface layer fluxes are strongly constrained by the surface boundary conditions, the corresponding results from ⟨*wθ*⟩^{KP} (row A.2) show a high bias (*R*_{M} = 1.12) and nearly a factor of 3 greater *δ*_{rms}; a quantitative reflection of the effects of variable forcing on the flux–profile relationships (25).

Quantitative evaluation of fluxes over various layers in the referenced figures; *F* profile (FP, middle) and *K* profile (KP, bottom) against LES (top). The root-mean-square difference is *δ*_{rms}, the ratio of the means *R*_{M} is the mean bias, and the depth–time correlation coefficient is *r*_{z,t}. These results are referenced by row number, which are grouped according to the flux; buoyancy (A.1–A.8), downwind momentum (B.1–B.8), and crosswind (C.1–C.8).

There is less fidelity in the more weakly constrained deeper boundary layer. For example, during regime *E*_{C}, the positive bias above about 70 m in Fig. 13c, leads to *R*_{M} = 1.24 (row A.4), while there is a low mean bias in Fig. 13b (*R*_{M} = 0.92) and about half the *δ*_{rms} (row A.3), but the respective correlations fall only slightly to 0.96 and 0.98. Prominent issues in Fig. 13c are the overly negative fluxes in regime SBL and excessive entrainment in regime *E*_{W}, which are quantified by *R*_{M} = 1.17, as well as by nearly twice the *δ*_{rms} and the reduced correlation (row A.6), than found for ⟨*wθ*⟩^{FP} (row A.5), whose regime *E*_{W} behavior is directly governed by

The full depth of regime DBL is represented only by ⟨*wθ*⟩^{FP}. Most importantly, these values are maintained at comparable levels to Fig. 13a for some time after the boundary layer collapse at *t*_{SU}, including at the depth of the maximum (Fig. 9d), despite the negative buoyancy flux nearer the surface (e.g., Fig. 10, hour 34). In particular, they display a similar decay to zero at all depths below the boundary layer by hour 36, as governed by *wθ*⟩^{KP} = 0 in the detrainment zone below the boundary layer is quantified by the statistics in Table 3 (row A.8). The correlation is only 0.77, compared to 0.95 for *F*-profile (row A.7), and *δ*_{rms} is about 25% more.

### d. Momentum flux components

Both the magnitude and orientation of the stress vector contribute to errors in parameterized momentum flux components. Clear manifestations of each being dominant are evident in regimes SBL and *E*_{W}. In particular, Fig. 14c values greater than 1 after hour 42 are due to very large shears in the calculations of *τ*^{KP} from (27) and (28). On the other hand, the tendency for _{W}) leads to the strange behavior of Fig. 15c below the surface layer, where there is a zero contour near 15 m with a *σ* = 0.5. Thus, both *K*-profile components are poor representations of these regimes, with low correlations (0.75 and 0.55) and sizable *δ*_{rms} (0.33 and 0.27) in Table 3 (rows B.6 and C.6). In contrast, Figs. 14a and 14b are more highly correlated (0.99), with a much lower *δ*_{rms} = 0.06, and little mean bias (row B.5). The small differences between _{τ}, do have more effect on the sine function of (31) than the cosine, so Figs. 15a and 15b are less correlated (0.61), with higher *δ*_{rms} = 0.12 and more mean bias (row C.5).

Comparison of (a) downwind momentum flux −⟨*wu*⟩_{τ} to parameterized distributions of (b) *h* (solid red) in (a) and *h*^{PAR} in (b) and (c).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of (a) downwind momentum flux −⟨*wu*⟩_{τ} to parameterized distributions of (b) *h* (solid red) in (a) and *h*^{PAR} in (b) and (c).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Comparison of (a) downwind momentum flux −⟨*wu*⟩_{τ} to parameterized distributions of (b) *h* (solid red) in (a) and *h*^{PAR} in (b) and (c).

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 14, but for the crosswind momentum flux ⟨*wυ*⟩_{τ} and the parameterized

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 14, but for the crosswind momentum flux ⟨*wυ*⟩_{τ} and the parameterized

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

As in Fig. 14, but for the crosswind momentum flux ⟨*wυ*⟩_{τ} and the parameterized

Citation: Journal of Physical Oceanography 51, 10; 10.1175/JPO-D-20-0308.1

Figures 14a and 14b both show the downwind momentum flux decreasing monotonically from _{τ}, are tracked much better by *E*_{C}, for example, show that both *F*-profile components (rows B.1 and C.1) are quite representative. In particular, the correlation coefficients are high (0.91) and *δ*_{rms} is no more than 0.20, though there is a low mean crosswind bias. The failure of *δ*_{rms}, despite the strong constraint on *τ*^{KP} from (25).

Below the surface layer of regime *E*_{C} the *F*-profiles (row B.3), as are those of

Momentum fluxes in the detrainment zone above *d*_{E}(*t*_{ED}) are short lived, and so perhaps negligible for many purposes. Should they become important, however, the qualitative agreement evident between Figs. 14a and 14b and between Figs. 15a and 15b, offers a path forward, with the angle

### e. Comparative case evaluation

Evaluations of boundary layer (*σ* < 1) fluxes, including stress magnitude, *τ*, from different cases are quantified in Table 4 by the bracketed *δ*_{rms}, and by the correlations in square brackets. For AprS, the higher fidelity of *F*-profile scheme in Table 3 is reflected in the smaller *δ*_{rms} and higher correlations, compared to the *K*-profile, for all four fluxes. This relative performance is closely mirrored in the buoyancy flux from the four other cases, but in the momentum flux only from the other two Stokes forced cases, except for the comparatively high D24S *δ*_{rms} = 0.48. This anomaly is a consequence of *K*-profile values that are as much as a factor of 4 too high from late in D24S. Thus, it appears as if the shear is not responding to the falling wind as fast as the stress, such that *τ*^{KP} significantly exceeds *τ* over much of the boundary layer and the substantial *δ*_{rms} = 0.84 in Table 4 is accompanied by a loss of correlation.

Evaluation of *F*-profile (FP) and *K*-profile (KP) fluxes throughout the boundary layer (*σ* = −*z*/*h* < 1) of different cases, as quantified by the root-mean-square difference from the LES, *δ*_{rms} (parentheses), and the depth–time correlation coefficient (square brackets). All regimes are included, except the June MRW period.

The Table 4 *F*-profile statistics from AprS compared to AprN and from JunS compared to JunN, quantify the improved performance of all four fluxes with Stokes forcing, over cases without. In the idealized, but unphysical absence of Stokes forcing during AprN and JunN, there is much closer alignment of the wind, stress, and shear vectors throughout the boundary layer. Therefore, (8) provides better estimates of ⟨*wυ*⟩^{KP} = Γ_{υ} than it does in the Stokes cases, especially near the surface. This improved fidelity is quantified by the decreases in *K*-profile *δ*_{rms} relative to the corresponding Stokes cases, in contrast to the increases in the *F*-profile values. Considering *τ* for example, a decrease from 0.53 in AprS to 0.38 in AprN is associated with an *F*-profile increase from 0.19 to 0.38. In general the AprN and JunN momentum fluxes from both schemes are comparable, though the *K*-profile *δ*_{rms} becomes the smaller for all three quantities in JunN.

## 7. Discussion and conclusions

The boundary layer depth is an important upper-ocean parameter, because it bounds the extent of large surface forced local and nonlocal turbulent transports through the shape function *G*_{C}, and it directly scales the mixing coefficients *K*_{m,s} (7). However, the multiple options in Table 1 can be problematic. Neither the inversion depth, nor a mixed layer depth is a viable proxy, because they do not track the collapse with stable buoyancy forcing, for example. The depth *h*^{L19} is not appropriate throughout much of regimes DBL and SBL when it lies beyond the *d*_{MKE} bound (Fig. 9b), and of regime *E*_{W} when Fig. 7b shows it is too deep to be consistent with the entrainment rule (17). Similarly, *h*^{KPP} is overly deep in regime *E*_{W} (Fig. 7c). In contrast, both the entrainment depth and buoyancy flux are robust diagnostics of LES buoyancy flux profiles, with *h* = *h*^{BL}, consistent with (26). However, the parameterizations of these depths are not straightforward, especially when the degree of equilibrium is a factor. The entrainment rule does give a parameterized *h*^{ER} and consistent *h* = *h*^{AM} from (21), as well as *d*_{MAX} (22) in regime DBL, but these empirical relationships from AprS and AprN may not be universal. Nonetheless, *h*^{PAR} = *h*^{AM} is appropriate in the regime SBL example of Fig. 10, though

In general, the momentum flux is aligned with neither the wind, nor the shear, though at depth the stress vector tends to remain more oriented with the wind than to the veering shear, such there can be a significant across-shear component. A major effect of Stokes forcing is to accentuate the misalignment between the stress and shear vectors, especially near the surface where the Stokes drift and shear are greatest. For example, during regimes SBL and *E*_{W} of AprS, the angle between the wind and shear Ω_{W} is greater than 45° near 4-m depth. The Hughes et al. (2020) observations of shear in the diurnal warm layer also show Ω_{W} > 45° below 3 m. Thus, the absence of across-shear momentum flux is a possible explanation of why not one of four turbulence parameterizations correlates with their observations. Hence, downgradient eddy viscosity does not appear to be a reliable assumption. Therefore, a fortunate finding is that throughout the boundary layer the angle Ω_{τ} of the stress from the wind is highly correlated with the angle _{τ} is better tracked by another empirical angle *wu*⟩^{KP} from (27), or with a *K*-profile stress magnitude, there would be no need to parameterize Γ_{υ}, and hence to improve (8) near the surface.

The development environment for the entrainment and detrainment rules is limited to the Southern Ocean in autumn, but there is a wide range of variable forcing. Thus, the detrainment rules (22) and (23) evaluated in Fig. 11a, include unstable periods of AprS and JunS, as well as the more familiar AprS detrainment below a stable boundary layer. The full range, from extreme wave conditions, including significant swell (L19), to idealized calm seas is involved in the development of the entrainment rule (17). Although the Coriolis parameter is fixed, a variety of rotational interactions, such as cos Ω^{PAR} in (30), is provided by the counterinertial wind rotation of the WRW period, by the subsequent inertial rotation and by the falling wind of D24S. This rule also applies to the very different entrainment of regimes *E*_{C} and *E*_{W}, but it should not be regarded as universal, pending further development. In particular, the function *F*(*δρ*) in (17) remains to be determined, but the 0.7 coefficient of (32) is consistent with the function decreasing to zero as the stratification at the entrainment depth vanishes. As a preliminary examination of shallower, much stronger stratification and dominant wind forcing, the entrainment rule has been applied to the Watkins and Whitt (2020) simulations of a hurricane over a coastal shelf. Compared to early in AprS and AprN, the stratification at the inversion is stronger by about a factor of 10, the entrainment forcing of (17) is about the same, even though the integrated TKE production to the inversion is about an order of magnitude higher. The entrainment *B*_{E} is order 100 times greater, and the boundary layer depth is only about one-tenth. Thus, these results would fit well within the spread of Fig. 6, with *F*(*δρ*) equal to about 10, at 10 times greater stratification. In the parlance of the dimensional analysis of similarity theory, this finding supports regarding surface layer TKE source integrals as independent variables (L19a), that determine the dependent variables associated with entrainment, at least in regime *E*_{C}. Alternatively, integrals to the inversion (Fig. 1) would yield consistent results with no change in *F*(*δρ*). However, these deeper integrals have yet to be parameterized.

From section 4, the fractions of positive boundary layer TKE production by Eulerian shear, Stokes shear and buoyancy that go to increasing potential energy rather than dissipating, are 0.020, 0.035, and 0.27, respectively. An interpretation is that buoyancy produces TKE at the largest scales and hence is most efficient at increasing potential energy by entrainment. Furthermore, Langmuir turbulence has larger scales and is more efficient than turbulent motions driven by Eulerian shear. Thus, the effects of smaller-scale, near-surface processes, such as wave breaking and solar penetration, may be relatively small, especially for deep boundary layers. In pure convection (*B*_{E}/*B*_{0} from (17), ranges from about 0.10 to 0.15, depending on the ratio (*h*/*d*_{E}) in Fig. 7a. Considering this range is an extrapolation from strong wind and wave forcing, with ocean stratification and *F*(*δρ*) = 1, it is roughly consistent with the empirical 0.2 from atmospheric convection (section 4).

The *K*-profile fluxes of section 6 are based on Monin–Obukhov similarity theory in the surface layer, including Stokes effects from L19a as updated in section 6, on rules for local and nonlocal (8) boundary layer transport developed in L19b, and on a parameterized boundary layer depth *h*^{PAR}. They do not yet extend into the detrainment zone below the boundary layer. The parameterizations are empirical and assume that the forcing, fluxes and local shear and stratification are in sufficient equilibrium for eddy diffusivity and viscosity concepts to be applicable. Accordingly, they appear to be most reliable during regimes SBL and *E*_{C} apart from the WRW and MRW periods, less so for the afternoon entrainment of regime *E*_{W}, and unreliable during regime DBL and late in D24S. The prospect of improving detrainment is not promising, because there is no signature of a distinct boundary layer depth in any turbulence distribution.

An interpretation of the differences between schemes in the Li et al. (2019) comparison is that realistic forcing has different effects when equilibrium assumptions may not be valid. In contrast, the *F* profiles are constructed to apply to nonequilibrium boundary layers and detrainment zones by incorporating some time history. Specific examples are *h*^{AM} (21), *d*_{MAX} (22), *B*_{MAX} (23), and *τ* profile into momentum flux components. In addition, *K*-profile fluxes at the entrainment depth where Figs. 8 and 13 show them to be most reliable, are used to determine *h*^{ER}, consistent with the entrainment rule (17). In principle, these values could be supplied by any boundary layer scheme, including STM. In any case, the empirical coefficients and criteria from limited cases may not always be applicable. Another possibility would be to use *d*_{MKE}, as a scale for the vertical penetration of the stress during unstable detrainment (Fig. 10; hour 32). The specific functions of Table 2 do serve as examples to demonstrate how curvature can accommodate nonequilibrium, but there could be more widely applicable alternatives, especially for counterinertial wind rotation, such as the April WRW period with

The *F* profiles are constructed with continuous first derivatives, so that accelerations and buoyancy tendencies are given directly at any depth, rather than as flux differences that strongly depend on gradients. The consequences for numerical stability, adaptive vertical grids and conservation could be substantial. Also, it would be straightforward to solve a heat equation very near surface, so that the evolving temperature would be most appropriate for bulk formula calculations of the surface sensible and latent heat fluxes. This temperature could be combined with an estimate of the cool skin effect (Fairall et al. 1996) to give the sea surface temperature governing the emission of longwave radiation.

Both parameterizations of section 6 have been applied with some success to three flavors of diurnal cycle. The most important is the one April day when solar radiation is sufficient for the surface buoyancy forcing to become stable and the boundary layer to collapse with detrainment below. Diurnal effects are negligible over the first April day and the second June day when the forcing remains unstable and strong, but there are discernible signals when the unstable forcing is weak during regime DBL of the first June day. However, additional cases with much stronger noon heating found in summer or at lower latitudes, with penetrating solar radiation, and with stronger wind are needed for widespread verification. Nonetheless, the prospect of a general mixing scheme applicable to all regimes is supported by the smooth and orderly time–depth flux variations of Figs. 13–15, with past history clearly involved (Figs. 3, 8, and 10), but not always eddy diffusivity and viscosity that rely on equilibrium with local stratification and shear.

## Acknowledgments

This work was made possible by support from the U.S. Department of Energy (DOE) under solicitation: DE-FOA-0001036, Climate and Earth System Modeling: SciDAC and Climate Variability and Change, Grant SC-00126005. The 1-h averaged data are available half-hourly from Version 1.0. UCAR/NCAR DASH Repository (https://doi.org/10.5065/87n8-9r86). The other principal investigators—Todd Ringler, Gokhan Danabasoglu, and Matt Long—are gratefully acknowledged, as are the contributions of Alice DuVivier and Justin Small. A special thanks to Dan Whitt for providing results from the Watkins and Whitt (2020) hurricane simulations archived at https://doi.org/10.6084/m9.figshare.12486056.v5 (irene-les-mean-2304 × 84all.nc). The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation. The four SOFS simulations were forced, initialized and evaluated with data sourced from Australia’s Integrated Marine Observing System (IMOS) - IMOS is enabled by the National Collaborative Research Infrastructure Strategy (NCRIS). They utilized resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-05CH11231. The idealized simulations utilized high-performance computing on Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Information Systems Laboratory.

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