3.1.1 The gauge-fixing condition

The Frèchet derivatives of the naive energy with respect to $\boldsymbol{v}$ and $b$ are trivial. We concentrate here on the variations of the Casimir written as in (3.3). The variation with respect to $b$ is given by

(3.5) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6FF}b}\unicode[STIX]{x1D6FF}b & = & \displaystyle (\unicode[STIX]{x1D6F9}_{,b}\unicode[STIX]{x1D6FF}b+\unicode[STIX]{x1D6F9}_{,q}\unicode[STIX]{x1D6FF}q)\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x1D6F9}_{,b}\unicode[STIX]{x1D6FF}b+\unicode[STIX]{x1D6F9}_{,q}(\unicode[STIX]{x1D735}(\unicode[STIX]{x1D6FF}b)\boldsymbol{\cdot }\unicode[STIX]{x1D74E}))\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D6F9}_{,b}\unicode[STIX]{x1D6FF}b-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D6F9}_{,q}\unicode[STIX]{x1D714})\unicode[STIX]{x1D6FF}b\nonumber\\ \displaystyle & = & \displaystyle [(\unicode[STIX]{x1D6F9}-q\unicode[STIX]{x1D6F9}_{,q})_{,b}-\unicode[STIX]{x1D6F9}_{,qq}(\unicode[STIX]{x1D735}q\boldsymbol{\cdot }\unicode[STIX]{x1D74E})]\unicode[STIX]{x1D6FF}b,\end{eqnarray}$$

while the variation with respect to $\boldsymbol{v}$ is given by

(3.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6F9}}{\unicode[STIX]{x1D6FF}\boldsymbol{v}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{v}=\unicode[STIX]{x1D6F9}_{,q}\frac{\unicode[STIX]{x1D6FF}q}{\unicode[STIX]{x1D6FF}\boldsymbol{v}}\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{v}=\unicode[STIX]{x1D6F9}_{,qq}(\unicode[STIX]{x1D735}q\times \unicode[STIX]{x1D735}b)\boldsymbol{\cdot }\unicode[STIX]{x1D6FF}\boldsymbol{v}.\end{eqnarray}$$

Note that to avoid notational clutter we use the comma derivative notation, i.e.

(3.7a,b ) $$\begin{eqnarray}f_{,a}\equiv \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}a},\quad f_{,ab}\equiv \frac{\unicode[STIX]{x2202}^{2}f}{\unicode[STIX]{x2202}a\unicode[STIX]{x2202}b},\ldots\end{eqnarray}$$

Given the mass distribution function $V(q,b)$ , the ground state and the gauge-fixed Casimir are found solving

(3.8a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}{\mathcal{H}}}{\unicode[STIX]{x1D6FF}b} & \cong & \displaystyle [-Z+(\unicode[STIX]{x1D6F9}-q\unicode[STIX]{x1D6F9}_{,q})_{,b}]-\unicode[STIX]{x1D6F9}_{,qq}(\unicode[STIX]{x1D735}q\boldsymbol{\cdot }\unicode[STIX]{x1D74E})=0,\end{eqnarray}$$

(3.8b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}{\mathcal{H}}}{\unicode[STIX]{x1D6FF}\boldsymbol{v}} & \cong & \displaystyle \boldsymbol{v}+\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D6F9}_{,q}\unicode[STIX]{x1D735}b)=0,\end{eqnarray}$$

(3.8c ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}{\mathcal{H}}}{\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}} & = & \displaystyle \int _{\mathbb{D}}[\unicode[STIX]{x1D703}(q(\boldsymbol{x})-q)\unicode[STIX]{x1D703}(b(\boldsymbol{x})-b)]dV=V(q,b).\end{eqnarray}$$

3.1.2 Non-quiescent ground states and natural coordinates

While our theory recovers Winters et al.’s quiescent ground state (i.e. having zero KE) as a special case, we are mainly interested in characterizing ground states with non-zero kinetic energy, and in which streamlines do not end in stagnation points, which requires $\unicode[STIX]{x1D6F9}_{\ast ,qq}\neq 0$ and $\unicode[STIX]{x1D735}q_{\ast }\times \unicode[STIX]{x1D735}b_{\ast }\neq 0$ or else from (3.8b ) $\boldsymbol{v}_{\ast }=0$ . Also, we assume that the mass distribution function is a continuous and differentiable function, i.e. buoyancy and potential vorticity are continuously distributed. More complicated cases consisting of layers where buoyancy and/or potential vorticity are interleaved with layers where one or the other are constant will not be considered here.

The form of the equations, and the fact that buoyancy and potential vorticity are an obvious choice for Lagrangian coordinates, suggest that the ground state is more conveniently expressed with a ‘natural’ set of coordinates which is given by buoyancy, potential vorticity and a third coordinate $t$ . The latter is defined so that the relationship between ‘geometric’ $(x^{1},x^{2},x^{3})$ and natural coordinates $(q,b,t)$ is such that the volume element becomes

(3.9) $$\begin{eqnarray}\sqrt{|g|}dx^{1}dx^{2}dx^{3}=\unicode[STIX]{x1D6F9}_{\ast ,qq}dbdqdt.\end{eqnarray}$$

This can be achieved if locally the relationship between geometric and natural coordinates is obtained by integrating the Pfaffian system

(3.10) $$\begin{eqnarray}dx^{i}=v_{\ast }^{i}dt,\quad i=1,\ldots ,3\end{eqnarray}$$

where $\boldsymbol{v}_{\ast }=-\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D6F9}_{\ast ,q}\unicode[STIX]{x1D735}b_{\ast })$ is the velocity field of the ground state, i.e. the solution of (3.8b ) and the subscript $_{\ast }$ reminds us that these are ground state fields (see § A.2 for a derivation of (3.9) from (3.10)). In the following sections, we explore some of the general properties of the ground states that are solutions of (3.8a )–(3.8c ). The analysis relies on the following general property: if $\boldsymbol{v}_{\ast },b_{\ast }$ are solutions of (3.8a )–(3.8c ), the Casimir $\unicode[STIX]{x1D6F9}_{\ast }$ of the ground state, is related to the Bernoulli head

(3.11) $$\begin{eqnarray}B_{\ast }(q,b)=\boldsymbol{v}_{\ast }(q,b)^{2}/2-bZ_{\ast }(q,b)+P_{\ast }(q,b)\end{eqnarray}$$

of the ground state expressed in natural coordinates, via

(3.12) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F9}}_{\ast }\equiv q\unicode[STIX]{x1D6F9}_{\ast ,q}-\unicode[STIX]{x1D6F9}_{\ast }=B_{\ast }(q,b),\end{eqnarray}$$

whose general solution is

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\ast }=q\int _{q_{min}}^{q}\frac{B_{\ast }(q^{\prime },b)}{{q^{\prime }}^{2}}dq^{\prime }\end{eqnarray}$$

(see § A.1).

3.1.3 Apedic ground states and their properties

As a first application of our framework, we wish to characterize under what conditions we recover Winters et al. APE definition.

Let us start with a general definition: an apedic (from the Greek   meaning level, flat) ground state is a ground state such that $b_{\ast }=b_{\ast }(Z)$ . By extension, the leaf in phase space to which an apedic ground state belongs is called an apedic leaf.

If we express the absolute vorticity of a ground state as

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D74E}_{\ast }=\unicode[STIX]{x1D714}^{t}\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}q+\unicode[STIX]{x1D714}^{b}\unicode[STIX]{x1D735}q\times \unicode[STIX]{x1D735}t+\unicode[STIX]{x1D714}^{q}\unicode[STIX]{x1D735}t\times \unicode[STIX]{x1D735}b,\end{eqnarray}$$

then the ground state is apedic if and only if

(3.15a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{,t}^{q} & = & \displaystyle 0,\end{eqnarray}$$

(3.15b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{,t}^{t} & = & \displaystyle 0,\end{eqnarray}$$

$t$

$t$

$\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D74E}=0$

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D735}Z\times \unicode[STIX]{x1D735}b+\unicode[STIX]{x1D714}_{,t}^{t}\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}q+\unicode[STIX]{x1D714}_{,t}^{q}\unicode[STIX]{x1D735}t\times \unicode[STIX]{x1D735}b=0,\end{eqnarray}$$

which the ground state must satisfy. This equation is satisfied if and only if (3.15a )–(3.15b ) hold, since $\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}q$ and $\unicode[STIX]{x1D735}t\times \unicode[STIX]{x1D735}b$ are independent vectors, and on an apedic manifold $\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}Z=0$ by definition.

Let us now explore some more properties of apedic ground states. Let $\unicode[STIX]{x1D735}_{\boldsymbol{v}_{\ast }}\boldsymbol{v}_{\ast }$ be the acceleration along streamlines, where $\unicode[STIX]{x1D735}$ is the Levi-Civita connection on the manifold, i.e.

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D735}_{\boldsymbol{v}_{\ast }}\boldsymbol{v}_{\ast }=\left[x_{,tt}^{j}+\unicode[STIX]{x1D6E4}_{mp}^{j}x_{,t}^{m}x_{,t}^{p}\right]\boldsymbol{\unicode[STIX]{x2202}}_{j},\end{eqnarray}$$

where the $x^{j}(t)$ are the coordinates along streamlines parameterized with $t$ , i.e. the solution to $dx^{i}=v_{\ast }^{i}dt$ and $\unicode[STIX]{x1D6E4}_{mp}^{i}$ are the Christoffel symbols of the Levi-Civita connection. Then it is a matter of straightforward algebra (see § A.3) to show that for a ground state to be apedic the conditions (3.15a )–(3.15b ) require

(3.18a ) $$\begin{eqnarray}\displaystyle \left(\frac{\boldsymbol{\unicode[STIX]{x2202}}}{\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{v}_{\ast }}\boldsymbol{v}_{\ast }\right)_{,q} & = & \displaystyle \left(\frac{\boldsymbol{\unicode[STIX]{x2202}}}{\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{q}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{v}_{\ast }}\boldsymbol{v}_{\ast }\right)_{,b},\end{eqnarray}$$

(3.18b ) $$\begin{eqnarray}\displaystyle \left(\frac{\boldsymbol{\unicode[STIX]{x2202}}}{\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{v}_{\ast }}\boldsymbol{v}_{\ast }\right)_{,t} & = & \displaystyle Ek_{\ast ,tb}.\end{eqnarray}$$

Thus, a sufficient condition for a ground state to be apedic is that streamlines are geodesics in an inertial frame and the kinetic energy depends only on either $q$ and $t$ or $q$ and $b$ .

Also, it is worth remarking that the notion of apedicity, depends on the local direction of the plumb line, and thus it is not frame independent. Thus, a flow in solid body rotation will appear apedic to an observer in solid body rotation with the flow, but not to an inertial observer.

3.1.4 The Casimir of inertial apedic ground states

Under the assumption that the ground state is apedic in an inertial system, we can make further analytic inroads. Let $Ek_{\ast }=g_{ij}x_{,t}^{i}x_{,t}^{j}/2=g_{tt}/2$ be the kinetic energy of the ground state. Assuming that apedicity is due to the trajectories being inertial, we have (see § A.3)

(3.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\ast }^{b} & = & \displaystyle Ek_{\ast ,q},\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\ast }^{q} & = & \displaystyle -Ek_{\ast ,b},\end{eqnarray}$$

that is

(3.21) $$\begin{eqnarray}\unicode[STIX]{x1D74E}_{\ast }=\unicode[STIX]{x1D735}\times (Ek_{\ast }\unicode[STIX]{x1D735}t)+\unicode[STIX]{x1D714}_{\ast }^{t}\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}q,\end{eqnarray}$$

which, substituted in (3.1a ) (recall that $\boldsymbol{v}_{\ast }$ is parallel to $\unicode[STIX]{x1D735}t$ )

(3.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}(P_{\ast })=b_{\ast }\unicode[STIX]{x1D735}Z=\unicode[STIX]{x1D735}\left(\int ^{Z}b_{\ast }\,\text{d}Z\right), & & \displaystyle\end{eqnarray}$$

indicates that the pressure distribution in apedic ground states is hydrostatic.

The definitions of local available potential energy proposed in the past that rely on a Casimir (e.g. Holliday & McIntyre Reference Holliday and McIntyre1981), usually, (but not always, see, e.g. Codoban & Shepherd (Reference Codoban and Shepherd2003)) include only the potential and pressure term, combined in $\int ^{b}Zdb$ . This is not surprising, since the traditional available energy approach is to define the background potential energy as the energy of the restratified flow. In our formalism, it can be recovered by enforcing only conservation of buoyancy when we seek the ground state. Indeed, in this case, $\unicode[STIX]{x1D6F9}_{\ast }$ depends only on buoyancy, and (3.8a )–(3.8b ) show that $Z_{\ast }=Z_{\ast }(b)$ and $\boldsymbol{v}_{\ast }=0$ . Of course, this is a solution of the full problem if the state $\mathfrak{q}$ has zero potential vorticity throughout. If we want to enforce both buoyancy and potential vorticity conservation, the corresponding ground state in general must have kinetic energy, and thus the system will have a lower available energy. Superficially, it would seem that enforcing potential vorticity conservation on the ground state would necessarily lead to ground states with kinetic energy, thus lowering the amount of available energy, unless $\mathfrak{q}$ has zero potential vorticity. In fact, the set of states whose ground state is described by the Winters et al. ground state is wider, and they will be discussed in the next section, where we apply the theory to simple channel geometries, both inertial and rotating.

3.1.5 Boussinesq channel flows

For arbitrary mass distribution functions and geometries, (3.8a )–(3.8c ) constitute a formidable nonlinear optimization problem for the ground state. It is therefore of advantage to solve this optimization problem by other means, for instance, integrating numerically the Euler equations with a holonomic brake to find the ground state. Once the pair $b_{\ast },\boldsymbol{v}_{\ast }$ is known, the Casimir can be calculated from its Bernoulli head.

Nevertheless, analytic progress can be made if we are willing to sacrifice geometric complexity. We consider simple ‘open’ channel geometries, which can be geometrically described by Cartesian coordinates. For definiteness, let $0\leqslant Z\leqslant L_{z}$ be the vertical direction, $0\leqslant y\leqslant L_{y}$ the spanwise direction and $0\leqslant x\leqslant L_{x}$ the streamwise direction. The bottom, top and side walls are free-slip boundaries, while the flow is periodic in the streamwise direction. For simplicity, we limit the search of ground states to states that are $2.5$ -dimensional: these are states defined by $Z=Z(b,q)$ and $y=y(b,q)$ , and such that the velocity, which depends on two dimensions $(b,q)$ is normal to the $Z,y$ plane (the $1/2$ dimension).

3.1.6 Apedic ground states in inertial channels

In this section, we prove that 2.5-dimensional ground states (as defined above) in inertial channels are apedic. Further if, for any value of $b$ , the mass distribution function $V$ is such that $V_{,qb}$ is an even function of $q$ , the kinetic energy of the ground state is zero.

The method to construct an apedic ground state for simple flow geometries is a natural extension of the technique introduced by Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995) and consists of three steps:

1. (i) Determine the resorted height $Z_{\ast }(b)$ solving (3.8c ) in the limit $q\rightarrow q_{min}$ . This coincides with the restratification of the fluid.

2. (ii) Compute the resorted distribution of potential vorticity along the spanwise direction $y_{\ast }(q)$ .

3. (iii) Finally, determine the kinetic energy $Ek_{\ast }$ of the ground state.

Later, we will modify this recipe to calculate the ground state and associated Casimir of rotating flows characterized by a low Rossby number ground state, with the Rossby number being defined as $Ro=O(|(\unicode[STIX]{x1D735}\times \boldsymbol{v})\boldsymbol{\cdot }\unicode[STIX]{x1D735}b|/\boldsymbol{f}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b)$ , in other words, flows in which the leading-order contribution to potential vorticity is provided by the frame vorticity.

It is straightforward to verify that 2.5-dimensional ground states in inertial channels are apedic. In geometric coordinates, the components of the velocity vector are $[0,0,v_{\ast }(b(Z,y),q(Z,y))]$ , thus $dx=v(b,q)dt$ and $dy=dZ=0$ , which implies that the velocity is uniform along geodesics. Rather than working in ‘natural’ coordinates, we work with ‘hybrid’ coordinates $(b,q,x)$ . Being apedic $Z_{\ast }=Z_{\ast }(b)$ , (here, as we have done before, quantities used as coordinates will not carry the $_{\ast }$ subscript) which can be calculated from the limit $q\rightarrow q_{min}$ of (3.8c )

(3.23) $$\begin{eqnarray}\displaystyle \int \unicode[STIX]{x1D703}(b_{\ast }(Z)-s)dZdydx & = & \displaystyle -L_{y}L_{x}\int _{b_{min}}^{b_{Max}}\unicode[STIX]{x1D703}(b-s)\frac{dZ_{\ast }}{db}db\nonumber\\ \displaystyle & = & \displaystyle V(q_{min},s),\end{eqnarray}$$

whose solution is

(3.24) $$\begin{eqnarray}Z_{\ast }(b)=L_{z}\left(1-\frac{V(q_{min},b)}{V(q_{min},b_{min})}\right).\end{eqnarray}$$

Its functional inverse $b_{\ast }(Z)$ is the buoyancy distribution of the ground state. From this, we can calculate the hydrostatic pressure of the ground state. Also, for later convenience, we define $N_{\ast }^{-2}\equiv Z_{\ast ,b}(b)$ .

Next, consider the volume element

(3.25) $$\begin{eqnarray}\mathfrak{V}=dZdydx=\left|\frac{\unicode[STIX]{x2202}(Z,y)}{\unicode[STIX]{x2202}(b,q)}\right|dbdqdx.\end{eqnarray}$$

Using again (3.8c )

(3.26) $$\begin{eqnarray}\left|\frac{\unicode[STIX]{x2202}(Z,y)}{\unicode[STIX]{x2202}(b,q)}\right|=\frac{V_{,qb}}{L_{x}},\end{eqnarray}$$

we obtain $y=y(q,b)$ solving

(3.27) $$\begin{eqnarray}\frac{V_{,qb}}{L_{x}}=\left|\frac{\unicode[STIX]{x2202}(Z,y)}{\unicode[STIX]{x2202}(b,q)}\right|=\left|\frac{V_{,b}(q_{min},b)}{L_{x}L_{y}}\frac{\unicode[STIX]{x2202}y}{\unicode[STIX]{x2202}q}\right|.\end{eqnarray}$$

The solution that spans the domain is

(3.28) $$\begin{eqnarray}y_{\ast }=L_{y}\frac{V_{,b}(q_{min},b)-V_{,b}(q,b)}{V_{,b}(q_{min},b)}.\end{eqnarray}$$

Note that in this apedic ground state, isopycnals are flat, whereas surfaces of constant potential vorticity are not and the relationship between natural and geometric coordinates is invariant under the dilation $V\rightarrow e^{s}V$ .

At this point, all we have left is to calculate the kinetic energy of the ground state. By construction, we must have $\boldsymbol{v}=F\boldsymbol{\unicode[STIX]{x2202}}/\boldsymbol{\unicode[STIX]{x2202}}x$ and $\unicode[STIX]{x1D735}\times \boldsymbol{v}=F_{,b}\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}x+F_{,q}\unicode[STIX]{x1D735}q\times \unicode[STIX]{x1D735}x$ . Using the definition of potential vorticity (A 6) and (3.26) we may try as a first ansatz a solution to

(3.29) $$\begin{eqnarray}F_{,q}=\frac{qV_{,qb}}{L_{x}},\end{eqnarray}$$

with the constant of integration set along planes of constant buoyancy so that the total momentum along the $x$ direction is conserved. In the special case

(3.30) $$\begin{eqnarray}V_{,qb}=\unicode[STIX]{x1D6FF}(q)\tilde{V}_{,b},\end{eqnarray}$$

which corresponds to a leaf characterized by a uniformly zero distribution of potential vorticity, $F$ and the kinetic energy of the ground state are zero. The Casimir then reduces (up to a factor $qf(b)$ which does not concern us) to

(3.31) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\ast }=\int _{b_{min}}^{b}Z(s)ds,\end{eqnarray}$$

and the ground state to a quiescent fluid restratified according to (3.24). Two-dimensional flows contained in a vertical plane have zero potential vorticity, and thus satisfy (3.30). In this case, as mentioned earlier, we recover the same definition of available energy used by Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995).

Consider a three-dimensional flow at time $t$ which evolved from an initially two-dimensional flow contained in the vertical plane seeded with random infinitesimal perturbations along the third dimension. At $t=0$ the mass distribution function is very well approximated by $V_{,q}=\widetilde{V}_{,b}\unicode[STIX]{x1D6FF}(q)$ . As the flow develops instabilities, the delta-like distribution in the mass distribution function develops tails. Let us assume that in the absence of any mechanism that breaks the symmetry, the potential vorticity that the diabatic processes generate ex nihilo is equally distributed around the origin. In other words, we assume that whatever mechanism generates potential vorticity, it is not biased against either positive or negative potential vorticity. While this hypothesis seems reasonable, and it is supported by analysis of numerical experiments of different flow configurations, where a stretched exponential appears to give a good fit (figure 2), we cannot prove it. Thus, it must be considered as a conjecture. Here, we explore its consequences. Without loss of generality, we can assume that at time $t$

(3.32) $$\begin{eqnarray}V_{,qb}\simeq \frac{L_{y}L_{x}N_{\ast }^{-2}(b)}{\unicode[STIX]{x1D70E}(b)}w\left(\frac{q}{\unicode[STIX]{x1D70E}(b)}\right),\end{eqnarray}$$

where $w(x)$ is a suitable even function. As long as it decays sufficiently rapidly, we can replace $p_{min}$ with $-\infty$ and $p_{Max}$ with $+\infty$ . $\unicode[STIX]{x1D70E}(b)$ measures the spread (i.e. the variance if the distribution was Gaussian) in potential vorticity near a given buoyancy level. Let $W(x)=\int _{-\infty }^{x}tw(t)dt$ and $\overline{W}_{n}=\int _{-\infty }^{\infty }W^{n}(t)dt$ . Integration of (3.29) yields for the velocity the following expression (setting the integration constant to zero for the moment)

(3.33) $$\begin{eqnarray}F(q,b)=\int _{-\infty }^{q}q\frac{V_{,qb}}{L_{x}}\,\text{d}q=L_{y}W\left(\frac{q}{\unicode[STIX]{x1D70E}(b)}\right)N_{\ast }^{-2}(b)\unicode[STIX]{x1D70E}(b),\end{eqnarray}$$

from which the kinetic energy per unit volume of the ground state can be easily calculated

(3.34) $$\begin{eqnarray}\displaystyle Ek_{\ast } & = & \displaystyle \int _{-\infty }^{+\infty }\left(\int _{b_{min}}^{b_{Max}}\frac{F^{2}}{2}V_{,bq}\,\text{d}b\right)\text{d}q\left/\int _{-\infty }^{+\infty }\left(\int _{b_{min}}^{b_{Max}}V_{,bq}\,\text{d}b\right)\,\text{d}q\right.\nonumber\\ \displaystyle & = & \displaystyle \frac{L_{y}^{2}}{2}\left(\frac{\overline{W}_{2}}{L_{z}}\int _{0}^{L_{z}}N_{\ast }^{-2}(b_{\ast }(Z)\unicode[STIX]{x1D70E}(b_{\ast }(Z)))^{2}dZ\right),\end{eqnarray}$$

the inescapable consequence of which is that the kinetic energy per unit volume of this point on the leaf grows with the width $L_{y}$ of the channel squared. While we have derived this result assuming that the mass distribution function has the particular form given by (3.32), it fundamentally rests on the fact that under the dilation $V\rightarrow e^{s}V$ we have $F\rightarrow e^{s}F$ and $Ek\rightarrow e^{2s}Ek$ . Therefore, what we have derived cannot be the ground state that we are seeking. To understand what went wrong, and to find a cure for it, in figure 3 we plot $F$ calculated using (3.29) where the mass distribution function was obtained from a direct numerical simulation of stratified Couette flow (Scotti Reference Scotti2015) (indicated with the label (1)).

Figure 2.

Value of $\log _{10}(V_{,qb})$ in stratified shear flows. (a) A stratified Couette flow: the inset shows $V_{,qb}(q,0.5)$ (blue line) superimposed to a stretched exponential $\exp -|q/0.3|^{0.6}$ (orange line). (b) A time-evolving shear layer, after the initial break-up of the Kelvin–Helmholtz instabilities. In this case too, a stretched exponential with a similar exponent fits the profile ( $\exp -|q/4.5|^{0.6}$ ). Note how, in both cases, profiles of $V_{qp}$ along lines of constant buoyancy are symmetric around the origin.

Figure 3.

(a) Value of $F(b_{0},q)$ versus $q$ where $F$ is obtained solving (3.29) in a time-evolving stratified shear layer, where $b_{0}=(b_{min}+b_{Max})/2$ is the midplane buoyancy. (b) Value of $F(b_{0},q)$ versus $y(b_{0},q)$ obtained using one (1), two (2) or four (4) charts to cover the domain.

In this set-up, the Lagrangian parcels with high negative potential vorticity are shunted on one side of the domain, most of the domain is filled by the (many) particles with little potential vorticity, while the parcels with large positive vorticity are pushed to the other side of the physical domain. We can see immediately two problems with this approach: first, having separated the large negative from the large positive potential vorticity particles, we have created a large, sustained shear that forces the velocity to reach a large value in the centre of the domain. Second, while this set-up does not violate a free-slip boundary condition at the side walls, it cannot accommodate periodic boundary conditions.

Indeed, no matter what the details of the mass distribution function are, a single $(b,q,x)\overset{\unicode[STIX]{x1D709}}{\rightarrow }(z,y,x)$ chart cannot cover the manifold if the latter is periodic in the $y$ direction. Let us explore then what happens if we allow two charts to cover the manifold. The first chart maps the $(b,q,x)$ space to the half of the channel such that $0\leqslant y\leqslant L_{y}/2$ , while the second covers the other half. Further, since the mass distribution function is an extensive quantity by definition, we require that each half of the physical domain contributes exactly one half to the total mass distribution function. In each half, we proceed exactly as we did before, and since $F(q,b)\overset{q\rightarrow \pm \infty }{\longrightarrow }0$ it is possible to stitch together the velocity at the edge of every subdomain to obtain the profile shown on figure 3(b) labelled (2). Because in each half the mass distribution function is half of the original one, the energy per unit volume in each half (an intensive quantity) is now 25 % of the energy per unit volume obtained using only one chart. As a bonus, the velocity field is now periodic. Again, what makes this possible is the fact that $V_{,qb}$ is an even function of $q$ .

Of course, there is no reason to stop at two charts. The procedure can be repeated with $4,8,\ldots$ , charts, each covering $1/4,1/8,\ldots$ of the channel, and each time obtaining a field with a kinetic energy per unit volume $1/16,1/64,\ldots$ of the previous (figure 3 shows the profiles obtained using one, two and four charts). Thus, we generate a sequence of velocity fields in which potential vorticity is interleaved at finer and finer scales, while at the same time reducing the overall kinetic energy by a factor $4$ with each iteration. Barring the existence of a cutoff scale, this shows how to construct a sequence of states which in the limit approaches a ground state with zero kinetic energy. These ground states are also obviously global minima on their leaf, since the only way to further lower the total energy would break the conservation of volume constraint.

Naturally, it is possible to consider initial conditions in which the symmetry in $q$ is broken ab initio, even in the absence of rotation. For example, consider filling a channel, which, as before, has dimensions $(L_{z},L_{y},L_{x})$ with a fluid whose buoyancy has a uniform gradient along the spanwise direction $\unicode[STIX]{x1D735}b=N_{0}^{2}\boldsymbol{\unicode[STIX]{x2202}}/\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{y}$ , and flowing (in a way consistent with Boussinesq dynamics) such that $\boldsymbol{v}=\unicode[STIX]{x0394}v_{0}(Z/L_{z})\boldsymbol{\unicode[STIX]{x2202}}/\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{x}$ , so that $\unicode[STIX]{x1D6FA}=-\unicode[STIX]{x0394}v_{0}/L_{z}\unicode[STIX]{x1D735}x\times \unicode[STIX]{x1D735}Z$ . Thus, the potential vorticity is uniform and equal to $-N_{0}^{2}\unicode[STIX]{x0394}v_{0}/L_{z}$ . We can handle this singular limit by ‘smearing’ the potential vorticity distribution around its constant value by an amount $O(\unicode[STIX]{x1D716})$ , carry out the calculation and finally taking the $\unicode[STIX]{x1D716}\rightarrow 0$ limit. In this case, the buoyancy of the ground state is such that $\unicode[STIX]{x1D735}b_{\ast }=N_{0}^{2}(L_{y}/L_{z})\boldsymbol{\unicode[STIX]{x2202}}/\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{Z}$ and $\boldsymbol{v}_{\ast }=\unicode[STIX]{x0394}v_{0}(y/L_{y})\boldsymbol{\unicode[STIX]{x2202}}/\boldsymbol{\unicode[STIX]{x2202}}\boldsymbol{x}$ . Interestingly, the ground state contains as much kinetic energy as the initial state, and thus the available energy per unit volume is simply the difference between the potential energy of the initial and ground state $(\unicode[STIX]{x0394}bL_{z}/12)$ . (It is important to realize that this ground state, like any other ground state as defined in our theory, is stable relative to perturbations that belong to the same leaf of the ground state. The ground state considered in the last example belongs to a class of flows whose stability has been studied by Deloncle, Chomaz & Billant (Reference Deloncle, Chomaz and Billant2007). The potential vorticity of the unstable modes is such that the unstable modes can be excited in the rest state only via the action of non-holonomic forces (forces that alter the potential vorticity distribution). This example confirms that non-holonomic forces can increase the available energy, which is used by the unstable modes to grow. In other words, the introduction of unstable modes moves the state to a different leaf, on which the ground state has a lower energy.) We must however emphasize that when the mass distribution function symmetry in $q$ is broken, the global apedic minimum on the leaf could be three-dimensional.

3.1.7 Rotating channels in the low Rossby number limit: geostrophic ground states

Another class of ground states that is amenable to analytic treatment are the ground states of flows contained in rotating channels where the frame vorticity is much larger than the relative vorticity. In other words, we look for ground states characterized by a small value of the Rossby number (as defined earlier). We also assume that the Burger number $Bu=(N_{\ast }H/fL)^{2}$ is kept finite, and as such we drop it from the asymptotics. We call low- $Ro$ ground states geostrophic ground states. We work with natural $(b,q,t)$ coordinates assuming that the ground state is 2.5-dimensional.

Under these assumptions, the main source of non-inertiality along trajectories in the ground state is due to the frame vorticity $\unicode[STIX]{x1D6EF}=f\unicode[STIX]{x1D735}y\times \unicode[STIX]{x1D735}x$ , which we write as

(3.35) $$\begin{eqnarray}\unicode[STIX]{x1D6EF}=\unicode[STIX]{x1D6EF}^{t}\unicode[STIX]{x1D735}b\times \unicode[STIX]{x1D735}q+\unicode[STIX]{x1D6EF}^{b}\unicode[STIX]{x1D735}q\times \unicode[STIX]{x1D735}t+\unicode[STIX]{x1D6EF}^{q}\unicode[STIX]{x1D735}t\times \unicode[STIX]{x1D735}b,\end{eqnarray}$$

with

(3.36a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}^{t} & = & \displaystyle f(y_{,b}x_{,q}-y_{,q}x_{,b}),\end{eqnarray}$$

(3.36b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}^{b} & = & \displaystyle fy_{,q}x_{,t},\end{eqnarray}$$

(3.36c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}^{p} & = & \displaystyle -fy_{,b}x_{,t},\end{eqnarray}$$

$f$

$y_{,t}=0$

Under the 2.5-dimensional assumption and in the limit of vanishing Rossby numbers, $x_{,t}=\sqrt{2Ek}$ , and the steady state version of (3.1a ) becomes (Pedlosky Reference Pedlosky1986, p. 45)

(3.37) $$\begin{eqnarray}\unicode[STIX]{x1D735}\overline{\unicode[STIX]{x1D6F9}}=f\sqrt{2Ek}\,\unicode[STIX]{x1D735}y-Z\unicode[STIX]{x1D735}b+O(Ro),\end{eqnarray}$$

where we made use of (3.36b )–(3.36c ). Thus, to $O(Ro)$ the integrability condition

(3.38) $$\begin{eqnarray}f\sqrt{2Ek}_{,Z}\unicode[STIX]{x1D735}Z\times \unicode[STIX]{x1D735}y=b_{,y}\unicode[STIX]{x1D735}Z\times \unicode[STIX]{x1D735}y,\end{eqnarray}$$

shows that the ground state is in the so-called thermal wind balance. As we did for flows in inertial channels, we endeavour to construct the ground state explicitly by seeking first the coordinate maps $Z_{\ast }=Z(b,q)$ and $y_{\ast }=y(b,q)$ . The solution can be expressed as (for details see § A.4)

(3.39a ) $$\begin{eqnarray}\displaystyle Z_{\ast } & = & \displaystyle \mathop{\sum }_{n=1}^{\infty }Z_{n}(q)(b-b_{min})^{n},\end{eqnarray}$$

(3.39b ) $$\begin{eqnarray}\displaystyle y_{\ast } & = & \displaystyle \mathop{\sum }_{n=0}^{\infty }y_{n}(q)(b-b_{min})^{n},\end{eqnarray}$$

(3.39c ) $$\begin{eqnarray}\displaystyle \sqrt{2Ek_{\ast }} & = & \displaystyle \mathop{\sum }_{n=0}^{\infty }v_{l}(q)(b-b_{min})^{n}.\end{eqnarray}$$

$V_{,bq}$

$b-b_{min}$

(3.40) $$\begin{eqnarray}V_{,qb}=L_{x}\mathop{\sum }_{n=0}^{\infty }F_{n}(q)(b-b_{min})^{n},\end{eqnarray}$$

the coefficients of (3.39a )–(3.39c ) are given by

(3.41a ) $$\begin{eqnarray}y_{n}=f^{-1}\int _{q_{min}}^{q}sF_{n}(s)ds,\end{eqnarray}$$

(3.41b,c ) $$\begin{eqnarray}Z_{1}=\frac{f}{q},\quad Z_{l+1}=\frac{\displaystyle lZ_{1,q}y_{l}-\mathop{\sum }_{n=2}^{l}(nZ_{n}y_{l+1-n,q}-(l+1-n)Z_{n,q}y_{l+1-n})}{(l+1)y_{0,q}},\end{eqnarray}$$

(3.41d ) $$\begin{eqnarray}v_{0}=v_{1}=0,\quad v_{l+1}=\frac{\displaystyle -Z_{l,q}+f\mathop{\sum }_{n=1}^{l}(nv_{l+1-n,q}y_{n}-(l+1-n)v_{l+1-n}y_{n,q})}{f(l+1)y_{0,q}}.\end{eqnarray}$$

With the complete solution in hand, it is now time to revisit the small- $Ro$ assumption made at the beginning. The goal is to characterize the mass distribution functions that lead to small- $Ro$ ground states. It is straightforward to calculate the relative vorticity of the ground state (It follows from the fact that the infinitesimal contribution to the circulation is $\boldsymbol{v}\boldsymbol{\cdot }d\boldsymbol{l}=\sqrt{2Ek}dx^{3}=2Ekdt$ , from which we can calculate the curl.)

(3.42) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\ast }=-(2Ek)_{,b}\unicode[STIX]{x1D735}t\times \unicode[STIX]{x1D735}b+(2Ek)_{,q}\unicode[STIX]{x1D735}q\times \unicode[STIX]{x1D735}t.\end{eqnarray}$$

Let us start from the special case that leads to (A 34). The most stringent condition is

(3.43) $$\begin{eqnarray}Ro=O\left(\frac{(\sqrt{2Ek_{\ast }})_{,Z}}{f}\right)=\frac{L_{z}}{f^{2}qF_{0}(q)}\ll 1.\end{eqnarray}$$

We can estimate

(3.44) $$\begin{eqnarray}O(qF_{0})=\overline{q}\frac{L_{y}L_{z}}{\unicode[STIX]{x0394}q\unicode[STIX]{x0394}b}=\overline{q}\frac{L_{y}}{\unicode[STIX]{x0394}pN^{2}},\end{eqnarray}$$

where $N^{2}=\unicode[STIX]{x0394}b/L_{z}$ . Here $\unicode[STIX]{x0394}q$ is a measure of the spread of potential vorticity, e.g. the root-mean-square (r.m.s.) of the fluctuations, $\unicode[STIX]{x0394}b$ is a measure of the spread of buoyancy and $\overline{q}$ the mean potential vorticity (which, when needed, can be estimated as $fN^{2}$ ). Thus, the ground state satisfies the low- $Ro$ condition if the following condition on the spread of potential vorticity

(3.45) $$\begin{eqnarray}\frac{\unicode[STIX]{x0394}q}{\overline{q}}\ll \frac{L_{y}L_{z}}{L_{R}^{2}},\end{eqnarray}$$

where $L_{R}\equiv L_{z}N/f$ is the internal Rossby radius of deformation, is satisfied.

Let us now consider the more general case of a mass distribution function which is not linear in the buoyancy, but which still satisfies (3.45). We have

(3.46) $$\begin{eqnarray}O(qF_{n})=Ro^{-1}\frac{L_{R}^{2}}{\unicode[STIX]{x0394}b}(\unicode[STIX]{x0394}b)^{-n},\end{eqnarray}$$

whence

(3.47a ) $$\begin{eqnarray}\displaystyle O(fy_{n}) & = & \displaystyle O(\unicode[STIX]{x0394}q\,qF_{n})=fL_{y}(\unicode[STIX]{x0394}b)^{-n},\end{eqnarray}$$

(3.47b ) $$\begin{eqnarray}\displaystyle O(fy_{n,q}) & = & \displaystyle O(qF_{n})=Ro^{-1}\frac{L_{R}^{2}}{\unicode[STIX]{x0394}b}(\unicode[STIX]{x0394}b)^{-n}.\end{eqnarray}$$

(3.48) $$\begin{eqnarray}O(Z_{2}(b-b_{min})^{2})=Ro\frac{L_{y}L_{z}}{L_{R}^{2}}L_{z}.\end{eqnarray}$$

Given the nature of the recurrence relation (A 29), it follows that all the other terms in the series for $Z$ are $O(Ro)$ , and thus

(3.49) $$\begin{eqnarray}Z_{\ast }=\frac{f}{q}(b-b_{min})+O(Ro).\end{eqnarray}$$

Along essentially the same lines we have that $\sqrt{2Ek_{\ast }}=O(Ro)$ , and thus to $O(Ro)$ the pressure in the ground state is hydrostatic. Hence

(3.50) $$\begin{eqnarray}\overline{\unicode[STIX]{x1D6F9}}=-\frac{f}{2}\frac{(b-b_{min})^{2}}{q}+O(Ro),\end{eqnarray}$$

and

(3.51) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\ast }=\frac{f}{4}\frac{(b-b_{min})^{2}}{q}+O(Ro),\end{eqnarray}$$

which shows that the Casimir of geostrophic leaves, defined as leaves characterized by a geostrophic ground state (i.e. a low- $Ro$ state) does not depend, to leading order, on the details of the mass distribution function that defines the leaf to which the ground state belongs. A field on a geostrophic leaf does not need to be in geostrophic equilibrium, but the ageostrophic component must be such that the overall spread in potential vorticity is small when looked over the entire field. As long as the diabatic dynamics does not cause a significant spread in potential vorticity, the flow remains on a geostrophic leaf and the diagnostic energy ${\mathcal{E}}$ is to leading order independent of the mass distribution function. Thus, the local diagnostic energy has, to $O(Ro)$ a universal character.