On the triviality of potential vorticity conservation in geophysical fluid dynamics

The equations of motion of physical systems may always be written in a covariant form because the laws governing their evolution do not depend on the choice of coordinates. To render covariance manifest, governing equations of classical fluid mechanics may be written in terms of 4-dimensional tensor components suitable for any curvilinear, non-inertial coordinate system in which time intervals are absolute (Charron et al 2014). This ensures that the underlying theory is relativistic (Charron et al 2015), obviously not in the traditional sense of describing fluids with velocities comparable to that of light but in the literal sense of 'obeying a principle of relativity'—in this case, the principle of Newtonian relativity.

In such formulations, conservation laws take the same form in all coordinate systems:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {x}^{\mu }}(\sqrt{g}{A}^{\mu })=\displaystyle \frac{\partial }{\partial t}(\sqrt{g}{A}^{0})+\displaystyle \frac{\partial }{\partial {x}^{i}}(\sqrt{g}{A}^{i})=0,\end{eqnarray} \tag{ 1.1 }$$

where ${A}^{\mu }$ is a conserved 4-current with A⁰ a scalar charge density and Aⁱ its flux density. The quantity g is the determinant of the covariant metric tensor g_μν of the coordinate system. Repeated Greek indices run from 0 to 3, and repeated Latin indices from 1 to 3. The coordinate x⁰ = t is the time, and the xⁱ's are any curvilinear spatial coordinates. An invariant spatial volume element is written ${{dx}}^{1}{{dx}}^{2}{{dx}}^{3}\sqrt{g}\equiv {d}^{3}x\sqrt{g}$.

The equations of motion of an inviscid classical fluid in an external gravitational potential Φ under adiabatic conditions may be expressed as

$$\begin{eqnarray}&&{{\rm{\Lambda }}}^{\mu }=0,\end{eqnarray} \tag{ 1.2 }$$

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{(\beta )}=0,\end{eqnarray} \tag{ 1.3 }$$

where the 4-vector ${{\rm{\Lambda }}}^{\mu }$ and the scalar Λ_(β) are

$$\begin{eqnarray}&&{{\rm{\Lambda }}}^{\mu }\equiv {(\rho {u}^{\mu }{u}^{\nu }+{h}^{\mu \nu }p)}_{:\nu }+\rho {h}^{\mu \nu }{{\rm{\Phi }}}_{,\nu },\end{eqnarray} \tag{ 1.4 }$$

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{(\beta )}\equiv -\rho {u}^{\mu }{s}_{,\mu }\end{eqnarray} \tag{ 1.5 }$$

(the parentheses around the symbol β indicate that it is not a space-time index—see Zadra and Charron (2015) where these equations are derived from a least action principle). The symbol ρ represents the fluid density, ${u}^{\mu }\equiv {{dx}}^{\mu }/{dt}$ the 4-velocity field with u⁰ = 1, ${h}^{\mu \nu }\equiv {g}^{\mu \nu }-{g}^{0\mu }{g}^{0\nu }/{g}^{00}$ a symmetric contravariant tensor for purely spatial distances with h^0μ = h^μ0 = 0, p the pressure, and s the fluid specific entropy. The absolute nature of time intervals in Newtonian mechanics imposes a constraint on the space-time metric tensor: the contravariant component g⁰⁰ must be a non-zero constant (taken here as unity). A comma followed by an index indicates an ordinary partial derivative: ${(\cdot )}_{,\mu }\equiv \partial (\cdot )/\partial {x}^{\mu }$. A material derivative is written $d{(\cdot )/{dt}\equiv {u}^{\mu }(\cdot )}_{,\mu }$. A colon followed by an index indicates a covariant derivative. For instance, given a 4-vector ${A}^{\mu }$, its covariant derivative ${{A}^{\mu }}_{:\nu }$ is

$$\begin{eqnarray}&&{{A}^{\mu }}_{:\nu }\equiv {{A}^{\mu }}_{,\nu }+{{\rm{\Gamma }}}_{\alpha \nu }^{\mu }{A}^{\alpha }.\end{eqnarray} \tag{ 1.6 }$$

The terms

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\alpha \nu }^{\mu }\equiv \displaystyle \frac{1}{2}{g}^{\mu \beta }({g}_{\alpha \beta ,\nu }+{g}_{\beta \nu ,\alpha }-{g}_{\nu \alpha ,\beta })\end{eqnarray} \tag{ 1.7 }$$

are Christoffel symbols of the second kind. Because time intervals are absolute in Newtonian mechanics, ${{\rm{\Gamma }}}_{\alpha \nu }^{0}=0$. Moreover, the absolute nature of time intervals implies that a contravariant tensor of rank n becomes a tensor of rank n − m (0 ≤ m ≤ n) when m of its indices are set to zero. For instance, if B^μν is a contravariant second-rank tensor, B^0ν is a contravariant first-rank tensor (a contravariant 4-vector), and B⁰⁰ is a zeroth-rank tensor (a scalar). Notice that, in general, this rule does not apply to covariant tensors.

The scalar equation of motion ${{\rm{\Lambda }}}^{0}={(\rho {u}^{\nu })}_{:\nu }=0$ represents mass continuity, Λⁱ = 0 the three momentum equations, and Λ_(β) = 0 the material conservation of specific entropy. Because $\sqrt{g}\,{{\rm{\Gamma }}}_{\nu \mu }^{\mu }={(\sqrt{g})}_{,\nu }$, the left-hand side of (1.1) may be rewritten ${(\sqrt{g}{A}^{\mu })}_{,\mu }=\sqrt{g}{{A}^{\mu }}_{:\mu }$ and (1.1) is equivalent to ${{A}^{\mu }}_{:\mu }=0$ since $\sqrt{g}\ne 0$. The formalism used in this paper is described in more details in Charron et al (2014).

A relation is said to hold on-shell when the equations of motion (1.2) and (1.3) are used to obtain the relation. It is said to hold off-shell when Λ^μ and Λ_(β) are not necessarily assumed to vanish.

Previous studies by Newcomb (1967), Bretherton (1970), Ripa (1981), Salmon (1982, 1988, 1998, 2013), Müller (1995), Padhye and Morrison (1996) and others associate the conservation of potential vorticity with a particle-relabeling symmetry transformation of the equations of motion—some of them, for instance Newcomb (1967), Müller (1995), Padhye and Morrison (1996), invoking Noether's first theorem. In the following, it is argued that this association cannot be made via Noether's first theorem because potential vorticity conservation (Ertel's theorem) is an algebraic identity that takes the form of a trivial law of the second kind, following the terminology of Olver (1993). Such algebraic identities are obtained independently of the equations of motion (i.e. off-shell) while conservation laws associated with a symmetry via Noether's first theorem exist on-shell only. The statement that potential vorticity conservation is a trivial law may perhaps surprise fluid dynamicists. Here, 'trivial' obviously does not mean dynamically uninteresting or useless. As will be seen below, it means in mathematical terms that the conserved current associated with potential vorticity is given by the divergence of an antisymmetric tensor.

In section 2, a specific sub-class of admissible coordinate systems called comoving coordinates is described. Comoving coordinates are characterized by a mesh that follows exactly the moving fluid elements. This sub-class of coordinates is introduced because it is used by authors interested in the particle-relabeling transformation. In section 3, the use of Noether's first theorem under particle-relabeling is revisited. A striking feature of particle-relabeling is that its associated constraints are unrelated to the actual dynamics obtained from minimizing the action functional. The triviality of potential vorticity conservation in arbitrary coordinates as an algebraic identity is then demonstrated in section 4. This establishes that Noether's first theorem is irrelevant to associate potential vorticity conservation with a symmetry of the equations of motion. Starting from arbitrary coordinates, it is straightforward to show the triviality of potential vorticity conservation in comoving coordinates. This is also done in section 4. Conclusions are drawn in section 5.

Some trivial conservation laws of the second kind may be associated with infinite-dimensional symmetries of the equations of motion via Noether's second theorem. These infinite-dimensional symmetries arise in under-determined dynamical systems. A well-known example of such laws is the conservation of electric charge associated with the U_em(1) group of local gauge symmetry of an electromagnetic field coupled to a fermionic field. In a follow-up paper, the applicability of Noether's second theorem to fluid dynamics described by Clebsch potentials will be discussed.

The tensor formalism introduced in the preceding section applies to any coordinate system admissible in classical fluid mechanics. In this section, a particular sub-class of reference frames called comoving coordinates is described. Such coordinate systems are 'attached' to the fluid elements and move with the fluid. Therefore, the three spatial coordinates associated with a given fluid element, often called its labels, do not change as time evolves. One may start from the tensor formalism in arbitrary coordinates described in section 1 and find specific relations that apply in comoving coordinates only. It is well known that such coordinates are useful to express a Lagrangian for fluid dynamics in a traditional form, i.e. kinetic minus potential energy (Herivel 1955).

Comoving coordinates will be referred to as ${\hat{x}}^{\mu }$, with ${\hat{x}}^{0}=\tau =t$ the time and ${\hat{x}}^{i}$ the labels. In the following, comoving coordinates and quantities expressed in comoving coordinates will always be indicated with hatted variables, except scalars—say f, for which $\hat{f}=f$ at all space-time points. Because the comoving coordinate mesh follows the fluid elements, the 4-velocity ${\hat{u}}^{\mu }\equiv d{\hat{x}}^{\mu }/{dt}=(1,0,0,0)$. The mass continuity equation and the material conservation of specific entropy, respectively (1.2) with μ = 0 and (1.3), take the form

$$\begin{eqnarray}&&\sqrt{\hat{g}}{{\rm{\Lambda }}}^{0}=\displaystyle \frac{\partial }{\partial {\hat{x}}^{0}}(\sqrt{\hat{g}}\rho {\hat{u}}^{0})+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{u}}^{i})=\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho )=0,\end{eqnarray} \tag{ 2.1 }$$

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{(\beta )}=-\rho \left({\hat{u}}^{0}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{0}}+{\hat{u}}^{i}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}\right)=-\rho \displaystyle \frac{\partial s}{\partial \tau }=0,\end{eqnarray} \tag{ 2.2 }$$

where $\hat{g}$ is the determinant of the covariant metric tensor ${\hat{g}}_{\mu \nu }$ in coordinates ${\hat{x}}^{\mu }$. The product $\sqrt{\hat{g}}\rho$ as well as specific entropy s are therefore time-independent on-shell in comoving coordinates. It may be shown from (1.4) that one form of the on-shell equations governing the covariant 4-velocity is written

$$\begin{eqnarray}&&\displaystyle \frac{{\hat{{\rm{\Lambda }}}}_{j}-{\hat{u}}_{j}{{\rm{\Lambda }}}^{0}}{\rho }=\displaystyle \frac{\partial {\hat{u}}_{j}}{\partial \tau }+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial p}{\partial {\hat{x}}^{j}}+\displaystyle \frac{\partial }{\partial {\hat{x}}^{j}}\left({\rm{\Phi }}-\displaystyle \frac{1}{2}{\hat{g}}_{00}\right)=0,\end{eqnarray} \tag{ 2.3 }$$

where

$$\begin{eqnarray}&&\displaystyle \frac{{\hat{{\rm{\Lambda }}}}_{j}-{\hat{u}}_{j}{{\rm{\Lambda }}}^{0}}{\rho }=\displaystyle \frac{\partial {x}^{\nu }}{\partial {\hat{x}}^{j}}{g}_{\mu \nu }\left(\displaystyle \frac{{{\rm{\Lambda }}}^{\mu }-{u}^{\mu }{{\rm{\Lambda }}}^{0}}{\rho }\right)=\displaystyle \frac{\partial {x}^{i}}{\partial {\hat{x}}^{j}}\left(\displaystyle \frac{{{\rm{\Lambda }}}_{i}-{u}_{i}{{\rm{\Lambda }}}^{0}}{\rho }\right)\end{eqnarray} \tag{ 2.4 }$$

with

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Lambda }}}_{i}-{u}_{i}{{\rm{\Lambda }}}^{0}}{\rho }=\displaystyle \frac{{{du}}_{i}}{{dt}}-\displaystyle \frac{1}{2}{g}_{\mu \nu ,i}{u}^{\mu }{u}^{\nu }+\displaystyle \frac{1}{\rho }{p}_{,i}+{{\rm{\Phi }}}_{,i}.\end{eqnarray} \tag{ 2.5 }$$

In a particle-like formulation of fluid dynamics, potential vorticity conservation has been associated with the particle-relabeling transformation by Newcomb (1967), Bretherton (1970), Ripa (1981), Salmon (1982, 1988, 1998, 2013), Müller (1995), Padhye and Morrison (1996) and others. Noether's first theorem is sometimes invoked to justify the association between the particle-relabeling transformation and potential vorticity conservation. It is shown below that Noether's first theorem is inadequate to establish this association.

In this section, a demonstration of Noether's first theorem under a particle-relabeling transformation is presented. Special attention is given to one crucial step in the demonstration; a step where identically vanishing terms must be identified or else one will be misled to unjustified conclusions.

In a particle-like formulation, the fluid elements may be labeled with any admissible curvilinear coordinate system $({\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})$ at a given time ${\hat{x}}^{0}=\tau$. The mass of a given fluid element dm is provided by $d{\hat{x}}^{1}d{\hat{x}}^{2}d{\hat{x}}^{3}\sqrt{\hat{g}}\rho ={d}^{3}\hat{x}\sqrt{\hat{g}}\rho$. Imposing that dm is time-independent ensures the a priori conservation of total mass and is equivalent to $\partial (\sqrt{\hat{g}}\rho )/\partial \tau =0$. In addition, the material conservation of specific entropy must also be assumed a priori, therefore $s\equiv s({\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})$ and ∂ s/∂ τ = 0. In other words, in a particle-like formulation of Hamilton's least action principle it is always assumed that Λ⁰ and Λ_(β) vanish a priori¹ . Therefore, in this section the expression 'off-shell' concerns the momentum equations only.

Consider Noether's first theorem for the particle-like formulation in comoving coordinates. The action functional is written

$$\begin{eqnarray}&&{ \mathcal S }[{x}^{1},{x}^{2},{x}^{3}]=\int d\tau \,{d}^{3}\hat{x}\,\sqrt{\hat{g}}\rho (K-I-{\rm{\Phi }}),\end{eqnarray} \tag{ 3.1 }$$

with K the scalar specific kinetic energy as calculated in an inertial frame, and I = I(ρ, s) the specific internal energy of the fluid. This action functional leads to the three momentum equations of motion but does not lead to mass and entropy conservation—which are assumed a priori. In this formulation, the dynamical fields are the spatial coordinates xⁱ in an arbitrary frame (but not comoving), and therefore

$$\begin{eqnarray*}&&K=\displaystyle \frac{1}{2}{\hat{g}}_{\mu \nu }{\hat{u}}^{\mu }{\hat{u}}^{\nu }=\displaystyle \frac{1}{2}{g}_{\mu \nu }{u}^{\mu }{u}^{\nu }=\displaystyle \frac{1}{2}{g}_{\mu \nu }\displaystyle \frac{\partial {x}^{\mu }}{\partial \tau }\displaystyle \frac{\partial {x}^{\nu }}{\partial \tau }\end{eqnarray*}$$

since

$$\begin{eqnarray}&&{u}^{\mu }=\displaystyle \frac{\partial {x}^{\mu }}{\partial {\hat{x}}^{\nu }}{\hat{u}}^{\nu }=\displaystyle \frac{\partial {x}^{\mu }}{\partial {\hat{x}}^{0}}{\hat{u}}^{0}+\displaystyle \frac{\partial {x}^{\mu }}{\partial {\hat{x}}^{i}}{\hat{u}}^{i}=\displaystyle \frac{\partial {x}^{\mu }}{\partial \tau }.\end{eqnarray} \tag{ 3.2 }$$

A particle-relabeling transformation will be interpreted as a passive coordinate transformation from $({\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})$ to $(\hat{x}{{\prime} }^{1},\hat{x}{{\prime} }^{2},\hat{x}{{\prime} }^{3})$ within the sub-class of comoving coordinates, where

$$\begin{eqnarray}&&\hat{x}{{\prime} }^{0}=\tau ^{\prime} ={\hat{x}}^{0}=\tau =t,\end{eqnarray} \tag{ 3.3 }$$

$$\begin{eqnarray}&&\hat{x}{{\prime} }^{i}={\hat{x}}^{i}+{\hat{\epsilon }}^{i}\end{eqnarray} \tag{ 3.4 }$$

are also comoving coordinates, and with ${\hat{\epsilon }}^{0}\,=\,0$ and ${\hat{\epsilon }}^{i}$ infinitesimally small. Because both $\hat{x}{{\prime} }^{\mu }$ and ${\hat{x}}^{\mu }$ are chosen to be comoving coordinates with $d\hat{x}{{\prime} }^{i}/{dt}=d{\hat{x}}^{i}/{dt}=0$, it means that ${\hat{\epsilon }}^{i}={\hat{\epsilon }}^{i}({\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})$ must be independent of τ. A passive variation of the action functional ${ \mathcal S }$ under such particle-relabeling is written

$$\begin{eqnarray}&&\tilde{\delta }{ \mathcal S }=\int d\tau {d}^{3}\hat{x}[\tilde{\delta }(\sqrt{\hat{g}}\rho )(K-I-{\rm{\Phi }})+\sqrt{\hat{g}}\rho (\tilde{\delta }K-\tilde{\delta }I-\tilde{\delta }{\rm{\Phi }})].\end{eqnarray} \tag{ 3.5 }$$

Under an infinitesimal particle-relabeling, the dynamical fields (the arbitrary, non-comoving coordinates) ${x}^{i}(\tau ,{\hat{x}}^{1},{\hat{x}}^{2},$ ${\hat{x}}^{3})$ become ${\tilde{x}}^{i}(\tau ^{\prime} ,\hat{x}{{\prime} }^{1},\hat{x}{{\prime} }^{2},\hat{x}{{\prime} }^{3})$. The symbol ${\tilde{x}}^{i}$ indicates a different functional form induced by the particle-relabeling, however the values of the actual arbitrary coordinates at a given point do not change under a particle-relabeling: ${x}^{i}(\tau ,{\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})={\tilde{x}}^{i}(\tau ^{\prime} ,\hat{x}{{\prime} }^{1},\hat{x}{{\prime} }^{2},\hat{x}{{\prime} }^{3})={\tilde{x}}^{i}(\tau ,{\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})+(\partial {x}^{i}/\partial {\hat{x}}^{j})\,{\hat{\epsilon }}^{j}$ to first order. The variation induced on the dynamical fields xⁱ by a particle-relabeling interpreted as a passive transformation is therefore given by

$$\begin{eqnarray}&&\tilde{\delta }{x}^{i}\equiv {\tilde{x}}^{i}(\tau ,{\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})-{x}^{i}(\tau ,{\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})=-\displaystyle \frac{\partial {x}^{i}}{\partial {\hat{x}}^{j}}{\hat{\epsilon }}^{j}.\end{eqnarray} \tag{ 3.6 }$$

Passive transformations in comoving coordinates leave ${d}^{3}\hat{x}$ unchanged. Since the mass of a fluid element is conserved a priori, the passive transformation of a mass element is $\tilde{\delta }({d}^{3}\hat{x}\sqrt{\hat{g}}\rho )=\tilde{\delta }(\sqrt{\hat{g}}\rho ){d}^{3}\hat{x}=0$, implying that

$$\begin{eqnarray}&&\tilde{\delta }(\sqrt{\hat{g}}\rho )=0.\end{eqnarray} \tag{ 3.7 }$$

Under a particle-relabeling transformation, since K − I − Φ is a scalar, the passive variation of $\sqrt{\hat{g}}\rho (K-I-{\rm{\Phi }})$ is

$$\begin{eqnarray}\tilde{\delta }(\sqrt{\hat{g}}\rho (K-I-{\rm{\Phi }})) & = & \sqrt{\hat{g}}\rho \tilde{\delta }(K-I-{\rm{\Phi }})=-\sqrt{\hat{g}}\rho \displaystyle \frac{\partial (K-I-{\rm{\Phi }})}{\partial {\hat{x}}^{i}}{\hat{\epsilon }}^{i},\\ & = & -\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}(K-I-{\rm{\Phi }}))+(K-I-{\rm{\Phi }})\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}).\end{eqnarray} \tag{ 3.8 }$$

The passive variation of $\sqrt{\hat{g}}\rho (K-I-{\rm{\Phi }})$ may also be expressed as

$$\begin{eqnarray}\tilde{\delta }(\sqrt{\hat{g}}\rho (K-I-{\rm{\Phi }})) & = & \sqrt{\hat{g}}\rho \left[\displaystyle \frac{1}{2}{u}^{\mu }{u}^{\nu }\tilde{\delta }{g}_{\mu \nu }+{u}_{j}\tilde{\delta }{u}^{j}-\tilde{\delta }I-\tilde{\delta }{\rm{\Phi }}\right],\\ & = & \sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\left[-\displaystyle \frac{1}{2}{u}^{\mu }{u}^{\nu }\displaystyle \frac{\partial {g}_{\mu \nu }}{\partial {\hat{x}}^{i}}-{u}_{j}\displaystyle \frac{\partial }{\partial \tau }\left(\displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}\right)+\displaystyle \frac{p}{{\rho }^{2}}\displaystyle \frac{\partial \rho }{\partial {\hat{x}}^{i}}+T\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}+\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {\hat{x}}^{i}}\right],\\ & = & \sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}\left[\displaystyle \frac{\partial {u}_{j}}{\partial \tau }-\displaystyle \frac{1}{2}{u}^{\mu }{u}^{\nu }\displaystyle \frac{\partial {g}_{\mu \nu }}{\partial {x}^{j}}+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial p}{\partial {x}^{j}}+\displaystyle \frac{\partial {\rm{\Phi }}}{\partial {x}^{j}}\right]\\ & & -\,\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left(\displaystyle \frac{p}{\rho }\right)+\sqrt{\hat{g}}\rho \displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}{u}_{j}\displaystyle \frac{\partial {\hat{\epsilon }}^{i}}{\partial \tau }+\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}T\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}\\ & & -\,\displaystyle \frac{\partial }{\partial \tau }\left(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}{u}_{j}\right),\\ & = & -\sqrt{\hat{g}}{{\rm{\Lambda }}}_{j}\tilde{\delta }{x}^{j}+\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho {u}_{j}\tilde{\delta }{x}^{j})+\sqrt{\hat{g}}\rho \displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}{u}_{j}\displaystyle \frac{\partial {\hat{\epsilon }}^{i}}{\partial \tau }\\ & & -\,\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{\epsilon }}^{i}p)+\displaystyle \frac{p}{\rho }\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i})+\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}T\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}},\end{eqnarray} \tag{ 3.9 }$$

where T is temperature. Equating (3.8) with (3.9), one obtains Noether's first theorem under passive transformations in comoving coordinates:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho {u}_{j}\tilde{\delta }{x}^{j})+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\left[K-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right]\right)\\ &&=\sqrt{\hat{g}}{{\rm{\Lambda }}}_{j}\tilde{\delta }{x}^{j}+\left(K-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right)\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i})-\sqrt{\hat{g}}\rho \displaystyle \frac{\partial {x}^{j}}{\partial {\hat{x}}^{i}}{u}_{j}\displaystyle \frac{\partial {\hat{\epsilon }}^{i}}{\partial \tau }-T\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}.\end{eqnarray} \tag{ 3.10 }$$

The three conditions

$$\begin{eqnarray}&&\tilde{\delta }(\sqrt{\hat{g}}\rho )=-\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i})=0,\end{eqnarray} \tag{ 3.11 }$$

$$\begin{eqnarray}&&\tilde{\delta }s=-\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}{\hat{\epsilon }}^{i}=0,\end{eqnarray} \tag{ 3.12 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\hat{\epsilon }}^{i}}{\partial \tau }=0\end{eqnarray} \tag{ 3.13 }$$

established by Padhye and Morrison (1996) must be satisfied to obtain a conservation equation on-shell. The expressions (3.11) and (3.12) are nothing but constraints on, respectively, mass² and entropy conservation under a passive comoving coordinate transformation. These two constraints impose that Λ⁰ and Λ_(β) continue to vanish a priori. They are unrelated to symmetries of the dynamics obtained from minimizing the action functional (3.1), which leads to the momentum equations only—not mass and entropy conservation.

Notice that (3.11) and (3.12) may be expressed as covariant equations in arbitrary coordinates:

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{\hat{g}}}\tilde{\delta }(\sqrt{\hat{g}}\rho )=-\displaystyle \frac{1}{\sqrt{\hat{g}}}\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{i})=-\displaystyle \frac{1}{\sqrt{\hat{g}}}\displaystyle \frac{\partial }{\partial {\hat{x}}^{\mu }}(\sqrt{\hat{g}}\rho {\hat{\epsilon }}^{\mu })=-{(\rho {\epsilon }^{\mu })}_{:\mu }=0,\end{eqnarray} \tag{ 3.14 }$$

$$\begin{eqnarray}&&\tilde{\delta }s=-\displaystyle \frac{\partial s}{\partial {\hat{x}}^{i}}{\hat{\epsilon }}^{i}=-\displaystyle \frac{\partial s}{\partial {\hat{x}}^{\mu }}{\hat{\epsilon }}^{\mu }=-{s}_{,\mu }{\epsilon }^{\mu }=0,\end{eqnarray} \tag{ 3.15 }$$

given that ${\hat{\epsilon }}^{\mu }={\hat{x}}_{\,,\nu }^{\mu }{\epsilon }^{\nu }$ and ${\hat{\epsilon }}^{0}={\epsilon }^{0}\,=\,0$. However, the condition (3.13) is not covariant under arbitrary coordinate transformations—its covariant form would be $\partial {\hat{\epsilon }}^{i}/\partial \tau +{\hat{{\rm{\Gamma }}}}_{0j}^{i}{\hat{\epsilon }}^{j}=0$. Therefore, (3.13) is satisfied in comoving coordinates only. This lack of covariance implies that (3.13) has no specific meaning in arbitrary coordinates. As is the case for (3.11) and (3.12), the constraint (3.13) is unrelated to a symmetry of the dynamics and only imposes that the transformed coordinates remain comoving with the fluid. The choice

$$\begin{eqnarray}&&{\hat{\epsilon }}^{i}=\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}\rho }\displaystyle \frac{\partial s}{\partial {\hat{x}}^{j}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{k}},\end{eqnarray} \tag{ 3.16 }$$

with $\zeta =\zeta ({\hat{x}}^{1},{\hat{x}}^{2},{\hat{x}}^{3})$ an arbitrary infinitesimal passive tracer, leads to an acceptable transformation in these coordinates since it satisfies the three constraints (3.11)–(3.13). The term ε^μναβ is the Levi-Civita symbol, and ${(\sqrt{\hat{g}})}^{-1}{\varepsilon }^{\mu \nu \alpha \beta }$ is a contravariant fourth-rank tensor with vanishing covariant derivative.

The term $\sqrt{\hat{g}}\rho {u}_{j}\tilde{\delta }{x}^{j}$ appearing in (3.10) may be rewritten as

$$\begin{eqnarray}&&\sqrt{\hat{g}}\rho {u}_{j}\tilde{\delta }{x}^{j}=\sqrt{\hat{g}}\rho q\zeta +\displaystyle \frac{\partial }{\partial {\hat{x}}^{j}}\left({\varepsilon }^{0{ijk}}{\hat{u}}_{i}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}\zeta \right)\end{eqnarray} \tag{ 3.17 }$$

from the definition of potential vorticity q:

$$\begin{eqnarray}&&q\equiv \displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}\rho }\displaystyle \frac{\partial {\hat{u}}_{j}}{\partial {\hat{x}}^{i}}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}.\end{eqnarray} \tag{ 3.18 }$$

Define

$$\begin{eqnarray}&&{\hat{A}}^{{ij}}\equiv \displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}\left(K-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right)\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}=-{\hat{A}}^{{ji}},\end{eqnarray} \tag{ 3.19 }$$

$$\begin{eqnarray}&&{\hat{B}}^{i}\equiv \displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}{\hat{u}}_{j}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}},\end{eqnarray} \tag{ 3.20 }$$

$$\begin{eqnarray}&&{\hat{b}}^{i}\equiv -\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}\displaystyle \frac{{\hat{{\rm{\Lambda }}}}_{j}}{\rho }\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}.\end{eqnarray} \tag{ 3.21 }$$

Off-shell and once the three constraints are explicitly satisfied, (3.10) reduces to

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho q\zeta )+\displaystyle \frac{\partial }{\partial {\hat{x}}^{j}}\left[\sqrt{\hat{g}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{i}}{\hat{A}}^{{ij}}-\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}{\hat{B}}^{j}\zeta )\right]=\sqrt{\hat{g}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{i}}{\hat{b}}^{i},\end{eqnarray} \tag{ 3.22 }$$

from the transformation (3.16), and from (2.1) and (2.2). The relation (3.22) is a conservation law in comoving coordinates on-shell (i.e. it has the form of a continuity equation in comoving coordinates when ${\hat{b}}^{i}=0$). However due to the presence of ζ, (3.22) on-shell is not Ertel's theorem.

One may then follow the line of thought presented in Padhye and Morrison (1996), and manipulate (3.22) to rewrite it as

$$\begin{eqnarray}&&\zeta \left[\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho q)+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{b}}^{i})\right]=\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left[\sqrt{\hat{g}}\zeta {\hat{b}}^{i}+\sqrt{\hat{g}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{j}}{\hat{A}}^{{ij}}+\zeta \displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}{\hat{B}}^{i})\right]\end{eqnarray} \tag{ 3.23 }$$

after using the condition ∂ζ / ∂τ = 0. Padhye and Morrison (1996), working on-shell (i.e. ${\hat{b}}^{i}=0$), integrate this equation over the labels, use the divergence theorem with suitable boundary conditions, and conclude from the du Bois-Reymond lemma that potential vorticity is materially conserved because ζ is arbitrary.

We however arrive at a fundamentally different conclusion. It will be shown in sub-section 4.3 that both the left-hand side and right-hand side of (3.23) vanish identically when mass and entropy are assumed to be conserved a priori. This implies that the arbitrariness of ζ, the du Bois-Reymond lemma, and therefore Noether's first theorem are irrelevant to the fact that the material derivative of potential vorticity vanishes on-shell. Given that (3.23) is nothing but the algebraic identity 0 = 0 when mass and entropy are conserved a priori, this relation does not represent a non-trivial conservation law that requires the equations of motion for momentum and specific symmetry conditions on the metric tensor and gravitational potential.

In this section, it will be demonstrated that the equation describing potential vorticity conservation, which is written in the form of a continuity equation as (1.1), is an algebraic identity off-shell and a trivial law.

4.1. Definition of a trivial conservation law of the second kind

First, consider a generic antisymmetric tensor F^μν = −F^νμ. Define a 4-vector c^μ as the covariant divergence of this antisymmetric tensor:

$$\begin{eqnarray*}&&{c}^{\mu }\equiv {{F}^{\mu \nu }}_{:\nu }.\end{eqnarray*}$$

By virtue of (A.2), the covariant divergence of c^μ identically vanishes:

$$\begin{eqnarray}&&{{c}^{\mu }}_{:\mu }={{F}^{\mu \nu }}_{:\nu :\mu }=0.\end{eqnarray} \tag{ 4.1 }$$

Following the terminology of Olver 1993, pp 264–265, (4.1) is a trivial conservation law of the second kind. Such trivial conservation laws are characterized by currents solely written in terms of the divergence of an antisymmetric tensor. They are algebraic identities obtained off-shell—i.e. the equations of motion (1.2) and (1.3) are not required to establish such trivial conservation laws.

Antisymmetric quantities such as F^μν are sometimes referred to as 'superpotentials'. Conserved currents may always be defined up to the divergence of a superpotential.

4.2. In arbitrary coordinates

In tensor notation, potential vorticity q is defined as

$$\begin{eqnarray}&&q\equiv \displaystyle \frac{{\omega }^{\mu }{s}_{,\mu }}{\rho },\end{eqnarray} \tag{ 4.2 }$$

where

$$\begin{eqnarray}&&{\omega }^{\mu }=\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}{u}_{\alpha :\nu }=\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}{u}_{\alpha ,\nu },\end{eqnarray} \tag{ 4.3 }$$

$$\begin{eqnarray}&&{u}_{\alpha }={g}_{\alpha \mu }{u}^{\mu }.\end{eqnarray} \tag{ 4.4 }$$

Here, the fields defining q are considered off-shell, i.e. they are not assumed to be governed by (1.2) and (1.3). The trivial nature of potential vorticity conservation may be demonstrated in arbitrary coordinates. It is shown that

$$\begin{eqnarray}&&{c}^{\mu }=\rho {u}^{\mu }q+\displaystyle \frac{{\omega }^{\mu }{{\rm{\Lambda }}}_{(\beta )}}{\rho }-\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{\nu }-{u}_{\nu }{{\rm{\Lambda }}}^{0}){s}_{,\alpha }}{\rho }={{F}^{\mu \nu }}_{:\nu },\end{eqnarray} \tag{ 4.5 }$$

where

$$\begin{eqnarray}&&{F}^{\mu \nu }={u}^{\mu }{B}^{\nu }-{u}^{\nu }{B}^{\mu }+\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}\left[\left(\displaystyle \frac{1}{2}{u}^{\sigma }{u}_{\sigma }-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right){s}_{,\alpha }+{u}_{\alpha }\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }\right],\end{eqnarray} \tag{ 4.6 }$$

$$\begin{eqnarray}&&{B}^{\mu }=\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}{u}_{\nu }{s}_{,\alpha }\end{eqnarray} \tag{ 4.7 }$$

(see appendix B). The fact that F^μν is antisymmetric ensures that c^μ in (4.5) is a trivially conserved 4-current of the second kind. The charge density c⁰ associated with that 4-current is ρ q off-shell because ε^00να = 0. The conservation law

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}(\sqrt{g}\rho q)+\displaystyle \frac{\partial }{\partial {x}^{i}}\left(\sqrt{g}\left[\rho {u}^{i}q+\displaystyle \frac{{\omega }^{i}{{\rm{\Lambda }}}_{(\beta )}}{\rho }-\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{j}-{u}_{j}{{\rm{\Lambda }}}^{0}){s}_{,k}}{\rho }\right]\right)=0,\end{eqnarray} \tag{ 4.8 }$$

which follows from taking the covariant divergence of (4.5), reduces to ${(\sqrt{g}\rho {u}^{\mu }q)}_{,\mu }=\sqrt{g}\rho {u}^{\mu }{q}_{,\mu }=\sqrt{g}\rho \,{dq}/{dt}=0$ on-shell—i.e. after using the equations of motion ${{\rm{\Lambda }}}_{j}={g}_{j\nu }{{\rm{\Lambda }}}^{\nu }=0$, Λ⁰ = 0 and Λ_(β) = 0. The conservation law (4.8) is an algebraic identity and is demonstrated without assuming that the equations of motion are satisfied. On-shell, the conservation equation

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\omega }^{\mu }{{\rm{\Lambda }}}_{(\beta )}}{\rho }-\displaystyle \frac{{\varepsilon }^{0\mu \nu \alpha }}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{\nu }-{u}_{\nu }{{\rm{\Lambda }}}^{0}){s}_{,\alpha }}{\rho }\right)}_{:\mu }=0\end{eqnarray} \tag{ 4.9 }$$

is a trivial law of the first kind (see Olver 1993, for a definition). In general, a trivial conservation law is a linear combination of trivial laws of the first and second kinds. Two conservation laws are said to be equivalent if they differ by a trivial conservation law (Olver 1993). The conservation laws ${(\rho {u}^{\mu }q)}_{:\mu }=0$ and ${{c}^{\mu }}_{:\mu }=0$ are equivalent and ${(\rho {u}^{\mu }q)}_{:\mu }=0$ is therefore trivial.

The triviality of potential vorticity conservation has been suggested in previous studies. For instance, Müller (1995) demonstrated that the evolution of the potential vorticity charge density ρ q is governed by a conservation equation which is a mathematical identity, although he still used Noether's first theorem to associate potential vorticity conservation with particle-relabelling. Rosenhaus and Shankar (2016) also analyzed the triviality of potential vorticity conservation in the context of incompressible flows.

4.3. In comoving coordinates

The antisymmetric tensor provided by (4.6) may be expressed in comoving coordinate systems with ${\hat{u}}^{i}\,=\,0$ as

$$\begin{eqnarray}&&{\hat{F}}^{00}=0,\end{eqnarray} \tag{ 4.10 }$$

$$\begin{eqnarray}&&{\hat{F}}^{0i}=\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}{\hat{u}}_{j}\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}=-{\hat{F}}^{i0},\end{eqnarray} \tag{ 4.11 }$$

$$\begin{eqnarray}&&{\hat{F}}^{{ij}}=\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}\left[\left(K-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right)\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}+{\hat{u}}_{k}\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }\right]=-{\hat{F}}^{{ji}}.\end{eqnarray} \tag{ 4.12 }$$

The 4-current (4.5) may also be expressed in comoving coordinates as

$$\begin{eqnarray}&&{\hat{c}}^{0}=\displaystyle \frac{1}{\sqrt{\hat{g}}}\displaystyle \frac{\partial }{\partial {\hat{x}}^{\nu }}(\sqrt{\hat{g}}{\hat{F}}^{0\nu })=\rho q,\end{eqnarray} \tag{ 4.13 }$$

$$\begin{eqnarray}&&{\hat{c}}^{i}=\displaystyle \frac{1}{\sqrt{\hat{g}}}\displaystyle \frac{\partial }{\partial {\hat{x}}^{\nu }}(\sqrt{\hat{g}}{\hat{F}}^{i\nu })=\displaystyle \frac{{\hat{\omega }}^{i}{{\rm{\Lambda }}}_{(\beta )}}{\rho }-\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}\displaystyle \frac{({\hat{{\rm{\Lambda }}}}_{j}-{\hat{u}}_{j}{{\rm{\Lambda }}}^{0})}{\rho }\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}.\end{eqnarray} \tag{ 4.14 }$$

The trivial conservation law of the second kind (4.8) for potential vorticity density is valid in arbitrary coordinates. In particular, it may be expressed in comoving coordinates with ${\hat{u}}^{i}\,=\,0$:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho q)+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left(\sqrt{\hat{g}}\left[\displaystyle \frac{{\hat{\omega }}^{i}{{\rm{\Lambda }}}_{(\beta )}}{\rho }-\displaystyle \frac{{\varepsilon }^{0{ijk}}}{\sqrt{\hat{g}}}\displaystyle \frac{({\hat{{\rm{\Lambda }}}}_{j}-{\hat{u}}_{j}{{\rm{\Lambda }}}^{0})}{\rho }\displaystyle \frac{\partial s}{\partial {\hat{x}}^{k}}\right]\right)=0.\end{eqnarray} \tag{ 4.15 }$$

On-shell, the terms Λ_(β), ${\hat{{\rm{\Lambda }}}}_{j}$, and Λ⁰ all vanish and this trivial conservation law becomes $\partial (\sqrt{\hat{g}}\rho q)/\partial \tau =0$. From (2.1), this reduces to ∂ q/∂ τ = 0, i.e. the material conservation of potential vorticity.

In comoving coordinates as in any other admissible coordinate system, the conservation of potential vorticity density is an off-shell trivial law of the second kind. Noether's first theorem, which leads to conservation laws on-shell only, is therefore irrelevant to associate a particle-relabeling transformation or any symmetry of the equations of motion to potential vorticity conservation. Particle-relabeling is a coordinate transformation within the sub-class of comoving coordinates whose associated constraints (3.11)–(3.13) are unrelated to the dynamics obtained from minimizing (3.1). Because the equations of motion are covariant, a particle-relabeling transformation does leave their form intact. However Noether's first theorem under particle-relabeling does not lead to a dynamically relevant conservation equation, in particular for potential vorticity. In addition and from a more general standpoint, it has been demonstrated by Rosenhaus (2002) that Noether's first theorem is not applicable for transformations involving arbitrary functions such as (3.16).

Consider now ${\hat{F}}^{{ij}}$ under the assumptions that Λ⁰ and Λ_(β) vanish a priori, as was required in section 3. It is nothing but ${\hat{A}}^{{ij}}$ provided by (3.19). Under the same assumptions, ${\hat{F}}^{0i}$ becomes ${\hat{B}}^{i}$ from (3.20), and ${\hat{c}}^{i}$ becomes ${\hat{b}}^{i}$ from (3.21). The left-hand side of (3.23) is therefore written

$$\begin{eqnarray}&&\zeta \left[\displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}\rho q)+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{b}}^{i})\right]=\zeta \left[\displaystyle \frac{\partial }{\partial {\hat{x}}^{0}}(\sqrt{\hat{g}}{\hat{c}}^{0})+\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{c}}^{i})\right]=\zeta \displaystyle \frac{{\partial }^{2}}{\partial {\hat{x}}^{\mu }\partial {\hat{x}}^{\nu }}(\sqrt{\hat{g}}{\hat{F}}^{\mu \nu })=0.\end{eqnarray} \tag{ 4.16 }$$

It vanishes identically due to the commutativity of ordinary derivatives and the antisymmetry of ${\hat{F}}^{\mu \nu }$. The right-hand side of (3.23) also vanishes identically:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left[\sqrt{\hat{g}}\zeta {\hat{b}}^{i}+\sqrt{\hat{g}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{j}}{\hat{A}}^{{ij}}+\zeta \displaystyle \frac{\partial }{\partial \tau }(\sqrt{\hat{g}}{\hat{B}}^{i})\right]=\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left[\sqrt{\hat{g}}\zeta {\hat{c}}^{i}+\sqrt{\hat{g}}\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{j}}{\hat{F}}^{{ij}}+\zeta \displaystyle \frac{\partial }{\partial {\hat{x}}^{0}}(\sqrt{\hat{g}}{\hat{F}}^{0i})\right],\\ &&\quad =\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{i}}\left[\sqrt{\hat{g}}{\hat{c}}^{i}+\displaystyle \frac{\partial }{\partial {\hat{x}}^{j}}(\sqrt{\hat{g}}{\hat{F}}^{{ji}})+\displaystyle \frac{\partial }{\partial {\hat{x}}^{0}}(\sqrt{\hat{g}}{\hat{F}}^{0i})\right]+\zeta \left[\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{c}}^{i})+\displaystyle \frac{{\partial }^{2}}{\partial {\hat{x}}^{0}\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{F}}^{0i})\right],\\ &&\quad =\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{i}}\left[\sqrt{\hat{g}}{\hat{c}}^{i}-\displaystyle \frac{\partial }{\partial {\hat{x}}^{\nu }}(\sqrt{\hat{g}}{\hat{F}}^{i\nu })\right]+\zeta \left[\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}(\sqrt{\hat{g}}{\hat{c}}^{i})+\displaystyle \frac{\partial }{\partial {\hat{x}}^{0}}(\sqrt{\hat{g}}{\hat{c}}^{0})\right],\\ &&\quad =\displaystyle \frac{\partial \zeta }{\partial {\hat{x}}^{i}}[\sqrt{\hat{g}}{\hat{c}}^{i}-\sqrt{\hat{g}}{\hat{c}}^{i}]+\zeta \displaystyle \frac{{\partial }^{2}}{\partial {\hat{x}}^{\mu }\partial {\hat{x}}^{\nu }}(\sqrt{\hat{g}}{\hat{F}}^{\mu \nu }),=0.\end{eqnarray} \tag{ 4.17 }$$

This implies that, given a priori conservation of mass and entropy, (3.23) is always true, whether the constraint (3.13) associated with particle-relabeling is satisfied or not. In a similar fashion, it may be verified that (3.22) identically reduces to

$$\begin{eqnarray}&&\sqrt{\hat{g}}{\hat{B}}^{i}\displaystyle \frac{\partial }{\partial {\hat{x}}^{i}}\left(\displaystyle \frac{\partial \zeta }{\partial \tau }\right)=0\end{eqnarray} \tag{ 4.18 }$$

when mass and entropy are conserved a priori, implying that Noether's first theorem under particle-relabeling is unrelated to potential vorticity conservation but only associates particle-relabeling with the material conservation of ζ.

The application of Noether's first theorem simply leads to the consistency check of a constraint, given that potential vorticity conservation is a trivial law of the second kind, i.e. an algebraic identity. Noether's first theorem under a particle-relabeling is a circular statement on the material conservation of ζ, not a statement on the material conservation of q. If one ignores the fact that (4.8) is a trivial conservation law, one will wrongly apply Noether's first theorem to associate particle-relabeling with potential vorticity conservation.

It has previously been mentioned that the symmetry associated with potential vorticity conservation is invisible in an Eulerian formulation but exists in a label formulation (see for example Shepherd 2015). The fact that the equations of motion for classical fluids may be written in a manifestly covariant form—which applies equally to Eulerian and comoving (or label) coordinates—clearly shows that this cannot be the case: the existence of a symmetry of the equations of motion does not depend on the choice of coordinates because the equations of motion themselves are covariant.

A trivial conservation law of the second kind is an off-shell identity obtained independently of the equations of motion. A conservation law cannot be trivial of the second kind, and at the same time exist on-shell as a consequence of the application of Noether's first theorem. In this paper, it was demonstrated that potential vorticity conservation is a trivial law of the second kind in arbitrary coordinates. Noether's first theorem is irrelevant in this case. Consequently, invoking Noether's first theorem to associate potential vorticity conservation with the particle-relabeling transformation is unjustified.

Particle-relabeling is naturally defined in terms of comoving coordinate transformations. One of its associated constraints is not covariant under arbitrary coordinate transformations. Continuous symmetries or constraints that are apparent only in a given sub-class of coordinate systems but are broken in arbitrary coordinates—i.e. resulting from non-covariant conditions—cannot give rise to dynamically relevant conservation laws. This is because the covariance of the equations of motion implies that if a symmetry of these equations exists, it must exist independently of the choice of coordinates—inertial, non-inertial, Eulerian, comoving, etc.

Particle-relabeling is parameterized by an arbitrary passive tracer ζ. From Noether's first theorem under particle-relabeling, the associated conserved charge density is ρ q ζ. Given that ρ q was shown to be the charge density of a trivial conservation law, it follows that the conservation law associated with particle-relabeling in the context of Noether's first theorem is simply equivalent to the material conservation of ζ, which does nothing but confirm the prior assumption that ζ was a passive tracer.

If potential vorticity conservation is associated with a symmetry, the symmetry must be infinite-dimensional and obtained via Noether's second theorem. This theorem is applicable in under-determined dynamical systems only. This question is currently under study in the context of a manifestly covariant field formulation of geophysical fluid dynamics. A key element to consider is whether the equations of motion describing classical fluids in an external gravitational potential are under-determined.

The authors are grateful to Christopher Subich, Stéphane Gaudreault, and Janusz Pudykiewicz for providing comments that led to an improved manuscript.

• 1.  
  Consider an antisymmetric tensor F^μν = −F^νμ. It will be shown that the scalar ${{F}^{\mu \nu }}_{:\nu :\mu }$ vanishes in a Riemannian space. The term $\sqrt{g}{{F}^{\mu \nu }}_{:\nu }$ is written

  $$\begin{eqnarray}\sqrt{g}{{F}^{\mu \nu }}_{:\nu } & = & \sqrt{g}{{F}^{\mu \nu }}_{,\nu }+\sqrt{g}\,{{\rm{\Gamma }}}_{\alpha \nu }^{\mu }{F}^{\alpha \nu }+\sqrt{g}\,{{\rm{\Gamma }}}_{\alpha \nu }^{\nu }{F}^{\mu \alpha },\\ & = & \sqrt{g}{{F}^{\mu \nu }}_{,\nu }+{(\sqrt{g})}_{,\nu }{F}^{\mu \nu },\\ & = & {(\sqrt{g}{F}^{\mu \nu })}_{,\nu }.\end{eqnarray} \tag{ A.1 }$$

  Moreover,

  $$\begin{eqnarray}{{F}^{\mu \nu }}_{:\nu :\mu } & = & {(\sqrt{g})}^{-1}{(\sqrt{g}{{F}^{\mu \nu }}_{:\nu })}_{,\mu },\\ & = & {(\sqrt{g})}^{-1}{(\sqrt{g}{F}^{\mu \nu })}_{,\nu ,\mu },\\ & = & 0\end{eqnarray} \tag{ A.2 }$$

  from (A.1), the antisymmetry of F^μν, and the commutativity of ordinary derivatives.
• 2.  
  One may verify the identity

  $$\begin{eqnarray}&&{\varepsilon }^{\alpha \mu \nu \sigma }{{A}^{\beta }}_{\alpha }+{\varepsilon }^{\beta \alpha \nu \sigma }{{A}^{\mu }}_{\alpha }+{\varepsilon }^{\beta \mu \alpha \sigma }{{A}^{\nu }}_{\alpha }+{\varepsilon }^{\beta \mu \nu \alpha }{{A}^{\sigma }}_{\alpha }={\varepsilon }^{\beta \mu \nu \sigma }{{A}^{\alpha }}_{\alpha },\end{eqnarray} \tag{ A.3 }$$

  contract it with ${\delta }_{\beta }^{0}$, and replace ${{A}^{\mu }}_{\alpha }$ by ${{u}^{\mu }}_{:\alpha }$ to obtain

  $$\begin{eqnarray}&&{\varepsilon }^{0\alpha \nu \sigma }{{u}^{\mu }}_{:\alpha }+{\varepsilon }^{0\mu \alpha \sigma }{{u}^{\nu }}_{:\alpha }+{\varepsilon }^{0\mu \nu \alpha }{{u}^{\sigma }}_{:\alpha }={\varepsilon }^{0\mu \nu \sigma }{{u}^{\alpha }}_{:\alpha }.\end{eqnarray} \tag{ A.4 }$$

  This holds because ${{u}^{0}}_{:\alpha }=0$ (recall that u⁰ = 1 and ${{\rm{\Gamma }}}_{\mu \nu }^{0}=0$). This is a tensor identity when multiplied by ${(\sqrt{g})}^{-1}$.

In this appendix, potential vorticity conservation expressed in arbitrary coordinates is shown to be a trivial law of the second kind. From the definition of potential vorticity,

$$\begin{eqnarray*}\rho {u}^{\alpha }q & = & \displaystyle \frac{{\varepsilon }^{0\mu \nu \sigma }}{\sqrt{g}}{u}_{\nu :\mu }{s}_{,\sigma }{u}^{\alpha },\\ & = & {\left(\displaystyle \frac{{\varepsilon }^{0\mu \nu \sigma }}{\sqrt{g}}{u}_{\nu }{s}_{,\sigma }{u}^{\alpha }\right)}_{:\mu }-\displaystyle \frac{{\varepsilon }^{0\mu \nu \sigma }}{\sqrt{g}}{u}_{\nu }{s}_{,\sigma }{{u}^{\alpha }}_{:\mu },\\ & = & {({u}^{\alpha }{B}^{\mu })}_{:\mu }+\displaystyle \frac{1}{\sqrt{g}}{u}_{\nu }{s}_{,\sigma }({\varepsilon }^{0\alpha \mu \sigma }{{u}^{\nu }}_{:\mu }+{\varepsilon }^{0\alpha \nu \mu }{{u}^{\sigma }}_{:\mu }-{\varepsilon }^{0\alpha \nu \sigma }{{u}^{\mu }}_{:\mu }),\\ & = & {\left({u}^{\alpha }{B}^{\mu }-{u}^{\mu }{B}^{\alpha }+\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\left[{s}_{,\sigma }K+{u}_{\sigma }\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }\right]\right)}_{:\mu }-{\omega }^{\alpha }\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }+{u}^{\mu }{{B}^{\alpha }}_{:\mu }-\displaystyle \frac{{\varepsilon }^{0\alpha \nu \mu }}{\sqrt{g}}{u}_{\nu }{u}^{\sigma }{s}_{,\sigma :\mu }\end{eqnarray*}$$

after using (A.4), (1.5), (4.3), (4.7) and K = u_ν u^ν /2. From

$$\begin{eqnarray}{u}^{\mu }{{B}^{\alpha }}_{:\mu } & & -\displaystyle \frac{{\varepsilon }^{0\alpha \nu \mu }}{\sqrt{g}}{u}_{\nu }{u}^{\sigma }{s}_{,\sigma :\mu }=\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}{u}^{\beta }{u}_{\mu :\beta }{s}_{,\sigma },\\ & = & \displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{\mu }-{u}_{\mu }{{\rm{\Lambda }}}^{0})}{\rho }{s}_{,\sigma }-{\left(\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}{s}_{,\sigma }{\rm{\Phi }}\right)}_{:\mu }-\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\displaystyle \frac{{p}_{,\mu }}{\rho }{s}_{,\sigma },\\ & = & \displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{\mu }-{u}_{\mu }{{\rm{\Lambda }}}^{0})}{\rho }{s}_{,\sigma }-{\left(\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}{s}_{,\sigma }{\rm{\Phi }}\right)}_{:\mu }-{\left(\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\left[I+\displaystyle \frac{p}{\rho }\right]{s}_{,\sigma }\right)}_{:\mu },\end{eqnarray} \tag{ B.1 }$$

one finally gets

$$\begin{eqnarray}\rho {u}^{\alpha }q & = & {\left({u}^{\alpha }{B}^{\mu }-{u}^{\mu }{B}^{\alpha }+\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\left[{s}_{,\sigma }\left(K-{\rm{\Phi }}-I-\displaystyle \frac{p}{\rho }\right)+{u}_{\sigma }\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }\right]\right)}_{:\mu }\\ & & -\,{\omega }^{\alpha }\displaystyle \frac{{{\rm{\Lambda }}}_{(\beta )}}{\rho }+\displaystyle \frac{{\varepsilon }^{0\alpha \mu \sigma }}{\sqrt{g}}\displaystyle \frac{({{\rm{\Lambda }}}_{\mu }-{u}_{\mu }{{\rm{\Lambda }}}^{0}){s}_{,\sigma }}{\rho },\end{eqnarray} \tag{ B.2 }$$

which proves (4.5).