## 1. Introduction

Small-scale mixing in the stratified ocean interior plays an important role in regulating the meridional overturning circulation (MOC) (see, e.g., Munk 1966; Munk and Wunsch 1998; Wunsch and Ferrari 2004), though recent evidence suggests that wind-driven upwelling in the Southern Ocean may be just as important as interior mixing in this respect (Toggweiler and Samuels 1998; Wolfe and Cessi 2009, 2010; Lumpkin and Speer 2007). Regardless, many geochemical and biological processes are impacted locally by mixing (Mann and Lazier 2006).

From a theoretical point of view, the problem has been traditionally framed in terms of the rate *ϵ*_{k} at which turbulent kinetic energy (TKE) is lost to friction; the background stratification, characterized by the value of the Brunt–Väisälä (BV) frequency *N*; and the mixing efficiency, which is intended to quantify the fraction of the overall energy lost to diabatic processes, which goes into irreversibly raising the center of mass of the system. From these quantities, length and time scales characteristic of the turbulence are derived. Vast theoretical, laboratory, and observational efforts have been devoted to provide robust estimates of these quantities in terms of the properties of the large-scale flow that drives the turbulence [see, e.g., Ivey et al. (2008), for a review], though in general a simple parameterization may not be possible (Mater et al. 2013; Mater and Venayagamoorthy 2014). In this note we focus on the use of the Thorpe scale *L*_{T} to diagnose rates of energy dissipation and mixing in flows that are overall stably stratified.^{1} The method holds the promise of estimating dissipation from relatively inexpensive vertical measurements of density obtained from CTD casts, free-falling profilers, or even moored sensors and is being widely used (see, e.g., Waterhouse et al. 2014). It has even been applied to atmospheric datasets (Wilson et al. 2010). The method relies on the assumption that *L*_{T} is a suitable proxy for the Ozmidov scale *L*_{O} ≡ (*ϵ*_{k}/*N*^{3})^{1/2}, which is supported by open-ocean measurements (Dillon 1982; Wesson and Gregg 1994; Moum 1996; Ferron et al. 1998).

In a companion paper, Mater et al. (2015, hereinafter MVSM) compare estimates of dissipation from microstructure profilers with estimates based on the Thorpe scale in field data and find that the latter can overestimate the former, that is, *L*_{T} ≫ *L*_{O}, when the vertical size of the overturns is large. Increasing the resolution of global circulation models makes them more sensitive to how mixing is parameterized (Jayne 2009), and it is therefore important to critically reassess the theoretical derivation of the standard recipe linking *L*_{T} to dissipation in order to establish its range of applicability. This is accomplished in this paper by combining theoretical arguments based on the turbulent decomposition of kinetic energy (KE) and available potential energy (APE) of Scotti and White (2014, hereinafter SW) with highly resolved direct numerical simulations (DNS) of stratified mixing. SW’s approach allows the separation of APE into turbulent and mean components and highlights the differences between flows in which the energy for mixing comes from the kinetic energy associated with the mean flow (shear-driven mixing) and flows in which the energy for mixing comes primarily from the available potential energy of the mean field (convective-driven mixing).

While a more precise definition will be given further below, we emphasize that the distinction is based on where the energy for mixing is initially contained in the large-scale field. For example, the classical scenario of mixing driven by the onset of Kelvin–Helmholtz instabilities is an example of shear-driven mixing according to our classification because even though the instabilities may build up APE in coherent structures (the billows), which is eventually transferred to turbulent KE and turbulent APE, the billows are *not* part of the large-scale flow but rather an instability of the latter. Conversely, in the appendix we provide an example from the reflection of internal wave beams off a sloping bottom where the large-scale flow develops, as part of its own dynamics (i.e., not because of instabilities), regions of elevated APE and little KE. Instabilities will then use a fraction of the APE to drive mixing. Clearly shear- and convective-driven mixing as defined here are idealizations. Any real mixing event in the ocean will combine elements of the two. However, it is important to recognize that they involve different energetics pathways. In particular, the standard recipe that gives turbulent dissipation and mixing in terms of *L*_{T} is appropriate for shear-driven turbulence. For convective-driven mixing, the use of the standard recipe overestimates the amount of dissipation. The crux of the matter is that the Thorpe scale measures the total amount of APE (mean plus turbulent), whereas for estimating dissipation and mixing it is the amount of APE associated with the turbulent eddies that matters. This is why in flows where the mean field has little or no APE (shear-driven turbulence) the standard recipe works. We show that if the length scale used is suitably redefined to isolate the contribution of the actively turbulent eddies, good agreement can be obtained even in the case of convective-driven mixing, though how to practically extract the component due to the mixing eddies may not be so easy.

MVSM assume that the mixing events in their datasets are shear-driven events and discuss the bias in *L*_{T}/*L*_{O} as a result of sampling “young” events. Our analysis highlights a second possibility for biases in *L*_{T}/*L*_{O}, related to the nature of the mixing event.

The rest of the paper is organized as follows: Section 2 reviews briefly SW’s theory and presents the theoretical analysis; the main hypotheses to be tested are discussed in section 3. Section 4 describes the details of the numerical simulations and contains a rationale for the choice of flows, followed by section 5, which uses the datasets to verify the assumptions at the heart of the analysis presented in section 2. Finally, in section 6 we summarize the results and discuss possible applications to real ocean data. An appendix discusses a scenario where convective-driven mixing, as defined in this paper, can be expected to occur in the ocean.

## 2. Theory

### a. Dimensional analysis

*N*; the background shear

*S*; and the rate of dissipation of turbulent kinetic energy

*ϵ*

_{k}. Two length scales can be derived from these quantities, the Ozmidov scalewhich is usually interpreted as the vertical size of the largest eddies that can overturn (Ozmidov 1965), and the Corrsin scalewhich bounds from below the size of eddies that are deformed by shear (Corrsin 1958). The ratio

*L*

_{C}/

*L*

_{O}= Ri

^{3/4}, where Ri ≡

*N*

^{2}/

*S*

^{2}is the gradient Richardson number, indicates that in shear-driven turbulent mixing flows, the Ozmidov scale is the largest of the two. If a suitable proxy for

*L*

_{O}can found, and

*N*is known,

*ϵ*

_{k}can be obtained from Eq. (1).

### b. Oceanographic practice

*L*

_{O}from CTD casts has gained considerable popularity. Briefly stated, the method starts from a vertical density profile obtained from a CTD cast, a free-falling instrument, or from a set of moored instruments. Under the assumption that turbulent eddies displace water parcels from their undisturbed position, the first step is to reconstruct the undisturbed profile. This is achieved through an algorithm that takes the observed profile

*ρ*(

*z*), which in general will have regions where

*dρ*/

*dz*> 0, and returns a profile

*ρ*

_{1}and

*ρ*

_{2}is the same in the measured profile and in the undisturbed profile. Practically,

*ρ*. Let

*z*be the in situ elevation of a parcel of density

*ρ*. The Thorpe displacement is defined asFinally, the Thorpe scale

*L*

_{T}is defined as the rms value of

*δT*over the profile, though Thorpe points out that the rms should be calculated over many profiles (see Thorpe 2005, his footnote 7). In the latter case,

*L*

_{O}and

*L*

_{T}are proportional in shear-dominated mixing (Dillon 1982, 1984; Dillon and Park 1987; Wesson and Gregg 1994; Ferron et al. 1998), and

*L*

_{T}has by now become the de facto estimate for the Ozmidov scale.

*ϵ*

_{k},

^{2}since substituting

*L*

_{T}for

*L*

_{O}in Eq. (1) above yieldsThe quantity

*C*is a dimensionless constant assumed

*O*(1), though a priori should depend on

*L*

_{C}/

*L*

_{O}, and

*N*is the Brunt–Väisälä frequency calculated from the resorted profile. Once

*ϵ*

_{k}is known, the diffusivity is recovered via the well-known relationwhere Γ is the ratio of turbulent buoyancy flux to

*ϵ*

_{k}, generally assumed to be constant and equal to 0.2 (Osborn 1980). Even though Thorpe (2005, p. 176, note 6) cautions against using it in situations other than purely shear-driven mixing, Eq. (4) has become the standard way to estimate

*ϵ*

_{k}from CTD casts regardless of the mechanism that sustains turbulence (Alford et al. 2006; Waterhouse et al. 2014). It has also been recently proposed as a way to parameterize turbulent mixing in numerical simulations of large-scale overturns where the resolution employed is not enough to capture the turbulent scales responsible for the actual mixing (Klymak and Legg 2010). MVSM provide strong field evidence that

*L*

_{T}/

*L*

_{O}can be (much) greater than one in flows characterized by large overturns, in which case Eq. (4) overestimates dissipation and mixing. It is therefore essential to determine the range of applicability of Eq. (4). Unfortunately, a derivation based solely on dimensional analysis is not enough. Equation (1) provides a length scale, for which we have a name, the Ozmidov scale, but dimensional analysis does not tell us how to calculate it without a priori knowledge of

*ϵ*

_{k}, an estimate of which of course is what we are seeking. In the next section, we provide an alternative derivation based on energetics. Its point of departure can be traced to the original work of Thorpe (1977), as expanded by Dillon (1984) and Dillon and Park (1987).

### c. An alternative approach based on energetics

Dillon’s seminal analysis relied on a number of assumptions, which were in part due to the limitations of the observational datasets available and in part due to the still early stage of development of the APE concept, which, since then, has been developed into a powerful tool to diagnose stratified flows [for a recent review, see Tailleux (2013), and references therein]. Particularly relevant to the problem at hand is the work recently published by SW, in which the authors show that just as the kinetic energy of a turbulent flow can be divided into a mean and a turbulent component, so can the APE. Such a split is possible because we can define the APE of a single parcel of fluid (though still dependent on a reference state) and then apply the machinery of Reynolds decomposition. Thus, we have now at our disposal a much more precise diagnostic tool. Further, direct numerical simulation, in its early infancy in the 1980s, provides datasets where we can test hypothesis without relying on ad hoc assumptions.

#### 1) Reynolds decomposition of kinetic and potential energy: Shear- and convective-driven mixing

*f*be a generic field. We denote its Reynolds decomposition

*S*

_{ij}be the strain rate tensor. In analogy, we define Θ

_{I}≡ ∂

_{i}

*b*, with

*b*≡

*g*(

*ρ*−

*ρ*

_{0})/

*ρ*

_{0}, where

*ρ*

_{0}is a reference density. Just as for the globally defined APE, we need to introduce a reference state, analogous to

*b*(

**x**,

*t*) to yield a field

**x**= (

*x*,

*y*,

*z*) and time

*t*asNote how the integrand

*b*in the isochorically resorted density field. Also, from now on we will drop the parametric dependence on

*t*for clarity. Like any other field variable, the APE can be Reynolds decomposed, and appropriate transport equations can be written for its mean and turbulent components, just as transport equations can be written for the mean and turbulent component of the KE [see, e.g., Tennekes and Lumley 1972, their Eq. (3.2.1)]. Figure 1 provides a schematic representation of how the total energy within a given volume (which can be infinitesimal, given the locality of the theory) is apportioned between the mean and turbulent KE and APE reservoirs. Each quantity is by definition positive and is connected to the other reservoirs by conversion terms. The values

*κ*being the diffusivity of the stratifying agent and

The turbulent KE and APE reservoirs are directly connected to mixing, whence it follows that quantification of mixing should depend on the amount of energy contained in these reservoirs and a suitably defined residence time. Energy is injected by the mean field into turbulence, where it is dissipated either as heat via *ϵ*_{k} or goes into raising the center of mass of the isochorically resorted system via *ϵ*_{p}.

*ϕ*

_{k}into turbulence. Figure 2 shows the resulting simplified flow of energy. There will be turbulent APE generated as a result of the instabilities, but generally the energy flow is out of mean KE into turbulence. Under the standard assumptions of quasi equilibrium between production and dissipation (Osborn 1980),

*ϵ*

_{p}, and the efficiencyis related to the flux Richardson numbervia the well-known relationWe call this the shear-driven mixing case. A well-studied example is the case of shear leading to turbulence via the Kelvin–Helmholtz route to instability (Smyth et al. 2001).

At the opposite end of the spectrum, we consider flows whose large-scale dynamics creates statically unstable localized regions with little or no shear. That is, the energy of the large-scale flow is contained in the mean APE reservoir. Within the unstable region, turbulence derives its energy from the mean APE reservoir, and the energetics in this case is shown in Fig. 3. Note that in this case, the turbulent buoyancy flux transfers energy from the mean APE to turbulent KE, and thus it is negative. Also, since we assume Ri_{f} ≪ −1,

According to these definitions, shear- and convective-driven mixing represent opposite ends of the spectrum of conditions under which mixing occurs in the ocean. While any real mixing event will likely encompass both shear and convective features, it provides a useful classification. In particular, we will show that Eq. (4) provides a valid estimate of dissipation for shear-driven mixing [i.e., *L*_{T}/*L*_{O} = *O*(1)] but likely overestimates mixing for convective-driven mixing (*L*_{T}/*L*_{O} ≫ 1).

#### 2) A displacement-based estimate of mean and turbulent APE

*l*

_{T}:so that to lowest orderSimple algebra [see Eq. (2.29)–(2.31) in SW] shows thatso that replacing

*b*in the volume under consideration straddles a region of rapid change in the derivative of the reference profile. But in this case, it is no more legitimate to treat

*l*

_{T}as a small quantity. Indeed, the precise formulation of the small parameter isand the APE averaged over the parcels in the volume under consideration is(from now on we drop the

*N*

^{2}for simplicity). In passing, if the resorted field is linear, then Eq. (11) is exact. Indeed,

*μ*measures the departure of the resorted profile from linearity over the range of vertical motions associated with the flow. Further, we can Reynolds decompose the field

*l*

_{T}(

*z*,

*b*) to obtain the mean

#### 3) From displacements to dissipation

*ϵ*

_{p}is the turbulent APE dissipation rate (Fig. 1). We assume thatwhere at this point

*α*is an as yet unspecified nondimensional quantity.

^{3}The equality of the dissipation times is based on the assumption that the loss of KE and APE is due to the action of the same eddies. Note that this does not imply that the dissipation rates are the same. However, it does imply that the ratio of turbulent APE dissipation rate to turbulent KE dissipation rate equals the ratio of turbulent APE to turbulent KECombining Eqs. (15) with (17), we obtainFinally, the turbulent diffusivity becomeswhich does not depend explicitly on Γ, though

*α*may still depend on it.

## 3. From theory to practice

In the preceding section we have derived an estimate for the dissipation within a turbulent stratified flow [Eq. (18)] that rests on two assumptions:

- Hypothesis A
_{1}: The term*μ*is small or equivalently Eq. (11) is a good approximation for the APE. - Hypothesis A
_{2}: The dissipative time scales for turbulent KE and turbulent APE are equal.

The second goal is to investigate when the formula for the dissipation in terms of the Thorpe scale Eq. (4) provides a reasonable estimate of the actual dissipation. MVSM frame this last problem in terms of the ratio *L*_{T}/*L*_{O}, which should be close to unity if Eq. (4) describes dissipation accurately. Here, in light of the theory developed in the preceding section, assuming for the time being that Eq. (18) is correct, we see that Eqs. (4) and (18) are equal if the following are met:

- Hypothesis B
_{1}: We identify the constant*C*with (2*α*Γ)^{−1}. - Hypothesis B
_{2}: Calculating the reference state by resorting the density profile given by each vertical cast as is oceanographic practice is statistically equivalent to using the isochorically restratified profileas defined in the APE theory. - Hypothesis B
_{3}: The Thorpe scale*L*_{T}coincides with the turbulent displacement.

*α*Γ on how turbulence is sustained will be addressed with the help of the numerical experiments described in the next sections. To be valid, estimates of dissipation based on the Thorpe scale require 2

*α*Γ to be at most a weak function of the parameters of the flow.

The second condition is necessary to ensure that the BV frequencies used in Eqs. (4) and (18) agree. In the numerical calculations, we have access to the full *b* field, and the domain contains all of the relevant scales. Therefore, the isochoric resorting provides the “true” background state (Winters et al. 1995). The situation when dealing with field measurements is different. For obvious practical reasons multiple vertical casts cannot be guaranteed to sample the same statistical ensemble since large-scale processes (e.g., internal tides) can change the background stratification between one cast and the next. Thus, using multiple casts to determine a single reference state would alias processes (e.g., displacements due to internal tides) that do not have anything to do with the turbulence responsible for the small-scale inversions, resulting in a biased estimate of the Thorpe scale. Rather, each cast must be seen as traversing a separate realization out of an ensemble of random processes, each characterized by a potentially different background state that can be approximated by resorting the vertical density profile measured during the cast (Dillon 1984). This hypothesis can be examined with the aid of the numerical datasets comparing the PDF obtained from the isochoric restratification with the PDF obtained from *n* vertical casts, as a function of *n*.

_{2}applies, whence it follows that statistically

*δT*and

*l*

_{T}are equivalent (assuming that a suitably large ensemble for

*δT*exists), and as suchwith the equality holding if and only if

*μ*and assuming statistical equivalence between

*δT*and

*l*

_{T}, the Thorpe scale measures the total APE in the region of interest. However, what we need is a measure of the fraction of APE of the eddies directly engaged in the turbulent process, that is the turbulent APE. As the rest of the paper will show, in mixing events that are closer to being convective than shear driven,

## 4. Numerical simulations

*b*≡

*g*(

*ρ*−

*ρ*

_{0})/

*ρ*

_{0}, where

*g*is the local gravity,

*ρ*is the density, and

*ρ*

_{0}is the reference density, and the concentration of the stratifying agentHere,

**u**is the velocity, and

*P*is the pressure. The nondimensional form of the equations depends on five parameters. Three are dynamical, namely, the Reynolds number Re, which measures the strength of diabatic processes; the bulk Richardson number Ri, which couples buoyancy to momentum; and the ratio of viscosity to diffusivity Pr. In addition, the geometry of the domain and possibly the initial conditions introduce two additional aspect ratios.

The equations are solved on a staggered grid, whereby the fluxes are computed at the corresponding cell edges, and scalar quantities (pressure and reduced gravity) are calculated at cell centers. The equations are solved in time using a projection method. Convergence is checked ensuring that the spectra of velocity and density increments capture the peak and rolloff. Details can found in Scotti (2008).

### a. Choice of flows

When considering numerical simulations, a choice out of necessity must be made regarding what scales can be realistically simulated as opposed to what scales need to be idealized or modeled. In this paper, we focus on the scales that experience turbulent motions, whose dynamics we want to capture realistically. To achieve a meaningful scale separation between the Ozmidov and the Kolmogorov scales, the number of grid points becomes quite elevated, reaching close to one billion nodes in one of the simulations (see Table 1). Yet, despite such a large number of grid points, we are still forced to consider very idealized driving conditions, detailed below, which are usually nowhere to be found in the real ocean. As a justification, we offer the following argument.

DNS parameters. The simulations are characterized by three nondimensional dynamical parameters: Pr is the ratio of viscosity to diffusivity of the stratifying agent; a bulk Richardson number Ri that measures how strongly buoyancy is coupled to the vertical momentum; and a Reynolds number Re that controls the strength of the viscous term. In the Couette flows they are based on the velocity and buoyancy difference across the channel and the channel height giving Re ≡ Δ*UH*/*ν* and Ri ≡ Δ*bH*/Δ*U*^{2}. In the convective case, we use *N*^{−1} as a time scale, and the thickness of the overturned layer *δ*_{o} as a length scale. Then

Any parameterization of turbulent processes, of which Eq. (18) is one, relies on the assumption that, in flows that share similar, though not necessarily identical, driving mechanisms, a degree of universality exists on the scales experiencing turbulence. It is of course possible to reject such an assumption, but since in the real ocean no two mixing events are ever identical, then no parameterization would be possible. Conversely, if we are willing to concede that the parameterization of turbulent processes is possible, the relevant question shifts from “does flow A exist in the ocean?” to “does flow A capture the essential features of a class of turbulent flows that exists in the ocean ?” The latter is the question we considered when choosing what flows to simulate.

### b. Shear-driven mixing

As an example of highly idealized shear-driven mixing, we consider the flow in a channel driven by a velocity differential applied at the lower and upper boundary (stratified Couette flow). Periodic conditions are imposed on the horizontal directions, and Dirichlet conditions are imposed on velocity and reduced gravity at the upper and lower boundary. In this configuration Ri = Δ*BH*/Δ*U*^{2} and Re = *H*Δ*U*/*ν*, where Δ*B* and Δ*U* are the reduced-gravity and velocity differences imposed at the boundaries, *H* is the height of the channel and *ν* is the viscosity. This type of flow must be considered as the simplest case of shear-driven mixing that can be maintained in steady state. The purpose is to test the theoretical prediction of the previous section, for which we need a flow that is turbulent, stratified, and in which a reasonable separation exists between the Ozmidov scale and the Kolmogorov scale on one hand and the Ozmidov scale and the boundaries on the other, while at the same time *μ* is small. Stratified Couette flows with the parameters considered here fit these requirements. We could have considered the case of a time-evolving mixing layer in which Kelvin–Helmholtz instabilities drive turbulence (Smyth et al. 2001). However, a Couette flow has several advantages: The flow is maintained at steady state by the injection of momentum, buoyancy, and energy at the boundaries, so we do not have to worry with external time scales imposed by the large-scale evolution of the flow. Moreover, unlike a time-evolving mixing layer, in stratified Couette flows we can separately vary the ratio *L*_{0}/*η*_{k}, where *η*_{k} is the Kolmogorov scale and the ratio Γ, which is roughly proportional to Ri (Scotti and White 2014, manuscript submitted to *J. Fluid Mech.*), which allows us to test the scaling ideas over a wider range of parameters. Finally, since the flow is homogeneous along the horizontal directions and in time, averages along the *x* and *y* directions and time *t* are used as a proxy for ensemble averages, resulting in very robust statistical estimates. Table 1 indicates the parameters of these simulations. The buoyancy Reynolds number Re_{b} ≡ (*L*_{O}/*η*_{k})^{4/3} of the simulations are representative of values encountered in the field (Smyth et al. 2001).

### c. Convective-driven mixing

As an example of convective-driven mixing, we consider a uniformly stratified fluid in which at *t* = 0 an isolated patch of unstably stratified fluid is released. The patch is obtained by exchanging parcels of fluids across the midplane near the center of the domain. Practically, this means that vertical resorting yields very close estimates to the actual reference state. The fluid is enclosed by no-slip boundaries at the top and bottom and is otherwise periodic in the remaining directions. A small-amplitude sinusoidal perturbation with a wavelength equal to the spanwise length of the domain is introduced along the *y* direction to ensure that the flow becomes three-dimensional. Figure 4 shows the initial condition in reduced gravity, the reduced-gravity profile at *t* = 0 along a vertical transect across the middle of the patch, and the reference reduced-gravity profile obtained by an isochoric restratification, which, as mentioned earlier, coincides with the vertically resorted profile by construction. In an unbounded domain, energy can radiate away as internal waves. This is prevented in our simulations by the solid boundaries at the top and bottom. Rather, we observe the development of a pattern of standing waves.

In these flows, we use *N* and the vertical scale *h* of the overturn to nondimensionalize the problem, given Ri = 1 and Re = *h*^{2}*N*/*ν* = 4000. This value is not far from what can be expected in overturned regions generated by internal tide beams reflecting off a sloping bottom in the weakly stratified ocean interior (see the appendix). MVSM interpret

The aspect ratio of the overturned patch *L*_{h}/*h* measures the horizontal extent of the patch. Here, we consider two cases: a high aspect ratio patch (*L*_{h}/*h* = 10), which we take as being representative of “oceanic” conditions, and an isotropic patch (*L*_{h}/*h* = 1).

This flow represents a highly idealized case of convective-driven mixing. We choose it because it represents a flow in which the energy for mixing is provided initially by mean APE. As the flow evolves, its energetics involve in a nontrivial way the four energy reservoirs in the form of a large-scale, largely adiabatic exchange between mean APE and mean KE due to the standing waves and a turbulent episode involving the transfer of energy to turbulent APE and turbulent KE. Note that in a more realistic setting, we would expect that mean APE and mean KE would instead be associated with radiating internal waves. In the appendix we provide an example of oceanic significance (reflection of internal tide beams of a sloping topography) that shows how thin, near-horizontal regions can develop where the stratification is locally unstable with little shear present. Moreover, we show that under the weakly stratified conditions often encountered below the main thermocline, the time scale of convective instabilities is long enough for the large-scale flow to grow the overturned region to finite size before we expect turbulence and mixing to substantially alter the flow.

Since the patch is localized in the *x*–*z* plane, and evolving in time, averages along the *y* direction are used as a proxy for the ensemble average.

## 5. Results

### a. Shear-driven flows

#### 1) Thorpe displacement versus

We discuss here the flow with the highest value of Re_{b}, but very similar results apply for smaller values of Re_{b} as well. Figure 5a shows the reduced-gravity profile along a randomly chosen vertical transect and, for comparison, both the isochorically resorted profile *z* < 0.2 and near *z* = 0.6), resulting in low Thorpe displacement in these regions (Fig. 5b). We have calculated *δT* from 1000 randomly chosen vertical casts, using the profile in each cast to calculate a reference state, according to the oceanographic practice outline earlier. If we compare the PDF of *δT* versus the PDF of *l*_{T}, we note that the PDF of *δT* reasonably follows the PDF of *l*_{T} even when only a relatively few casts are included, though the agreement obviously improves with the number of casts because of the departure from Gaussianity along the tails of the distribution. However, there is a persistent bias favoring displacements smaller than 0.05 [Fig. 6(left)]. If we now calculate for each cast *L*_{T}, its PDF peaks at about 80% of the rms value of *l*_{T} [Fig. 6 (right)]. Considering the asymmetric shape of the PDF of *L*_{T}, the expected value of *L*_{T} is close to 90% of the value calculated from the actual reference state, obtained from an isochoric restratification. While not perfect, the agreement is certainly very good, and therefore we can conclude that the statistics derived using

#### 2) Dissipation time scale and turbulent KE to turbulent APE ratios

As expected from our analysis, away from the viscous layers near the boundaries where *N*^{2} varies rapidly and *μ* is not small, the mean turbulent APE profile (Fig. 5c) is very well approximated using *l*_{T}. This validates hypothesis A_{1}. Because the flow is at steady state, *τ* is the same for KE and APE (A_{2}). Further, *Nτ*Γ^{−1/2} appears to be constant to first order [Fig. 7 (bottom)], and we find that *α* ≃ 4.5Γ^{1/2}. It is oceanographic practice to assume a value close to 1/5 for Γ; hence, (2*α*Γ)^{−1} = 1.2, slightly larger than the range of values reported in the literature (0.6–0.9) (Dillon 1982; Ferron et al. 1998), though since *α* tends to increase somewhat with the buoyancy Reynolds number, the discrepancy is not significant.

In the core of the flow, the dissipation obeys a lognormal distribution. When normalized with the value given by Eq. (18), the PDFs collapse nicely. The peak of the distribution appears to be slightly less than what is predicted by Eq. (18), but overall the dissipation is well represented (Fig. 8).

Note that as the background stratification weakens, the rms value of *l*_{T} increases, but it never exceeds more than 15% of the channel depth. Thus, we can confidently assume that the physical boundaries do not artificially constrain the size of the largest mixing eddies.

Overall, in shear-driven mixing flows the conditions under which the quadratic approximation to APE holds and turbulent KE and turbulent APE have the same dissipative time scale are satisfied (hypothesis A_{1}–A_{2}). Likewise, (2*α*Γ)^{−1} is found to be close to unity (hypothesis B_{1}) and individually resorted profiles are statistically close to the actual reference state (hypothesis B_{2}). Hypothesis B_{3} is satisfied by definition in shear-driven mixing.

### b. Convective-driven mixing

_{2}is trivially verified. The subsequent development of turbulence may invalidate locally the symmetry along the midplane, but we find that the differences are small. Thus, as in the shear-driven case, we can assume that for statistical purposes

The energetics of the adjustment process of a localized region where the density field has been overturned involves a nontrivial interaction between the four reservoirs (Fig. 3). The mean APE and KE exchange energy as standing waves, characterized by a time scale that is *O*[tan(*A*) *N*^{−1}], where *A* is the aspect ratio of the overturned patch (Fig. 9a). There is also a single turbulent episode, which occurs near the beginning of the simulation (*tN* = 20; see Fig. 10) where part of the energy initially contained in the mean APE is transferred to the turbulent APE and turbulent KE. The time scale of the turbulent episode is *O*(*N*^{−1}). The turbulent episode shows up clearly when we consider the evolution of the background potential energy (BPE), defined as the difference between the potential energy *bz* and the APE defined in Eq. (6). Before turbulence sets in, and after the turbulence disappears, the BPE is constant [once the trivial diffusive component given by *E*_{BPE} to the initial amount of energy contained in the system (all in the APE reservoir at *t* = 0) measures the overall mixing efficiency of the process, that is, how much of the available energy is lost to irreversibly raising the center of mass of the system. For both aspect ratios, we find an efficiency slightly larger than 10%. Note that while in shear-driven mixing this definition of mixing efficiency coincides with the flux Richardson number, it does not in the present case. In addition to losing energy to turbulent APE and from there to mixing, mean APE loses energy to turbulent KE and dissipation at a rate equal or slightly larger in the high aspect ratio and somewhat smaller (but comparable) in the isotropic case (Fig. 9b). Therefore, roughly three-quarters of the APE initially contained in the overturned region remain available at the end of the turbulent episode. The substantial equality between energy lost to mixing and energy lost to turbulent kinetic energy dissipation is also found in Rayleigh–Benard convection (SW) and experiments of mixing induced by Rayleigh–Taylor (RT) instabilities in an otherwise stratified medium (Lawrie and Dalziel 2011, their Fig. 8); hence, Γ ≃ 1. It can be explained by noticing that the rate at which mean APE is fed into turbulent KE is *N*^{2} is the BV frequency of the resorted field (positive by definition) (Fig. 1). Within the overturn, the gradient of the mean reduced gravity is directed upward and the magnitude

### c. Time scales

As before, we define the dissipative time scale during the turbulent episode as the ratio of turbulent APE or turbulent KE to the corresponding dissipation rates. Figure 10b shows the evolution of the respective ratios as a function of time. It remains remarkably constant for the kinetic energy, whereas when calculated with the APE it is initially larger but within the same order of magnitude and eventually settles to a constant value during the decaying phase. Regardless, the two values are reasonably close so that we can consider a single value equal to 0.8 buoyancy periods as representative. This value is also consistent with the overall duration of the turbulent event, about 1.6 buoyancy periods.^{4} It is also largely independent of the initial aspect ratio of the overturn, unlike the standing wave period, which depends on the aspect ratio. This is consistent with the notion that the dissipative time scale is set by the physics of the local instabilities.

As is the case for steady-state shear-driven mixing, the energy partition ratio mirrors the ratio of the corresponding fluxes, in this case close to unity. This can be seen clearly in the time evolution of turbulent KE and APE (Fig. 10a).

Snapshots of *l*_{T} scale as the mean and turbulent APE calculated from the DNS (Fig. 12a). Likewise, the dissipation during the turbulent episode estimated with Eq. (18), assuming Γ = 1 (i.e., equipartition of the energy lost to dissipation and mixing) and *α* = 5, agrees very well at all times, whereas estimating the dissipation using the standard formula based on the Thorpe scale [Eq. (4)] consistently overestimates the actual dissipation (Fig. 12b). Thus, in this flows hypotheses A_{1} and A_{2} are satisfied, but neither B_{1}, since (2*α*Γ) ≃ 10^{−1}, nor B_{3}.

Given the disparity in magnitude between the mean and turbulent displacements, the need to base the estimation of dissipation on the turbulent component of *l*_{T} becomes apparent. Using Eq. (18) with *α* = 5 and Γ = 1 to normalize the PDF of turbulent KE dissipation, we obtain a good collapse of the PDFs (Fig. 13), with the peak being reasonably close to unity.

Likewise, the ratio of the Thorpe to Ozmidov scales in these flows is always larger and at times much larger than one. However, we must bear in mind that this ratio cannot be interpreted as an “age” of the event, as is the case for the shear-driven instability. In the latter case, the instability initially transfers energy to APE, but until secondary instabilities energize the dissipative scales, dissipation is small. Thus, *L*_{T}/*L*_{O} is large at the beginning and decreases with time as turbulent dissipation increases, while the APE frontloaded at the beginning is quickly lost. Here, *L*_{T} is a measure of the total APE and not of the APE that is part of the turbulent processes. Thus, the ratio *L*_{T}/*L*_{O}, while always greater than one, does not decrease monotonically during the event. It decreases initially, mostly because as turbulence develops *L*_{O} increases, but as the turbulent event comes to an end, *L*_{T}/*L*_{O} increases (Fig. 12c). Thus, the mixing context matters when interpreting the ratio *L*_{T}/*L*_{O}.

## 6. Discussion

*L*

_{T}to dissipation

*ϵ*

_{k}and stratification

*N*,is expected to be valid.

Traditionally, the justification for using the Thorpe scale to estimate dissipation lies in the assumption that it approximates the Ozmidov scale. In this note, we show that an energetics argument can be used to derive an estimate of turbulent KE dissipation and mixing using a suitably defined length scale and the background value of the BV frequency. The theory assumes the knowledge of the true reference state in order to calculate how far a parcel is displaced from its rest position *l*_{T}. In oceanographic practice, *l*_{T} is approximated by the Thorpe displacement *δT*, in which the rest state is calculated by resorting the vertical density cast along a profile. We show that as long as sufficient, many casts are included, statistically speaking *δT* ≃ *l*_{T}.

The alternative approach is based on the following assumptions:

- The displacement are small relative to the vertical scale of variation of
*N*^{2}. - The dissipation time scale for kinetic energy and available potential energy are the same.
- The dissipation time scales as
*N*^{−1}. - The partition ratio of turbulent APE dissipation to turbulent KE dissipation is known and, because of assumption 2, equals the ratio of turbulent APE to turbulent KE.

While in the standard formula, the length scale is usually defined as the rms value of the Thorpe displacement *δT* (i.e., the Thorpe scale); in our approach, we rely on a Reynolds decomposition of the displacement *l*_{T}. For a shear-driven flow, the Thorpe scale coincides with the rms value of *l*_{T} since the mean value of *l*_{T}, that is, the component due to the mean flow, is zero. Thus, *L*_{T}/*L*_{O} = *O*(1), which validates the standard oceanographic practice.

However, for mixing driven by a large-scale buildup of APE resulting in statically unstable regions (convective-driven mixing), the mean component of *l*_{T} can be much larger than the turbulent component. Since *l*_{T} and *δT* are statistically equivalent, the Thorpe scale is dominated by the mean component of *l*_{T}, and following the standard oceanographic practice overestimates the amount of mixing, that is, *L*_{T} ≫ *L*_{O}. It would amount to assuming that the entire amount of APE in the statically unstable region is dissipated locally within a few buoyancy periods. However, the large-scale dynamics must still retain a degree of control on the mean APE, part of which may be transferred back to the mean KE. An incorrect estimation of dissipation and mixing can skew important parameters, such as the conversion rate from barotropic to baroclinic energy at topographic features (Alford et al. 2011).

### Diagnostic versus prognostic use of the Thorpe scale

*δT*is calculated in the usual way. To define the approximated mean Thorpe scale,

*δT*is convolved with a top-hat filter

*z*) of width equal to

*δ*

_{0}/5, where

*δ*

_{0}is the initial thickness of the overturned region:The factor 5 was chosen based on a visual inspection of the data to separate the turbulent fluctuations from the large-scale overturn. From field data, it would be necessary to determine if a scale separation exists between the large vertical displacement associated with the mean field and the small-scale displacements due to turbulence and select the width of the filter accordingly. We can define an approximated turbulent APE asWhen integrated along the transect, the approximated turbulent APE compares favorably with the actual turbulent APE once the turbulent patch develops (Fig. 14), especially considering the typical scatter involved in field data.

The situation is different if the Thorpe scale is used prognostically to parameterize unresolved mixing in regions that are statically unstable (Klymak and Legg 2010). In this case, by definition we do not have access to even a single realization of the ensemble, but only to the mean value of the Thorpe scale. The present work suggests that using *L*_{T} to estimate dissipation via Eq. (4) may overestimate the dissipation and consequently provides at best an upper limit to the turbulent diffusivity but offers no robust guidance on how to provide a more accurate parameterization, since the fraction of APE actually involved in mixing, which is 10% in the present simulations, may be different in other flows. Likewise, the time scale for mixing, measured in buoyancy periods, may be dependent on other factors as well.

In summary, SW’s energetics approach indicates that mixing events inhabit a continuum bracketed by two idealized conditions: shear- and convective-driven mixing. This framework can be used to justify the use of a displacement scale to estimate dissipation and mixing, provided that the displacement scale used reflects only the contribution of turbulent fluctuations. For shear-driven flows at typical values of the flux Richardson number, our approach in practice coincides with the oceanographic practice because in such flows the Thorpe scale is a good measure of the displacements due to turbulent eddies and thus can be used to approximate the turbulent APE.

For flows driven by an excess of mean APE (convective-driven mixing), using the standard oceanographic approach leads to potentially overestimating the dissipation. As a diagnostic tool, we have shown that a suitable displacement scale can be obtained from the Thorpe displacement *δT* even with relatively few data points, using a vertical spatial filter as a proxy for the ensemble average. A final remark is in order. The two flows considered here represent limiting cases of what in realistic situations is a continuous spectrum. As such, a vertical filter applied to the Thorpe displacements should always be used to separate the turbulent contribution. However, determination of dissipation and eddy diffusivity depends also on two nondimensional coefficients *C*_{ϵ} ≡ (2*α*Γ)^{−1} and *C*_{mix} ≡ (2*α*)^{−1}. The coefficient *C*_{mix} varies between 0.1 for pure convective-driven flows and 0.25 for pure shear-driven flows, a rather modest spread. The coefficient *C*_{ϵ} on the other hand varies from 0.1 to 1.2, a more substantial variation. Therefore, if the goal is to diagnose the turbulent diffusivity, the crucial quantity is

While this paper is based solely on theoretical considerations backed by numerical calculations, comparison of estimates based on Eq. (4) with actual measurements from microstructure profilers in field data presented in a companion paper (MVSM), as well as turbulence-resolving simulations of overturns driven by the interaction of the internal tide with the bottom (Chalamalla and Sarkar 2015), reach similar conclusions, namely, that using Eq. (4) can overestimate *ϵ*_{k} by an order of magnitude when the vertical size of the overturned region is large.

The author thanks B. D. Mater, L. St. Laurent, S. K. Venayagamoorthy, and J. Moum for engaging discussions on the subject and the three anonymous reviewers for their constructive criticism. This work was supported by ONR under Grant N00014-09-1-0288.

# APPENDIX

## Large Overturns during the Reflection of Internal Wave Beams

The purpose of this appendix is to show that reflecting internal wave beams can generate overturns in which most of the energy of the mean field is contained as mean APE over a significant period. The intent is not to discuss the physics of these events but to show a “realistic” example of convective-driven mixing.

*A*

_{c}, the maximum

^{5}slope of the isopycnals locally approaches infinity (hence, the local BV frequency

*N*

^{2}→ 0), while at the same time the shear

*S*approaches zero in such a way that the gradient Richardson number has a discontinuity at the critical amplitudewhere

*A*is the amplitude of the wave. From a kinematic point of view, locally the flow is approximated by a solid body rotation (since

*S*= 0) that tilts isopycnals past their stability point. From an instability point of view, locally the flow becomes statically unstable without going, during the runup, through a stage where shear instabilities can grow.

The reflection of an internal wave beam off a slope provides a striking example of this phenomena, namely, the occurrence of zero-shear statically unstable regions that originate away from boundaries. They have been observed in laboratory experiments (De Silva et al. 1997) and DNS (Chalamalla et al. 2013) and possibly over continental shelves (Nash et al. 2007). Scotti (2011) obtained an analytical solution for the reflection of beams in the time domain that shows how the development of such regions occurs even for relatively weak incoming beams. This analytical solution forms the basis for the analysis presented here. It applies to an inviscid fluid, and outside the near-wall boundary layer it was found to be in excellent agreement with fully nonlinear simulations (Chalamalla et al. 2013). We consider a sloping bottom near the equator (hence, rotation is negligible) making an angle *γ* = 7° with the horizontal, upon which a semidiurnal beam (*ω*_{b} = 1.45 × 10^{−4} s^{−1}) impinges and reflects. The ambient stratification is *N* = 8 × 10^{−4} s^{−1}, and the along-beam velocity is ≃1 cm s^{−1}. Let *R* ≡ (sin*α* + sin*γ*)/sin*γ*, where *α* is the angle the beam axis makes with the horizontal and signed so that *α* < 0 means the group velocity of the incoming beam is propagating downward (Fig. A1). For the present case, *R* = −½, so that the reflected beam propagates upslope. The incoming beam is described by a streamfunction whose maximum amplitude *ψ*_{max} obeys *ψ*_{max}*k*^{2}/*N*_{∞} = 0.01, where *N*_{∞} is the background BV frequency and *k* is the dominant wavenumber of the beam. The width of the beam is about one wavelength.

Figure A1 shows the region along the bottom centered on the incoming beam axis at three different times. The overturned region of interest (marked with A) is the one that forms off wall. Initially small, it grows over time as a flattened region (i.e., with a high aspect ratio), slowly moving upslope and getting closer to the wall, eventually merging with the statically unstable region that is attached to the wall. Note how with the possible exception of a very thin layer at the edge of the overturned region, the gradient Richardson number is everywhere greater than ¼ except within the overturned region itself, where it is negative.

To study the energetics within the statically unstable patch, we integrate the APE and the KE within such a region (KE is referenced to a frame of reference that moves with the mean velocity of the patch). For a newly born patch, over 95% of the energy is contained as APE, decreasing over the next several hours to 80% (Fig. A2). This shows how a large-scale flow can create and maintain regions characterized by a high ratio of APE to KE over a significant period of time that are not due to instabilities [which are not allowed in the Scotti (2011) solution]. Of course, higher-order nonlinearities not accounted in the analytic solution will clearly lead to a more complicated evolution, but in general we expect that energy for mixing will flow out of the (mean) APE reservoir where the mean (i.e., large scale) flow is injecting it.

*H*

_{p}. Referring the reader to Chandrasekhar (1981, p. 447, Table XLVI) for details, the essential point is that the time scale of the most unstable RT modes is

*T*

_{RT}~ (

*ν*/

*g*′

^{2})

^{1/3}, where

*g*′ is the reduced gravity across the unstable layer and

*ν*is the viscosity that, translated in terms of the properties of the patch, giveswhere

*N*

_{p}is the value of the BV frequency within the resorted patch andThe coefficient 50 that appears in the formula is appropriate for RT instabilities in the low Atwood number regime. For the particular case considered here, where Re = 7200,

*T*

_{RT}≃ 2 h, small enough to be able to extract APE from the patch before the external dynamics undoes the overturning, yet slow enough so that sufficient APE can accumulate during the initial evolution of the patch. Interestingly, if we apply the scaling Eq. (A2) to our simulations, for which Re = 4000, we obtain

*T*

_{RT}

*N*

_{p}≃ 3, not too far from the dissipative time scale that was calculated from the simulation.

## REFERENCES

Alford, M. H., , M. C. Gregg, , and M. A. Merrifield, 2006: Structure, propagation, and mixing of energetic baroclinic tides in Mamala Bay, Oahu, Hawaii.

,*J. Phys. Oceanogr.***36**, 997–1018, doi:10.1175/JPO2877.1.Alford, M. H., and Coauthors, 2011: Energy flux and dissipation in Luzon Strait: Two tales of two ridges.

,*J. Phys. Oceanogr.***41**, 2211–2222, doi:10.1175/JPO-D-11-073.1.Chalamalla, V. K., , and S. Sarkar, 2015: Mixing, dissipation rate, and their overturn-based estimates in a near-bottom turbulent flow driven by internal tides.

,*J. Phys. Oceanogr.***45**, 1969–1987, doi:10.1175/JPO-D-14-0057.1.Chalamalla, V. K., , B. Gayen, , A. Scotti, , and S. Sarkar, 2013: Turbulence during the reflection of internal gravity waves at critical and near-critical slopes.

,*J. Fluid Mech.***729**, 47–68, doi:10.1017/jfm.2013.240.Chandrasekhar, S., 1981:

Dover, 652 pp.*Hydrodynamics and Hydromagnetic Stability.*Corrsin, S., 1958: Local isotropy in turbulent shear flow. NACA Research Memo. RM 58B11, 15 pp.

De Silva, I. P. D., , J. Imberger, , and G. N. Ivey, 1997: Localized mixing due to a breaking internal wave ray at a sloping bed.

,*J. Fluid Mech.***350**, 1–27, doi:10.1017/S0022112097006939.Dillon, T. M., 1982: Vertical overturns: A comparison of Thorpe and Ozmidov length scales.

,*J. Geophys. Res.***87**, 9601–9613, doi:10.1029/JC087iC12p09601.Dillon, T. M., 1984: The energetics of overturning structures: Implication for the theory of fossil turbulence.

,*J. Phys. Oceanogr.***14**, 541–549, doi:10.1175/1520-0485(1984)014<0541:TEOOSI>2.0.CO;2.Dillon, T. M., , and M. M. Park, 1987: The available potential energy of overturns as an indicator of mixing in the seasonal thermocline.

,*J. Geophys. Res.***92**, 5345–5353, doi:10.1029/JC092iC05p05345.Ferron, B., , H. Mercier, , K. Speer, , A. Gargett, , and K. Polzin, 1998: Mixing in the Romanche Fracture Zone.

,*J. Phys. Oceanogr.***28**, 1929–1945, doi:10.1175/1520-0485(1998)028<1929:MITRFZ>2.0.CO;2.Gargett, A. E., , and T. Garner, 2008: Determining Thorpe scales from ship-lowered CTD density profiles.

,*J. Atmos. Oceanic Technol.***25**, 1657–1670, doi:10.1175/2008JTECHO541.1.Ivey, G. N., , K. B. Winters, , and J. R. Koseff, 2008: Density stratification, turbulence, but how much mixing?

,*Annu. Rev. Fluid Mech.***40**, 169–184, doi:10.1146/annurev.fluid.39.050905.110314.Jayne, S. R., 2009: The impact of abyssal mixing parameterizations in an ocean general circulation model.

,*J. Phys. Oceanogr.***39**, 1756–1775, doi:10.1175/2009JPO4085.1.Klymak, J. M., , and S. M. Legg, 2010: A simple mixing scheme for models that resolve breaking internal waves.

,*Ocean Modell.***33**, 224–234, doi:10.1016/j.ocemod.2010.02.005.Lawrie, A. G., , and S. B. Dalziel, 2011: Rayleigh–Taylor mixing in an otherwise stable stratification.

,*J. Fluid Mech.***688**, 507–527, doi:10.1017/jfm.2011.398.Lumpkin, R., , and K. Speer, 2007: Global ocean meridional overturning.

,*J. Phys. Oceanogr.***37**, 2550–2562, doi:10.1175/JPO3130.1.Mann, K., , and J. Lazier, 2006:

Blackwell, 496 pp.*Dynamics of Marine Ecosystems: Biological-Physical Interactions in the Oceans.*Mater, B. D., , and S. K. Venayagamoorthy, 2014: A unifying framework for parameterizing stably stratified shear-flow turbulence.

,*Phys. Fluids***26**, 036601, doi:10.1063/1.4868142.Mater, B. D., , S. M. Schaad, , and S. K. Venayagamoorthy, 2013: Relevance of the Thorpe length scale in stably stratified turbulence.

,*Phys. Fluids***25**, 076604, doi:10.1063/1.4813809.Mater, B. D., , S. K. Venayagamoorthy, , L. S. Laurent, , and J. N. Moum, 2015: Biases in Thorpe scale estimates of turbulence dissipation. Part I: Assessments from large-scale overturns in oceanographic data.

,*J. Phys. Oceanogr.***45**, 2497–2521, doi:10.1175/JPO-D-14-0128.1.Moum, J. N., 1996: Energy-containing scales of turbulence in the ocean thermocline.

,*J. Geophys. Res.***101**, 14 095–14 109, doi:10.1029/96JC00507.Munk, W., 1966: Abyssal recipes.

,*Deep-Sea Res. Oceanogr. Abstr.***13**, 707–730, doi:10.1016/0011-7471(66)90602-4.Munk, W., , and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing.

,*Deep-Sea Res.***45**, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.Nash, J. D., , M. H. Alford, , E. Kunze, , K. Martini, , and S. Kelly, 2007: Hotspots of deep ocean mixing on the Oregon continental slope.

,*Geophys. Res. Lett.***34**, L01605, doi:10.1029/2006GL028170.Osborn, T., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements.

,*J. Phys. Oceanogr.***10**, 83–89, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.Ozmidov, R. V., 1965: On the turbulent exchange in a stably stratified ocean.

,*Izv. Akad. Sci. USSR Atmos. Oceanic Phys.***1**, 861–871.Scotti, A., 2008: A numerical study of gravity currents propagating on a free-slip boundary.

,*Theor. Comput. Fluid Dyn.***22**, 383–402, doi:10.1007/s00162-008-0081-6.Scotti, A., 2011: Inviscid critical and near-critical reflection of internal waves in the time domain.

,*J. Fluid Mech.***674**, 464–488, doi:10.1017/S0022112011000097.Scotti, A., , and B. White, 2014: Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy.

,*J. Fluid Mech.***740**, 114–135, doi:10.1017/jfm.2013.643.Smyth, W., , J. Moum, , and D. Caldwell, 2001: The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations.

,*J. Phys. Oceanogr.***31**, 1969–1992, doi:10.1175/1520-0485(2001)031<1969:TEOMIT>2.0.CO;2.Tailleux, R. G. J., 2013: Available potential energy and exergy in stratified fluids.

,*Annu. Rev. Fluid Mech.***45**, 35–58, doi:10.1146/annurev-fluid-011212-140620.Tennekes, H., , and J. L. Lumley, 1972:

MIT Press, 300 pp.*A First Course in Turbulence.*Thorpe, S. A., 1977: Turbulence and mixing in a Scottish loch.

,*Philos. Trans. Roy. Soc. London***A286**, 125–181, doi:10.1098/rsta.1977.0112.Thorpe, S. A., 1987: On the reflection of a train of finite-amplitude internal waves from a uniform slope.

,*J. Fluid Mech.***178**, 279–302, doi:10.1017/S0022112087001228.Thorpe, S. A., 2005:

Cambridge University Press, 439 pp.*The Turbulent Ocean.*Toggweiler, J., , and B. Samuels, 1998: On the ocean’s large-scale circulation near the limit of no vertical mixing.

,*J. Phys. Oceanogr.***28**, 1832–1852, doi:10.1175/1520-0485(1998)028<1832:OTOSLS>2.0.CO;2.Tseng, Y.-H., , and J. Ferziger, 2001: Mixing and available potential energy in stratified flows.

,*Phys. Fluids***13**, 1281–1293, doi:10.1063/1.1358307.Waterhouse, A. F., and Coauthors, 2014: Global patterns of diapycnal mixing from measurements of the turbulent dissipation rate.

,*J. Phys. Oceanogr.***44**, 1854–1872, doi:10.1175/JPO-D-13-0104.1.Wesson, J. C., , and M. C. Gregg, 1994: Mixing at the Camarinal Sill in the Strait of Gibraltar.

,*J. Geophys. Res.***99**, 9847–9878, doi:10.1029/94JC00256.Wilson, R., , H. Luce, , F. Dalaudier, , and J. Lefrre, 2010: Turbulence patch identification in potential density or temperature profiles.

,*J. Atmos. Oceanic Technol.***27**, 977–993, doi:10.1175/2010JTECHA1357.1.Winters, K. B., , P. N. Lombard, , J. J. Riley, , and E. A. D’Asaro, 1995: Available potential energy and mixing in density stratified fluids.

,*J. Fluid Mech.***289**, 115–128, doi:10.1017/S002211209500125X.Wolfe, C. L., , and P. Cessi, 2009: Overturning circulation in an eddy-resolving model: The effect of the pole-to-pole temperature gradient.

,*J. Phys. Oceanogr.***39**, 125–142, doi:10.1175/2008JPO3991.1.Wolfe, C. L., , and P. Cessi, 2010: What sets the strength of the middepth stratification and overturning circulation in eddying ocean models?

,*J. Phys. Oceanogr.***40**, 1520–1538, doi:10.1175/2010JPO4393.1.Wunsch, C., , and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans.

,*Annu. Rev. Fluid Mech.***36**, 281–314, doi:10.1146/annurev.fluid.36.050802.122121.

^{1}

By “overall” we mean that locally the stratification can be statically unstable, even on scales much larger than the turbulent scales but that such statically unstable regions are isolated and immersed in an otherwise stably stratified environment.

^{2}

Note, however, that in regions where the salinity plays an important role in determining the density, there are significant technical hurdles that need to be addressed (see, e.g., Gargett and Garner 2008).

^{3}

The idea that *N* provides a time scale for the problem is already present in Dillon (1984). Here, we introduce the further hypothesis that the time scale is the same for turbulent KE and APE dissipation.

^{4}

One buoyancy period is equal to 2*πN*^{−1}.

^{5}

The maximum is evaluated over a wave cycle.