2.1. Local APE density

To quantify the portion of a flow's potential energy that is available to perform ‘useful’ work, it is necessary to define a reference state that minimises potential energy. Such a construction is analogous to reference states of minimum internal energy or maximum entropy in thermodynamics. The reference state is found from a volume-preserving and adiabatic rearrangement of the buoyancy field into a state that is in stable hydrostatic equilibrium. As discussed in § 1.2, if all horizontal sections of the domain are path connected (cf. figure 2(a), which shows a domain whose horizontal sections are not path connected), then the reference state is unique. Under the rearrangement, a parcel of fluid at location $\boldsymbol {x}=(x,y,z)$ with buoyancy $b(\boldsymbol {x})$ is mapped to the vertical position $z_{*}(b(\boldsymbol {x}))$ and the rearranged buoyancy field is denoted $b_{*}$, which means that $b_{*}(z_{*}(b))=b$.

The APE $A(\varOmega )$ associated with a buoyancy field in a domain $\varOmega$ of volume V accounts for the potential energy $(z_{*}-z)b$ associated with displacing each parcel of fluid from the (unique) reference equilibrium state of minimum potential energy that was described in the preceding paragraph (Winters et al. Reference Winters, Lombard, Riley and D'Asaro1995), and is therefore positive definite,

(2.1)\begin{equation} {A(\varOmega)=}\int\limits_{\varOmega}(z_{*}-z)b\,\mathrm{d} V\geqslant 0. \end{equation}

By definition, the minimum value of zero in (2.1) is produced by the reference field itself, for which $b=b_{*}$ and $z_{*}=z$.

Pointwise, however, $(z_{*}-z)b$ is not positive definite because $b$ does not necessarily have the same sign as $(z_{*}-z)$ (Roullet & Klein Reference Roullet and Klein2009). But, inclusion of the energy associated with the rearrangement of the reference state that would accompany the movement of a single parcel from $z_{*}$ to $z$, leads to

(2.2)\begin{equation} a = (z_{*}-z)b - \underbrace{\int\limits_{z}^{z_{*}}b_{*}(\eta)\,\textrm{d}\eta}_{a_{*}}, \end{equation}

which, as illustrated in figure 3, is positive definite if $b_{*}$ is monotonic and $b_{*}(z_{*})=b$ (Holliday & Mcintyre Reference Holliday and Mcintyre1981; Scotti et al. Reference Scotti, Beardsley and Butman2006), and is therefore a viable thermodynamic quantity analogous to exergy (Tailleux Reference Tailleux2013a). For a detailed discussion about how (2.2) emerges from a gauge-fixing condition in terms of Casimir functions (invariants), the reader is referred to Scotti & Passaggia (Reference Scotti and Passaggia2019). Conversely, the local APE (2.2) can be regarded as the Legendre transformation of $a_{*}$ (which is convex because $b_{*}$ increases monotonically) if $z_{*}$ is regarded as a free parameter, such that

(2.3)\begin{equation} a = \max_{z_{*}}\left\{ (z_{*}-z)b-a_{*}\right\}, \end{equation}

and the maximising condition that $b_{*}(z_{*})=\textrm {d}a_{*}/\textrm {d}z_{*}=b$ emerges naturally as the ‘force’ that is conjugate to the displacement $z_{*}-z$ releasing the maximum potential energy.

Figure 3.

Geometric ‘proof’ that (2.2) is positive definite when $b_{*}$ is a monotonically increasing function with $b_{*}(z_{*})=b$. Panel (a) illustrates a case for which $b_{*}(z)< b$ (and therefore $z< z_{*}(b)$) and (b) illustrates a case for which $b< b_{*}(z)$ (and therefore $z_{*}(b)< z$).

As noted by Tailleux (Reference Tailleux2013a), the thermodynamic nature of APE in (2.2) is not obvious from the integral definition (2.1) because, since the operation leading to $z_{*}$ is a volume preserving and adiabatic rearrangement of fluid in the domain,

(2.4)\begin{equation} \int\limits_{\varOmega}\int\limits_{\xi(z)}^{\xi(z_{*})}g(\eta)\,\mathrm{d}\eta \,\mathrm{d} V=0, \end{equation}

for any functions $g$ and $\xi$; hence, the volume integral of $a_{*}$ (for which $g$ corresponds to $b_{*}$ and $\xi$ corresponds to the identity function) over $\varOmega$ vanishes. To understand why, it is useful to regard $z_{*}$ as a permutation of fluid parcels of finite volume, which can be decomposed into a finite number of cycles (see e.g. MacLane & Birkhoff Reference MacLane and Birkhoff1999). Using a suitable change of variables to account for $\xi$, the integral of $g(\eta )$ over each cycle vanishes because it can be decomposed into an integral over $[z,z_{*}]$ followed by an integral over $[z_{*},z]$, which necessarily cancel because integrals of $g$, which depend only on the vertical coordinate, are path independent. Winters & Barkan (Reference Winters and Barkan2013) provide a more detailed explanation for the particular case corresponding to $a_{*}$ in (2.2) and Scotti & Passaggia (Reference Scotti and Passaggia2019) provide a more general explanation in terms of the symmetry associated with the system's Casimir functions.

To account for (2.4), which plays a fundamental role in the decomposition described in § 2.2, an equivalence relation (see, for e.g. MacLane & Birkhoff Reference MacLane and Birkhoff1999), denoted $\stackrel {\varOmega }{\cong }$, can be defined for expressions whose difference vanishes upon integration over a given domain $\varOmega$. For example, application of (2.4) to (2.2) allows us to write

(2.5)\begin{equation} a\stackrel{\varOmega}{\cong} (z_{*}-z)b, \end{equation}

to express the equality of $a$ and $(z_{*}-z)b$ to within terms whose integral over $\varOmega$ is zero.

2.2. The APE of subvolumes

The construction of APE described in § 2.1 involves a reference state computed from the buoyancy field in $\varOmega$. However, if the volume $\varOmega$ is partitioned into subvolumes $\omega _{i}\subset \varOmega$, then each subvolume has its own reference state $b_{*}(\omega _{i})$ corresponding to the rearrangement of the buoyancy field within $\omega _{i}$ to minimise the potential energy of $\omega _{i}$, as illustrated in figure 1. Together, the reference states of each subvolume have a potential energy that is at least as large as the unconstrained reference state associated with the parent volume $\varOmega$. Consequently, the sum of each subvolume's APE will be less than, or at best equal to, the maximum energy that can be released through reconfiguration of the buoyancy field over $\varOmega$. In such cases, the physical or notional boundaries that separate the subvolumes provide ‘forces’ that are conjugate to the additional displacements that would be required to release the full APE over $\varOmega$.

Starting from (2.2), consider the local APE in a subvolume $\omega \subset \varOmega$ with respect to the reference state of the parent volume $\varOmega$,

(2.6)\begin{equation} a(\varOmega) = \left[z_{*}(\varOmega)-z\right]b - \int\limits_{z}^{z_{*}(\varOmega)} b_{*}(\varOmega)|_{\eta}\,\mathrm{d}\eta. \end{equation}

To be explicit, and to avoid subscripts, the notation $z_{*}(\varOmega )$, $b_{*}(\varOmega )$ will be used to define the functions that map a buoyancy $b$ to the vertical position $z_{*}(\varOmega )|_{b}$ and a height $z$ to the corresponding reference buoyancy $b_{*}(\varOmega )|_{z}$, respectively. The notation ‘$|_{\cdot }$’ (such as $|_{\eta }$, $|_{b}$ and $|_{z}$ above) should be read as ‘evaluated at $\cdot$’, but will be omitted for brevity when arguments to functions are clear from the context.

Following the examples in § 1.2, it is useful to account separately for the APE of fluid within $\omega$ relative to: $(i)$ the parent reference height $z_{*}(\varOmega )$ and $(ii)$ the subvolume reference height $z_{*}(\omega )$. Specifically, it is useful to represent the available energy released during a two-stage sorting process $z\mapsto z_{*}(\omega )\mapsto z_{*}(\varOmega )$, in which a parcel of fluid is moved to its equilibrium position in $\omega$, before being moved to its final equilibrium position in $\varOmega$. To express the decomposition mathematically, (2.6) can be refactored as

(2.7)\begin{equation} a(\varOmega) = \underbrace{\left[z_{*}(\varOmega)-z_{*}(\omega)\right]b}_{(i)}+\underbrace{\left[z_{*}(\omega)-z\right]b}_{(ii)}- \int\limits_{z}^{z_{*}(\varOmega)}b_{*}(\varOmega)|_{\eta}\,\mathrm{d}\eta. \end{equation}

But (2.7) is unsatisfactory, because terms $(i)$ and $(ii)$ are not individually positive definite and the integral of $b_{*}$ is composite, involving both $z\mapsto z_{*}(\omega )$ and $z_{*}(\omega )\mapsto z_{*}(\varOmega )$. However, (2.4) can be used to reformulate (2.7) by adding and subtracting terms that vanish upon integration over the subvolume $\omega$, enabling the integral of $b_{*}$ in (2.7) to be appropriately distributed between terms $(i)$ and $(ii)$.

Although the reference state $b_{*}(\varOmega )$ is not necessarily the same as $b_{*}(\omega )$, (2.4) usefully implies that the net work associated with rearranging $b_{*}(\varOmega )$ over $\omega$ is zero, modulo integration over $\omega$ (cf. Scotti et al. Reference Scotti, Beardsley and Butman2006; Scotti & Passaggia Reference Scotti and Passaggia2019), such that

(2.8)\begin{equation} \int\limits_{z}^{z_{*}(\omega)}b_{*}(\varOmega)|_{\eta}\,\mathrm{d}\eta \stackrel{\omega}{\cong} 0. \end{equation}

The physical meaning of the decomposition (2.7) is therefore made explicit by invoking (2.8) to repartition the contributions to $(i)$ and $(ii)$,

(2.9)\begin{equation} a(\varOmega) \stackrel{\omega}{\cong} \underbrace{\left[z_{*}(\varOmega)-z_{*}(\omega)\right]b-\overbrace{\int\limits_{z_{*}(\omega)}^{z_{*}(\varOmega)} b_{*}(\varOmega)|_{\eta}\,\mathrm{d}\eta}^{a_{*}(\varOmega|\omega)}}_{a(\varOmega|\omega)\geqslant 0}+\underbrace{\left[z_{*}(\omega)-z\right]b-\overbrace{\int\limits_{z}^{z_{*}(\omega)}b_{*}(\omega)|_{\eta}\,\mathrm{d}\eta}^{a_{*}(\omega)\stackrel{\omega}{\cong} 0}}_{a(\omega)\geqslant 0}. \end{equation}

The contribution $a(\omega )$ corresponds to the APE inside the subvolume $\omega$ without reference to the context provided by $\varOmega$. In contrast, the contribution $a(\varOmega |\omega )$ corresponds to the additional potential energy of fluid within $\omega$ that can be made available by removing the partitions of $\varOmega$. Both contributions are positive definite because they have the same form as (2.2) and will therefore play a crucial role in accounting for context in § 4.3, with regards to the transport of APE between subvolumes. Although the right-hand side of (2.9) does not correspond to the traditional definition of APE density (2.6) in a pointwise sense, it articulates the two-stage sorting process and is equivalent to (2.6) under integration over the subvolume $\omega$. The decomposition (2.9) appears to be similar to the mean/eddy decomposition of APE density for a compressible fluid described by Tailleux (Reference Tailleux2018, § 4.1) to account for ‘non-resting’ reference states (cf. Andrews Reference Andrews2006). A notable difference, however, is that (2.9) was able to make use of (2.8) to remove sign-indefinite contributions, modulo integration over $\omega$.

Integration of (2.9) provides a decomposition for the APE of $\omega$, given $\varOmega$, whose constituent parts add together to give the integral of the traditional local APE density defined in (2.6),

(2.10)\begin{equation} A(\omega|\varOmega) = \underbrace{\int\limits_{\omega} a(\varOmega|\omega) \,\mathrm{d} V}_{A(\varOmega|\omega)\geqslant 0} + \underbrace{\int\limits_{\omega} a(\omega) \,\mathrm{d} V}_{A(\omega)\geqslant 0}=\int\limits_{\omega}a(\varOmega)\,\mathrm{d} V. \end{equation}

For the reasons stated below (2.9), $A(\varOmega |\omega )$ is referred to as ‘outer’ APE and $A(\omega )$ is referred to as ‘inner’ APE. The total APE $A(\omega |\varOmega )$ associated with $\omega$ is the sum of its inner and outer parts (for $A(\omega |\varOmega )$ read ‘the APE of $\omega$ given $\varOmega$’ and for $A(\varOmega |\omega )$ read ‘the APE of $\varOmega$ given $\omega$’). The outer APE $A(\varOmega |\omega )$ corresponds to the positive energy required to create the subvolume reference state $b_{*}(\omega )$ by rearranging the global/parent reference state $b_{*}(\varOmega )$, while the inner APE $A(\omega )$ corresponds to the positive energy required to create the buoyancy field $b$ by rearranging the subvolume reference state $b_{*}(\omega )$ over $\omega$. For mathematical readers, the choice of terminology comes from noting that, for a Boussinesq fluid, an adiabatic rearrangement of the buoyancy field is a permutation and that permutations leaving the contents of each subvolume unchanged can be regarded as subgroups (cf. the closed-loop cycles in § 4 of Winters & Barkan Reference Winters and Barkan2013). In this way the outer APE involves permutations that belong to the quotient of $\varOmega$-permutations with the (inner) $\omega$-permutations, which is reminiscent of the relationship between inner and outer automorphisms from group theory (cf. MacLane & Birkhoff Reference MacLane and Birkhoff1999).

The (global) APE $A(\varOmega )$ of the parent domain $\varOmega$ is obtained by summing (2.10) over all subvolumes $\omega$ belonging to a partition $\mathscr {P}(\varOmega )$ of $\varOmega$,

(2.11)\begin{equation} A(\varOmega) = \sum_{\omega\in\mathscr{P}(\varOmega)} A(\varOmega|\omega) + \sum_{\omega\in\mathscr{P}(\varOmega)}A(\omega). \end{equation}

In figure 1, for example, $\mathscr {P}(\varOmega )=\{\omega _{1},\omega _{2},\omega _{3},\omega _{4},\omega _{5}\}$. If $A(\varOmega )=0$, then $A(\varOmega |\omega )=0$ and $A(\omega )=0$ for all $\omega \subset \varOmega$ because $A(\varOmega |\omega )$ and $A(\omega )$ are positive definite. Application of (2.4) shows that the combined contribution from the integrals of $a_{*}(\varOmega |\omega )$ within $A(\varOmega |\omega )$ over each subvolume vanishes in (2.11) because, in summation, the integrals of $a_{*}(\varOmega |\omega )$ over $\omega$ correspond to an integral over $\varOmega$,

(2.12)\begin{align} \sum_{\omega\in\mathscr{P}(\varOmega)} A(\varOmega|\omega)=\sum_{\omega\in\mathscr{P}(\varOmega)}\int\limits_{\omega}\left[z_{*}(\varOmega)-z_{*}(\omega) \right]b\,\mathrm{d} V-\underbrace{\sum_{\omega\in\mathscr{P}(\varOmega)}\int\limits_{\omega} \overbrace{\int\limits_{z_{*}(\omega)}^{z_{*}(\varOmega)}b_{*}(\varOmega)|_{\eta}\,\mathrm{d}\eta}^{a_{*}( \varOmega|\omega)}\,\mathrm{d} V}_{0}, \end{align}

which shows that (2.11) is equivalent to the global APE (2.1). As explained in § 6.2, (2.11) can be applied recursively to subsets of $\varOmega$ down to infinitesimal subvolumes (i.e. individual points $\boldsymbol {x}$), for which the inner APE $A(\boldsymbol {x})$ necessarily vanishes and the outer APE $A(\omega |\boldsymbol {x})$ corresponds to the local APE density (2.6) with respect to $\omega$.

2.3. Background potential energy

The background potential energy (BPE) associated with a buoyancy field in $\varOmega$ is the potential energy of the reference state $b_{*}(\varOmega )$. The BPE is therefore equal to the difference between the system's potential energy and the global APE $A(\varOmega )$. As described in § 2.2 for APE, the BPE of a particular subvolume $\omega \subset \varOmega$ can be split between outer and inner contributions that do and do not make reference to the context provided by $\varOmega$, respectively. Formally, the definitions follow from subtracting the right-hand side of (2.9) from the potential energy $-bz$ and integrating over the subvolume $\omega$,

(2.13)\begin{equation} B(\omega|\varOmega) ={-} A(\omega|\varOmega) -\int\limits_{\omega}bz\,\mathrm{d} V = \underbrace{-A(\varOmega|\omega)}_{B(\varOmega|\omega)}\underbrace{-\int\limits_{\omega}bz_{*}(\omega)\,\mathrm{d} V}_{B(\omega)}. \end{equation}

The inner BPE $B(\omega )$ accounts for the subvolume reference state $b_{*}(\omega )$, while the outer BPE $B(\varOmega |\omega )$ accounts for the eventual position of the reference state $b_{*}(\omega )$ within the parent volume $\varOmega$. The outer components of BPE are necessarily equal in magnitude and opposite in sign to the outer components of APE $A(\varOmega |\omega )$, which means that $B(\varOmega |\omega )\leqslant 0$; hence

(2.14)\begin{equation} B(\varOmega) = \sum_{\omega\in\mathscr{P}(\varOmega)}B(\varOmega|\omega)+\sum_{\omega\in\mathscr{P}(\varOmega)}B(\omega)\leqslant \sum_{\omega\in\mathscr{P}(\varOmega)}B(\omega). \end{equation}

Equation (2.14) implies that the sum of the inner BPE in each subvolume is bounded from below by the global BPE $B(\varOmega )$, which corresponds to the fact that $b_{*}(\varOmega )$ is the global minimiser of potential energy. More generally, the inner BPE is subadditive under the union of disjoint subsets of $\varOmega$,

(2.15)\begin{equation} B(\omega_{1}\cup\omega_{2}) \leqslant B(\omega_{1}) + B(\omega_{2}), \end{equation}

which implies, and is implied by, the inner APE being superadditive under the union of disjoint subsets of $\varOmega$,

(2.16)\begin{equation} A(\omega_{1}\cup\omega_{2}) \geqslant A(\omega_{1}) + A(\omega_{2}), \end{equation}

as observed in possible formulations of APE in the presence of topographical barriers by Stewart et al. (Reference Stewart, Saenz, Hogg, Hughes and Griffiths2014). Any gap between the left- and right-hand sides of (2.16) results from the potential energy that can be made available by joining $\omega _{1}$ and $\omega _{2}$ and is a direct consequence of the positivity of the outer APE $A(\varOmega |\omega _{1})$ and $A(\varOmega |\omega _{2})$.

In providing a lower bound on the APE $A(\omega _{1}\cup \omega _{2})$, (2.16) resembles the result obtained by Andrews (Reference Andrews1981), and explained by Tailleux (Reference Tailleux2013b), that the APE defined with respect to alternative reference states is bounded from below by the APE corresponding to the unique (Lorenz) reference state obtained from an adiabatic transformation of the buoyancy field. As seen in (2.15), the reason for the resemblance is that constraints on the reference state increase BPE and therefore reduce available potential energy. The difference, however, is that in (2.15) and (2.16) the constraint comes from the isolation of subvolumes, rather than the adiabatic constraint embodied by the Lorenz reference state.

4.1. Budgets for inner and outer APE

Using (4.1)–(4.3) and the relation (2.4), the budget for the inner APE $A(\omega )$ is obtained by applying $\textrm {d}/\textrm {d}t$ and $\nabla ^{2}$ to the local APE,

(4.4)\begin{equation} a(\omega) = \left[z_{*}(\omega)-z\right]b - \int\limits_{z}^{z_{*}(\omega)} b_{*}(\omega)|_{\eta}\,\mathrm{d}\eta, \end{equation}

defined relative to the subvolume $\omega$, and integrating over $\omega$ (see e.g. Scotti & White Reference Scotti and White2014),

(4.5)\begin{align} \dfrac{\mathrm{d} A(\omega)}{\mathrm{d} t} &={-}Y(\omega)-\overbrace{\int\limits_{\omega}\int\limits_{b}^{b_{*}(\omega)} \left.\dfrac{\partial z_{*}(\omega)}{\partial t}\right|_{\beta}\,\mathrm{d}\beta\,\mathrm{d} V}^{T(\omega)=0}+\overbrace{ \int\limits_{\omega}w\left[b_{*}(\omega)-b\right]\textrm{d} V}^{C(\omega)}\nonumber\\ &\quad -\underbrace{\frac{1}{Pe}\int_{\omega}|\boldsymbol{\nabla} b|^{2}\dfrac{\partial z_{*}(\omega)}{\partial b}- \dfrac{\partial b_{*}(\omega)}{\partial z}\,\mathrm{d} V}_{D(\omega)}-\underbrace{\frac{2}{Pe} \int\limits_{\omega}\left[\dfrac{\partial b_{*}(\omega)}{\partial z}-\dfrac{\partial b}{\partial z}\right]\textrm{d} V}_{2L(\omega)}, \end{align}

where

(4.6)\begin{equation} Y(\omega)=\int\limits_{\partial\omega}\left(\boldsymbol{u}-\frac{\boldsymbol{\nabla}}{Pe}\right)a(\omega)\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d} S, \end{equation}

accounts for ‘transport’ across the subvolume boundary $\partial \omega$ (relative to which the unit normal $\boldsymbol {n}$ points outwards from the surface element $\mathrm{d}S$) and will be discussed in § 4.3. As shown by Scotti & Passaggia (Reference Scotti and Passaggia2019, § 2), the integral contribution in (4.5) from temporal changes in the background state, $T(\omega )$, vanishes (to see why, replace $\xi$ with $b_{*}$ and $g$ with $\partial _{t}z_{*}$ in (2.4), which expresses the fact that $\omega$-integrated functionals of the reference state are not affected by the adiabatic sorting $z\mapsto z_{*}(\omega )$). Equation (4.5) is equivalent to the global budgets discussed in Winters et al. (Reference Winters, Lombard, Riley and D'Asaro1995), who considered closed domains (such that $\boldsymbol {u}=\boldsymbol {0}$ on $\partial \omega$) and factored the linear terms in $Y(\omega )$, $D(\omega )$ and $2L(\omega )$ slightly differently.

Following the definition in (2.10) of local APE relative to $\varOmega$, replacement of $\omega$ with $\varOmega$ in (4.5) (except in specifying the domain of integration $\omega$) yields an equivalent expression for the budget of the total (outer plus inner) APE $A(\omega |\varOmega )=A(\varOmega |\omega )+A(\omega )$. The budget of the outer APE $A(\varOmega |\omega )$ therefore corresponds to the difference between the budgets of the total APE and the inner APE,

(4.7)\begin{align} \dfrac{\mathrm{d} A(\varOmega|\omega)}{\mathrm{d} t} &={-}\sum Y(\varOmega|\omega)-\overbrace{\int\limits_{\omega}\int\limits_{b_{*}(\omega)}^{b_{*} (\varOmega)}\left.\dfrac{\partial z_{*}(\varOmega)}{\partial t}\right|_{\beta}\,\mathrm{d}\beta\,\mathrm{d} V}^{T(\varOmega|\omega)}+\overbrace{ \int\limits_{\omega}w[b_{*}(\varOmega)-b_{*}(\omega)]\,\mathrm{d} V}^{C(\varOmega|\omega)}\nonumber\\ &\quad -\underbrace{\frac{1}{Pe}\int\limits_{\omega}|\boldsymbol{\nabla} b|^{2}\left[\dfrac{\partial z_{*}(\varOmega)}{\partial b}- \dfrac{\partial z_{*}(\omega)}{\partial b}\right]-\left[\dfrac{\partial b_{*}(\varOmega)}{\partial z}-\dfrac{\partial b_{*}(\omega)}{\partial z}\right]\textrm{d} V}_{D(\varOmega|\omega)}\nonumber\\ &\quad-\underbrace{\frac{2}{Pe}\int\limits_{\omega}\left[\dfrac{\partial b_{*}(\varOmega)}{\partial z}-\dfrac{\partial b_{*}(\omega)}{\partial z}\right] \textrm{d} V}_{2L(\varOmega|\omega)}, \end{align}

where

(4.8)\begin{equation} Y(\varOmega|\omega)=\int\limits_{\partial\omega}\left(\boldsymbol{u}-\frac{\boldsymbol{\nabla}}{Pe}\right)a(\varOmega|\omega)\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d} S. \end{equation}

Equation (4.7), unlike (4.5), contains a contribution from temporal changes in the reference state $T(\varOmega |\omega )$ that does not vanish upon integration over $\omega$ because it makes reference to the context provided by $\varOmega$.

4.2. Conversion between inner and outer APE due to mixing

The total dissipation $D(\omega |\varOmega )$ of APE within subvolume $\omega \subset \varOmega$ is

(4.9)\begin{equation} D(\omega|\varOmega) = D(\varOmega|\omega)+D(\omega) = \frac{1}{Pe}\int\limits_{\omega}|\boldsymbol{\nabla} b|^{2}\dfrac{\partial z_{*}(\varOmega)}{\partial b}-\dfrac{\partial b_{*}(\varOmega)}{\partial z}\,\mathrm{d} V, \end{equation}

which is equivalent to the integral of $G_{p}$ in Scotti & White (Reference Scotti and White2014). At the global minimum in potential energy, the total dissipation $D(\omega |\varOmega )$, along with its outer and inner components $D(\varOmega |\omega )$ and $D(\omega )$, respectively, vanishes for each subvolume $\omega$ because, when $A(\varOmega )=0$, $\partial _{z}b=\partial _{z}b_{*}(\varOmega )$ everywhere. However, the same cannot be said about a local minimum in potential energy (cf. figure 2a), for which $\partial _{z}b=\partial _{z}b_{*}(\omega )\neq \partial _{z}b_{*}(\varOmega )$ in general.

With the exception of somewhat pathological cases, constructed by squeezing part of a reference state $b_{*}$ horizontally to obtain a local buoyancy field $b$ that satisfies $|\boldsymbol {\nabla } b| = \partial _{z}b<\partial _{z}b_{*}$, the inner dissipation $D(\omega )$ and the total dissipation $D(\omega |\varOmega )$ are typically positive. In contrast, the sign of the outer component of the APE ‘dissipation’,

(4.10)\begin{equation} D(\varOmega|\omega) = \frac{1}{Pe}\int\limits_{\omega}|\boldsymbol{\nabla} b|^{2}\left[\dfrac{\partial z_{*}(\varOmega)}{\partial b}-\dfrac{\partial z_{*}(\omega)}{\partial b}\right]-\left[ \dfrac{\partial b_{*}(\varOmega)}{\partial z}-\dfrac{\partial b_{*}(\omega)}{\partial z}\right]\textrm{d} V, \end{equation}

depends on the reference state of both $\omega$ and $\varOmega$. Because $|\boldsymbol {\nabla } b|^{2}\geqslant 0$, stirring will generally make the first term in the integrand of (4.10) dominant, such that the sign of $\partial _{b}z_{*}(\varOmega )-\partial _{b}z_{*}(\omega )$ determines the sign of $D(\varOmega |\omega )$. In figure 2(b), for example, stirring that leads to mixing of the fluid in the left subvolume $\omega _{1}$ would raise the background potential energy associated with the reference state $z_{*}(\omega _{1})$ such that $\partial _{b}z_{*}(\varOmega )-\partial _{b}z_{*}(\omega _{1})<0$ and $D(\varOmega |\omega )<0$. Stirring (followed by mixing) would therefore result in the conversion of inner APE $A(\omega _{1})$ to outer APE $A(\varOmega |\omega _{1})$. The physical explanation relates to the constraint provided by the solid partition, which enables part of the ‘irreversible’ work $D(\omega |\varOmega )$ to be stored as outer (context dependent) APE $A(\varOmega |\omega _{1})$ that can be subsequently released by connecting $\omega _{1}$ and $\omega _{2}$.

In summary, the context that $\varOmega$ provides to a subvolume $\omega$ is crucial in determining the effect that irreversible mixing within $\omega$ has on the subvolume's outer APE. This is noteworthy because, as will be seen in § 5, outer APE is primarily responsible for driving flow between subvolumes separated by physical partitions. It is therefore useful to regard $D(\varOmega |\omega )$ as a conversion between inner and outer APE, as shown schematically on the left-hand side of figure 7. In the following section another mechanism for conversion between inner and outer APE is described.

Figure 7.

Diagram illustrating two ways in which inner APE can be converted into outer APE. The first, $D(\varOmega |\omega )$, is discussed in § 4.2 and accounts for diapycnal mixing that increases the outer APE at the expense of inner APE in example 2(b). The second, $X(\sigma ,\omega )$, accounts for the change in context that occurs when fluxes $J(\varOmega |\sigma ,\varOmega |\omega )$ and $J(\sigma ,\omega )$ transport heat or mass from $\sigma$ into $\omega$. The conversion $X(\omega ,\sigma )$ is dashed because if all pointwise fluxes from $\sigma$ to $\omega$ are positive then $X(\omega ,\sigma )=0$, as described in the paragraph below (4.12). An example of $X$ can be found in the large-scale circulation driven by outer APE that is discussed in § 5 (see figure 8), whose transport of buoyancy creates (or removes) inner APE from individual subvolumes.

4.3. Conversion between inner and outer APE due to transport

The terms $Y(\omega )$ and $Y(\varOmega |\omega )$ defined in (4.6) and (4.8), respectively, need to be considered carefully because, since they do not involve the APE of adjacent subvolumes, they are not proper transport terms. Instead, it desirable to define transport terms that account for APE transported into $\omega$ from an adjacent subvolume $\sigma$.

On the boundary between $\omega$ and $\sigma$ the inner and outer APE densities $a(\omega )$, $a(\sigma )$, $a(\varOmega |\omega )$ and $a(\varOmega |\sigma )$ that feature in (4.6) and (4.8) are not the same, because $\omega$ and $\sigma$ provide different local contexts. However, the local potential energy $-zb$ and the local APE density $a(\varOmega )$ (i.e. with respect to the parent volume $\varOmega$) in (2.9) have a single value anywhere in $\varOmega$, which makes it possible to isolate those parts of $Y(\omega )$ and $Y(\varOmega |\omega )$ that correspond to transport.

The definition of local APE density $a(\omega )$ in (2.9) in terms of $a(\sigma )$ at a point on the boundary $\partial \omega$ between $\omega$ and $\sigma$, noting that the potential energy $-zb$ is uniquely defined on $\partial \omega$, is

(4.11)\begin{equation} a(\omega)= a(\sigma)+\underbrace{\left[z_{*}(\omega)-z_{*}(\sigma)\right]b+a_{*}(\sigma)-a_{*}(\omega)}_{\varDelta(\sigma,\omega)}, \end{equation}

which enables $Y(\omega )$ to be expressed as the sum of a flux $J$ and a conversion term $X$,

(4.12) \begin{equation} Y(\omega) = \underbrace{\int\limits_{\partial\omega}\left(\boldsymbol{u}-\frac{\boldsymbol{\nabla}}{Pe}\right) a(\theta)\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d} S}_{\displaystyle\sum_{\sigma\in\mathscr{P}(\varOmega)} -J(\sigma,\omega)}+ \underbrace{\int\limits_{\partial\omega}\left(\boldsymbol{u}-\frac{\boldsymbol{\nabla}}{Pe}\right) \varDelta(\theta,\omega)\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d} S}_{\displaystyle\sum_{\sigma\in\mathscr{P}(\varOmega)} X(\sigma,\omega)},\end{equation}

where $\theta$, which depends on $\boldsymbol {u}$ and position on the boundary $\partial \omega$, is the subvolume from which $\boldsymbol {u}$ points outwards (for definiteness, this description focuses on cases for which $Pe\gg 1$ and therefore ignores the direction associated with $-\boldsymbol {\nabla } a$). Transport $J(\sigma ,\omega )$ corresponds to the flux of inner APE density from any subvolume $\sigma \in \mathscr {P}(\varOmega )$ into $\omega \in \mathscr {P}(\varOmega )$, and is skew–symmetric such that $J(\sigma , \omega )=-J(\omega ,\sigma )$. Therefore, over parts of the boundary $\partial \omega$ where $\theta \neq \omega$ in (4.12), $J(\sigma ,\omega )>0$ is a flux into $\omega$, while in cases where $\theta =\omega$, $J(\sigma ,\omega )=-J(\omega ,\sigma )< 0$ is a flux out of $\omega$ (under the assumption that $Pe\gg 1$). If $\omega$ is not connected to $\sigma$ then $J(\sigma ,\omega )=0$ because the subvolume $\sigma$ will not feature in the integral of fluxes over $\partial \omega$. Alongside $J$, $X$ accounts for the change in context that occurs when APE is transported into $\omega$ from $\sigma$. Unlike $J$, $X$ is not skew–symmetric because fluxes into $\omega$ from $\sigma$ (i.e. $\theta =\sigma$) contribute to $X(\sigma ,\omega )$ but do not contribute to $X(\omega ,\sigma)$.

Since the local APE density,

(4.13)\begin{equation} a(\varOmega)=a(\varOmega|\omega)+a(\omega)=a(\varOmega|\sigma)+a(\sigma), \end{equation}

is uniquely defined at boundaries, the conversion terms $X(\sigma ,\omega )$ in (4.12) appear with opposite sign in the expression for $Y(\varOmega |\omega )$,

(4.14) \begin{equation} Y(\varOmega|\omega) = Y(\omega|\varOmega)-Y(\omega)=\underbrace{\int\limits_{\partial\omega}\left(\boldsymbol{u}-\frac{\boldsymbol{\nabla}}{Pe}\right) a(\varOmega|\theta)\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d} S}_{\displaystyle\sum_{\sigma\in\mathscr{P}(\varOmega)}-J(\varOmega|\sigma, \varOmega|\omega)}-\sum_{\sigma\in\mathscr{P}(\varOmega)}X(\sigma,\omega). \end{equation}

When fluid (or thermal energy) is transported into a subvolume, the conversion term $X$ repartitions the inner and outer APE to account for the new context. For example, the transport of fluid from a subvolume $\sigma$ with uniform buoyancy, and therefore zero inner APE ($A(\sigma )=0$), can produce inner APE ($A(\omega )>0$) in an adjacent subvolume of different buoyancy, at the expense of a reduction in the outer APE $A(\sigma \cup \omega |\omega )$. The decomposition therefore quantifies the effect that transport across the boundary of an open system (i.e. a subvolume) has on its relationship with the surrounding environment. The role of $X$ is illustrated schematically alongside the effects of mixing that were discussed in § 4.2 in figure 7. The resulting budgets from (4.5) and (4.7) are

(4.15)\begin{align} \hspace{-16pt}\dfrac{\mathrm{d} A(\omega)}{\mathrm{d} t} &= C(\omega)+\overbrace{D(\varOmega|\omega)-D(\omega|\varOmega)}^{{-}D(\omega)}-2L(\omega)+\sum_{\sigma\in\mathscr{P}(\varOmega)}\left[ J(\sigma,\omega) - X(\sigma,\omega)\right],\nonumber\\\end{align}

(4.16)\begin{align} \dfrac{\mathrm{d} A(\varOmega|\omega)}{\mathrm{d} t} &= C(\varOmega|\omega)-D(\varOmega|\omega)-2L(\varOmega|\omega)-T(\varOmega|\omega) +\sum_{\sigma\in\mathscr{P}(\varOmega)} \left[J(\varOmega|\sigma,\varOmega|\omega)+X(\sigma,\omega)\right]. \end{align}

5.1. Buoyancy field evolution

Figure 8 displays the buoyancy field at different times during the simulation. The positive buoyancy of fluid in $\omega _{1}$ and $\omega _{3}$, relative to $\omega _{2}$, leads to counter-rotating circulation around each partition (clockwise over $\omega _{1}\cup \omega _{2}$ and counter-clockwise over $\omega _{2}\cup \omega _{3}$). The initial configuration of buoyancy within $\omega _{1}$, which is unstable, is not the same as it is in $\omega _{3}$, destroying the symmetry implied by the average buoyancy of each subvolume (equal to $3/4$ in $\omega _{1}$ and $\omega _{3}$). Regions of mixing are indicated by blue ($b\approx 0.25$) and yellow ($b\approx 0.75$). By $t=10$ the structure of the initial large-scale circulations and transport between subvolumes is less pronounced and has been replaced with vigorous mixing within each subvolume.

Figure 8.

The instantaneous buoyancy field from the numerical simulation described in § 5 and figure 4 with geometrical parameters $L=3$ and $\gamma =\mu =\nu =1/3$, and initial conditions $b_{1}=1/2$ and $b_{2}=3/4$. The colour map is from ‘3 Wave Colormaps’ at ‘https://sciviscolor.org’ and was selected to highlight regions of mixing around $b=0.25$ and $b=0.75$ in blue and yellow, respectively (see, e.g. Samsel et al. Reference Samsel, Wolfram, Bares, Turton and Bujack2019).

Figure 9 displays histograms corresponding to the probability density functions for buoyancy in $\omega _{1}$, $\omega _{2}$ and $\omega _{3}$, in which $V_{*}(\omega _{i})(b)$ is the volume of $\omega _{i}$ below $z_{*}(\omega _{i})(b)$. At $t=0$, the distribution of buoyancy consists of spikes at $b=1/2$ and $b=1$ for $\omega _{1}$, $b=0$ for $\omega _{2}$ and $b=3/4$ for $\omega _{3}$. Panels corresponding to later times illustrate that the distribution of buoyancy within each subvolume gets wider as the system evolves because diapycnal mixing creates intermediate values of buoyancy that were not present in the initial distributions. For example, panels ( f–l) show that an increasing proportion of fluid in $\omega _{3}$ has buoyancy $b\approx 3/8=0.375$ due to transport of fluid with $b\approx 0$ from $\omega _{2}$ into $\omega _{1}$. The particular evolution of the subvolume buoyancy distributions therefore depends on the system's topology.

Figure 9.

Probability density functions for the buoyancy field shown in each panel of figure 8 in each subvolume. Note that the histograms corresponding to $\omega _{1}$, $\omega _{2}$ and $\omega _{3}$ represent probability densities because the subvolumes, unlike their sum $\varOmega$, have unit volume. Also note that the scale used on the vertical axes is different in each row of the figure.

The probability densities shown in figure 9 are crucial in modulating the local and global dissipation of APE that results from diapycnal mixing. In (4.5) the dissipation $D(\omega )$ of inner APE accounts for the distribution of buoyancy within $\omega$ (i.e. $\partial _{b}z_{*}(\omega )$) in product with the dissipation of buoyancy variance $|\boldsymbol {\nabla } b|^{2}/Pe$. In (4.7), however, the dissipation $D(\varOmega |\omega )$ of outer APE necessarily accounts for the context provided by $\varOmega$ via the global reference distribution $\partial _{b} z_{*}(\varOmega )$.

5.2. APE evolution

Figure 10 displays the temporal evolution of the APE decomposition in each subvolume. For reference, the global APE $A(\varOmega )=A(\omega _{1}\cup \omega _{2}\cup \omega _{3})$ divided by three is shaded. Dashed lines, which correspond to the total APE $A(\omega |\varOmega )$ for each subvolume and therefore sum to $A(\varOmega )$, indicate whether the total APE of a given subvolume is above or below the average for $\varOmega$. As can be deduced analytically (see the tabulated expressions in the Appendix), at $t=0$, $A(\omega _{1})=1/8$, $A(\omega _{2})=0$, $A(\omega _{3})=0$, $A(\varOmega |\omega _{1})=1/24$, $A(\varOmega |\omega _{2})=7/48$ and $A(\varOmega |\omega _{3})=3/48$; hence (at $t=0$),

(5.1)\begin{equation} A(\varOmega)=\sum A(\varOmega|\omega_{i}) + \sum A(\omega_{i})= \frac{1}{8}+\frac{1}{24}+\frac{7}{48}+\frac{3}{48}=\frac{3}{8}. \end{equation}

Figure 10.

The temporal evolution of outer APE $A(\varOmega |\omega )$, inner APE $A(\omega )$ and their sum $A(\omega |\varOmega )$ for subvolumes $\omega _{1}$, $\omega _{2}$ and $\omega _{3}$. The shaded region corresponds to $A(\varOmega )/3$ and therefore represents the volume-averaged global APE to which contributions within each subvolume can be compared.

In $\omega _{1}$ the initial inner APE $A(\omega _{1})=1/8$ is significantly larger than the initial outer APE $A(\varOmega |\omega _{1})=1/24$, and both contributions diminish as the system evolves. In contrast, the inner APE $A(\omega _{2})$ and $A(\omega _{3})$, which are initially zero, increase due to the transport of buoyancy between volumes, and the conversion between outer and inner APE (described by $X$ in § 4.3), which will be discussed further in § 5.3. In this regard, the outer APE decays more rapidly than the inner APE because the outer APE drives the large-scale circulation between subvolumes that is ultimately responsible for generating inner APE in $\omega _{2}$ and $\omega _{3}$. After approximately $5$ time units the sum of inner APE $\sum A(\omega _{i})$ dominates the sum of outer APE $\sum A(\varOmega |\omega _{i})$, which marks the onset of significant stirring followed by mixing within each subvolume, as can be seen qualitatively in figure 8(d–l).

The largest contribution to the outer APE comes from $A(\varOmega |\omega _{2})$, due to the context provided by $\omega _{1}$ and $\omega _{3}$. Indeed, at $t=0$, $A(\varOmega |\omega _{2})$ is comparable to the total APE $A(\varOmega |\omega _{1})+A(\omega _{1})$ of $\omega _{1}$, in spite of the fact that $b\equiv 0$ in $\omega _{2}$. At around $t=10$, the total APE in $\omega _{2}$, which becomes dominated by the inner APE $A(\omega _{2})$, is larger than the inner APE in $\omega _{1}$.

5.3. Subvolume APE budgets

Figure 11 displays each term of the inner APE budget (4.15) with respect to time. Due to the flow's high Reynolds and Péclet number, diffusion down the inner reference states $L(\omega )$ is relatively small. For the same reason, the diffusive contribution to the transport terms $J$ and $X$ is excluded, which makes their calculation significantly simpler and their physical meaning more transparent. The residual numerical error shown in figure 11 is relatively small, which implies that (4.15) has been derived and implemented correctly. The error is not identically zero because pointwise contributions to the APE dissipation $D(\omega )$, which involves the product of three gradients, from the small number of computational cells that are adjacent to the corners of the solid boundaries between subvolumes are difficult to calculate accurately (see, e.g. figure 13 for a picture of the local APE dissipation).

Figure 11.

Terms in the budget of inner APE (4.15) for each subvolume (for the definitions of individual terms see (4.5) and (4.12)). The light grey shaded region corresponds to the temporal derivative of inner APE and the dark shaded region corresponds to the residual numerical error that is left when the terms in (4.15) are added together. The vertical line corresponds to the upper limit of the time domain used in figure 12.

In each subvolume the dissipation of inner APE $D(\omega )$ reaches a maximum after approximately 8 units of time. During the first 4 units of time the transport $J(\omega _{2}, \omega _{1})<0$ (which is the only term contributing to $\sum J$) corresponds to the inner APE that is lost from $\omega _{1}$ as relatively dense (unstable) parcels of fluid are transported out of the volume at high level and replaced with relative dense (stable) parcels of fluid at low level. This effect provides an example of how outer APE can drive flows that reduce inner APE within a subvolume. The loss of inner APE in $\omega _{1}$ due to $J(\omega _{2},\omega _{1})<0$ corresponds exactly to the gain in inner APE in $\omega _{2}$ due to $J(\omega _{1},\omega _{2})>0$, which justifies regarding $J$, defined in (4.12), as transport. However, the parcels of fluid with initial buoyancy $b_{1}$, which were negatively buoyant in $\omega _{1}$, become stable at the top of $\omega _{2}$. This change in context is quantified by $-X(\omega _{1},\omega _{2})$ in (4.12), which corresponds to a conversion from the inner APE $A(\omega _{2})$ into the outer APE $A(\varOmega |\omega _{2})$, and therefore appears as a sink in figure 11(b).

The most significant remaining term in figure 11 is the buoyancy work $C$, defined relative to the reference state of each subvolume, which represents the well-known reversible conversion between APE and kinetic energy. Its relatively large negative contribution in $\omega _{1}$ at $t\approx 6$ corresponds to the instability within $\omega _{1}$ arising from the unstable initial buoyancy state. In contrast, $C(\omega _{2})$ is typically positive, due to circulation within $\omega _{2}$ that transports relatively buoyant fluid entering $\omega _{2}$ at high level downwards and lifts relatively dense fluid upwards (see, e.g. figure 8e).

Figure 12 displays the temporal evolution of terms in the outer APE budget (4.16). The dominant term in all subvolumes is the (outer) conversion $C(\varOmega |\omega )$. In contrast to $C(\omega )$, which accounts for $\omega$ without context, $C(\varOmega |\omega )$ accounts for work over the reference buoyancy field and can only be non-zero when there is a mean vertical motion in a subvolume (because neither $b_{*}(\varOmega )$ nor $b_{*}(\omega )$ in (4.7) depend on $x$), resulting from high- and low-level openings in this particular problem. Consequently, figure 12 shows that outer APE is lost in each subvolume, because $C(\varOmega |\omega )$ is negative, to drive the large-scale circulation between subvolumes.

Figure 12.

Terms in the budget of outer APE (4.16) for each subvolume (for the definitions of individual terms see (4.7), (4.12) and (4.14)). The light grey shaded region corresponds to the temporal derivative of outer APE and the dark shaded region corresponds to the residual numerical error that is left when the terms in (4.16) are added together. Note that $D$ corresponds to the outer dissipation $D(\varOmega |\omega )=D(\omega |\varOmega )-D(\omega )$ in (4.16) and that $X$ corresponds to $\sum X(\sigma ,\omega )$ in (4.16), and therefore appears in this figure with an opposite sign to the $-X$ that appears in figure 11.

A further noteworthy feature of figure 12(b) is the positive contribution from $X(\omega _{1},\omega _{2})$ (cf. $-X(\omega _{1},\omega _{2})$ in figure 11b), which corresponds to the (reversible) conversion between inner and outer APE due to transport between subvolumes, as discussed above in relation to figure 11; the advection of parcels of fluid that are unstable with respect to $\omega _{1}$, but stable with respect to $\omega _{2}$, effectively converts inner APE $A(\omega _{1})$ into outer APE $A(\varOmega |\omega _{2})$ via transport $J(\omega _{1},\omega _{2})$, in accordance with figure 7.

5.4. Local APE dissipation

Figure 13(a) displays the local inner APE dissipation, whose integral over each subvolume is $D(\omega )$, as defined in (4.5). As can be seen from the colour scale used in figure 13(a), APE dissipation is dominated by large contributions from thin filaments, where both $|\boldsymbol {\nabla } b|$ and $\partial z_{*}/\partial b$ are large.

Figure 13.

The local (pointwise) contribution to the (inner) APE dissipation $D(\omega )$ in each subvolume $\omega$ is displayed in (a) and the (outer) APE dissipation $D(\varOmega |\omega )=D(\omega |\varOmega )-D(\omega )$ is displayed in (b) (note the factor 10 difference in the scales used for the colormaps). The local (pointwise) contributions to $D(\omega )$ and $D(\varOmega |\omega )$ can be found in (4.5) and (4.7), respectively. The (outer) APE dissipation $D(\varOmega |\omega )$ represents a conversion between inner and outer APE (see figure 7 for reference). All variables are plotted at $t=9.8$, after the initial condition shown in figure 4 (cf. figure 8g). The colour map for $(a)$ is part of the ‘Outlier-Focused Colormaps’ found at ‘https://sciviscolor.org’, which provides the ‘Colormoves’ interface that was used to construct the colour map in (b) (see, e.g. Samsel et al. Reference Samsel, Wolfram, Bares, Turton and Bujack2019).

As discussed in § 4.2, although the inner and total APE dissipation ($D(\omega )$ and $D(\omega |\varOmega )$, respectively) are typically positive, the sign of the ‘dissipation’ associated with the outer APE, $D(\varOmega |\omega )=D(\omega |\varOmega )-D(\omega )$, depends on context. Indeed, figure 13(b), which displays local contributions to the outer APE dissipation $D(\varOmega |\omega )$, highlights positive and negative contributions that correspond to a loss or gain of outer APE, respectively, due to local mixing. In this particular problem, for which figure 12 indicates that the integral $D(\varOmega |\omega )$ is small for all $\omega$, the positive and negative contributions appear to occur in approximately equal proportion within each subvolume.