2.1 CGL model

The physical variables which describe CGL plasmas are the fluid velocity $\boldsymbol {u}$, fluid density $\rho$, magnetic field induction $\boldsymbol {B}$, the anisotropic pressure components $p_\parallel$ and $p_\perp$ and the entropy $S$. The ideal CGL plasma equations are similar to the MHD equations and consist of: the mass continuity equation

(2.1)\begin{equation} \frac{\partial \rho}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot} (\rho\boldsymbol{u})=0; \end{equation}

the momentum equation (in semi-conservative form)

(2.2)\begin{equation} \frac{\partial}{\partial t}(\rho\boldsymbol{u})+\boldsymbol{\nabla}\boldsymbol{\cdot}\left( \rho \boldsymbol{u}\boldsymbol{u}+{\mathsf{{p}}} + {\mathsf{{M}}}_B \right) ={-}\rho\boldsymbol{\nabla}\varPhi, \end{equation}

in which ${\mathsf{{p}}}$ is the gyrotropic pressure tensor (which replaces the isotropic gas pressure $p{\mathsf{{I}}}$ used in MHD) given by

(2.3a-c)\begin{equation} {\mathsf{{p}}}=p_\perp{\mathsf{{I}}}+(p_\parallel{-}p_\perp)\boldsymbol{\tau}\boldsymbol{\tau}, \quad \boldsymbol{\tau}=\frac{\boldsymbol{B}}{B}, \quad B= |{\boldsymbol{B}}| , \end{equation}

and ${\mathsf{{M}}}_B$ is the magnetic pressure tensor

(2.4)\begin{equation} {\mathsf{{M}}}_B = \frac{1}{\mu_0}\left( \frac{B^2}{2} {\mathsf{{I}}} -\boldsymbol{BB} \right) = \frac{B^2}{\mu_0}\left(\frac{1}{2} {\mathsf{{I}}} -\boldsymbol{\tau}\boldsymbol{\tau} \right); \end{equation}

the entropy transport equation

(2.5)\begin{equation} \frac{\partial S}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} S=0 ; \end{equation}

Faraday's equation and Gauss's law

(2.6)\begin{gather} \frac{\partial \boldsymbol{B}}{\partial t}-\boldsymbol{\nabla}\times\left(\boldsymbol{u}\times \boldsymbol{B}\right) =0, \end{gather}

(2.7)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{B}= 0 ; \end{gather}

along with the pressure equations

(2.8)\begin{gather} \frac{\partial p_\parallel}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}\left(p_\parallel\boldsymbol{u}\right) + 2{\mathsf{{p}}}^\parallel\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{u} = 0, \end{gather}

(2.9)\begin{gather}\frac{\partial p_\perp}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}\left(p_\perp \boldsymbol{u}\right) + {\mathsf{{p}}}^\perp\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{u} = 0, \end{gather}

where ${\mathsf{{p}}}^\parallel$ and ${\mathsf{{p}}}^\perp$ are the terms comprising the gyrotropic pressure tensor

(2.10a-c)\begin{equation} {\mathsf{{p}}}^\parallel{=} p_\parallel \boldsymbol{\tau}\boldsymbol{\tau}, \quad {\mathsf{{p}}}^\perp{=} p_\perp \left({\mathsf{{I}}} - \boldsymbol{\tau}\boldsymbol{\tau}\right), \quad {\mathsf{{p}}} = {\mathsf{{p}}}^\parallel{+} {\mathsf{{p}}}^\perp . \end{equation}

Through Faraday's equation (2.6) and the mass continuity equation (2.1), the pressure equations (2.8)–(2.9) can be expressed as the double adiabatic equations

(2.11a,b)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\frac{p_\parallel B^2}{\rho^3}\right)=0, \quad \frac{{\rm d}}{{\rm d}t}\left(\frac{p_\perp}{\rho B}\right)=0, \end{equation}

which represent conservation of the particle magnetic moment and the second longitudinal adiabatic moment of the particles, where ${\rm d}/{\rm d}t=\partial /\partial t+\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla }$ is the Lagrangian time derivative following the flow.

In the momentum equation (2.2), $\varPhi$ is the potential of an external source of gravity (for example, in the case of the solar wind, it could represent the gravitational potential field of the Sun).

If one counts the number of evolution equations in (2.1)–(2.11a,b) there are 10 equations for the 10 variables $\boldsymbol {u}$, $\boldsymbol {B}$, $\rho$, $S$, $p_\parallel$ and $p_\perp$. If one does not need to know $S$, then there are nine equations for nine unknowns.

The non-canonical Poisson bracket formulation of MHD in Morrison & Greene (Reference Morrison and Greene1982) uses Faraday's law (2.6) in the form $\partial \boldsymbol {B}/\partial t-\boldsymbol {\nabla }\times (\boldsymbol {u}\times \boldsymbol {B}) + \boldsymbol {u}\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}=0$ for the mathematical case in which $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}\neq 0$ (see also Webb Reference Webb2018). In fact, the possibility of $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}\neq 0$ arises in numerical MHD due to numerical errors in the Gauss’ law (2.7).

To determine the relationship between the entropy $S$, the internal energy density $\varepsilon$ and the pressure components $p_\parallel$ and $p_\perp$ in (2.1)–(2.11a,b) we first note that $\varepsilon$ is given by

(2.12)\begin{equation} \varepsilon=\frac{p_\parallel{+}2p_\perp}{2}. \end{equation}

A different, related approach comes from the entropy law for ideal gases. We consider and compare both approaches.

One link between the approaches is the observation that integration of the double adiabatic equations (2.11a,b) yields

(2.13a,b)\begin{equation} p_\parallel{=}\exp\left({\bar{S}}_\parallel\right) \frac{\rho^3}{B^2}, \quad p_\perp{=}\exp\left({\bar{S}}_\perp\right)\rho B, \end{equation}

where the quantities ${\bar {S}}_\parallel$ and ${\bar {S}}_\perp$ are dimensionless forms of entropy integration constants arising from integrating the double adiabatic equations (2.11a,b). In principle, ${\bar {S}}_\parallel$ and ${\bar {S}}_\perp$ must be scalars advected with the flow. In particular, these quantities could be functions of the entropy $S$ satisfying (2.5). They could also be functions of other advected invariants of the flow, such as $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla } S/\rho$ (e.g. Tur & Yanovsky Reference Tur and Yanovsky1993; Webb et al. Reference Webb, Dasgupta, McKenzie, Hu and Zank2014a). For the sake of simplicity, we assume that $S_\parallel$ and $S_\perp$ depend only on $S$. The overbars on ${\bar {S}}$, ${\bar {S}}_\parallel$ and ${\bar {S}}_\perp$ denote dimensionless versions of these quantities.

If these $S_\parallel$ and $S_\perp$ are functions solely of the entropy $S$, then the internal energy density $\varepsilon$ will have the functional form $\varepsilon (\rho,S,B)$. This form also arises from the first law of thermodynamics coming from the transport equation for $\varepsilon$ when reversible thermodynamic processes are considered. Strictly speaking, we should use normalised or dimensionless variables in (2.13a,b), i.e. we should have used the variables

(2.14a-d)\begin{equation} {\bar{p}}_\parallel{=}\frac{p_\parallel}{p_{\parallel0}}, \quad {\bar{p}}_\perp{=}\frac{p_\perp}{p_{\perp0}}, \quad {\bar{\boldsymbol{B}}}=\frac{\boldsymbol{B}}{B_0}, \quad \bar{\rho}=\frac{\rho}{\rho_0}, \end{equation}

where the subscript zero quantities are dimensional constants. In a convenient abuse of notation, we have dropped the overbar superscripts in (2.13a,b).

2.2 Thermodynamic formulation

A physical description of $p_\parallel$ and $p_\perp$ and their relation to the internal energy density $\varepsilon$ was developed first by Holm & Kupershmidt (Reference Holm and Kupershmidt1986), which in part uses the work of Volkov (Reference Volkov1966), and also later by Hazeltine et al. (Reference Hazeltine, Mahajan and Morrison2013), which discusses both reversible and irreversible thermodynamics. Here we concentrate on the ideal reversible dynamics case.

An analysis of reversible work done on the plasma reveals that the internal energy per unit mass of the CGL plasma, $e=\varepsilon /\rho$, obeys the Pfaffian equation

(2.15)\begin{equation} {\rm d}e=T \,{\rm d}S+\frac{p_\parallel}{\rho^2}{\rm d}\rho -\frac{p_{\varDelta}}{\rho B} {\rm d}B. \end{equation}

where $T$ is the adiabatic temperature in the plasma. The expression (2.12) for the internal energy density gives the relations

(2.16a,b)\begin{equation} e=\frac{\varepsilon}{\rho}=\frac{(p_\parallel{+}2p_\perp)}{2\rho}, \quad p_{\varDelta}=p_\parallel{-}p_\perp, \end{equation}

(see also Holm & Kupershmidt Reference Holm and Kupershmidt1986) and

(2.17)\begin{equation} p \equiv \frac{(p_\parallel{+}2p_\perp)}{3} =\frac{2\varepsilon}{3}, \end{equation}

which is the gas pressure defined by one-third of the trace of the CGL pressure tensor (2.3a–c).

The Pfaffian equation (2.15) constitutes the first law of thermodynamics for a CGL plasma. As outlined in Hazeltine et al. (Reference Hazeltine, Mahajan and Morrison2013), it can be derived from the transport equation for $\varepsilon$ under thermodynamic processes that involve reversible work done on the plasma described by the gyrotropic part of the Vlasov distribution function. The derivation can be generalised to include irreversible work comprising a dissipative term and a further term due to gyro-viscosity, which is a non-dissipative term obtained in the limit of no scattering. Equation (2.15) is equivalent to the non-dissipative internal energy equation (2.51) in which a source term $\rho T\,{\rm d}S/{\rm d}t$ has been added to the right hand-side.

The above interpretation of (2.15) implies $e=e(\rho,S,B)$ along with the thermodynamic relations

(2.18a-c)\begin{equation} T =e_S = \varepsilon_S/\rho, \quad p_\parallel{=}\rho^2 e_\rho = \rho\varepsilon_\rho-\varepsilon, \quad p_{\varDelta}={-}\rho B e_B ={-}B\varepsilon_B . \end{equation}

Thus, if $e(\rho,S,B)$ is known, i.e. an equation of state has been specified, then the relations (2.18a–c) give $p_\parallel$ and $p_\perp$ as functions of $\rho,S,B$. Consistency must hold with the double adiabatic equations (2.11a,b), which constrains the possible expressions for $e(\rho,S,B)$.

To derive the constraints, we first substitute the relations (2.18a–c) into the internal energy (2.17), giving

(2.19)\begin{equation} e=\frac{1}{\rho}\left(\frac{3 p_\parallel}{2}-p_{\varDelta}\right) =\frac{3}{2}\rho e_\rho+B e_B. \end{equation}

This is a first-order partial differential equation for $e(\rho,S,B)$ which is easily solved by using the method of characteristics (e.g. Sneddon Reference Sneddon1957):

(2.20)\begin{equation} \frac{{\rm d}\rho}{(3\rho/2)}=\frac{{\rm d}B}{B}=\frac{{\rm d}e}{e}. \end{equation}

Integrating these differential equations yields the general solution

(2.21a,b)\begin{equation} e=\rho^{2/3} f(\xi,S),\quad \xi=B\rho^{{-}2/3}, \end{equation}

where $f(\xi, S)$ is an arbitrary function of $S$ and the similarity variable $\xi$. Now, the relations (2.18a–c) imply that

(2.22)\begin{equation} T =\rho^{2/3} f_S , \end{equation}

and

(2.23a,b)\begin{equation} p_\parallel{=}\tfrac{2}{3} \rho^{5/3} \left( f- \xi f_\xi\right), \quad p_\perp{=}\tfrac{1}{3} \rho^{5/3}\left( 2 f+\xi f_\xi \right). \end{equation}

Substitution of expressions (2.23a,b) into the double adiabatic equations (2.11a,b) followed by use of the relation $B=\rho ^{2/3}\xi$ then yields

(2.24a,b)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left( \frac{2}{3} \xi^2f -\frac{2}{3} \xi^3 f_\xi \right) =0, \quad \frac{{\rm d}}{{\rm d}t}\left( \frac{2}{3\xi}f+ \frac{1}{3} f_\xi \right) =0, \end{equation}

with ${\rm d}f/{\rm d}t = f_\xi \,{\rm d}\xi /{\rm d}t$ for any process in which $S$ is adiabatic (cf. the entropy equation (2.5)). However, from Faraday's equation (2.6) and the mass continuity equation (2.1), we find that $\xi$ obeys the transport equation

(2.25)\begin{equation} \frac{{\rm d}}{{\rm d}t}\xi = \left(-\frac{1}{3}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} + \boldsymbol{\tau}\boldsymbol{\tau}\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{u} \right)\xi . \end{equation}

Thus, $\xi$ is not an advected quantity, which implies that the adiabatic equations (2.24a,b) for $f(\xi,S)$ reduce to

(2.26a,b)\begin{equation} \frac{2}{3} \xi^2f -\frac{2}{3} \xi^3 f_\xi = c_\parallel(S), \quad \frac{2}{3\xi}f+ \frac{1}{3} f_\xi = c_\perp(S) . \end{equation}

The general solution of this pair of equations is given by

(2.27)\begin{equation} f(\xi,S) = c_\perp(S) \xi + c_\parallel(S)\frac{1}{2\xi^2} . \end{equation}

Consequently, expressions (2.23a,b) yield the equation of state

(2.28a,b)\begin{equation} p_\parallel{=} c_\parallel(S) \rho^3/B^2, \quad p_\perp{=}c_\perp(S) \rho B, \end{equation}

along with the relations

(2.29a,b)\begin{equation} {\bar{S}}_\parallel{=} \ln c_\parallel(S), \quad {\bar{S}}_\perp{=} \ln c_\perp(S) \end{equation}

from the double adiabatic integrals (2.13a,b). The corresponding internal energy (2.21a,b) and pressure (2.17) have the explicit form

(2.30)\begin{equation} e=c_\perp(S) B + c_\parallel(S) \frac{\rho^2}{2B^2} =\exp({\bar{S}}_\perp) B + \exp({\bar{S}}_\parallel) \frac{\rho^2}{2B^2} , \end{equation}

and

(2.31)\begin{equation} p=c_\perp(S) \frac{2\rho B}{3} + c_\parallel(S) \frac{\rho^3}{3B^2} =\exp({\bar{S}}_\perp) \frac{2\rho B}{3} + \exp({\bar{S}}_\parallel) \frac{\rho^3}{3B^2} . \end{equation}

Note that the expressions for $e$ and $p$ in terms of the double adiabatic integrals $\bar {S}_\parallel$ and $\bar {S}_\perp$ hold independently of any form of equation of state. Finally, either (2.22) or (2.18a–c), both of which rely on $e=e(\rho,S,B)$, yield the plasma temperature:

(2.32)\begin{equation} T = c_\perp'(S) B + c_\parallel'(S) \frac{\rho^2}{2B^2} =\frac{{\rm d}{\bar{S}}_\perp}{{\rm d}S} \exp({\bar{S}}_\perp) B + \frac{{\rm d}{\bar{S}}_\parallel}{{\rm d}S} \exp({\bar{S}}_\parallel) \frac{\rho^2}{2B^2} . \end{equation}

Hazeltine et al. (Reference Hazeltine, Mahajan and Morrison2013) arrived at a similar but less-general result under the assumption that $f$ is separable in $S$ and $\xi$. This leads to $c_\parallel (S) = Q(S) d_\parallel$ and $c_\perp (S) = Q(S) d_\perp$, where $d_\perp$ and $d_\parallel$ are constants. The assumption of separability includes, for example, the case where the gas pressure $p$ satisfies the ideal gas law:

(2.33)\begin{equation} p=\rho R T\quad\mbox{or}\quad T=\frac{p}{\rho R}, \end{equation}

where $R$ is the gas constant. In this case, from the relation (2.17), it follows that $f$ must satisfy the equation $\partial f/\partial S=({2}/{3R}) f$. Then it further follows that $Q(S)=\exp (2S/(3R))$.

Conditions on the derivatives of the internal energy $e$ for thermodynamic stability are discussed in Hazeltine et al. (Reference Hazeltine, Mahajan and Morrison2013), but these considerations lie beyond the scope of the present work.

2.2.1 Ideal gas law and entropy

The double adiabatic equations (2.8) for a CGL plasma can be combined in the form

(2.34)\begin{equation} \frac{{\rm d}}{{\rm d}t}\left(\frac{p_\parallel p_\perp^2}{\rho^5}\right)=0 \end{equation}

which suggests the advected quantity $p_\parallel p_\perp ^2/\rho ^5$ may be viewed as a function of the entropy $S$. A specific functional relation can be motivated by considering the MHD limit for a non-relativistic gas, where the gas entropy is given by the standard formula

(2.35a,b)\begin{equation} S=C_v\ln\left(\frac{p}{\rho^\gamma}\right), \quad \gamma=\frac{C_p}{C_v}=\frac{5}{3}, \end{equation}

with $C_v$ and $C_p$ being the specific heats at constant volume and pressure, respectively. In a CGL plasma, the gas pressure $p$ is given by expression (2.17) in terms of the pressure components $p_\parallel$ and $p_\perp$ which are equal in the MHD limit, because the pressure tensor (2.3a–c) must become isotropic. This results in the relation $p_\parallel =p_\perp =p$, and as a consequence $p_\parallel p_\perp ^2/\rho ^5 = p^3/\rho ^5 = \exp ( 3S/C_v)$ by assuming the formula (2.35a,b). Generalising this relation away from the MHD limit then suggests the formula

(2.36)\begin{equation} S=C_v \ln \left(\frac{p_\parallel^{1/3}p_\perp^{2/3}}{\rho ^{5/3}}\right) \end{equation}

for the entropy of a CGL plasma (see, e.g., Abraham-Shrauner Reference Abraham-Shrauner1967; Du et al. Reference Du, Zank, Li and Guo2020).

Furthermore, because the double adiabatic integrals (2.13a,b) give $p_\parallel p_\perp ^2/\rho ^5=\exp (\bar {S}_\parallel +2\bar {S}_\perp )$ where $\bar {S}_\parallel$ and $\bar {S}_\perp$ are each advected, the entropy formula (2.36) can be expressed as $S = (C_v/3) (\bar {S}_\parallel + 2\bar {S}_\perp )$. This relation now suggests that the entropy is a sum of components

(2.37)\begin{equation} S= S_{{\parallel}} + S_{{\perp}} \quad\hbox{where}\ S_{{\parallel}} \equiv C_{v\parallel}{\bar{S}}_\parallel \quad\text{and}\quad S_\perp{\equiv} C_{v\perp}{\bar{S}}_\perp, \end{equation}

with

(2.38a,b)\begin{equation} C_{v_\parallel}\equiv\tfrac{1}{3} C_v, \quad C_{v_\perp}\equiv\tfrac{2}{3} C_v. \end{equation}

Correspondingly, the internal energy density (2.16a,b) of the plasma can be split into a sum of densities

(2.39)\begin{equation} \varepsilon=\frac{p_\parallel{+}2 p_\perp}{2} =\frac{p_\parallel}{(\gamma_\parallel{-}1)} +\frac{p_\perp}{(\gamma_\perp{-}1)} \equiv \varepsilon_\parallel{+}\varepsilon_\perp, \end{equation}

with effective adiabatic indices:

(2.40a,b)\begin{equation} \gamma_\parallel{=}3, \quad \gamma_\perp{=}2 . \end{equation}

Further note that

(2.41a-c)\begin{equation} \varepsilon=\frac{p}{\gamma-1}, \quad p=\frac{1}{3}(p_\parallel{+}2 p_\perp), \quad \gamma=\frac{5}{3} \end{equation}

in terms of the gas pressure $p$.

In addition, by applying the standard ideal gas law (2.33) to each of the pressure components $p_\parallel$ and $p_\perp$, we can define corresponding temperatures:

(2.42a,b)\begin{equation} T_\parallel{\equiv} p_\parallel{/}(\rho R) \quad\text{and}\quad T_\perp{\equiv} p_\perp{/}(\rho R) .\end{equation}

Then the internal energy density (2.39) can be expressed as

(2.43)\begin{equation} \varepsilon = R\rho\left(\tfrac{1}{2}T_\parallel{+} T_\perp\right) , \end{equation}

which yields the relations

(2.44a-c)\begin{equation} C_{v_\parallel} = e_{T_\parallel} = \tfrac{1}{2} R, \quad C_{v_\perp} = e_{T_\perp} = R, \quad C_v = \tfrac{3}{2} R \end{equation}

from the specific heats (2.38a,b). Finally, by substituting the double adiabatic integrals (2.13a,b) into the quantities (2.42a,b) and using expressions (2.37) and (2.44a–c), we obtain

(2.45a,b)\begin{equation} \frac{\partial T_\parallel}{\partial S_\parallel} = 2 T_\parallel{/}R, \quad \frac{\partial T_\perp}{\partial S_\perp} = T_\perp{/}R, \end{equation}

which implies the expected thermodynamic relations:

(2.46a,b)\begin{equation} e_{S_\parallel} = T_\parallel, \quad e_{S_\perp} = T_\perp. \end{equation}

The preceding approach to the entropy of CGL plasmas has been adopted by Du et al. (Reference Du, Zank, Li and Guo2020). It is equivalent to the thermodynamic approach specialised to the case of an ideal gas in which the gas law is assumed to hold for all three pressures $p$, $p_\parallel$ and $p_\perp$.

The net upshot of this discussion is that we take $e=e(\rho,S,B)$ as the equation of state for the gas in the rest of the paper. Note, however, that $S_\parallel =S_\parallel (S)$ and $S_\perp (S)=S-S_\parallel (S)$ are functions of $S$. Here $e=e(\rho,S,B)$ is the form of the equation of state used by Holm & Kupershmidt (Reference Holm and Kupershmidt1986). The CGL Poisson bracket of Holm & Kupershmidt (Reference Holm and Kupershmidt1986) uses the entropy variable $\sigma =\rho S$ to describe the complicated thermodynamics of the CGL plasma.

2.3 Total mass and energy conservation laws

The mass continuity equation (2.1) can be expressed in the familiar co-moving form

(2.47)\begin{equation} \frac{\partial \rho}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla} \rho ={-}\rho \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} . \end{equation}

On a volume $V(t)$ moving with the fluid, the corresponding mass integral is conserved:

(2.48)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{V(t)} \rho \,{\rm d}^3x =0 . \end{equation}

A co-moving equation for the internal energy density (2.12) is obtained by combining the parallel and perpendicular pressure equations (2.8)–(2.9), which yields

(2.49)\begin{equation} \frac{\partial \varepsilon}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}(\varepsilon \boldsymbol{u}) +(p_\parallel{-}p_\perp)\boldsymbol{\tau}\boldsymbol{\tau}\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{u} +p_\perp\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0. \end{equation}

This equation can be written in terms of the gyrotropic pressure tensor ${\mathsf{{p}}}$ by noting from expression (2.10a–c) that

(2.50)\begin{equation} {\mathsf{{p}}}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u}=p_\perp \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} +(p_\parallel{-}p_\perp) \boldsymbol{\tau}\boldsymbol{\tau}\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{u} . \end{equation}

Then the co-moving equation (2.49) takes the form

(2.51)\begin{equation} \frac{\partial \varepsilon}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\varepsilon \boldsymbol{u} +{\mathsf{{p}}}\boldsymbol{\cdot}\boldsymbol{u}\right)=\boldsymbol{u}\boldsymbol{\cdot}( \boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}), \end{equation}

which is analogous to the internal energy density equation in fluid dynamics and MHD.

The total kinetic energy equation for the plasma is obtained by taking the scalar product of the momentum equation (2.2) with $\boldsymbol {u}$, which yields

(2.52)\begin{equation} \frac{\partial}{\partial t}\left(\frac{1}{2}\rho u^2+\rho\varPhi\right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\boldsymbol{u}\left(\frac{1}{2}\rho u^2+\rho\varPhi\right)\right) ={-}\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}} +\boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{E}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{B} \frac{\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B}}{\mu_0}, \end{equation}

where $u = |\boldsymbol {u}|$, and

(2.53a,b)\begin{equation} \boldsymbol{E}={-}\boldsymbol{u}\times\boldsymbol{B}\quad \hbox{and}\quad \boldsymbol{J} =\frac{\boldsymbol{\nabla}\times\boldsymbol{B}}{\mu_0}, \end{equation}

are the electric field $\boldsymbol {E}$ and the electric current $\boldsymbol {J}$.

Using Maxwell's equations (2.6)–(2.7), we obtain Poynting's theorem (the electromagnetic energy equation) in the form:

(2.54)\begin{equation} \frac{\partial}{\partial t}\left(\frac{1}{2\mu_0}B^2\right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\frac{1}{\mu_0}\boldsymbol{E}\times \boldsymbol{B}\right) ={-}\boldsymbol{J}\boldsymbol{\cdot}\boldsymbol{E} -\frac{1}{\mu_0} (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B})(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{B}). \end{equation}

The equation for the total energy in conserved form is obtained by adding the internal energy equation (2.51), the kinetic energy equation (2.52) and the electromagnetic energy equation (2.54), which gives

(2.55)\begin{equation} \frac{\partial}{\partial t}\left(\frac{1}{2}\rho u^2+\rho\varPhi +\varepsilon+\frac{1}{2\mu_0}B^2\right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\left(\frac{1}{2}\rho u^2+\rho \varPhi+\varepsilon\right) \boldsymbol{u} +{\mathsf{{p}}}\boldsymbol{\cdot}\boldsymbol{u} \!+\!\frac{1}{\mu_0} \boldsymbol{E}\times\boldsymbol{B}\right)\!=\!0. \end{equation}

It is useful here to note that the Poynting electromagnetic energy flux is

(2.56)\begin{equation} \boldsymbol{F}_{\text{em}} = \frac{1}{\mu_0} \boldsymbol{E}\times\boldsymbol{B} = \frac{B^2}{2\mu_0} \boldsymbol{u} + \boldsymbol{u}\boldsymbol{\cdot}{\mathsf{{M}}}_B. \end{equation}

The resulting energy balance equation on a volume $V(t)$ moving with the fluid is given by

(2.57)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{V(t)} \left(\frac{1}{2}\rho u^2+\rho\varPhi +\varepsilon+\frac{1}{2\mu_0}B^2\right){\rm d}^3x ={-}\oint_{\partial V(t)} \boldsymbol{u}\boldsymbol{\cdot} \left({\mathsf{{p}}}+ {\mathsf{{M}}}_B\right)\boldsymbol{\cdot}\hat{\boldsymbol{n}}\, {\rm d}A \end{equation}

in terms of the magnetic pressure tensor (2.4). The flux terms in (2.57) represent the rate at which the total pressure tensor ${\mathsf{{p}}}+ {\mathsf{{M}}}_B$ does work on the fluid. Note that there is no contribution due to advection of the total energy density. Further discussion is provided in § 6.1.1 (see also Padhye Reference Padhye1998; Anco & Dar Reference Anco and Dar2009, Reference Anco and Dar2010). In addition, note that if the total pressure tensor has no perpendicular component at the moving boundary, i.e. $({\mathsf{{p}}}+{\mathsf{{M}}}_B)\boldsymbol {\cdot }\hat {\boldsymbol {n}}=0$, then the moving energy integral will be conserved (i.e. a constant of the motion).

2.4 Cross-helicity

The cross-helicity transport equation for a CGL plasma is obtained by taking the scalar product of $\boldsymbol {u}$ with Faraday's equation (2.6) plus the scalar product of $\boldsymbol {B}$ with the momentum equation (2.2), in the following form. By a standard cross-product identity, Faraday's equation can be written

(2.58)\begin{equation} \frac{\partial}{\partial t}\boldsymbol{B} - \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} + \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{B} + (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u})\boldsymbol{B} =0. \end{equation}

The momentum equation minus $\boldsymbol {u}$ times the mass continuity equation (2.1) yields the velocity equation:

(2.59)\begin{equation} \frac{{\rm d}}{{\rm d}t}\boldsymbol{u}={-}\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot}\left({\mathsf{{p}}} +{\mathsf{{M}}}_B\right)-\boldsymbol{\nabla}\varPhi , \end{equation}

where $-(1/\rho )\boldsymbol {\nabla }\boldsymbol {\cdot }{\mathsf{{p}}}$ is the acceleration due to the anisotropic pressure. Forming and adding the respective scalar products with $\boldsymbol {u}$ and $\boldsymbol {B}$ then gives the equation for cross-helicity density $\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {B}$ as

(2.60)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{B}) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{B})\boldsymbol{u} +\left(\varPhi -\frac{1}{2} u^2\right)\boldsymbol{B}\right] ={-}\boldsymbol{B}\boldsymbol{\cdot}\left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}\right) , \end{equation}

where $u = |\boldsymbol {u}|$.

The thermodynamic form of this transport equation (2.60) is obtained by using the equation

(2.61)\begin{equation} {-}\boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}=\boldsymbol{B}\times (\boldsymbol{\nabla}\times\boldsymbol{\varOmega}) -\boldsymbol{\varOmega}(\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B})+\rho(T\boldsymbol{\nabla} S-\boldsymbol{\nabla} h), \end{equation}

which relies on the Pfaffian equation (first law of thermodynamics) (2.15) for a CGL plasma with an internal energy $e(\rho,B,S)$, where, from the thermodynamic relations (2.18a–c), $T=e_S$ is the gas temperature and $h=\varepsilon _\rho =e+\rho e_\rho$ is the enthalpy of the fluid:

(2.62)\begin{equation} h= \frac{3p_\parallel{+}2p_\perp}{2\rho} . \end{equation}

Here

(2.63)\begin{equation} \boldsymbol{\varOmega}=\frac{p_{\varDelta}}{B} \boldsymbol{\tau} . \end{equation}

Note that the case $\boldsymbol {\varOmega }=0$ corresponds to the MHD limit in which the anisotropy vanishes, $p_{\varDelta }=p_\parallel -p_\perp =0$. A derivation of the pressure divergence equation (2.61) is given in Appendix B. An alternative derivation is provided within the EP formulation of the CGL equations in § 4.1 and Appendix E. The scalar product of this equation with $({1}/{\rho })\boldsymbol {B}$ reduces to

(2.64)\begin{equation} {-}\boldsymbol{B}\boldsymbol{\cdot} \left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}\right) =\boldsymbol{B}\boldsymbol{\cdot}\left(T\boldsymbol{\nabla} S\right) -\boldsymbol{\nabla}\boldsymbol{\cdot}\left(h\boldsymbol{B}\right) \end{equation}

using Gauss’ law (2.7). As a result, the cross-helicity density transport equation in thermodynamic form is simply

(2.65)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{u\cdot B}) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{B})\boldsymbol{u} +\left(\varPhi+h-\frac{1}{2} u^2\right)\boldsymbol{B}\right] =\boldsymbol{B}\boldsymbol{\cdot}(T \boldsymbol{\nabla} S) . \end{equation}

The term on the right-hand side of the cross-helicity transport equation (2.65) can be written in a conserved form through use of the equation

(2.66)\begin{equation} \frac{\partial}{\partial t}\left(\phi\boldsymbol{\nabla} S\boldsymbol{\cdot}\boldsymbol{B}\right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\left[\left(\phi\boldsymbol{\nabla} S\boldsymbol{\cdot}\boldsymbol{B}\right)\boldsymbol{u}\right] = \left(\boldsymbol{\nabla}S\boldsymbol{\cdot}\boldsymbol{B}\right)\frac{{\rm d}}{{\rm d}t}\phi, \end{equation}

which holds for any scalar variable $\phi$ and follows from Faraday's equation (2.6) and the entropy equation (2.5). This leads to the non-local cross-helicity density conservation law

(2.67)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{w}\boldsymbol{\cdot}\boldsymbol{B}) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[(\boldsymbol{w}\boldsymbol{\cdot}\boldsymbol{B})\boldsymbol{u} +\left(\varPhi+h-\frac{1}{2} u^2\right)\boldsymbol{B}\right]=0, \end{equation}

in terms of

(2.68)\begin{equation} \boldsymbol{w}=\boldsymbol{u}+r \boldsymbol{\nabla} S, \end{equation}

where $r$ is a non-local variable obtained by integrating the temperature $T$ back along the path of the Lagrangian fluid element. Specifically,

(2.69a,b)\begin{equation} \frac{{\rm d}r}{{\rm d}t}={-}T, \quad r={-}\int^t T\, {\rm d}t, \end{equation}

where ${\rm d}/{\rm d}t=\partial /\partial t+\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla }$ is the Lagrangian time derivative.

The non-local conservation law (2.67) yields a moving cross-helicity balance equation (cf. Appendix C):

(2.70)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{V(t)} \boldsymbol{w}\boldsymbol{\cdot}\boldsymbol{B} \,{\rm d}^3x ={-}\oint_{\partial V(t)} \left(\varPhi+h-\frac{1}{2} u^2\right)\boldsymbol{B}\boldsymbol{\cdot}\hat{\boldsymbol{n}}\, {\rm d}A \end{equation}

on a volume $V(t)$ moving with the fluid. Note that if $\boldsymbol {B}$ is perpendicular to the boundary, i.e. $\boldsymbol {B}\boldsymbol {\cdot }\hat {\boldsymbol {n}}=0$, then the cross-helicity integral $\int _{V(t)} \boldsymbol {w}\boldsymbol {\cdot }\boldsymbol {B} \,{\rm d}^3\boldsymbol {x}$ is conserved in the flow.

An alternative form of the non-local conservation law (2.67) is obtained by directly taking the divergence of the pressure tensor ${\mathsf{{p}}}$ using the gyrotropic expression (2.10a–c), and combining it with the gradient of the enthalpy (2.62). As shown in Appendix B, this gives

(2.71)\begin{align} \boldsymbol{B}\boldsymbol{\cdot}\left(\frac{1}{\rho} \boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}-\boldsymbol{\nabla} h\right) & ={-}\left( \frac{p_\parallel}{2\rho} \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln c_\parallel(S) + \frac{p_\perp}{\rho} \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}\ln c_\perp(S) \right)\nonumber\\ & ={-}\left( \frac{p_\parallel}{2\rho} \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}{\bar{S}}_\parallel{+} \frac{p_\perp}{\rho} \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla}{\bar{S}}_\perp \right) \end{align}

in terms of the adiabatic integrals (2.13a,b) and (2.37). Expression (2.71) is equivalent to $-\boldsymbol {B}\boldsymbol {\cdot }(T\boldsymbol {\nabla } S)$ through the temperature expression (2.32). Instead, if we use the gas law temperatures (2.49) and the specific heats (2.51) associated with $p_\parallel$ and $p_\perp$, then we can write

(2.72)\begin{equation} \boldsymbol{B}\boldsymbol{\cdot}\left(\frac{1}{\rho} \boldsymbol{\nabla}\boldsymbol{\cdot}{\mathsf{{p}}}-\boldsymbol{\nabla} h\right) ={-}\left( T_\parallel \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} S_\parallel{+} T_\perp \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} S_\perp \right) \end{equation}

in term of the gas law entropies (2.37). Substituting expression (2.72) into the cross-helicity density transport equation (2.65) yields

(2.73)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{B}) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[(\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{B})\boldsymbol{u} +\boldsymbol{B}\left(\varPhi+h-\frac{1}{2} u^2\right)\right] =\boldsymbol{B}\boldsymbol{\cdot}(T_\parallel \boldsymbol{\nabla} S_\parallel{+}T_\perp\boldsymbol{\nabla} S_\perp) . \end{equation}

This result (2.73) can be also derived using the Newcomb (Reference Newcomb1962) action principle for CGL plasmas shown in § 4.

The corresponding non-local cross-helicity density conservation law is given by

(2.74)\begin{equation} \frac{\partial}{\partial t}(\tilde{\boldsymbol{w}}\boldsymbol{\cdot}\boldsymbol{B}) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[(\tilde{\boldsymbol{w}}\boldsymbol{\cdot} \boldsymbol{B})\boldsymbol{u} +\boldsymbol{B}\left(\varPhi+h-\frac{1}{2} u^2\right)\right]=0, \end{equation}

where

(2.75)\begin{equation} \tilde{\boldsymbol{w}}=\boldsymbol{u}+r_\parallel \boldsymbol{\nabla} S_\parallel{+}r_\perp \boldsymbol{\nabla} S_\perp, \end{equation}

with $r_\parallel$ and $r_\perp$ being non-local variables defined by the equations:

(2.76a,b)\begin{equation} \frac{{\rm d}r_\parallel}{{\rm d}t}={-}T_\parallel, \quad \frac{{\rm d}r_\perp}{{\rm d}t}={-}T_\perp. \end{equation}

The non-local cross-helicity density conservation law for MHD analogous to equation (2.67) was developed in Webb et al. (Reference Webb, Dasgupta, McKenzie, Hu and Zank2014a,Reference Webb, Dasgupta, McKenzie, Hu and Zankb), Webb (Reference Webb2018), Webb & Anco (Reference Webb and Anco2019), Yahalom (Reference Yahalom2017a,Reference Yahalomb) and Yahalom & Qin (Reference Yahalom and Qin2021). A topological interpretation of the generalised cross-helicity conservation law has been found in terms of an MHD Aharonov–Bohm effect in Yahalom (Reference Yahalom2017a,Reference Yahalomb). Yahalom (Reference Yahalom2013) discussed a topological interpretation of magnetic helicity as an Aharonov Bohm effect in MHD.

If $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla } S=0$, then the non-local conservation law (2.67) reduces to a local cross- helicity density conservation law given by $r= 0$ and $\boldsymbol {w}=\boldsymbol {u}$. This result is analogous to the local cross-helicity conservation law in MHD (e.g. Webb Reference Webb2018; Webb & Anco Reference Webb and Anco2019).

It is interesting to note that the velocity equation (2.59) can be written in the suggestive form:

(2.77)\begin{equation} \frac{{\rm d}\boldsymbol{u}}{{\rm d}t}=T\boldsymbol{\nabla} S-\boldsymbol{\nabla} h +\frac{\tilde{\boldsymbol{J}}\times \boldsymbol{B}}{\rho}-\boldsymbol{\nabla}\varPhi , \end{equation}

where

(2.78a,b)\begin{equation} \tilde{\boldsymbol{J}}\equiv \boldsymbol{J}-\boldsymbol{\nabla}\times\boldsymbol{\varOmega} =\frac{\boldsymbol{\nabla}\times\tilde{\boldsymbol{B}}}{\mu_0}, \quad \tilde{\boldsymbol{B}}\equiv \boldsymbol{B}\left(1-\frac{\mu_0p_{\varDelta}}{B^2}\right). \end{equation}

This (2.77) turns out to arise directly from the non-canonical Hamiltonian formulation presented in § 5.1.

2.5 Magnetic helicity

In ideal MHD and in ideal CGL plasmas, Faraday's equation (2.6) written in terms of the electric field

(2.79)\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t}+\boldsymbol{\nabla}\times \boldsymbol{E}=0 \end{equation}

can be uncurled to give $\boldsymbol {E}=-\boldsymbol {u}\times \boldsymbol {B}$ in the form

(2.80)\begin{equation} \boldsymbol{E}={-}\boldsymbol{\nabla}\phi_E-\frac{\partial \boldsymbol{A}}{\partial t}={-}(\boldsymbol{u}\times\boldsymbol{B})\quad \hbox{where}\ \boldsymbol{B}=\boldsymbol{\nabla}\times \boldsymbol{A}. \end{equation}

The uncurled form of Faraday's equation (2.80) implies

(2.81)\begin{equation} \frac{\partial \boldsymbol{A}}{\partial t}+\boldsymbol{E}+\boldsymbol{\nabla}\phi_E=0. \end{equation}

Combining the scalar product of Faraday's equation (2.79) with $\boldsymbol {A}$ plus the scalar product of the uncurled equation (2.81) with $\boldsymbol {B}$ yields the magnetic helicity transport equation

(2.82)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{B})+\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\boldsymbol{E}\times \boldsymbol{A}+\phi_E \boldsymbol{B}\right)=0 , \end{equation}

which can also be written in the form

(2.83)\begin{equation} \frac{\partial}{\partial t}(\boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{B})+\boldsymbol{\nabla}\boldsymbol{\cdot} \left[(\boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{B})\boldsymbol{u}+(\phi_E-\boldsymbol{u}\boldsymbol{\cdot} \boldsymbol{A}) \boldsymbol{B}\right]=0. \end{equation}

The total magnetic helicity for a volume $V(t)$ moving with the fluid is given by

(2.84)\begin{equation} H^{{M}}=\int_{V(t)} \boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{B}\,{\rm d}^3x . \end{equation}

It satisfies the moving balance equation (cf. Appendix C)

(2.85)\begin{equation} \frac{{\rm d}H^{{M}}}{{\rm d}t}={-}\oint_{\partial V(t)} (\phi_E -\boldsymbol{A}\boldsymbol{\cdot} \boldsymbol{u})\boldsymbol{B}\boldsymbol{\cdot}\hat{\boldsymbol{n}}\, {\rm d}A , \end{equation}

where $\hat {\boldsymbol {n}}$ is the outward unit normal of the moving boundary $\partial V(t)$. Thus, if $\boldsymbol {B}$ is perpendicular to the boundary, i.e. $\boldsymbol {B}\boldsymbol {\cdot }\hat {\boldsymbol {n}}=0$, then $H^{{M}}$ is conserved in the flow. This discussion applies both to ideal MHD and also to ideal CGL plasmas. (e.g. Kruskal & Kulsrud Reference Kruskal and Kulsrud1958; Woltjer Reference Woltjer1958; Moffatt Reference Moffatt1969; Berger & Field Reference Berger and Field1984; Moffatt & Ricca Reference Moffatt and Ricca1992; Arnold & Khesin Reference Arnold and Khesin1998).

It is beyond the scope of the present exposition to discuss the issues of how to define the relative magnetic helicity for volumes $V$ for which $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {n}\neq 0$ on $\partial V$ (e.g. Berger & Field Reference Berger and Field1984; Finn & Antonsen Reference Finn and Antonsen1985, Reference Finn and Antonsen1988; Webb et al. Reference Webb, Hu, Dasgupta and Zank2010). Similar considerations apply to field line magnetic helicity (e.g. Prior & Yeates Reference Prior and Yeates2014) and absolute magnetic helicity (e.g. Low Reference Low2006, Reference Low2011; Berger & Hornig Reference Berger and Hornig2018).

It is of interest to note that, under gauge transformations (i.e. $\boldsymbol {A}\to \boldsymbol {A}+\boldsymbol {\nabla }\chi$ and $\phi _E\to \phi _E -\partial \chi /\partial t$ for an arbitrary function $\chi (t,\boldsymbol {x})$), the moving balance equation is gauge invariant, but the magnetic helicity integral changes by addition of a boundary integral: $H^{{M}}\to H^{{M}} + \oint _{\partial V(t)} \chi \boldsymbol {B}\boldsymbol {\cdot }\hat {\boldsymbol {n}}\, {\rm d}A$. However, if the electric field potential $\phi _E$ satisfies the gauge

(2.86)\begin{equation} \phi_E=\boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{u}, \end{equation}

then the moving flux of the magnetic helicity vanishes and the resulting (gauge-dependent) magnetic helicity $H^{{M}}$ is conserved in the flow. This gauge turns out to hold precisely when $\boldsymbol {A}$ is advected by the flow, which is shown in the following subsection.

2.6 Lie dragged (advected) quantities

A quantity, $a$, is Lie dragged by the fluid flow if its advective Lie derivative vanishes:

(2.87)\begin{equation} \left(\frac{\partial}{\partial t}+{\mathcal{L}}_{\boldsymbol{u}}\right) a=0, \end{equation}

where ${\mathcal {L}}_{\boldsymbol {u}}$ is the ordinary Lie derivative with respect to the vector field $\boldsymbol {u}$ (e.g. Tur & Yanovsky Reference Tur and Yanovsky1993; Webb et al. Reference Webb, Dasgupta, McKenzie, Hu and Zank2014a; Anco & Webb Reference Anco and Webb2020). For a scalar quantity, its advective Lie derivative reduces to its material derivative

(2.88)\begin{equation} \frac{{\rm d}}{{\rm d}t} = \frac{\partial}{\partial t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}. \end{equation}

For quantities that are vector fields or differential forms, the advective Lie derivative also contains a rotation-shear term which involves $\boldsymbol {\nabla } \boldsymbol {u}$.

In CGL plasmas, $S$ is an advected scalar, as are the double adiabatic integrals (2.13a,b). There are two basic advected non-scalar quantities: the differential form ${\boldsymbol {\alpha }}=\boldsymbol {\nabla } S\boldsymbol {\cdot }{\rm d}\boldsymbol {x}\equiv \boldsymbol {\nabla }_i S \,{{\rm d} x}^i$ and the vector field $\boldsymbol {b}=(1/\rho )\boldsymbol {B}\equiv b^i\ \boldsymbol {\nabla }_i$ (note the vector field $\boldsymbol {b}$ is a directional derivative operator). The contraction of the vector field $\boldsymbol {b}$ with the co-vector or differential form ${\boldsymbol {\alpha }}$ is a scalar (note that $\boldsymbol {\nabla }_i\rfloor \,{{\rm d} x}^j=\delta ^i_j$, and $\boldsymbol {b}\rfloor {\boldsymbol {\alpha }}=b^i \alpha _i\equiv b^i\boldsymbol {\nabla }_iS$). Note that $\boldsymbol {b}$ is Lie dragged with the flow, (this statement is equivalent to Faraday's equation, when one takes into account the mass continuity equation: i.e. $(\partial _t+{\mathcal {L}}_{\boldsymbol {u}})\boldsymbol {b}=0$ which implies $\partial _t \boldsymbol {b}+[\boldsymbol {u},\boldsymbol {b}]=0$). Here ${\boldsymbol {\alpha }}$ is a one-form or co-vector, that is Lie dragged with the flow. The inner product of the vector field $\boldsymbol {b}$ with the one-form ${\boldsymbol {\alpha }}$ is a scalar invariant which is advected with the flow (e.g. Tur & Yanovsky Reference Tur and Yanovsky1993). Thus, the quantities

(2.89a-c)\begin{equation} \frac{\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} S}{\rho}, \quad \frac{\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} \bar{S}_\parallel}{\rho}, \quad \frac{\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} \bar{S}_\perp}{\rho} \end{equation}

are advected scalars in CGL plasmas.

The gauge (2.86) in which the magnetic helicity integral is conserved is equivalent to Lie dragging the one-form $\alpha =\boldsymbol {A}\boldsymbol {\cdot }{\rm d} \boldsymbol {x}$ with the fluid flow:

(2.90)\begin{equation} \left(\frac{\partial}{\partial t}+{\mathcal{L}}_{\boldsymbol{u}}\right)(\boldsymbol{A}\boldsymbol{\cdot}{\rm d}\boldsymbol{x}) =\left(\frac{\partial \boldsymbol{A}}{\partial t}-\boldsymbol{u}\times \boldsymbol{B}+\boldsymbol{\nabla}(\boldsymbol{A} \boldsymbol{\cdot}\boldsymbol{u})\right)\boldsymbol{\cdot}{\rm d}\boldsymbol{x}=0 , \end{equation}

which vanishes due to the uncurled form of Faraday's equation (2.80) combined with equation (2.86). Holm & Kupershmidt (Reference Holm and Kupershmidt1983a,Reference Holm and Kupershmidtb) used this advected-$\boldsymbol {A}$ gauge in the formulation of non-canonical Poisson brackets for MHD and for multi-fluid plasmas (see also Gordin & Petviashvili Reference Gordin and Petviashvili1987, Reference Gordin and Petviashvili1989; Padhye & Morrison Reference Padhye and Morrison1996a,Reference Padhye and Morrisonb; Padhye Reference Padhye1998).

3.1 Map formulae

We now formulate the maps between the CGL plasma variables $\rho$, $S$, $\boldsymbol {B}$, $p_\parallel$ and $p_\perp$ and their Lagrangian counterparts $\rho _0(\boldsymbol {x}_0)$, $S_0(\boldsymbol {x}_0)$, $\boldsymbol {B}_0(\boldsymbol {x}_0)$, $p_\parallel (\boldsymbol {x}_0)$ and $p_\perp (\boldsymbol {x}_0)$, respectively.

The density $\rho _0$ of a fluid element with label $\boldsymbol {x}_0$ is related to the Eulerian density by the mass conservation equation

(3.7)\begin{equation} \rho \,{\rm d}^3x=\rho_0\,{\rm d}^3x_0, \end{equation}

where ${\rm d}^3x = J \,{\rm d}^3x_0$. This implies

(3.8)\begin{equation} \rho=\frac{\rho_0}{J} . \end{equation}

From Newcomb (Reference Newcomb1962) the Cartesian components of $B^i$ in the Eulerian frame are related to the Lagrangian magnetic field component $B_0^k$ by the equation:

(3.9)\begin{equation} B^i=\frac{X_{ik} B_0^k}{J}. \end{equation}

The derivation of (3.9) follows by noting

(3.10)\begin{gather} B^i\,{\rm d}\sigma_i=B^k_0\,{\rm d}\sigma_{0k}, \end{gather}

(3.11)\begin{gather}{\rm d}^3x={{\rm d}x}^i\,{\rm d}\sigma_i=J \,{\rm d}^3x_0=J\,{\rm d}x^k_0 \,{\rm d}\sigma_{ok} \equiv JY_{ki} \,{{\rm d} x}^i\,{\rm d}\sigma_{0k}. \end{gather}

Equation (3.10) describes the conservation of magnetic flux, where ${\rm d}\sigma _i$ is the flux tube area normal to the $x^i$ coordinate surface, and ${\rm d}\sigma _{0k}$ is the flux tube area normal to the $x^k_0$ surface in the Lagrangian frame. Equation (3.11) relates the volume elements ${\rm d}^3x$ to ${\rm d}^3x_0$. From (3.11) one obtains

(3.12)\begin{equation} {\rm d}\sigma_i=A_{ik}\,{\rm d}\sigma_0^k, \end{equation}

for the transformation between the area elements ${\rm d}\sigma _i$ and ${\rm d}\sigma _0^k$. The magnetic flux conservation equation (3.10) now gives the transformation $B_0^k=A_{ik} B^i$ which, in turn, implies the transformation (3.9).

An alternative form of the relation (3.9) arises from the Lie dragged or frozen-in vector field

(3.13)\begin{equation} \boldsymbol{b}=\frac{B^i}{\rho} \frac{\partial}{\partial x^i}=\frac{B_0^k}{\rho_0} \frac{\partial}{\partial x_0^k}\equiv \frac{B_0^k}{\rho_0} X_{ik}\frac{\partial}{\partial x^i}, \end{equation}

which implies the transformation (3.9) between $B^i$ and $B_0^k$. In (3.13) we use the modern differential geometry notion of a vector field as a directional derivative operator (e.g. Misner, Thorne & Wheeler Reference Misner, Thorne and Wheeler1973).

From the relation (3.9), we have

(3.14)\begin{equation} B^2=\frac{\zeta^2 B_0^2}{J^2} \quad\hbox{where}\ \zeta^2=X_{ij}\tau_0^j X_{ik}\tau_0^k \end{equation}

in terms of the unit vector $\tau _0=B_0^{-1}\boldsymbol {B}_0$ along the magnetic field $\boldsymbol {B}_0$ (cf. (2.3a–c)), with $B_0=|\boldsymbol {B}_0|$ being the magnetic field strength at $\boldsymbol {x}_0$. Hence, we obtain

(3.15a,b)\begin{equation} B=\frac{\zeta\,B_0}{J}, \quad \tau^i =\frac{B^i}{B} =\frac{X_{ij} \tau_0^j}{\zeta} . \end{equation}

The latter relation can be inverted to obtain $\boldsymbol {\tau }_0$ in terms of $\boldsymbol {\tau }$ through the formulae (3.5a–c):

(3.16)\begin{equation} \tau_0^i = \frac{\zeta}{J} A_{ji} \tau^j . \end{equation}

Next, from the double adiabatic conservation laws (2.11a,b), we have the frozen-in quantities

(3.17a,b)\begin{equation} \frac{p_\perp}{\rho B}=\frac{p_{{\perp} 0}}{\rho_0B_0}, \quad \frac{p_\parallel B^2}{\rho^3}=\frac{p_{{\parallel} 0} B_0^2}{\rho_0^3} . \end{equation}

By combining these equations with (3.15a,b) and (3.8), we obtain

(3.18a,b)\begin{equation} p_\parallel{=}\frac{p_{{\parallel} 0}}{J\zeta^2}, \quad p_\perp{=}\frac{p_{{\perp} 0}\zeta}{J^2}, \end{equation}

which gives the respective relations between the pressures $p_{\parallel 0}$ and $p_{\perp 0}$ at a fluid element with label $\boldsymbol {x}_0$ and the Eulerian pressures $p_\parallel$ and $p_\perp$.

Finally, we note that the Eulerian entropy $S$ is equal to the entropy $S_0$ at a fluid element with label $\boldsymbol {x}_0$, since $S$ is frozen-in by the transport equation (2.5).

3.2 Lagrangian variational principle and Euler–Lagrange equations

The Lagrangian variational principle for MHD and for CGL plasmas was obtained in Newcomb (Reference Newcomb1962):

(3.19)\begin{equation} {\mathcal{A}}=\int_{t_0}^{t_1} \int_{V} L\, {\rm d}^3x\,{\rm d}t = \int_{t_0}^{t_1}\int_{V_0} L_0\, {\rm d}^3x_0\,{\rm d}t,\end{equation}

where $L$ is the Lagrangian in Eulerian variables and $L_0$ is its counterpart arising through the Lagrangian map. Here $V$ is a fixed spatial domain and $[t_0,t_1]$ is a fixed time interval. In general, $L$ will be given by the kinetic energy density minus the potential energy density for a fluid element. The kinetic energy density for both MHD and CGL plasmas is simply $\frac {1}{2}\rho u^2$, with $u=|\boldsymbol {u}|$. Subtracting this expression from the total energy density for a CGL plasma (cf. the moving energy balance equation (2.57))

(3.20)\begin{equation} H \equiv \frac{1}{2}\rho u^2+\rho\varPhi +\varepsilon+\frac{1}{2\mu_0} B^2 \end{equation}

yields the potential energy density $\rho \varPhi +\varepsilon +({1}/{2\mu _0})B^2$, where $\varepsilon$ is the internal energy density (2.12). Thus, a CGL plasma has the Lagrangian

(3.21)\begin{equation} L=\frac{1}{2}\rho u^2 -\rho\varPhi -\varepsilon -\frac{1}{2\mu_0}B^2 , \end{equation}

where, for the sake of completeness, we have included the energy contributed by an external gravitational potential $\varPhi (\boldsymbol {x})$ (e.g. which would arise from the gravitational field of the Sun in the case of solar and interplanetary physics). The corresponding Lagrangian $L_0$ is obtained from the relation $L\,{\rm d}^3x=L_0\,{\rm d}^3x_0 = LJ \,{\rm d}^3x_0$, which gives

(3.22)\begin{equation} L_0=J L . \end{equation}

Substituting the Eulerian expressions (3.21), (3.18a,b), (3.14), (3.8) and (3.2) into $L_0$, and then using expression (2.12) for the internal energy density in terms of the pressures $p_\parallel$ and $p_\perp$, we obtain

(3.23)\begin{equation} L_0 =\rho_0(\boldsymbol{x}_0)\left(\frac{1}{2} |\dot{\boldsymbol{X}}|^2-\varPhi(\boldsymbol{X})\right) -\left(\frac{p_{{\parallel} 0}(\boldsymbol{x}_0)}{2\zeta^2} +\frac{p_{{\perp} 0}(\boldsymbol{x}_0)\zeta}{J}\right) -\frac{\zeta^2 B_0(\boldsymbol{x}_0)^2}{2\mu_0 J} . \end{equation}

The Euler–Lagrange equations of the resulting action principle (3.19) for a CGL plasma are obtained by variation of the Cartesian components $X^i$ of the Lagrangian map (3.2), modulo boundary terms: $\delta {\mathcal {A}}/\delta X^i =0$. In essence, we work in a reference frame moving with the fluid flow, where the dynamics is described in terms of how the Eulerian position $\boldsymbol {x}$ of the frame varies with the fluid labels $\boldsymbol {x}_0$ and the time $t$ through the Lagrangian map (3.2). Equivalently, the Euler–Lagrange equations are given by applying to $L_0$ the Euler operator (variational derivative) $E_{X^i}$ in the calculus of variations, where $X^i(x_0^j,t)$ is a function of $x^j_0$ and $t$. Thus,

(3.24)\begin{equation} \frac{\delta{\mathcal{A}}}{\delta X^i} \equiv E_{X^i}(L_0) =\frac{\partial L_0}{\partial X^i} -\frac{\partial}{\partial t}\left(\frac{\partial L_0}{\partial \dot{X}^i}\right) -\frac{\partial}{\partial x_0^j}\left(\frac{\partial L_0}{\partial X_{ij}}\right)=0 , \end{equation}

where $\partial /\partial _{t}$ and $\partial /\partial _{x_0^j}$ act as total derivatives. (See § 6 for more details of variational calculus.) In Appendix C, we show that when (3.24) is simplified and expressed in terms of Eulerian variables, it reduces to the Eulerian momentum equation (2.2).

3.3 Hamilton's equations

We now derive the corresponding Hamilton formulation of the Euler–Lagrange equation (3.24) for a CGL plasma. Here we write $X^i=x^i$ for simplicity.

First, the generalised momentum is defined as

(3.25)\begin{equation} {\rm \pi}_i=\frac{\partial L_0}{\partial \dot{x}^i}=\rho_0 \dot{x}^i. \end{equation}

Next, the Hamiltonian density is defined by the Legendre transformation:

(3.26)\begin{equation} H_0={\rm \pi}_k\dot{x}^k -L_0. \end{equation}

After substitution of the CGL Lagrangian (3.23) along with $\dot {x}^i$ in terms of the momentum (3.25), we then obtain the expression

(3.27)\begin{equation} H_0=\frac{{\rm \pi}_i {\rm \pi}_i}{2\rho_0}+\rho_0\varPhi(\boldsymbol{x}) + \frac{p_{{\parallel} 0}}{2\zeta^2}+\frac{p_{{\perp} 0}\zeta}{J} +\frac{X_{ij}X_{ik}B_0^jB_0^k}{2\mu_0 J} . \end{equation}

In Eulerian variables, this expression is given by

(3.28)\begin{equation} H_0 = J H, \end{equation}

where $H$ is the Eulerian total energy density (3.20). The resulting Hamiltonian equations of motion consist of

(3.29a,b)\begin{equation} \dot{x}^i=\frac{\delta {\mathcal{H}}}{\delta{\rm \pi}^i}, \quad \dot{\rm \pi}^i ={-}\frac{\delta {\mathcal{H}}}{\delta x^i} , \end{equation}

which yield the momentum relation (3.25) and the Euler–Lagrange equation (3.24), where the Hamiltonian function is defined as

(3.30)\begin{equation} {\mathcal{H}}=\int_{V} H_0\, {\rm d}^3x_0 =\int_{V} H\, {\rm d}^3x . \end{equation}

As in ordinary Hamiltonian mechanics, with the use of the canonical variables $q^i= x^i$ and $p^i={\rm \pi} ^i$, Hamilton's equations (3.29a,b) can be written in a canonical Poisson bracket formulation:

(3.31)\begin{equation} \dot{{\mathcal{F}}}=\left\{{\mathcal{F}},{\mathcal{H}}\right\} =\int\left( {\mathcal{F}}_{q^i} {\mathcal{H}}_{p^i} -{\mathcal{F}}_{q^i}{\mathcal{H}}_{p^i} \right) {\rm d}^3x_0 . \end{equation}

Here ${\mathcal {F}}$ is a general functional of $q^i$ and $p^i$ (see Morrison Reference Morrison1982). There is an equivalent, non-canonical Poisson bracket that employs only Eulerian variables, which we present in § 5.

4.1 The EP equation

For CGL plasmas, the EP action principle is given by

(4.18)\begin{equation} {\mathcal{J}}=\int_{t_0}^{t_1} \int_{V} L\, {\rm d}^3x\, {\rm d}t , \end{equation}

where $L$ is the Lagrangian density (3.21). Here $V$ is a fixed spatial domain and $[t_0,t_1]$ is a fixed time interval. The stationary points of this action principle, $\delta {\mathcal {J}}=0$, under the variations (4.17), (4.10), (4.11) and (4.13) of the respective variables $\boldsymbol {u}$, $S$, $\rho$ and $\boldsymbol {B}$, turn out to yield the Eulerian momentum equation (2.2), as we now show.

The derivation of the stationary points is non-trivial because of the form of the variations in terms of $\eta$ given by (4.5) and (4.6). Specifically, the variation of ${\mathcal {J}}$ needs to be put into the form

(4.19)\begin{equation} \delta{\mathcal{J}}=\int_{t_0}^{t_1} \left\langle \eta, F\right\rangle {\rm d}t =\int_{t_0}^{t_1} \int_{V} {\boldsymbol{\eta}}\boldsymbol{\cdot} \boldsymbol{F}\,{\rm d}^3x\, {\rm d}t \end{equation}

modulo boundary terms, where $F=\boldsymbol {F}\boldsymbol {\cdot } \partial _{\boldsymbol {x}}$ is both a left-invariant vector field in the tangent space of $G$ and the directional derivative along a vector field $\boldsymbol {F}$ in Euclidean space. Then, because ${\boldsymbol {\eta }}$ is an arbitrary vector field (corresponding to an arbitrary variation $\delta g$), the equation yielding the stationary points of ${\mathcal {J}}$ is $\boldsymbol {F}=0$.

We now proceed to find $\delta {\mathcal {J}}$ and $\boldsymbol {F}$. A general expression is available for $\boldsymbol {F}$ in Holm et al. (Reference Holm, Marsden and Ratiu1998), which involves the diamond operator, $\diamond$, defined by property

(4.20)\begin{equation} \left\langle\frac{\delta L}{\delta a}\diamond a,\eta\right\rangle ={-}\left\langle\frac{\delta L}{\delta {a}},{\mathcal{L}}_{\eta}(a)\right\rangle , \end{equation}

where, as earlier, $a$ is an advected quantity. It will be instructive to instead show how to obtain $\boldsymbol {F}$ from the variation (4.17), (4.10), (4.11) and (4.13) directly using the Eulerian variables.

From the Lagrangian density (3.21), we obtain

(4.21)\begin{equation} \left.\begin{gathered} \frac{\delta L}{\delta\boldsymbol{u}} =\rho\boldsymbol{u}, \quad \frac{\delta L}{\delta S} ={-}\varepsilon_S ={-}\rho T, \quad \frac{\delta L}{\delta\rho} =\frac{1}{2}u^2-\varepsilon_\rho-\varPhi =\frac{1}{2}u^2-h-\varPhi,\\ \frac{\delta L}{\delta\boldsymbol{B}} ={-}\varepsilon_B \boldsymbol{\tau} -\frac{\boldsymbol{B}}{\mu_0} =\left(\frac{p_{\varDelta}}{B^2} -\frac{1}{\mu_0}\right) \boldsymbol{B} \equiv{-} \frac{1}{\mu_0}\tilde{\boldsymbol{B}}, \end{gathered}\right\}\end{equation}

using the thermodynamic relations (2.18a–c), where $T$ is the temperature and $h$ is the enthalpy (2.62) and $p_{\varDelta }=p_\parallel -p_\perp$ is the pressure anisotropy. Here we have also used the identity $\delta B = \boldsymbol {\tau }\boldsymbol {\cdot }\delta \boldsymbol {B}$, with ${\boldsymbol {\tau }}=\boldsymbol {B}/B$ being the unit vector along $\boldsymbol {B}$. Note that $\tilde {\boldsymbol {B}}$ is related $\boldsymbol {B}$ by (2.78a,b). The main difference in these variational derivative expressions (4.21) compared with the MHD case is the addition of the anisotropy term in $\delta L/\delta \boldsymbol {B}$, i.e. $\boldsymbol {B}$ is replaced by $\tilde {\boldsymbol {B}}$.

The total variation of $L$ is given by combining the products of the variational derivatives (4.21) and the corresponding variations (4.17), (4.10), (4.11) and (4.13). This yields

(4.22)\begin{align} \delta L& = \frac{\delta L}{\delta\boldsymbol{u}}\boldsymbol{\cdot}\delta\boldsymbol{u} + \frac{\delta L}{\delta S}{\delta S} + \frac{\delta L}{\delta\rho}{\delta\rho} + \frac{\delta L}{\delta\boldsymbol{B}}\boldsymbol{\cdot}\delta\boldsymbol{B} \nonumber\\ & = \rho\boldsymbol{u}\boldsymbol{\cdot} \left( \dot{\boldsymbol{\eta}} - {\mathcal{L}}_{\boldsymbol{\eta}} \boldsymbol{u}\right) -\rho T \left( -{\boldsymbol{\eta}}\boldsymbol{\cdot}\boldsymbol{\nabla} S \right) +\left( \frac{1}{2}u^2-h-\varPhi\right)\left(-\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho{\boldsymbol{\eta}}) \right)\nonumber\\ & \quad -\frac{1}{\mu_0}\tilde{\boldsymbol{B}}\boldsymbol{\cdot} \left( \boldsymbol{\nabla}\times({\boldsymbol{\eta}}\times\boldsymbol{B}) \right) . \end{align}

The next step is to bring each term into the form (4.19), modulo a total time derivative and a total divergence, using integration by parts. The first term in expression (4.22) expands out to give

(4.23)\begin{align} \rho\boldsymbol{u}\boldsymbol{\cdot} \dot{\boldsymbol{\eta}} +\rho\boldsymbol{u}\boldsymbol{u}\boldsymbol{:}\boldsymbol{\nabla}\boldsymbol{\eta} -\rho (\boldsymbol{\eta}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u})\boldsymbol{\cdot}\boldsymbol{u} & = \boldsymbol{\eta} \boldsymbol{\cdot}\left( -\frac{\partial (\rho\boldsymbol{u})}{\partial t} -\frac{1}{2} \rho \boldsymbol{\nabla}(u^2) -\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol{u}\boldsymbol{u}) \right)\nonumber\\ & \quad +\frac{\partial (\rho\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\eta})}{\partial t} +\boldsymbol{\nabla}\boldsymbol{\cdot}\left( \rho (\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\eta}) \boldsymbol{u} \right) . \end{align}

Similarly, the third term in (4.22) yields

(4.24)\begin{equation} {\boldsymbol{\eta}}\boldsymbol{\cdot} \left( \rho\boldsymbol{\nabla} \left(\tfrac{1}{2}u^2-h-\varPhi\right) \right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left( -\left(\tfrac{1}{2}u^2-h-\varPhi\right)\rho{\boldsymbol{\eta}} \right) . \end{equation}

The second term in (4.22) simply gives

(4.25)\begin{equation} {\boldsymbol{\eta}}\boldsymbol{\cdot}\left( \rho T \boldsymbol{\nabla} S \right) . \end{equation}

The last term in (4.22) can be rearranged by cross-product identities:

(4.26)\begin{equation} (\boldsymbol{B}\times {\boldsymbol{\eta}})\boldsymbol{\cdot}\left( \frac{1}{\mu_0}\boldsymbol{\nabla}\times \tilde{\boldsymbol{B}} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\left( \frac{1}{\mu_0}\tilde{\boldsymbol{B}} \times({\boldsymbol{\eta}}\times\boldsymbol{B}) \right) , \end{equation}

and

(4.27)\begin{equation} (\boldsymbol{B}\times {\boldsymbol{\eta}})\boldsymbol{\cdot}\left( \frac{1}{\mu_0}\boldsymbol{\nabla}\times \tilde{\boldsymbol{B}} \right) = {\boldsymbol{\eta}}\boldsymbol{\cdot}\left( -\frac{1}{\mu_0} \boldsymbol{B}\times\left( \boldsymbol{\nabla}\times \tilde{\boldsymbol{B}}\right) \right) . \end{equation}

Now, combining the four terms (4.27), (4.25), (4.24) and (4.23) and discarding the total derivatives, we obtain

(4.28)\begin{equation} \delta L = {\boldsymbol{\eta}}\boldsymbol{\cdot}\left( -\frac{\partial (\rho\boldsymbol{u})}{\partial t}-\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol{u}\boldsymbol{u}) +\rho( T \boldsymbol{\nabla} S -\boldsymbol{\nabla} h-\boldsymbol{\nabla} \varPhi ) -\frac{1}{\mu_0}\boldsymbol{B}\times\left( \boldsymbol{\nabla}\times\tilde{\boldsymbol{B}} \right) \right) \end{equation}

modulo total time derivatives and total divergences. This yields the desired relation

(4.29)\begin{equation} \int_{V} \delta L \,{\rm d}^3x =\int_{V} \boldsymbol{\eta}\boldsymbol{\cdot} \boldsymbol{F} \,{\rm d}^3x = \left\langle \eta, F\right\rangle, \end{equation}

with

(4.30)\begin{equation} \boldsymbol{F} ={-}\frac{\partial (\rho\boldsymbol{u})}{\partial t} -\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho\boldsymbol{u}\boldsymbol{u}) +\rho( T \boldsymbol{\nabla} S -\boldsymbol{\nabla} h-\boldsymbol{\nabla} \varPhi ) -\frac{1}{\mu_0}\boldsymbol{B}\times\left( \boldsymbol{\nabla}\times\tilde{\boldsymbol{B}} \right) . \end{equation}

The equation $\boldsymbol {F}=0$ resulting from the variational principle (4.19) is called the EP equation. It is equivalent to the Eulerian momentum equation (2.2), after the pressure divergence identity (2.61) is used. A direct derivation of this identity is provided in Appendix B. Alternatively, (4.30) can be viewed as showing how the pressure divergence identity arises from the variational principle (4.19). Appendix E derives the identity (2.61) and the EP equation $\boldsymbol {F}=0$ using the approach of Holm et al. (Reference Holm, Marsden and Ratiu1998).

Holm & Kupershmidt (Reference Holm and Kupershmidt1986) studied a corresponding Hamiltonian form of the CGL plasma equations for both relativistic and non-relativistic flows; however, details of the EP formulation were not covered.

5.1 Non-canonical Hamiltonian equations

A non-canonical Poisson bracket (5.10) provides a Hamiltonian formulation once a Hamiltonian functional ${\mathcal {H}}$ is chosen. The formulation involves expressing the dynamical variables ${\mathsf{{Z}}}(\boldsymbol {x},t)$ formally as functionals

(5.13)\begin{equation} {\mathcal{Z}}(\boldsymbol{x}',t) \equiv \int_{V} {\mathsf{{Z}}}(\boldsymbol{x}',t) \delta(\boldsymbol{x}-\boldsymbol{x}')\, {\rm d}^3x \end{equation}

and writing Hamilton's equations in the form

(5.14)\begin{equation} \frac{\partial \mathcal{Z}}{\partial t} = \{ {\mathcal{Z}}, {\mathcal{H}} \}. \end{equation}

Using the explicit expression (5.10) for the bracket yields

(5.15)\begin{equation} \frac{\partial {\mathsf{{Z}}}}{\partial t} = \boldsymbol{\mathcal{D}} \frac{\delta {\mathcal{H}}}{\delta {\mathsf{{Z}}}} \end{equation}

in terms of the Hamiltonian operator $\boldsymbol {\mathcal {D}}$.

The appropriate Hamiltonian for describing CGL plasmas is given by the conserved total energy (cf. § 2.3):

(5.16)\begin{equation} {\mathcal{H}}=\int_{V} H\, {\rm d}^3x , \end{equation}

where $H$ is the Eulerian total energy density (3.20). Substituting this Hamiltonian into the non-canonical Hamilton's equations (5.15) can be shown to yield the CGL plasma equations (2.1)–(2.6) and (2.8)–(2.9).

5.2 Casimirs

In a non-canonical Hamiltonian system, a functional ${\mathcal {C}} = \int _{V} C\, \textrm {d}^3x$ satisfying the equation

(5.17)\begin{equation} \{{\mathcal{C}},{\mathcal{F}}\}\equiv 0\quad\text{ for all functionals ${\mathcal{F}}$ } \end{equation}

is called a Casimir. Existence of a non-trivial Casimir ${\mathcal {C}}\not \equiv 0$ indicates that the Poisson bracket is degenerate. Correspondingly, the Hamiltonian operator $\boldsymbol {\mathcal {D}}$ in the cosymplectic form of the bracket (5.10) will have a non-trivial kernel:

(5.18)\begin{equation} \boldsymbol{\mathcal{D}} {\mathcal{C}}_{{\mathsf{{Z}}}} \equiv 0, \end{equation}

where ${\mathcal {C}}_{{\mathsf{{Z}}}} \equiv {\delta {\mathcal {C}}}/{\delta {\mathsf{{Z}}}}$. Note that a Casimir is a conserved integral, because the time evolution of any functional is given by the Hamiltonian equations (5.14):

(5.19)\begin{equation} \frac{{\rm d}}{{\rm d}t}{\mathcal{C}} = \frac{\partial {\mathcal{C}}}{\partial t}=\{{\mathcal{C}},{\mathcal{H}}\}=0. \end{equation}

Casimirs are useful in stability analysis of steady flows and plasma equilibria (e.g. Holm et al. Reference Holm, Marsden, Ratiu and Weinstein1985; Hameiri Reference Hameiri2004). The conservation (5.19) holds modulo boundary integrals, and an investigation of boundary conditions is needed for a Casimir to be a strictly conserved integral (i.e. a constant of motion).

All Casimirs can be determined by solving (5.17), or alternatively (5.18) (see, e.g., Hameiri (Reference Hameiri2004) and Padhye & Morrison (Reference Padhye and Morrison1996a,Reference Padhye and Morrisonb) for the MHD case). For CGL plasmas, we obtain the Casimir determining equations:

(5.20)\begin{equation} \left.\begin{gathered} \boldsymbol{\nabla}^j(\rho {\mathcal{C}}_{M^j}) = \boldsymbol{\nabla}\boldsymbol{\cdot}(\rho {\mathcal{C}}_{\boldsymbol{M}})=0, \quad \boldsymbol{\nabla}^j(\sigma {\mathcal{C}}_{M^j})= \boldsymbol{\nabla}\boldsymbol{\cdot}(\sigma {\mathcal{C}}_{\boldsymbol{M}})=0,\\ \rho \boldsymbol{\nabla}^i {\mathcal{C}}_{\rho} + \sigma \boldsymbol{\nabla}^i {\mathcal{C}}_{\sigma} + M^j\boldsymbol{\nabla}^i {\mathcal{C}}_{M^j} + \boldsymbol{\nabla}^j ( M^i {\mathcal{C}}_{M^j}) + B^j\boldsymbol{\nabla}^i {\mathcal{C}}_{B^j} - B^j \boldsymbol{\nabla}_j {\mathcal{C}}_{B^i}\\ \equiv\rho \boldsymbol{\nabla} {\mathcal{C}}_{\rho} + \sigma \boldsymbol{\nabla} {\mathcal{C}}_{\sigma} + (\boldsymbol{\nabla} {\mathcal{C}}_{\boldsymbol{M}})\boldsymbol{\cdot}\boldsymbol{M} + \boldsymbol{\nabla}\boldsymbol{\cdot}({\mathcal{C}}_{\boldsymbol{M}}\boldsymbol{M}) + (\boldsymbol{\nabla} {\mathcal{C}}_{\boldsymbol{B}})\boldsymbol{\cdot}\boldsymbol{B} - \boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{\nabla} {\mathcal{C}}_{\boldsymbol{B}} =0 ,\\ \boldsymbol{\nabla}^j(B^i {\mathcal{C}}_{M^j}) - \boldsymbol{\nabla}_j(B^j {\mathcal{C}}_{M^i}) = \boldsymbol{\nabla}\times(\boldsymbol{B}\times{\mathcal{C}}_{\boldsymbol{M}}) =0 , \end{gathered}\right\} \end{equation}

where $\boldsymbol {\nabla }$ ($\boldsymbol {\nabla }^k$) acts as the total derivative with respect to $\boldsymbol {x}$ ($x^k$). This is an overdetermined system of linear partial differential equations for

(5.21)\begin{equation} {\mathcal{C}} = \int_{V} C(t,\boldsymbol{x},{\mathsf{{Z}}},\boldsymbol{\nabla}{\mathsf{{Z}}},\ldots,\boldsymbol{\nabla}^l{\mathsf{{Z}}})\,{\rm d}^3x, \end{equation}

with ${\mathsf{{Z}}}$ denoting the dynamical variables (5.11). The system can, in principle, be integrated to find ${\mathcal {C}}$ explicitly, once a differential order $l$ for the dependence of ${\mathcal {C}}$ on derivatives of the variables is chosen. Note that solutions of the divergence form $C=\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {F}$ lead to ${\mathcal {C}}$ being a boundary integral which can be assumed to be trivial if suitable boundary conditions are imposed. This classification problem is beyond the scope of the present work.

5.3 Mass, cross-helicity and magnetic helicity Casimirs

The well-known Casimirs for ideal barotropic MHD are the mass integral, the cross-helicity integral and the magnetic helicity integral. CGL plasmas with an isentropic equation of state $e=e(\rho,B)$ possess these same Casimirs:

(5.22a-c)\begin{equation} {\mathcal{C}}_1=\int_V \rho\, {\rm d}^3x, \quad {\mathcal{C}}_2=\int_V \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{B}\, {\rm d}^3x, \quad {\mathcal{C}}_3=\int_V \boldsymbol{A}\boldsymbol{\cdot}\boldsymbol{B}\, {\rm d}^3x. \end{equation}

In the more physically realistic case with an equation of state $e=e(\rho,S,B)$, ${\mathcal {C}}_1$ and ${\mathcal {C}}_3$ still hold as Casimirs, but ${\mathcal {C}}_2$ turns out to be a Casimir only in the case $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla } S=0$, as we show shortly.

Physically, the mass integral ${\mathcal {C}}_1$ is the total mass of the plasma; the cross-helicity integral ${\mathcal {C}}_2$ describes the linking of the fluid vorticity and magnetic field flux tubes; and the magnetic helicity integral ${\mathcal {C}}_3$ describes the knotting, linking and twist and writhe of the magnetic flux tubes (see,e.g., Berger & Field Reference Berger and Field1984; Moffatt & Ricca Reference Moffatt and Ricca1992; Hameiri Reference Hameiri2004). (Also see, e.g., Yoshida (Reference Yoshida2016), for interesting applications). In a fixed volume $V$ with a boundary $\partial V$, ${\mathcal {C}}_1$ is conserved, $\partial {\mathcal {C}}_1/\partial t=0$, if $\boldsymbol {u}$ has no normal component at the boundary (cf. § 2.3). Conservation of ${\mathcal {C}}_3$, $\partial {\mathcal {C}}_3/\partial t=0$, holds if, in addition, $\boldsymbol {B}$ has no normal component at the boundary, whereas the cross-helicity integral ${\mathcal {C}}_2$ is conserved, $\partial {\mathcal {C}}_2/\partial t=0$, only with the further condition $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla } S=0$ (cf. §§ 2.4 and 2.5).

5.4 Advected Casimirs

In general, Casimirs can be sought by looking among advected scalars, $\theta$, because the corresponding scalar integral $\int _{V(t)} \rho \theta \,\textrm {d}^3x$ will be conserved on volumes $V(t)$ moving with the flow. This implies that, on a fixed volume $V$, the integral $\int _{V} \rho \theta \,\textrm {d}^3x$ will be conserved up to a flux integral $-\oint _{\partial V} \rho \theta \boldsymbol {u}\boldsymbol {\cdot }\hat {\boldsymbol {n}}\,\textrm {d}A$ that vanishes if $\boldsymbol {u}$ has no normal component at the boundary $\partial V$ and, therefore, $\int _{V} \rho \theta \,\textrm {d}^3x$ will satisfy the Casimir property $\partial {\mathcal {C}}/\partial t=0$. Note that this property is necessary but not sufficient for $\int _{V} \rho \theta \,\textrm {d}^3x$ to be a Casimir, because there are conserved integrals such as the total energy and angular momentum that do not belong to the kernel of the Poisson bracket.

For CGL plasmas, the basic advected scalars are $S$, $\bar {S}_\parallel$, $\bar {S}_\perp$. Additional advected scalars are provided by Ertel's theorem: if $\theta$ is an advected scalar, then so is ${\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\theta }/{\rho }$. This yields the advected quantities (2.89a–c) shown in § 2.6.

One can show that

(5.23)\begin{equation} {\mathcal{C}}_4 =\int_V \rho f(S,\theta)\, {\rm d}^3\boldsymbol{x} \quad\text{where}\quad \theta=\frac{\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} S}{\rho} , \end{equation}

is an advected Casimir, for any function $f(S,\theta )$. We omit the proof.

6.1 Noether's theorem and evolutionary symmetries

To use Noether's theorem, we need to obtain the Lie point symmetries of the Lagrangian variational principle (3.19). As this variational principle employs the variables $x^i=X^i(x_0^j,t)$ given by the Cartesian components of the Lagrangian map (3.2), a Lie point symmetry acting on the coordinate space $(t,x_0^j,x^i)$ has the form

(6.1a-c)\begin{equation} t\to t + \epsilon \xi^t + O(\epsilon^2), \quad x^i\to x^i +\epsilon \xi^i + O(\epsilon^2), \quad x_0^i\to x_0^i +\epsilon \xi_0^i + O(\epsilon^2) \end{equation}

with $\epsilon$ denoting the parameter in the point symmetry transformation, where $\xi ^i$, $\xi _0^i$, $\xi ^t$ are functions of $t,x_0^j,x^i$. The infinitesimal transformation corresponds to the generator

(6.2)\begin{equation} {X} =\xi^i\frac{\partial}{\partial x^i} +\xi_0^i\frac{\partial}{\partial x_0^i}+\xi^t\frac{\partial}{\partial t} \end{equation}

whereas the finite transformation (6.1a–c) is given by exponentiation of the generator, $(t,x^i,x_0^i)\to \exp (\epsilon {X})(t,x^i,x_0^i)$.

To be a symmetry, a generator (6.2) must leave the variational principle invariant modulo boundary terms. This is equivalent to the condition that the change in the Lagrangian must satisfy (Ovsjannikov Reference Ovsjannikov1978; Ibragimov Reference Ibragimov1985)

(6.3)\begin{equation} \text{pr}\,{X} L_0 = \xi^t D_t L_0 + \xi_0^i D_{x^i_0} L_0 + D_t \varLambda_0^t + D_{x_0^i} \varLambda_0^i \end{equation}

where the operators $D_t=\partial /\partial _{t} + \dot {x}^i\partial /\partial _{x^i} +\cdots$ and $D_{x_0^i} = \partial /\partial _{x_0^i} + x^{ji}\partial /\partial _{x^j} +\cdots$ denote total derivative operators with respect to the independent variables $t$ and $x_0^i$ and $D_{x^i}$ denotes the partial derivative with respect to $x^i$ keeping $\boldsymbol {x}_0$ and $t$ constant. Here $\varLambda ^t_0$ and $\varLambda ^i_0$ are arbitrary potentials, that arise in Noether's theorem, because the variational derivative of a perfect derivative term has zero variational derivative. The first two terms on the right-hand side of (6.3) represent the Lie derivative ${\mathcal {L}}_{\boldsymbol {X}} L_0$ using the chain rule for differentiation. The terms involving total derivatives of $L_0$ on the right-hand side of (6.3) can be understood to arise from the change in the spatial domain $V_0$ and the time interval $[t_0,t_1]$ in the variational principle (3.19) under the action of an infinitesimal point transformation (6.1a–c). On the left-hand side of (6.3), $\textrm {pr}$ denotes prolongation to the extended coordinate space $(t,x_0^j,x^i,\dot {x}^i,x^{ij})$ (i.e. jet space) in which $\dot {x}^i$ and $x^{ij}$ are coordinates that correspond to $\dot {X}^i(x_0^j,t)$ and $\partial X^i(x_0^j,t)/\partial x_0^j$ on solutions $x^i=X^i(x_0^j,t)$ of the Euler–Lagrange equations (3.24) of the variational principle. An explicit formula for the components of $\textrm {pr}\,{X}$ will not be needed if we express the invariance condition (6.3) by using the characteristic form of the generator in which only the dependent variables $x^i$ undergo a transformation

(6.4a,b)\begin{equation} \hat{{X}} =\hat{\xi}^i \frac{\partial}{\partial x^i}, \quad \hat{\xi}^i = \xi^i -\xi^t \dot{x}^i - \xi_0^j x^{ij} ,\end{equation}

which arises from how ${X}$ acts on solutions $x^i=X^i(x_0^j,t)$. The relationship between the two forms (6.4a,b) and (6.2) is that $\textrm {pr}\,{X}= \textrm {pr}\,\hat {{X}} + \xi ^t D_t + \xi _0^i D_{x_0^i}$. Hence, (6.3) becomes (Olver Reference Olver1993; Bluman & Anco Reference Bluman and Anco2002)

(6.5)\begin{equation} \text{pr}\,\hat{{X}} (\hat{L}_0) = D_t \varLambda_0^t + D_{x_0^i} \varLambda_0^i, \end{equation}

where

(6.6)\begin{equation} \hat{L}_0 = \left.L_0\right|_{\boldsymbol{X}=\boldsymbol{x}} \end{equation}

is a function of $t,x_0^j,x^i,\dot {x}^i,x^{ij}$. The prolongation $\textrm {pr}\,\hat {{X}}$ is given by simply extending $\hat {{X}}$ to act on $\dot {x}^i$ and $x^{ij}$ through the total derivative relations $\textrm {pr}\,\hat {{X}}(\dot {x}^i) = D_t(\hat {{X}} x^i) = D_t \xi ^t$ and $\textrm {pr}\,\hat {{X}}(x^{ij}) = D_{x_0^j}(\hat {{X}} x^i) = D_{x_0^j}\xi ^i$.

In turn, this form (6.5) of the invariance condition can be expressed succinctly as

(6.7)\begin{equation} E_{x^i}(\text{pr}\,\hat{{X}}(\hat{L}_0)) =0 \end{equation}

using the Euler–Lagrange operator $E_{x^i}$ which has the property that it annihilates a function iff the function is given by a total divergence with respect to $t$ and $x_0^j$. The formulation (6.7) can be used as a determining equation to find all Lie point symmetries of the variational principle (3.19). (See Olver (Reference Olver1993) and Bluman & Anco (Reference Bluman and Anco2002) for a general discussion.)

Each Lie point symmetry (6.1a–c) gives rise to a conservation law through combining the condition (6.5) and Noether's identity

(6.8)\begin{equation} \text{pr}\,\hat{{X}}(\hat{L}_0) =\hat{\xi}^i E_{x^i} (L_0) +D_t W^t + D_{x_0^i} W^i, \end{equation}

where

(6.9a,b)\begin{equation} W^t = \hat{\xi}^j \frac{\partial \hat{L}_0}{\partial \dot{x}^j}, \quad W^i= \hat{\xi}^j \frac{\partial \hat{L}_0}{\partial x^{ji}} . \end{equation}

Thus, we obtain the following statement of Noether's theorem.

Proposition 6.1 If the Lagrangian variational principle (3.19) is invariant up to boundary terms under an infinitesimal point transformation (6.1a–c), then the Euler–Lagrange equations (3.24) of the variational principle possess a conservation law

(6.10)\begin{equation} D_t \mathcal{I}^t_0 +D_{x_0^i} {\mathcal{I}}^i_0 =0 \end{equation}

in which the conserved density and the spatial flux are given by

(6.11a,b)\begin{equation} \mathcal{I}^t_0 = \left.\left(\varLambda_0^t -W^t\right)\right|_{x=\boldsymbol{X}} , \quad \mathcal{I}^i_0 = \left.\left(\varLambda_0^i -W^i\right)\right|_{x=\boldsymbol{X}} , \end{equation}

using (6.5) and (6.9a,b).

Explicit expressions for $\partial \hat {L}_0/\partial \dot {x}^j = (\partial L_0/\partial \dot {X}^j)|_{\boldsymbol {X}=\boldsymbol {x}}$ and $\partial \hat {L}_0/\partial x^{ji} = (\partial L_0/\partial X^{ji})|_{\boldsymbol {X}=\boldsymbol {x}}$ are provided by (C1) in Appendix C. Note that the conservation law is a local continuity equation which holds when $x^i=X^i(x_0^j,t)$ satisfies the Euler–Lagrange equations (3.24). On solutions, $D_t|_{x=\boldsymbol {X}} = \partial /\partial t$ and $D_{x_0^i}|_{x=\boldsymbol {X}} = \partial /\partial x_0^i$ (acting as total derivatives).

We remark that (6.11a,b) can be derived alternatively using the canonical form of a symmetry generator (6.2), which requires computing the prolongation. See Webb & Zank (Reference Webb and Zank2007) and Webb & Anco (Reference Webb and Anco2019) for the MHD case. The canonical symmetry approach to conservation laws for the CGL system is also described in Appendix H.

6.1.1 Eulerian form of a Lagrangian conservation law

A Lagrangian conservation law (6.11a,b) can be expressed equivalently as an Eulerian conservation law (Padhye Reference Padhye1998)

(6.12)\begin{equation} \partial_t \varPsi^t +\boldsymbol{\nabla}_i \varPsi^i=0 \end{equation}

whose conserved density and spatial flux are given by

(6.13a,b)\begin{equation} \varPsi^t =\frac{1}{J} \left.\mathcal{I}_0^t\right|_{x=\boldsymbol{X}} , \quad \varPsi^i =\frac{1}{J}\left.\left( u^i \mathcal{I}_0^t +X_{ik} \mathcal{I}_0^k \right)\right|_{x=\boldsymbol{X}} \end{equation}

on solutions $x^i=X^i(x_0^j,t)$ of the Euler–Lagrange equations (3.24). This form (6.12)–(6.13a,b) of a conservation law is appropriate for considering conserved integrals

(6.14)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{V} \varPsi^t \,{\rm d}^3x ={-} \oint_{\partial V} {\boldsymbol{\varPsi}}\boldsymbol{\cdot} \hat{\boldsymbol{n}}\,{\rm d}A \end{equation}

on a fixed spatial domain $V$ in the CGL plasma.

In general, an Eulerian conservation law is a continuity equation of the form (6.12) holding on the solution space of the CGL plasma equations (2.1)–(2.10a–c). The corresponding conserved integral (6.14) has the physical content that the rate of change of the integral quantity $\int _{V} \varPhi ^t\, \textrm {d}^3x$ in $V$ is balanced by the net flux leaving the boundary of $V$. It is often physically useful to consider instead a spatial domain $V(t)$ that moves with the plasma. The form of the conservation law for moving domains is given by

(6.15)\begin{equation} \frac{{\rm d}}{{\rm d}t}\int_{V(t)} \varPsi^t \,{\rm d}^3x ={-} \oint_{\partial V(t)} {\boldsymbol{\varGamma}}\boldsymbol{\cdot} \hat{\boldsymbol{n}}\,{\rm d}A \end{equation}

in terms of the moving flux

(6.16)\begin{equation} {\boldsymbol{\varGamma}} = {\boldsymbol{\varPsi}} - \boldsymbol{u}\varPsi^t = \frac{1}{J} \boldsymbol{X}\boldsymbol{\cdot}{\boldsymbol{\mathcal{I}}}_0 . \end{equation}

Note that the moving integral quantity $\int _{V(t)} \varPsi ^t\,\textrm {d}^3x$ will be an invariant (i.e. a constant of motion) when the net moving flux vanishes on the boundary $\partial V (t)$. (See Anco & Webb (Reference Anco and Webb2020) for a discussion of moving domain conservation laws and invariants in fluid mechanics.) The equivalence between the conservation laws (6.14) and (6.15) can be derived by writing the continuity equation (6.12) in terms of the material (co-moving) derivative (2.88): $\textrm {d}\varPhi ^t/\textrm {d}t = -(\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u})\varPhi ^t - \boldsymbol {\nabla }\boldsymbol {\cdot }(\boldsymbol {\varPhi }-\varPhi ^t\boldsymbol {u})$, where $(\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u})$ represents the expansion or contraction of an infinitesimal volume $\textrm {d}^3x$ moving with the fluid (see, e.g., Anco & Dar Reference Anco and Dar2009, Reference Anco and Dar2010).

Now, rather than seeking to find all Lie point symmetries, we consider two main classes: kinematic and fluid relabelling. Kinematic symmetries are characterised by the generator (6.1a–c) having $\xi _0^i=0$, with $\xi ^t$ and $\xi ^i$ being functions only of $t,x^i$. Fluid relabelling symmetries have $\xi ^i=\xi ^t=0$ in the generator (6.1a–c), with $\xi _0^i$ being a function only of $t,x_0^i$.

A useful general remark is that any Lie point symmetry of a variational principle corresponds to a Lie point symmetry of the Euler–Lagrange equations, because invariance of a variational principle means that its extremals are preserved. Thus, one way to find all Lie point symmetries of a variational principle is by first obtaining the Lie point symmetries of the Euler–Lagrange equations and, second, checking which of those symmetries leaves invariant the variational principle.

Before deriving conservation laws using the above analysis, it is useful to note that there are three basic steps in the analysis.

1. (i) First it is necessary to determine for a given Lie symmetry, whether the Lie invariance condition (6.5) for the action can be satisfied by choosing the potentials $\varLambda ^t_0$ and $\varLambda ^i_0$. Here the left-hand side of (6.5) for the action $\textrm {pr}\,\hat {X}(L_0)$ is evaluated for the symmetry operator $X$.

2. (ii) Determine the surface vector components $W^t$ and $W^i$ that occur in the Noether identity (6.8), where $W^t$ and $W^i$ are given by (6.9a,b) (recall $\dot {x}^j=\partial x^j(\boldsymbol {x}_0,t)/\partial t=u^j$ and $x^{ji}=\partial x^j/\partial x_0^i$). Then using Proposition 6.1 one can obtain the Lagrangian conservation law (6.10) with conserved density $\mathcal {I}_0^t$ and flux $\mathcal {I}_0^i$ given in (6.11a,b).

3. (iii) Determine the Eulerian form of the conservation law using the results of Padhye (Reference Padhye1998) described by (6.12) and (6.13a,b).

6.1.2 Lie invariance condition

The Lie invariance condition for the action in (6.5) may be written in the form

(6.17)\begin{align} & J\left\{ \boldsymbol{\nabla}\boldsymbol{\cdot}(\rho \hat{\boldsymbol{\xi}}) \left[\varPhi+h-\frac{1}{2}u^2\right] +\rho \boldsymbol{u}\boldsymbol{\cdot} \left(\frac{{\rm d}\hat{\boldsymbol{\xi}}}{{\rm d}t} -\hat{\boldsymbol{\xi}}\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}\right) +\rho T\hat{\boldsymbol{\xi}}\boldsymbol{\cdot}\boldsymbol{\nabla} S\right.\nonumber\\ & \quad -\frac{\tilde{\boldsymbol{B}}}{\mu_0}\boldsymbol{\cdot} \left[\boldsymbol{\nabla}\times(\hat{\boldsymbol{\xi}}\times\boldsymbol{B}) -\hat{\boldsymbol{\xi}} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B}\right]\nonumber\\ & \quad \left.+\boldsymbol{\nabla}\boldsymbol{\cdot}\left\{\rho \hat{\boldsymbol{\xi}} \left[\frac{1}{2} u^2 -(h+\varPhi)\right] +\hat{\boldsymbol{\xi}}\boldsymbol{\cdot}\left({\mathsf{{p}}}+{\mathsf{{M}}}_{{\mathsf{{B}}}}\right) +\frac{1}{\mu_0} \left(\hat{\boldsymbol{\xi}}\times\boldsymbol{B}\right) \times\tilde{\boldsymbol{B}}\right\}\right\}\nonumber\\ & =D_t\varLambda^t_0+ D_{x_0^j} \left(\varLambda^j_0\right). \end{align}

Here we use the notation $\hat {\boldsymbol {\xi }}\equiv \hat {\boldsymbol {\xi }}_0$ in the fluid re-labelling symmetry case. The Lie invariance condition (6.17) also applies for the general Lie symmetry case, including the Lie point Galilean symmetry cases, the fluid relabelling symmetry cases and other more general cases. The derivation of (6.17) from (6.5) is outlined in Appendix G. In the general case ${\hat {\xi }}^i=\xi ^i-(\xi ^tD_t+\xi ^s_0 D_{x_0^s})x^i$. The Lie invariance condition (6.17) is similar to the Eulerian Lie invariance condition for the action used by Webb & Anco (Reference Webb and Anco2019) for the case of MHD. Here the evolutionary form of the symmetry operator $X$ is used rather than the canonical symmetry operator used by Webb & Anco (Reference Webb and Anco2019).

6.1.3 Galilean symmetries

The Eulerian form of the CGL plasma equations (2.1)–(2.10a–c) clearly indicates that they possess the Galilean group of Lie point symmetries when $\varPhi =0$, which are generated by (Rogers & Ames Reference Rogers and Ames1989; Fuchs Reference Fuchs1991)

(6.18a-d)\begin{equation} P_0=\frac{\partial}{\partial t}, \quad P_i=\frac{\partial}{\partial x^i}, \quad K_i=t\frac{\partial}{\partial x^i}+\frac{\partial}{\partial u^i}, \quad J_i=\epsilon_{ijk}\left(x^j\frac{\partial}{\partial x^k}+u^j\frac{\partial}{\partial u^k}+B^j \frac{\partial}{\partial B^k} \right). \end{equation}

These generators describe a time translation ($P_0$), space translations ($P_i$, $i=1,2,3$), Galilean boosts ($K_i$, $i=1,2,3$) and rotations ($J_i$, $i=1,2,3$), respectively, about the $x$, $y$ and $z$ axes. As shown in appendix F, the only additional Lie point symmetries admitted by (2.1) to (2.10a–c) consist of scalings. Hence, Galilean symmetries and scaling symmetries comprise all kinematic Lie point symmetries of the CGL plasma equations.

The Galilean symmetries (6.18a–d) have a corresponding Lagrangian form

(6.19a-d)\begin{equation} {X}_{P_0} = \frac{\partial}{\partial t}, \quad {X}_{P_i} = \frac{\partial}{\partial x^i}, \quad {X}_{K_i}=t\frac{\partial}{\partial x^i}, \quad {X}_{J_i}=\epsilon_{ijk} x^j\frac{\partial}{\partial x^k} . \end{equation}

Note that the prolongation of these Lagrangian generators to $\dot {x}^i=\dot {X}^i = u^i$ through the Lagrangian relation (3.1) for the fluid velocity yields the corresponding Eulerian generators (6.18a–d) acting on the variables $(t,x^i,u^i)$. As $\xi _0^i=0$ for all of the generators (6.19a–d), they are of kinematic type.

In the case when the gravitational potential $\varPhi$ is non-zero, then the preceding generators must satisfy the condition ${X} \varPhi (x^i) =0$ to be admitted as symmetries. For example, if $\varPhi$ is invariant under $z$-translation, then ${X}_{P_3}$ and ${X}_{K_3}$ are symmetries; if $\varPhi$ is invariant under $z$-rotation, then ${X}_{J_3}$ is a symmetry; and if $\varPhi$ is spherically symmetric, then ${X}_{J_i}$, $i=1,2,3$, are symmetries.

6.1.4 Galilean conservation laws

It is straightforward to show that each Galilean generator (6.19a–d) satisfies the condition (6.7) for invariance of the Lagrangian variational principle (3.19) when $\varPhi =0$ and thereby yields a conservation law (6.10).

Specifically, time translation ($X_{P_0}$) has $\hat {\xi }^i=-\dot {x}^i$ and, thus, $\textrm {pr}\,\hat {{X}}_{P_0} L_0 = - \ddot x^i \partial L_0/\partial _{\dot {x}^i} - \dot {x}^{ij}\partial L_0/\partial _{x^{ij}} = -D_t L_0$ due to $\partial L_0/\partial _t =0$. This gives $\varLambda _0^t = -L_0$, $\varLambda _0^i=0$, which yields conservation of energy

(6.20a,b)\begin{equation} \mathcal{I}^t_0 = \dot{X}^j \frac{\partial L_0}{\partial \dot{X}^j} -L_0 =H_0 , \quad \mathcal{I}^i_0 = \dot{X}^j \frac{\partial L_0}{\partial X^{ji}} , \end{equation}

with $H_0$ being the Hamiltonian (3.27).

Space translations ($X_{P_j}$) have $\hat {\xi }^i=\delta ^i{}_j$ and, thus, $\textrm {pr}\,\hat {{X}}_{P_j} L_0 = 0$ due to $\partial L_0/\partial _{x^i}=0$ (where we neglect gravity) combined with $D_t\hat {\xi }^i=0$ and $D_{x_0^j}\hat {\xi }^i=0$. Hence, $\varLambda _0^t =0$, $\varLambda _0^i=0$, which yields conservation of momentum

(6.21a,b)\begin{equation} \mathcal{I}^t_0 ={-}\frac{\partial L_0}{\partial \dot{X}^j}, \quad \mathcal{I}^i_0 ={-}\frac{\partial L_0}{\partial X^{ji}} . \end{equation}

Galilean boosts ($X_{K_j}$) have $\hat {\xi }^i=t\delta ^i{}_j$ and, thus, $\textrm {pr}\,\hat {{X}}_{K_j} L_0 = \rho _0 \dot {x}^j = D_t (\rho _0 x^j)$. This gives $\varLambda _0^t =\rho _0 x^j$, $\varLambda _0^i=0$, yielding conservation of Galilean momentum (centre of mass)

(6.22a,b)\begin{equation} \mathcal{I}^t_0 = \int \frac{\partial L_0}{\partial \dot{X}^j}{\rm d}t -t\frac{\partial L_0}{\partial \dot{X}^j}, \quad \mathcal{I}^i_0 ={-}t\frac{\partial L_0}{\partial X^{ji}} . \end{equation}

Rotations ($X_{J_j}$) have $\hat {\xi }^i=\epsilon ^i{}_{kj}x^k$. This leads to $\textrm {pr}\,\hat {{X}}_{J_j} L_0 = 0$, because $L_0$ depends on $x^i$ only through the scalars $|\dot {x}^i|$ and $|X_{ij}\tau _0^j|$ which are rotationally invariant. Thus $\varLambda _0^t =0$, $\varLambda _0^i=0$, which yields conservation of axial angular momentum

(6.23a,b)\begin{equation} \mathcal{I}^t_0 = \epsilon_{jkl}x^k \frac{\partial L_0}{\partial \dot{X}^l}, \quad \mathcal{I}^i_0 = \epsilon_{jkl}x^k\frac{\partial L_0}{\partial X^{li}} . \end{equation}

The set of Galilean conservation laws (6.20a,b) to (6.23a,b) can be written in a unified form through use of the observation that

(6.24a,b)\begin{equation} \varLambda_0^t =\dot{\xi}^j \int \frac{\partial L_0}{\partial \dot{X}^j}{\rm d}t -\xi^t L_0, \quad \varLambda_0^i= 0 , \end{equation}

with

(6.25a,b)\begin{equation} \xi^t = a_0, \quad \xi^i = a_1^i + a_2^i t + a_3^k \epsilon_{ijk}x^j , \end{equation}

where $a_0$, $a_1^i$, $a_2^i$ and $a_3^i$ ($i=1,2,3$) are arbitrary constants (parameterising the respective Galilean symmetry generators (6.19a–d)). Hence, we have

(6.26a,b)\begin{equation} \mathcal{I}_0^t = \dot{\xi}^j \int \frac{\partial L_0}{\partial \dot{X}^j}{\rm d}t -\xi^t L_0 -(\xi^j -\xi^t\dot{X}^j) \frac{\partial L_0}{\partial \dot{X}^j}, \quad \mathcal{I}_0^i ={-}(\xi^j -\xi^t\dot{X}^j) \frac{\partial L_0}{\partial X^{ji}} . \end{equation}

The resulting set of Eulerian conservation laws (6.12)–(6.13a,b) takes the form

(6.27)\begin{gather} \varPsi^t = H \xi^t -\rho u^j \xi^j +\rho x^j \dot{\xi}^j, \end{gather}

(6.28)\begin{gather}\varPsi^i = u^i \varPsi^t -(\xi^j -\xi^t u^j) \left(p^{ij} +M_B^{ij}\right), \end{gather}

where we have used

(6.29a-c)\begin{equation} \dfrac{\partial L_0}{\partial\dot{X}^j} =\rho_0 u^j, \quad \int \dfrac{\partial L_0}{\partial \dot{X}^j}{\rm d}t =\rho_0 x^j, \quad \dfrac{\partial L_0}{\partial X^{jk}} = \left(p^{jl}+M_B^{jl}\right) A_{lk} , \end{equation}

which follow from expressions (C1) and (C6) in Appendix C, along with the formulae (3.5a–c) and (3.8). Here $p^{ij}$ and $M_B^{ij}$ represents the components of the CGL non-isotropic pressure tensor (2.3a–c) and the magnetic pressure tensor (2.4), respectively. Note that $H =\rho |\boldsymbol {u}|^2 -L$ is the Hamiltonian (3.20).

In the MHD case, $p^{ij}$ is replaced by the components of the isotropic MHD gas pressure tensor $p\,\delta ^{ij}$.

The preceding derivations may be summarised as follows.

1. (i) Time translation symmetry ($a_0$) yields energy conservation, for which

   (6.30)\begin{equation} \left.\begin{gathered} \varPsi^t = H = \rho \left( \frac{1}{2}|\boldsymbol{u}|^2+\varPhi(\boldsymbol{x}) \right) +\varepsilon +\frac{B^2}{2\mu_0} , \\ {\boldsymbol{\varPsi}} = \left(\frac{1}{2}\rho |\boldsymbol{u}|^2 + \varepsilon+\rho\varPhi\right)\boldsymbol{u} + {\mathsf{{p}}}\boldsymbol{\cdot}\boldsymbol{u} +\frac{1}{\mu_0}\boldsymbol{E}\times\boldsymbol{B} ,\\ {\boldsymbol{\varGamma}} = {\boldsymbol{\varPsi}} -\boldsymbol{u}H =\left({\mathsf{{p}}}+{\mathsf{{M}}}_B\right)\boldsymbol{\cdot} \boldsymbol{u} , \end{gathered}\right\} \end{equation}
   where $\boldsymbol {E}=-\boldsymbol {u}\times \boldsymbol {B}$ is the electric field strength, and $\boldsymbol {E}\times \boldsymbol {B}/\mu _0$ is the Poynting flux.

2. (ii) Space translation symmetry ($a_1^i, i=1,2,3$) yields momentum conservation with

   (6.31)\begin{equation} \left.\begin{gathered} \varPsi^t ={-}\boldsymbol{M} ={-}\rho \boldsymbol{u}, \quad {\boldsymbol{\varPsi}} ={-}{\mathsf{{T}}} ,\\ {\boldsymbol{\varGamma}} = {\boldsymbol{\varPsi}} -\boldsymbol{u}{\boldsymbol{\varPsi}}^t ={-}\left({\mathsf{{p}}} +{\mathsf{{M}}}_B\right) , \end{gathered}\right\} \end{equation}
   where
   (6.32)\begin{equation} {\mathsf{{T}}} = \rho \boldsymbol{u}\boldsymbol{u} + {\mathsf{{p}}}+ {\mathsf{{M}}}_B \end{equation}
   is the CGL plasma stress tensor.

3. (iii) Galilean boost symmetry yields centre of mass conservation, with

   (6.33)\begin{equation} \left.\begin{gathered} \varPsi^t = \rho \boldsymbol{x} -t\boldsymbol{M} = \rho (\boldsymbol{x} -t\boldsymbol{u}) , \quad {\boldsymbol{\varPsi}} = \rho \boldsymbol{u} \boldsymbol{x} -t{\mathsf{{T}}},\\ {\boldsymbol{\varGamma}} = {\boldsymbol{\varPsi}}-\boldsymbol{u}(\rho \boldsymbol{x} -t\boldsymbol{M}) ={-}t \left({\mathsf{{p}}} +{\mathsf{{M}}}_B\right) .\end{gathered}\right\} \end{equation}

4. (iv) Rotational symmetry yields angular momentum conservation, with

   (6.34)\begin{equation} \left.\begin{gathered} \varPsi^t = \boldsymbol{x}\times \boldsymbol{M} = \rho \boldsymbol{x}\times \boldsymbol{u}, \quad {\boldsymbol{\varPsi}} = \boldsymbol{x}\times {\mathsf{{T}}} ,\\ {\boldsymbol{\varGamma}} = {\boldsymbol{\varPsi}} -\boldsymbol{u}(\boldsymbol{x}\times \boldsymbol{M}) = \boldsymbol{x}\times \left({\mathsf{{p}}} + {\mathsf{{M}}}_B\right) , \end{gathered}\right\} \end{equation}
   where $\boldsymbol {M} =\rho \boldsymbol {u}$ is the momentum density vector.

6.1.5 Fluid relabelling symmetries

Fluid relabelling symmetries correspond to transformations (6.1a–c) that change the Lagrangian fluid labels $\boldsymbol {x}_0$, but leave the Eulerian variables and $\boldsymbol {x}$ and $t$ invariant: $x_0^i \to x_0^i +\epsilon \xi _0^i +O(\epsilon ^2)$, where $\xi _0^i=\xi _0^i(t,x_0^j)$. The symmetry generator has the form

(6.35)\begin{equation} {X} =\xi_0^i\frac{\partial}{\partial x_0^i} . \end{equation}

For evaluating the condition for invariance of the Lagrangian variational principle (3.19), we need the characteristic form of the generator:

(6.36a,b)\begin{equation} \hat{{X}} =\hat{\xi}_0^i\partial_{x^i} , \quad \hat{\xi}_0^i ={-} \xi_0^j x^{ij} . \end{equation}

To proceed we use the formulation (6.5) instead of (6.7), which allows for consideration of functions $\varLambda _0^t$, $\varLambda _0^i$ that can involve non-localpotentials. This generality is necessary to derive the cross-helicity conservation law (2.67)–(2.68) which involves the temperature potential (2.69a,b).

6.1.6 Noether's theorem for fluid-relabelling conservation laws

The Lie invariance condition (6.17) in the fluid relabelling symmetry cases, may be written in the form

(6.37)\begin{align} & J\left\{\boldsymbol{\nabla}\boldsymbol{\cdot}(\rho \hat{\boldsymbol{\xi}}) \left[\varPhi+h-\frac{1}{2}u^2\right] +\rho \boldsymbol{u}\boldsymbol{\cdot} \left(\frac{{\rm d}\hat{\boldsymbol{\xi}}}{{\rm d}t} -\hat{\boldsymbol{\xi}}\boldsymbol{\cdot}\boldsymbol{\nabla} \boldsymbol{u}\right) -\frac{\tilde{\boldsymbol{B}}}{\mu_0}\boldsymbol{\cdot} \left[\boldsymbol{\nabla}\times(\hat{\boldsymbol{\xi}}\times\boldsymbol{B}) -\hat{\boldsymbol{\xi}} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B}\right]\right\}\nonumber\\ & \quad =D_t\left(\varLambda^t_0-\rho_0r {\boldsymbol{\xi}}_0\boldsymbol{\cdot}\boldsymbol{\nabla}_0S\right) +D_{x_0^i}\left(\varLambda^i_0-A_{ki} G^k\right), \end{align}

where

(6.38)\begin{equation} \boldsymbol{G}=\rho \hat{\boldsymbol{\xi}} \left[\frac{1}{2} u^2 -(h+\varPhi)\right] +\hat{\boldsymbol{\xi}}\boldsymbol{\cdot}\left({\mathsf{{p}}}+{\mathsf{{M}}}_{{\mathsf{{B}}}}\right) +\frac{1}{\mu_0} \left(\hat{\boldsymbol{\xi}}\times\boldsymbol{B}\right) \times\tilde{\boldsymbol{B}}. \end{equation}

For cross-helicity conservation, $\hat {\boldsymbol {\xi }}=-\boldsymbol {B}/\rho$ and ${\boldsymbol {\xi }}_0=\boldsymbol {B}_0/\rho _0$. Setting $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {B}=0$ (Gauss's law), the left-hand side of (6.37) vanishes. The condition that the right-hand side of (6.37) vanish is satisfied by the choices

(6.39a,b)\begin{equation} \varLambda^t_0=\rho_0r{\boldsymbol{\xi}}_0\boldsymbol{\cdot}\boldsymbol{\nabla} S, \quad \varLambda^i_0=A_{ki} G^k, \end{equation}

where $\boldsymbol {G}$ is given by (6.38). Evaluating formulae (6.38) and (6.39a,b) gives the formulae

(6.40)\begin{gather} \boldsymbol{G}=\boldsymbol{B}\left[e+\varPhi-\frac{1}{2}u^2+\frac{B^2}{2\mu_0\rho}\right], \end{gather}

(6.41)\begin{gather}\varLambda^t_0=r \boldsymbol{B}_0\boldsymbol{\cdot}\boldsymbol{\nabla}_0 S \equiv r (J \boldsymbol{B}\boldsymbol{\cdot} \boldsymbol{\nabla} S), \end{gather}

(6.42)\begin{gather}\varLambda^i_0=B_0^i\left[h+\varPhi-\frac{1}{2}u^2+\frac{B^2}{2\mu_0\rho}- \frac{p_\parallel}{\rho}\right], \end{gather}

where $h=e+p_\parallel /\rho$ is the enthalpy of the CGL plasma. Here $\hat {\boldsymbol {\xi }}=-\boldsymbol {B}/\rho$ and ${\boldsymbol {\xi }}_0=\boldsymbol {B}_0/\rho _0$, which gives rise to the cross-helicity conservation laws described by (6.45a,b) and (6.46).

Noether's theorem given by Proposition 6.1 now gives the following main result.

Proposition 6.2 The Lagrangian variational principle (3.19) is invariant up to boundary terms under the infinitesimal fluid relabelling transformation

(6.43)\begin{equation} x_0^i \to x_0^i + \epsilon B_0^i/\rho_0 +O(\epsilon^2) . \end{equation}

The resulting conservation law (6.10)–(6.11a,b) of the Euler–Lagrange equations (3.24)) is obtained by using (6.41)–(6.42) for $\varLambda ^t_0$ and $\varLambda ^i_0$ and using the results

(6.44a,b)\begin{equation} W_0^t = \hat{\xi}_0^j \rho_0 \dot{x}^j ={-}J \boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{u} , \quad W_0^i = \hat{\xi}_0^j A_{ki}\left(p^{jk} +M_B^{jk}\right) ={-}\frac{1}{\rho} \left(p_\parallel{-}\frac{B^2}{2\mu_0}\right) B_0^i . \end{equation}

Here $W^t_0$ and $W^i_0$ are given by (6.9a,b), in which the derivatives of $L_0$ are given by (G2a–c). This yields the conserved density and the flux

(6.45a,b)\begin{equation} \mathcal{I}^t_0 = J \boldsymbol{B}\boldsymbol{\cdot}(\boldsymbol{u} + r\boldsymbol{\nabla} S) , \quad \mathcal{I}^i_0 = B_0^i \left( h+\varPhi -\tfrac{1}{2}u^2 \right) . \end{equation}

The corresponding Eulerian conservation law (6.12)–(6.13a,b) is given by

(6.46)\begin{equation} \left.\begin{gathered} \varPsi^t = \boldsymbol{B}\boldsymbol{\cdot}(\boldsymbol{u} + r\boldsymbol{\nabla} S) ,\quad {\boldsymbol{\varPsi}} = \left( h+\varPhi -\tfrac{1}{2}u^2 \right) \boldsymbol{B} + \left( \boldsymbol{B}\boldsymbol{\cdot}(\boldsymbol{u} + r\boldsymbol{\nabla} S) \right)\boldsymbol{u} ,\\ {\boldsymbol{\varGamma}} = \left( h+\varPhi -\tfrac{1}{2}u^2 \right) \boldsymbol{B} . \end{gathered}\right\} \end{equation}

This is the cross-helicity density conservation law (2.67)–(2.69a,b).

6.2 Classical Noether approach

In this subsection we use a classical version of Noether's theorem (see, e.g., Bluman & Kumei (Reference Bluman and Kumei1989) and Appendix H) to derive conservation laws for the CGL plasma action (3.19) that uses the canonical Lie symmetry operator ${\tilde X}=\textrm {pr}\,X$ rather than the evolutionary form $\textrm {pr}\,{\hat X}$. The canonical symmetry operator form of Noether's theorem was used by Webb & Zank (Reference Webb and Zank2007) and Webb & Anco (Reference Webb and Anco2019) for the MHD fluid case. In this approach one searches for Lie transformations and divergence transformations that leave the action (6.1a–c) invariant, where

(6.47)\begin{equation} L_0'=L_0+\epsilon D_\alpha{\bar{\varLambda}}^\alpha_0+O(\epsilon^2), \end{equation}

is the divergence transformation. Here $D_0=\partial /\partial t$ and $D_i=\partial /\partial x_0^i$ are total partial derivatives with respect to $t$ and $x_0^i$. Note we use ${\bar {\varLambda }}^\alpha _0$ and ${\bar {\varLambda }}^\alpha$ to denote the potentials, in order to distinguish them from the potentials used for the evolutionary potentials in § 6.1.

The condition for the action to remain invariant under (6.1a–c) and (6.47) may be written in the form (cf. Bluman & Kumei Reference Bluman and Kumei1989):

(6.48)\begin{equation} {\rm pr}\,X L_0+L_0\left[D_t\xi^t+D_{x_0^j}\left(\xi^j_0\right)\right] +D_t{\bar{\varLambda}}^0_0+D_{x_0^j}{\bar{\varLambda}}^j_0=0, \end{equation}

where

(6.49)\begin{equation} {\rm pr}\,X L_0=\xi^t\frac{\partial}{\partial t}+\xi^s_0\frac{\partial}{\partial x_0^s}+\xi^k\frac{\partial}{\partial x^k}+\xi^{x^k_t}\frac{\partial}{\partial x^k_t} +\xi^{x_{kj}}\frac{\partial}{\partial x^{kj}}+\cdots, \end{equation}

is the prolonged, canonical Lie symmetry operator.

Here $\textrm {pr}\,X$ is related to $\textrm {pr}\,{\hat X}$ by the equations (Ovsjannikov Reference Ovsjannikov1978; Ibragimov Reference Ibragimov1985; Bluman & Kumei Reference Bluman and Kumei1989)

(6.50)\begin{equation} \left.\begin{gathered} {\rm pr}\,X={\rm pr}\,{\hat{X}}+\xi^\alpha_0 D_\alpha,\\ {\rm pr}\,{\hat{X}}={\hat{\xi}}^k\frac{\partial}{\partial x^k}+D_\alpha\left({\hat{\xi}}^k\right) \frac{\partial}{\partial x^k_\alpha}+D_{\alpha} D_{\beta}\left({\hat{\xi}}^k\right) \frac{\partial}{\partial x^k_{\alpha\beta}}+\cdots, \end{gathered}\right\}\end{equation}

where the evolutionary symmetry generator ${\hat {\xi }}^i$ is given by (6.4a,b) and $\xi ^t\equiv \xi ^0_0$.

Noether's theorem follows from Noether's identity:

(6.51)\begin{equation} {\rm pr}\,X L_0+L_0 D_\alpha\xi^\alpha_0+D_\alpha{\bar{\varLambda}}^\alpha_0 =\hat{\xi}^i E_{x^i}\left(L_0\right) +D_\alpha\left(W^\alpha+L_0\xi^\alpha_0\right) +D_\alpha\left({\bar{\varLambda}}^\alpha_0\right), \end{equation}

where $E_{x^i}(L_0)\equiv \delta {\mathcal {J}}/\delta x^i$ is the variational derivative of the action ${\mathcal {J}}$ with respect to $x^i$. For the case of CGL plasmas, the surface terms $W^\alpha$ are given by

(6.52a,b)\begin{equation} W^t\equiv W^0=\hat{\xi}^j \frac{\partial L_0}{\partial x^j_t}, \quad W^i=\hat{\xi}^j\frac{\partial L_0}{\partial x^{ji}} \end{equation}

(see Bluman & Kumei (Reference Bluman and Kumei1989) and Ibragimov (Reference Ibragimov1985) for more general cases).

If the Lie invariance condition (6.48) is satisfied, then the left-hand side of (6.51) vanishes and, consequently, the right-hand side of (6.51) must vanish. If, in addition, the Euler–Lagrange equations $E_{x^i}(L_0)=0$ are satisfied, then (6.51) implies

(6.53)\begin{equation} D_\alpha\left(W^\alpha+L_0\xi^\alpha_0+{\bar{\varLambda}}^\alpha_0\right)=0, \end{equation}

which is the conservation law of Noether's first theorem, which applies if the Euler–Lagrange equations $E_{x^i}(L_0)=0$ are independent, which is the case for a finite Lie algebra of Lie point symmetries.

In the more general case where the symmetries depend on continuous functions $\{ \phi ^k (\boldsymbol {x}_0,t): 1\leqslant k\leqslant N\}$, then the Lie pseudo-algebra of symmetries is infinite dimensional. In this case, Noether's second theorem implies that the Euler–Lagrange equations are not all independent, and that there exists differential relations between the Euler–Lagrange equations (see, e.g., Olver (Reference Olver1993) and Hydon & Mansfield (Reference Hydon and Mansfield2011) for details). Noether's second theorem, in some cases results in mathematically trivial conservation laws. Charron & Zadra (Reference Charron and Zadra2018) discuss Ertel's theorem and Noether's second theorem.

For the CGL plasma model, the $W^\alpha$ from (6.52a,b) are given by

(6.54a,b)\begin{equation} W^t={\hat{\xi}}^j\rho_0 u^j, \quad W^i=\hat{\xi}^j \left(p^{js}+M_B^{js}\right) A_{si}. \end{equation}

Substitution of (6.54a,b) for the $W^\alpha$ into Noether's theorem (6.53) gives the Lagrangian conservation law

(6.55)\begin{equation} \frac{\partial {\mathcal{I}}^0}{\partial t}+\frac{\partial {\mathcal{I}}^i}{\partial x_0^i}=0, \end{equation}

where

(6.56)\begin{equation} \left.\begin{gathered} {\mathcal{I}}^0=\rho_0u^j \hat{\xi}^j +\xi^tL_0+{\bar{\varLambda}}^t_0, \\ {\mathcal{I}}^i=\hat{\xi}^j\left(p^{js} +M_B^{js}\right) A_{si}+\xi^i_0 L_0+{\bar{\varLambda}}^i_0, \end{gathered}\right\}\end{equation}

are the conserved density and flux.

The Lagrangian conservation law (6.55) corresponds to an Eulerian conservation law of the form (6.12) (e.g. Padhye Reference Padhye1998) with conserved density $\varPsi ^t$, and flux $\varPsi ^j$ given by

(6.57)\begin{equation} \left.\begin{gathered} \varPsi^t=\rho u^k\hat{\xi}^k+\xi^t L+{\bar{\varLambda}}^0, \\ \varPsi^j= \hat{\xi}^k\left(T^{jk}-L\delta^{jk}\right)+\xi^j L+{\bar{\varLambda}}^j, \end{gathered}\right\} \end{equation}

where

(6.58)\begin{equation} \left.\begin{gathered} T^{jk}=\rho u^j u^k+p^{jk}+M_B^{jk}, \\ {\bar{\varLambda}}^0={\bar{\varLambda}}^0_0/J,\quad {\bar{\varLambda}}^j=\left(u^j {\bar{\varLambda}}^0_0+x^{js} {\bar{\varLambda}}^s_0\right)/J. \end{gathered}\right\} \end{equation}

The Lagrangian and Eulerian conservation laws for the Galilean group (§§ 6.1.2–6.1.3) and the cross-helicity conservation law associated with the fluid relabelling symmetry $\hat {\boldsymbol {\xi }}=-\boldsymbol {B}/\rho$ (§§ 6.1.4–6.1.5) now follow from (6.56)–(6.58) for appropriate choices of the potentials ${\bar {\varLambda }}^t_0$ and ${\bar {\varLambda }}^j_0$ and of the symmetry generators $\xi ^i$, $\xi ^i_0$ and $\xi ^t$. Note that the evolutionary symmetry potentials used in § 6.1 are different than those used in this subsection (§ 6.2).

Example 6.3 The time translation symmetry of the Lagrangian action (3.19), satisfies the Lie invariance condition (6.48) by choosing

(6.59a-e)\begin{equation} \xi^t=1,\quad \xi^i=0, \quad \xi^s_0=0,\quad \hat{\xi}^{i}={-} u^i, \quad {\bar{\varLambda}}_0^\alpha=0, \end{equation}

where $i,s=1,2,3,$ and $\alpha =0,1,2,3$ is a variational symmetry of the action (3.19). The corresponding conservation law using Noether's theorem results (6.57) is the energy conservation law:

(6.60)\begin{align} & \frac{\partial}{\partial t}\left[\frac{1}{2}\rho |\boldsymbol{u}|^2+\varepsilon +\frac{B^2}{2\mu_0} +\rho\varPhi(\boldsymbol{x})\right]\nonumber\\ & \quad +\boldsymbol{\nabla}\boldsymbol{\cdot}\left[\rho \boldsymbol{u}\left(\frac{1}{2}|\boldsymbol{u}|^2 +\varPhi(\boldsymbol{x})\right) +\frac{\boldsymbol{E}\times\boldsymbol{B}}{\mu_0} +\varepsilon \boldsymbol{u}+{\mathsf{{p}}}\boldsymbol{\cdot}\boldsymbol{u}\right]=0, \end{align}

where ${\mathsf{{p}}}$ is the CGL pressure tensor (2.3a–c), $\boldsymbol {E}=-\boldsymbol {u}\times \boldsymbol {B}$ is the electric field strength and $\boldsymbol {E}\times \boldsymbol {B}/\mu _0$ is the Poynting flux.

Example 6.4 If the gravitational potential $\varPhi (\boldsymbol {x})$ is independent of $x^{j_1}$ say, then the Lie invariance condition (6.48) for a divergence symmetry of the action is satisfied, by the choice

(6.61a-d)\begin{equation} \xi^i=\delta^{ij_1},\quad \xi^s_0=\xi^t=0,\quad {\hat{\xi}}^{i}=\delta^{ij_1}, \quad {\bar{\varLambda}}_0^0={\bar{\varLambda}}_0^i=0, \end{equation}

and condition (6.48) reduces to the equation

(6.62)\begin{equation} {-}\rho_0\frac{\partial \varPhi}{\partial x^{j_1}}=0. \end{equation}

Then using (6.47) we obtain the momentum conservation equation in the $x^{j_1}$ direction in the form

(6.63)\begin{equation} \left\{\frac{\partial}{\partial t}\left(\rho \boldsymbol{u}\right) +\boldsymbol{\nabla}\boldsymbol{\cdot}\left(\rho \boldsymbol{u}\otimes \boldsymbol{u}+{\mathsf{{p}}} +\frac{B^2}{2\mu_0} {\mathsf{{I}}} -\frac{\boldsymbol{B}\otimes \boldsymbol{B}}{\mu_0}\right)\right\}^{j_1}=0. \end{equation}

In the case where gravity can be neglected (i.e. $\varPhi =0$) the superscript $j_1$ can be dropped in (6.63). Technically, the conservation of momentum law appears more complicated in non-Cartesian coordinates (e.g. in spherical geometry), where the metric tensor, the covariant derivative and the affine connection play an important role.

Example 6.5 The Galilean boost symmetry, with infinitesimal generators

(6.64a-e)\begin{equation} \xi^i=\varOmega^i t, \quad \xi^s_0=0,\quad \xi^t=0, \quad {\bar{\varLambda}}^0_0={-}\rho_0(\boldsymbol{x}_0)\boldsymbol{\varOmega}\boldsymbol{\cdot}\boldsymbol{x}, \quad {\bar{\varLambda}}_0^i=0, \end{equation}

($i,s=1,2,3$) and (6.57) gives rise to the centre of mass conservation law

(6.65)\begin{equation} \frac{\partial}{\partial t}[\boldsymbol{\varOmega}\boldsymbol{\cdot}\rho (\boldsymbol{u}t-\boldsymbol{x})] +\boldsymbol{\nabla}\boldsymbol{\cdot} \left[\boldsymbol{\varOmega}\boldsymbol{\cdot} \left\{ \rho (\boldsymbol{u}t-\boldsymbol{x})\otimes\boldsymbol{u} +t\left[{\mathsf{{p}}}+\frac{B^2}{2\mu_0} {\mathsf{{I}}} -\frac{\boldsymbol{B}\otimes\boldsymbol{B}}{\mu_0}\right] \right\} \right]=0, \end{equation}

provided

(6.66)\begin{equation} \boldsymbol{\varOmega}t\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi\equiv {\boldsymbol{\xi}}\boldsymbol{\cdot} \boldsymbol{\nabla}\varPhi=0. \end{equation}

Thus, one obtains the centre of mass conservation law for Galilean boosts perpendicular to the external gravitational field. Thus, for a spherically symmetric gravitational potential (e.g. for the Sun), a Galilean boost conservation law exists for a boost ${\boldsymbol {\xi }}=\boldsymbol {\varOmega }t$ perpendicular to the radial direction.

Example 6.6 The Lie transformation generators

(6.67a-d)\begin{equation} \xi^{i}=\epsilon_{ijk} \varOmega^j x^k,\quad \xi^s_0=\xi^t=0,\quad \hat{\xi}^{i}=\xi^{x^i},\quad {\bar{\varLambda}}^{\alpha}_0=0, \end{equation}

give rise to the angular momentum conservation equation

(6.68)\begin{equation} \frac{\partial}{\partial t}\left[\boldsymbol{\varOmega}\boldsymbol{\cdot}(\boldsymbol{x}\times \boldsymbol{M})\right] +\boldsymbol{\nabla} \left[ \boldsymbol{\varOmega}\boldsymbol{\cdot}(\boldsymbol{x}\times {\mathsf{{T}}}\right]=0, \end{equation}

where

(6.69a,b)\begin{equation} \boldsymbol{M}=\rho \boldsymbol{u}\quad \hbox{and}\quad \left(\boldsymbol{x}\times {\mathsf{{T}}}\right)^{pj}=\epsilon_{pqk} x^q T^{kj}, \end{equation}

define the mass flux or momentum density $\boldsymbol {M}$ and $\boldsymbol {x}\times {\mathsf{{T}}}$ respectively. The invariance condition (6.48) for a divergence symmetry of the action, requires

(6.70)\begin{align} {\tilde X} L_0& ={-}\rho_0(\boldsymbol{\varOmega}\times\boldsymbol{x})\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi +\epsilon_{kps}\varOmega^p \left[p^{sk}+\frac{B^2}{2\mu_0}\delta^{sk}- \frac{B^s B^k}{\mu_0}\right]\nonumber\\ & \equiv{-}\rho_0(\boldsymbol{\varOmega}\times\boldsymbol{x})\boldsymbol{\cdot}\boldsymbol{\nabla}\varPhi=0. \end{align}

Note that the second term in the first line of (6.70) vanishes because the term in large square brackets is symmetric in $s$ and $k$.

For the case where $\boldsymbol {\varOmega }=\varOmega \boldsymbol {e}_z$ the condition (6.70) reduces to the equation

(6.71)\begin{equation} {\tilde{X}} L_0={-}\rho_0 \varOmega r\sin\theta \frac{\partial \varPhi}{\partial \phi}=0, \end{equation}

where $(r,\theta,\phi )$ are spherical polar coordinates. Condition (6.71) is satisfied for $\varPhi =\varPhi (r,\theta )$.

Example 6.7 The fluid relabelling symmetry transformations

(6.72a-d)\begin{equation} \xi^i=0,\quad\xi^t=0,\quad \xi^s_0={-}\frac{B_0^s}{\rho_0},\quad \hat{\xi}^i =\frac{B^i}{\rho}\equiv b^i, \end{equation}

and the choices

(6.73a-c)\begin{equation} {\bar{\varLambda}}^0=r \left(\boldsymbol{B}\boldsymbol{\cdot}\boldsymbol{\nabla} S\right), \quad {\bar{\varLambda}}^i= u^i {\bar{\varLambda}}^0, \quad \frac{{\rm d}r}{{\rm d}t}={-}T, \end{equation}

for the gauge potentials ${\bar {\varLambda }}^\alpha$ ($\alpha =0,1,2,3$), leave the action invariant (i.e. (6.48) is satisfied). Using (6.57) to calculate $\varPsi ^t$ and $\varPsi ^j$ gives the results (6.46) for the cross-helicity conservation law (2.67)–(2.69a,b).

The above examples illustrate the use of the classical version of Noether's first theorem in obtaining conservation laws of the CGL equations.