2.1. Background and available potential energy

For an initially stable density profile, mixing of density raises the centre of mass of the volume and thus requires a source of energy. However, not all instantaneous increases in the centre of mass of the system are due to mixing. For a general volume, we need a method for separating changes in the potential energy due to irreversible molecular diffusion from those due to reversible processes. Following Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995), we start with gravitational potential energy in the flow, integrated over the volume $V$ and with gravitational acceleration $g$ ,

and partition $E_{p}$ into background potential energy and that which is available to do work.

For an incompressible, Boussinesq flow, the background potential energy (BPE) is the potential energy that is not available to do work in the system,

where $\hat{{\it\rho}}(z)$ is the sorted density profile that is created from ${\it\rho}(x,y,z)$ by an adiabatic, volume-conserving rearrangement of fluid parcels such that $\hat{{\it\rho}}(z)$ is everywhere statically stable and $\text{d}\hat{{\it\rho}}/\text{d}z\leqslant 0$ (Peltier & Caulfield Reference Peltier and Caulfield2003). The BPE corresponds to the lowest gravitational potential energy reachable by an adiabatic rearrangement of fluid parcels (Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995). The expression in (2.2) is equivalent to that given by Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995), $E_{b}=g\int _{V}\hat{z}{\it\rho}\,\text{d}V,$ where $\hat{z}$ is the vertical position of a parcel of fluid of density ${\it\rho}$ in the rearranged (sorted) state. Note that the BPE is unavailable to do work in systems where motions are constrained to be adiabatic. There exist flows where these assumptions would not be appropriate; for example, in some double-diffusive flows the BPE can be a source of available energy through differences in internal energy (for more on this see Tailleux Reference Tailleux2009); we shall not consider this possibility here.

Upon a change of variables, the BPE can also be written using ${\it\Psi}({\it\rho})$ , the p.d.f. of density defined across a volume of vertical height $2H$ , and an origin at the midpoint,

which is used throughout this paper to calculate changes in $E_{b}$ ( ${\it\rho}_{\ast }$ is a dummy variable). The p.d.f. of density uniquely determines the BPE, which is independent of the actual spatial distribution of density in the flow. In a closed system, only molecular diffusion can alter the p.d.f., reduce variance of density and change $E_{b}$ . We may therefore measure energy that is no longer available due to irreversible mixing by calculating the change in $E_{b}$ between the initial and final states (Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995). Note that the change in $E_{b}$ cannot be calculated from just the change in p.d.f., knowledge of the initial or final p.d.f. is also required. The magnitudes of both the BPE and PE depend on where the origin is defined to be; therefore an individual fluid parcel cannot be said to have PE or BPE, unless an origin is defined. Even then, calculation of the BPE can only occur in a defined volume and requires knowledge of the three-dimensional density structure within that volume and so this calculation cannot be considered to be local to a fluid parcel.

It is important to distinguish between the physical homogenisation of fluid parcels by molecular diffusion and the energetic consequences of such mixing. The process of mixing can be described as a change in the p.d.f. of density or as a reduction in the density variance. In a stratified flow, mixing has energetic consequences and this ‘energy change due to mixing’ is what we are measuring when we calculate the change in BPE. It must be stressed that this ( ${\rm\Delta}E_{b}$ ) is not a direct measure of the amount of mixing.

The available potential energy (APE) of a volume of fluid is the difference between the gravitational potential energy and the background potential energy,

and represents the energy stored in the buoyancy field if the fluid volume is not in gravitational equilibrium (Lorenz Reference Lorenz1955). Although a fluid parcel can be thought of as having APE (as the magnitude of the APE is independent of where the origin is defined), calculation of the APE requires knowledge of the entire three-dimensional density field ${\it\rho}(x,y,z)$ , as $\hat{{\it\rho}}$ (or ${\it\Psi}({\it\rho})$ ) can only be calculated within a defined volume.

The total mechanical energy that is available to do work at a given time includes any kinetic energy $E_{k}$ (KE) in the system, in addition to the APE. This total available energy is

We will calculate the mixing efficiency as a cumulative quantity between some initial and final state. For a turbulent event, the cumulative mixing efficiency is

where ${\rm\Delta}$ represents the change between the initial and final states of the system. This definition has been widely used in experimental and computational studies (e.g. Ivey & Imberger Reference Ivey and Imberger1991; Peltier & Caulfield Reference Peltier and Caulfield2003; Dalziel et al. Reference Dalziel, Patterson, Caulfield and Coomaraswamy2008; Prastowo et al. Reference Prastowo, Griffiths, Hughes and Hogg2009; Hult, Troy & Koseff Reference Hult, Troy and Koseff2011a , among others). All the examples described in this paper will be assumed to have quiescent initial and final conditions ( ${\rm\Delta}E_{k}=0$ ) such that buoyancy instability provides the source for energy in the flow.

2.2. Alternative measures of mixing efficiency

We will now compare ${\it\eta}$ to some other commonly used measures of the efficiency of stratified mixing. Consider a closed system, where there is no net flux of energy or density into or out of the domain. We will continue to assume that the flow is incompressible and Boussinesq, and initialise the flow with some amount of available potential energy and kinetic energy (i.e. buoyancy and mechanical forcing). If the equation of state is linear, we can ignore net changes in the internal energy. With these assumptions, mixing results in a decrease in available energy and an increase in background potential energy. For a detailed discussion of the role of internal energy when these assumptions are relaxed, see Tailleux (Reference Tailleux2009).

In our simplified picture of mixing, any available energy that is initially present in the flow is either dissipated by viscosity or consumed by an increase in the BPE of the flow. Kinetic energy is converted to APE when fluid parcels are carried away from their neutral buoyancy level. Any increases in potential energy that result from movements of fluid parcels will be temporary unless diffusion acts to alter the density of the parcel and its surroundings, a process which increases the BPE (Peltier & Caulfield Reference Peltier and Caulfield2003). Under this model, energy within a volume is exchanged between several reservoirs, a process that can be described by the volumetric equations

where ${\it\varepsilon}$ is the rate of viscous dissipation, integrated across the domain. The buoyancy flux $b$ represents the net rate of transfer between APE and KE at an instant in time and integrated across the domain, while $m$ is the rate of irreversible energy transfer to BPE. Energy is transferred to BPE as a consequence of mixing. We will assume that the rate of transfer of internal energy to BPE ( $D_{{\it\rho}}$ ) due to laminar diffusion down the background gradient is negligible compared to the rate of increase in BPE due to turbulent mixing. The process described by these equations is illustrated by the diagram in figure 2. For a more detailed derivation of these equations see Peltier & Caulfield (Reference Peltier and Caulfield2003).

Figure 2.

A simplified diagram of energy fluxes between kinetic energy (KE), available potential energy (APE), internal energy (IE) and background potential energy (BPE) in an incompressible, Boussinesq flow with a linear equation of state. In a closed system the source fluxes will be zero over the time interval defined by ${\it\Delta}$ (i.e. the forcing provides the initial energy for mixing, but is zero for $t>0$ ).

The cumulative mixing efficiency ${\it\eta}$ in (2.6) is also equal to

where $T$ is a time scale that encapsulates a mixing event (Peltier & Caulfield Reference Peltier and Caulfield2003). An instantaneous mixing efficiency can be defined along the same lines,

In general, ${\it\eta}$ is not equal to the average instantaneous mixing efficiency, ${\it\eta}\neq (1/T)\int _{0}^{T}{\it\eta}_{i}\,\text{d}t$ . Moreover, a mixing event will usually occur over some finite time period and in this case the meaning of an instantaneous mixing efficiency would be somewhat questionable.

So far we have described only global, volumetric, energy-based measures to characterise the efficiency of mixing. Osborn (Reference Osborn1980) defined the local flux Richardson number as ‘the ratio of the buoyancy flux to the turbulent production’. Upon scaling and averaging the point-wise turbulent energy equations, Osborn (Reference Osborn1980) wrote the local flux Richardson number as

where $b^{\prime }$ and ${\it\varepsilon}^{\prime }$ are defined at a point in space, rather than for a volume (see also Turner Reference Turner1973; Linden Reference Linden1979; Ivey & Imberger Reference Ivey and Imberger1991). Although the flux Richardson number $Ri_{f}$ and the instantaneous mixing efficiency ${\it\eta}_{i}$ share similarities in form, they are equal only in specific circumstances. The energy transfer $b^{\prime }$ does not differentiate between reversible and irreversible conversions to potential energy. Calculation of the efficiency of irreversible mixing is inherently non-local as it requires knowledge of the density field for the entire volume (so that we can find ${\it\Psi}({\it\rho})$ and calculate ${\rm\Delta}E_{b}$ ). However, we expect the volume average of $Ri_{f}$ to be equal to the instantaneous mixing efficiency ${\it\eta}_{i}$ (and ${\it\eta}$ ) for the special case of a domain that remains in a statistically steady state (Peltier & Caulfield Reference Peltier and Caulfield2003).

Mixing in ocean models is frequently parameterised in terms of the dissipation of local kinetic energy using the flux coefficient (also known as the mixing coefficient)

In general, ${\it\gamma}$ is defined as a local quantity, where $b^{\prime }$ and ${\it\varepsilon}^{\prime }$ are averaged at a point in space (Osborn Reference Osborn1980). Again, ${\it\gamma}$ is not guaranteed to differentiate between irreversible and reversible changes in the potential energy. We can relate the local $Ri_{f}$ and ${\it\gamma}$ as ${\it\gamma}=Ri_{f}/(1-Ri_{f})$ .

Confusingly, all three of $Ri_{f}$ , ${\it\gamma}$ and ${\it\eta}$ have been referred to as ‘the mixing efficiency’ in various papers, a situation not helped by a typical $Ri_{f}$ for mixing in a stably stratified shear flow being between $0.1$ and $0.2$ , a value small enough to make ${\it\gamma}\approx Ri_{f}$ . However, neither the flux Richardson number nor the flux coefficient are guaranteed to measure the irreversible increase of the potential energy associated with molecular diffusion. Moreover, the flux coefficient ${\it\gamma}$ does not correspond to an efficiency (on purely physical grounds) as it is not equal to the energy used for a particular purpose divided by the energy supplied to the system (instead $0\leqslant {\it\gamma}<\infty$ , whereas an efficiency $0\leqslant {\it\eta}\leqslant 1$ ). We therefore refer throughout this paper to ${\it\eta}$ as the mixing efficiency.

3.1. Mixing efficiency depends on regions remote from where mixing takes place

We will now return to the two idealised experiments plotted in figure 1. In the first experiment, we set up in a tank the unstable density profile shown in figure 1(a). The final density profile, measured when there is no remaining kinetic energy, will be the dashed line in figure 1(a) (known from the results of both laboratory experiments and simulations, as has been described above). Calculating the mixing efficiency from the initial and final density profiles, we find ${\it\eta}=3/4$ .

Now we will perform a second experiment, keeping the density gradients and discontinuity the same, but using a smaller tank (as plotted in figure 1 b). We reduce the volume of the experiment to only contain the mixing region, leaving out all the parts of the volume which do not change between the initial and final state. The initial density profile matches the profile in figure 1(a) over $-z_{p}\leqslant z\leqslant z_{p}$ and the final state is well-mixed, exactly as it was for the same range in the first experiment. When we calculate the mixing efficiency of this second experiment, we find ${\it\eta}=2/3$ .

Our two experiments have identical net changes in gravitational potential energy and therefore the viscous dissipation must be the same for both flows. The mixing is also identical (i.e. there is the same change in the volume of each density class) and yet, despite these similarities, the two flows have different values of mixing efficiency. We would observe the same behaviour if, instead of using a smaller tank, we replaced the stratified layers outside the mixing region with stable layers composed of densities outside the range of those present in the mixing region and which were not involved in the turbulent flow.

We could perform a series of experiments, each in a different sized tank (where the tank boundaries are situated at $z=\pm H$ ). If $H$ is varied, the mixing efficiency ${\it\eta}$ changes as shown in figure 3(b). For these calculations, the density profile is sorted across $\pm H$ to find the BPE. For $|H|<z_{p}$ the flow changes in a fundamental way as the amount of mixing and change in potential energy no longer remain the same; we shall not pursue this limit further. The efficiency is seen to increase with $H$ until $H=z_{n}$ , beyond which there is no further dependence. These examples demonstrate that mixing efficiency is influenced by regions of the flow (here, $z_{p}<|z|\leqslant z_{n}$ ) that do not change from the initial state and are not part of the turbulent mixing. This effect could be removed by always calculating the mixing efficiency only in the volume that changes from the initial conditions. However, calculating the mixing efficiency only across the mixing region artificially restricts the available energy that could be released. Thus the minimum calculation height for the mixing efficiency is the neutral buoyancy height,  $z_{n}$ .

There are more general examples of flows where turbulent motions occur for $|z|>z_{n}$ , requiring extension of the principles above. In this case, the calculation height must be increased beyond $z_{n}$ to include all the region that can change from the initial conditions. One example is a three-layer flow with one Rayleigh–Taylor unstable interface and one stable interface (Jacobs & Dalziel Reference Jacobs and Dalziel2005). Only the two Rayleigh–Taylor unstable layers are swapped in the sorted density profile. However, Jacobs & Dalziel (Reference Jacobs and Dalziel2005) found that, in experiments, turbulence generated in the unstable region entrained fluid across the stable interface and the final mixing region extended slightly further than $z_{n}$ . In this case, the mixing efficiency should be based on the region that has mixed, but only because this is larger than  $z_{n}$ .

Another example is when the initial conditions contain kinetic energy as well as APE. The above principle suggests that the calculation volume for ${\it\eta}$ should be the larger of $H=z_{n}$ and the height to which the turbulent motions could extend (aided by the initial KE). A priori determination of this height is a complicated function of the flow. However, in both examples ${\it\eta}$ should become independent of $H$ for sufficiently large $H$ , and this can provide a posteriori validation of the calculation.

There is class of flows for which $z_{n}$ is defined by the height of the domain. Examples include a buoyant plume rising in a homogeneous (or ‘weakly’ stratified) fluid, convection in either a homogeneous fluid or a sufficiently weak stratification and horizontal convection (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013; Gayen, Griffiths & Hughes Reference Gayen, Griffiths and Hughes2014). In these cases, the flow is contained by the domain boundaries and the mixing efficiency calculation must involve (and is therefore a function of) the entire domain.

We can construct flows with identical mixing and identical dissipation of kinetic energy and yet different mixing efficiencies because of how we defined ‘energy used in mixing’. The background potential energy is a function of the global density profile, regardless of whether a region is turbulent or participates directly in the flow. To explain this further we will use a second example.

3.2. Removal of APE in regions remote from mixing

We have seen in § 3.1 that the mixing efficiency depends on the density profile outside the region where mixing physically occurs. We will examine this further by considering an initial (general polynomial) profile of the form

where $q>0$ (see figure 4(a) for a range of profiles with $q$ between $0.2$ and $2$ ) and ${\it\rho}^{\prime }$ ( ${>}0$ ) is the maximum perturbation from the mean density $\overline{{\it\rho}}$ .

Figure 4.

(a) Density profile with varying $0.2\leqslant q\leqslant 2$ . (b) The variation of mixing efficiency with $q$ , over the neutral buoyancy height $\pm z_{n}$

We assume that the final density profile is the perfect mixing profile described earlier (a result consistent with the experiments of Davies Wykes & Dalziel (Reference Davies Wykes and Dalziel2014)), with a constant-density region that is symmetric around the initially unstable interface and a perfect mixing height of $z_{p}=z_{n}\times 2^{-1/q}$ . The mixing efficiency calculated for $H\geqslant z_{n}$ is

where ${\rm\Gamma}$ is the Gamma function. The variation of ${\it\eta}_{z_{n}}$ with $q$ is plotted on figure 4(b). As $q\rightarrow 0$ and the mixing region becomes very thin ( $z_{p}\rightarrow 0$ ), ${\it\eta}_{z_{n}}$ tends to 1. As $q\rightarrow \infty$ , the mixing efficiency ${\it\eta}_{z_{n}}$ tends to $1/2$ (the maximum mixing efficiency for a monotonic, gravitationally unstable profile, see the Appendix).

In the examples shown in figures 3(a) and 4(a), there are parcels of fluid (outside the mixing region) that do not move or change in density between the initial and final states and whose contribution to APE in the initial state (from the gravitational instability) vanishes in the globally stable final state. The removal of APE and increase in BPE is normally viewed as a consequence of diffusion of solute from one fluid parcel to another, a process involving internal energy (Tailleux Reference Tailleux2013). However, the removal of APE in parts of the volume which are remote from the turbulent mixing region, a process we term ‘relabelling’, does not involve this pathway, does not require the local fluid to move and is a consequence of physical mixing elsewhere. This initially available energy becomes dynamically inaccessible to the flow because the profile becomes stable before that energy can be released. The example in figure 4(a) shows how, as the height of the mixing region becomes small relative to the neutral buoyancy height of the stratification (i.e. $q\rightarrow 0$ ), the mixing efficiency tends to ${\it\eta}=1$ . Thus as $q$ decreases, a greater proportion of the initial available potential energy is relabelled and becomes inaccessible to the flow.