2.1. Model equations

The starting point for deriving the model equations in the CCS is the same as that used by Tronci (Reference Tronci2010) and Amano (Reference Amano2018), i.e. a set of multifluid equations governing the dynamics of the thermal components accompanied by the Vlasov equation for the energetic particles, which provide self-consistent closure to Maxwell's equations:

(2.1)\begin{gather} \partial_t n_s+\boldsymbol{\nabla}\boldsymbol{\cdot} (n_s{\boldsymbol V}_s)=0, \end{gather}

(2.2)\begin{gather}m_sn_s(\partial_t{\boldsymbol V}_s+{\boldsymbol V}_s\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol V}_s)= e_sn_s({\boldsymbol E}+{\boldsymbol V}_s\times{\boldsymbol B})-\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_s, \end{gather}

(2.3)\begin{gather}\partial_t f_p={-}\boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla} f_p-\frac{e_p}{m_p} ({\boldsymbol E}+\boldsymbol{v}\times{\boldsymbol B})\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} f_p, \end{gather}

(2.4)\begin{gather}\partial_t{\boldsymbol B}={-}\boldsymbol{\nabla}\times{\boldsymbol E}, \end{gather}

(2.5)\begin{gather}\partial_t{\boldsymbol E}=\epsilon_0^{{-}1}\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-\epsilon_0^{{-}1}{\boldsymbol J}, \end{gather}

(2.6a,b)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol E}=\sigma/\epsilon_0,\quad \boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol B}=0, \end{gather}

(2.7a–c)\begin{gather}\sigma= \sigma_f+\sigma_k,\quad \sigma_f=\sum_s e_s n_s, \quad \sigma_k=\sum_p e_p \int \, \textrm{d}^{3}v f_p, \end{gather}

(2.8a–c)\begin{gather}{\boldsymbol J}={\boldsymbol J}_f+{\boldsymbol J}_k,\quad {\boldsymbol J}_f=\sum_s e_s n_s {\boldsymbol V}_s , \quad {\boldsymbol J}_k=\sum_p e_p \int \, \textrm{d}^{3}v f_p \boldsymbol{v} , \end{gather}

where $\boldsymbol{\mathsf{P}}_s$ is the pressure tensor of the thermal species $s, f_p=f_p({\boldsymbol x},\boldsymbol {v},t)$ is the Vlasov distribution function, i.e. the particle density in phase space $({\boldsymbol x},\boldsymbol {v})$ for the particle species $p$. In this paper, we assume that the kinetic species consist of energetic ions, e.g. populations of alpha particles or resonant ions. Note that the system (2.1)–(2.8a–c) is not fully kinetic because it has been assumed that the moment hierarchy that yields the fluid equations (2.1) and (2.2) has been truncated in view of some appropriate fluid closure that results in independent expressions for the pressure tensors $\boldsymbol{\mathsf{P}}_s$ of the fluid species. For a two-fluid plasma bulk, the subscript $s$ is $s=i,e$. Note that $\boldsymbol {v}$ is the microscopic particle velocity, and therefore,

(2.9)\begin{equation} \frac{\partial v_i}{\partial x_j}=0,\quad \forall \, i,j. \end{equation}

In the low-frequency limit, it is legitimate to impose quasineutrality in a strict manner, i.e. setting $\sigma =0$. The quasineutrality assumption renders the displacement current term negligible and the Gauss law redundant. In addition, when the electron inertial effects can be ignored but the electron pressure is comparable to the magnetic pressure, the electron momentum equation results in the following generalized Ohm's law:

(2.10)\begin{equation} {\boldsymbol E}={-}{\boldsymbol V}_e\times {\boldsymbol B}-\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e}. \end{equation}

A rigorous derivation of (2.10) requires an appropriate ordering of the various terms in the electron fluid momentum equation. We may employ an Alfvén normalization by introducing the dimensionless quantities below:

(2.11)\begin{equation} \left.\begin{array}{c@{}} \displaystyle \tilde{\boldsymbol{\nabla}}=\ell_0 \boldsymbol{\nabla},\quad \tilde{t}=\dfrac{t}{\ell_0/{V}_A},\quad \tilde{n}_s=\dfrac{n_s}{n_0},\\ \displaystyle \tilde{B}=\dfrac{B}{B_0},\quad \tilde{{V}}_s=\dfrac{{V}_s}{{V}_A}, \quad \tilde{\boldsymbol{\mathsf{P}}}_s=\dfrac{\boldsymbol{\mathsf{P}}_s}{B_0^{2}/\mu_0},\\ \displaystyle \tilde{J}=\dfrac{J}{B_0/(\ell_0\mu_0)}, \quad \tilde{E}=\dfrac{E}{B_0{V}_A}, \end{array}\right\} \end{equation}

where ${V}_A=B_0/\sqrt {\mu _0 m_in_0}$ is the Alfvén speed, and $B_0, n_0$ and $\ell _0$ are the characteristic magnetic field, number density and length, respectively. The electron momentum equation can then be written in the following non-dimensional form:

(2.12)\begin{equation} {\boldsymbol E}={-}{\boldsymbol V}_e\times{\boldsymbol B}-d_i\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{n_e}-\frac{d_e^{2}}{d_i}\frac{\textrm{d} {\boldsymbol V}_e}{\textrm{d} t}.\end{equation}

Here, the tildes have been dropped and the parameters $d_e$ and $d_i$ are the relative electron and ion skin depths, respectively. Neglecting ${{O}} (d_e^{2})$ terms (note that for an electron-proton plasma $d_e/d_i\sim 0.023$), thus eliminating electron length scales, the electron inertial term is removed, while the electron pressure tensor survives. Then, restoring dimensions in (2.12) yields (2.10). This result stems from the assumption that the electron pressure scales as the magnetic pressure and hence it is legitimate for plasmas with high electron $\beta$. In an alternative ordering scheme, e.g. for very low $\beta$, the order of the electron pressure term would be lower than the order of the Hall term and it could thus be neglected.

As regards the form of $\boldsymbol{\mathsf{P}}_e$, because of the specific ordering described above, we may consider a gyrotropic electron pressure. To see this, let us write the electron pressure equation as it emerges from the moment hierarchy of the Vlasov equation (e.g. see Hunana et al. Reference Hunana, Tenerani, Zank, Khomenko, Goldstein, Webb, Cally, Collados, Velli and Adhikari2019),

(2.13)\begin{align} & m_e[\partial_t \boldsymbol{\mathsf{P}}_e+\boldsymbol{\nabla}\boldsymbol{\cdot} ({\boldsymbol V}_e\boldsymbol{\mathsf{P}}_e + \boldsymbol{\mathsf{Q}}_e)+\boldsymbol{\mathsf{P}}_e\boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol V}_e+(\boldsymbol{\mathsf{P}}_e\boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol V}_e)^{\top}]\nonumber\\ & \quad =e[{\boldsymbol B}\times \boldsymbol{\mathsf{P}}_e+({\boldsymbol B}\times\boldsymbol{\mathsf{P}}_e)^{\top}], \end{align}

where the superscript $\top$ denotes transpose and $\boldsymbol{\mathsf{Q}}_e$ is the electron heat flux tensor. Performing the Alfvén normalization (2.11), the electron pressure equation becomes

(2.14)\begin{align} & \frac{d_e^{2}}{d_i}[\partial_t \boldsymbol{\mathsf{P}}_e+\boldsymbol{\nabla}\boldsymbol{\cdot} ({\boldsymbol V}_e\boldsymbol{\mathsf{P}}_e + \boldsymbol{\mathsf{Q}}_e)+\boldsymbol{\mathsf{P}}_e\boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol V}_e+(\boldsymbol{\mathsf{P}}_e\boldsymbol{\cdot} \boldsymbol{\nabla} {\boldsymbol V}_e)^{\top}]\nonumber\\ & \quad =[{\boldsymbol B}\times \boldsymbol{\mathsf{P}}_e+({\boldsymbol B}\times\boldsymbol{\mathsf{P}}_e)^{\top}]. \end{align}

Neglecting electron length scales $(d_e\rightarrow 0)$, only the right-hand side of (2.13) survives, and thus $\boldsymbol{\mathsf{P}}_e$ satisfies ${\boldsymbol B}\times \boldsymbol{\mathsf{P}}_e+({\boldsymbol B}\times \boldsymbol{\mathsf{P}}_e)^{\top }=0$. A general solution of this equation is given by the gyrotropic pressure tensor

(2.15)\begin{equation} \boldsymbol{\mathsf{P}}_e=\frac{P_{e\parallel}-P_{e\perp}}{B^{2}} {\boldsymbol B}{\boldsymbol B}+P_{e\perp}\boldsymbol{\mathsf{I}},\end{equation}

where

(2.16)\begin{equation} \left. \begin{array}{c@{}} P_{e\parallel}=\boldsymbol{\mathsf{P}}_e\boldsymbol{:}{\boldsymbol b}{\boldsymbol b},\\ P_{e\perp}=\tfrac{1}{2}\boldsymbol{\mathsf{P}}_e\boldsymbol{:} (\boldsymbol{\mathsf{I}}-{\boldsymbol b}{\boldsymbol b}). \end{array}\right\} \end{equation}

Here, ${\boldsymbol b}:={\boldsymbol B}/|{\boldsymbol B}|$ and $\boldsymbol{\mathsf{A}}\boldsymbol {:}\boldsymbol{\mathsf{B}}$ indicates double contraction between the second-order tensors $\boldsymbol{\mathsf{A}}$ and $\boldsymbol{\mathsf{B}}$, i.e. $\boldsymbol{\mathsf{A}}\boldsymbol {:}\boldsymbol{\mathsf{B}}=A_{ij}B_{ij}$. We consider this specific form of electron pressure throughout the rest of the paper, because it is not only legitimate in the small electron length-scale limit, but, as will be seen in the next section, facilitates also the identification of an appropriate Hamiltonian structure.

Now, using $\boldsymbol {\nabla }\times {\boldsymbol B}=\mu _0{\boldsymbol J}$ and (2.8a–c), the electron velocity can be written as

(2.17)\begin{equation} {\boldsymbol V}_e=\frac{n_i}{n_e}{\boldsymbol V}-\frac{1}{en_e}\left(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-\sum_p e_p\int \, \textrm{d}^{3}v\, \boldsymbol{v} f_p\right), \end{equation}

where ${\boldsymbol V}\equiv {\boldsymbol V}_i$. Inserting (2.17) in the Ohm's law (2.10), we obtain

(2.18)\begin{equation} {\boldsymbol E}={-}\frac{n_i}{n_e}{\boldsymbol V}\times{\boldsymbol B}+\frac{1}{en_e}(\mu_0^{{-}1}\boldsymbol{\nabla}\times {\boldsymbol B}-{\boldsymbol J}_k)\times{\boldsymbol B}-\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e}.\end{equation}

Note that owing to the energetic particle component, quasineutrality does not imply $n_i = n_e$, but $n_i=n_e-e^{-1}\sigma _k$. In view of (2.18), Faraday's law (2.4) leads to the following induction equation:

(2.19)\begin{equation} \partial_t{\boldsymbol B}=\boldsymbol{\nabla}\times\left[\frac{n_i}{n_e}{\boldsymbol V}\times{\boldsymbol B}- \frac{1}{en_e}(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-{\boldsymbol J}_k) \times{\boldsymbol B}+\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e}\right]. \end{equation}

Inserting the generalized Ohm's law (2.18) into the ion momentum equation (2.2) and the Vlasov equation (2.3), we obtain respectively

(2.20)\begin{gather} mn_i(\partial_t{\boldsymbol V}+{\boldsymbol V}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol V})= \left[\frac{n_i}{n_e}\sigma_k {\boldsymbol V} +\frac{n_i}{n_e} ({\boldsymbol J}-{\boldsymbol J}_k )\right]\times{\boldsymbol B}-\frac{n_i}{n_e} \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e-\boldsymbol{\nabla} P_i, \end{gather}

(2.21)\begin{gather}\partial_t f_p+\boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla} f_p+\frac{e_p}{m_p} \left\{\left[\boldsymbol{v}-\frac{n_i}{n_e}{\boldsymbol V}+\frac{1}{en_e} ({\boldsymbol J}-{\boldsymbol J}_k)\right]\times{\boldsymbol B} -\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e} \right\}\boldsymbol{\cdot} \boldsymbol{\nabla}_{\boldsymbol{v}} f_p=0, \end{gather}

where, from now on, $m\equiv m_i$. Here, we have assumed that the thermal ions have isotropic pressure. For a magnetized plasma, a consistent consideration of thermal pressure effects for the ions within the two-fluid framework requires the inclusion of anisotropic (and in particular non-gyrotropic) pressure effects. This will be the topic of future research. Presently, it suffices for our purposes to consider an isotropic pressure $P_i$ assuming low temperature ions because thermal ions that deviate from the isotropic pressure description can be incorporated in the kinetic component described by the Vlasov equation.

The continuity equation for the ions remains unchanged, while for the electron fluid, inserting (2.17) into (2.1), we find

(2.22)\begin{equation} \partial_t n_e={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (n_i{\boldsymbol V})-\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_k}{e}, \end{equation}

that is,

(2.23)\begin{equation} e\partial_t(n_i-n_e)=\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_k, \end{equation}

which is an equation for electric charge conservation. The complete system of the dynamical equations consists of the ion continuity equation, the induction equation (2.19), the momentum equation (2.20), the Vlasov equation (2.21), and the electron continuity equation (2.22) or equivalently the charge conservation equation (2.23). Computing the first-order velocity moment of the Vlasov equation, one finds

(2.24)\begin{equation} \partial_t \sigma_k+\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_k=0, \end{equation}

which, combined with (2.23), gives the local charge conservation. Thus, for $e(n_{i}-n_{e})+\sigma _{k}=0$ at $t=0$, the quasineutrality constraint, which is invoked in various derivations in this paper, is preserved by the dynamics.

2.2. Hamiltonian structure

Let us now follow the procedure introduced by Tronci (Reference Tronci2010) to derive the Hamiltonian structure of the above model. The difference here is that we apply the procedure in a more general model, by considering a two-fluid bulk plasma and generalizing also for electron pressure anisotropy. The starting point for this derivation is the combination of the Hall MHD noncanonical bracket, derived by Holm (Reference Holm1987), with the particle Poisson bracket. This can be done by directly adding the two brackets, which are written, however, in terms of the canonical momentum density $\bar {{\boldsymbol M}}:=\rho {\boldsymbol V}+({e}/{m})\rho {\boldsymbol A}$, where $\rho :=m n_i$, and the canonical particle momenta ${\boldsymbol \pi }_p=m_p\boldsymbol {v}+e_p{\boldsymbol A}$, where ${\boldsymbol A}$ is the vector potential.

Before we proceed to this construction, let us recapitulate here some basic notions of noncanonical Hamiltonian dynamics. The Lagrange–Euler map from the Lagrangian to the Eulerian description, renders the Poisson brackets explicitly dependent on the dynamical variables of the system, say $\boldsymbol {\xi }$, i.e. it has the form

(2.25)\begin{equation} \{F,G\}=\langle F_{\xi_i}, {\mathcal{J}} _{ij}(\boldsymbol{\xi}) G_{\xi_j}\rangle, \end{equation}

where $F_\xi$ represents the functional derivative of $F$ with respect to $\xi$ and ${\mathcal {J}} (\boldsymbol {\xi })$ is the so-called Poisson operator.

This noncanonical Poisson bracket still satisfies the antisymmetry condition and the Jacobi identity:

(2.26)\begin{gather} \{F,G\}={-}\{G,F\}, \end{gather}

(2.27)\begin{gather}\{F,\{G,H\}\}+\{H,\{F,G\}\}+\{G,\{H,F\}\}=0, \end{gather}

where $F,G,H$ are functionals that are defined on the functional phase space. Owing to the explicit dependence on the dynamical variables, there exist non-trivial functionals ${\mathcal {C}}$ satisfying

(2.28)\begin{equation} \{ {\mathcal{C}} ,F\}=0,\quad \forall \, F. \end{equation}

These functionals ${\mathcal {C}}$ are called Casimirs. The equations of motion result from the following Hamilton equations:

(2.29)\begin{equation} \partial_t \xi_i=\{\xi_i,{\mathcal{H}}\}, \end{equation}

where ${\mathcal {H}}$ is the Hamiltonian of the model. Obviously, the Hamiltonian is conserved in view of the antisymmetry of the Poisson bracket and the Casimirs are constants owing to their commutative property. Such Hamiltonian structures have been identified in the context of fluid mechanics and ordinary magnetohydrodynamics (Morrison & Greene Reference Morrison and Greene1980; Morrison Reference Morrison1982,  Reference Morrison1998), Hall MHD (Holm Reference Holm1987; Lingam et al. Reference Lingam, Morrison and Miloshevich2015), extended-MHD (Abdelhamid, Kawazura & Yoshida Reference Abdelhamid, Kawazura and Yoshida2015; Lingam et al. Reference Lingam, Morrison and Miloshevich2015), and Vlasov–Poisson and Vlasov–Maxwell theories (Morrison Reference Morrison1980) (with a correction for Vlasov–Maxwell in the papers by Weinstein & Morrison (Reference Weinstein and Morrison1981), Marsden & Weinstein (Reference Marsden and Weinstein1982), Morrison (Reference Morrison1982) and a limitation to the correction pointed out by Morrison (Reference Morrison1982), which was followed up more recently by Morrison (Reference Morrison2013); Heninger & Morrison (Reference Heninger and Morrison2020) and then by Lainz, Sardón & Weinsten (Reference Lainz, Sardón and Weinsten2019)). For a more comprehensive presentation of the noncanonical Hamiltonian dynamics emerging in the Eulerian description of fluids and other continuum theories, the reader is referred to the paper by Morrison (Reference Morrison1998). For Vlasov models, the interested reader can consult the paper by Morrison (Reference Morrison2009).

Regarding our model, the sum of the Hall MHD and the particle brackets expressed in terms of canonical momenta is given by

(2.30)\begin{align} \{F,G\}& =\int \, \textrm{d}^{3}x\left\{\vphantom{\sum_p} \bar{\boldsymbol{M}}\boldsymbol{\cdot} (G_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{\bar{\boldsymbol{M}}}- F_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{\bar{\boldsymbol{M}}})\right.\nonumber\\ & \quad+\rho(G_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{\rho}-F_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{\rho}) -e^{{-}1}(G_{{\boldsymbol A}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_e}-F_{{\boldsymbol A}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_e})\nonumber\\ & \quad-\left.\frac{1}{en_e}(\boldsymbol{\nabla}\times{\boldsymbol A})\boldsymbol{\cdot} (F_{{\boldsymbol A}}\times G_{{\boldsymbol A}})+ \sum_p \int \, \textrm{d}^{3}{\boldsymbol \pi} f_p [\![F_{\bar{f}_p},G_{\bar{f}_p}]\!]_{{\boldsymbol \pi}_p} \right\}, \end{align}

where

(2.31)\begin{equation} [\![g,h]\!]_{{\boldsymbol \pi}_p}=\boldsymbol{\nabla} g \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\boldsymbol \pi}_p} h-\boldsymbol{\nabla} h \boldsymbol{\cdot} \boldsymbol{\nabla}_{{\boldsymbol \pi}_p} g, \end{equation}

and $\bar {f}_p$ is the distribution function expressed in terms of ${\boldsymbol \pi }_p$, i.e. $\bar {f}_p=\bar {f}_p({\boldsymbol x},{\boldsymbol \pi }_p,t)$.

The above Poisson bracket, along with the appropriate Hamiltonian, which is the sum of the Hall MHD and particle Hamiltonians expressed in terms of $\bar {\boldsymbol {M}}, {\boldsymbol \pi }$, describe correctly the dynamics of the system. However, the canonical momentum variables are not convenient because they mix up the velocity and the magnetic field potential. Ultimately, we would like to have a system of equations that treats velocity and magnetic field variables separately, i.e. like the system we derived by the standard approach in the previous subsection. To this end, following Tronci (Reference Tronci2010) and Marsden & Weinstein (Reference Marsden and Weinstein1982),we may perform the following change of variables:

(2.32a,b)\begin{equation} \bar{\boldsymbol{M}}\rightarrow {\boldsymbol V},\quad \bar{f}_p({\boldsymbol x},{\boldsymbol \pi}_p,t)=\bar{f}_p({\boldsymbol x},m_p\boldsymbol{v}+ e_p{\boldsymbol A},t)\rightarrow f_p({\boldsymbol x},\boldsymbol{v},t), \end{equation}

where

(2.33)\begin{equation} {\boldsymbol V}=\rho^{{-}1}\bar{\boldsymbol{M}}-\frac{e}{m}{\boldsymbol A}. \end{equation}

For convenience, this change of variables will be performed in two stages. First, we can write the bracket in terms of $f_p$ and then we can complete the change transforming from $\bar {\boldsymbol {M}}$ to ${\boldsymbol V}$. For the first change, we can directly use a result from Marsden & Weinstein (Reference Marsden and Weinstein1982), that is, for any functional $F[{\boldsymbol A},f_p]$ and functions $g({\boldsymbol x},\boldsymbol {v}), h({\boldsymbol x},\boldsymbol {v})$, we have $F[{\boldsymbol A},f_p({\boldsymbol x},\boldsymbol {v},t)]=\bar {F}[{\boldsymbol A},\bar {f}_p({\boldsymbol x},{\boldsymbol \pi }_p,t)], g({\boldsymbol x},\boldsymbol {v})=\bar {g}({\boldsymbol x},{\boldsymbol \pi }_p), h({\boldsymbol x},\boldsymbol {v})=\bar {h}({\boldsymbol x},{\boldsymbol \pi }_p)$, and the following transformation rules must hold:

(2.34)\begin{gather} [\![\bar{g},\bar{h}]\!]_{{\boldsymbol \pi}_p}= \frac{1}{m_p}[\![g,h]\!]+\frac{e_p}{m_p^{2}}(\boldsymbol{\nabla}\times{\boldsymbol A}) \boldsymbol{\cdot} (\nabla_{\boldsymbol{v}} g\times\nabla_{\boldsymbol{v}} h), \end{gather}

(2.35)\begin{gather}\bar{F}_{{\boldsymbol A}}=F_{{\boldsymbol A}}-\sum_p \frac{e_p}{m_p}\int \, \textrm{d}^{3}v f_p\nabla_{\boldsymbol{v}} F_{f_p}, \end{gather}

where

(2.36)\begin{equation} [\![g,h]\!]:=\boldsymbol{\nabla} g \boldsymbol{\cdot} \nabla_{\boldsymbol{v}} h-\boldsymbol{\nabla} h \boldsymbol{\cdot} \nabla_{\boldsymbol{v}} g. \end{equation}

Using (2.34) and (2.35), the bracket (2.30) takes the following form:

(2.37)\begin{align} \{F,G\}& =\int \, \textrm{d}^{3}x \left\{\vphantom{\sum_p}\bar{\boldsymbol{M}}\boldsymbol{\cdot} (G_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{\bar{\boldsymbol{M}}}-F_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{\bar{\boldsymbol{M}}})\right.\nonumber\\ & \quad +\rho(G_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{\rho}-F_{\bar{\boldsymbol{M}}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{\rho}) -e^{{-}1}(G_{{\boldsymbol A}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_e}-F_{{\boldsymbol A}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_e})\nonumber\\ & \quad -\frac{1}{en_e}(\boldsymbol{\nabla}\times{\boldsymbol A})\boldsymbol{\cdot} (F_{{\boldsymbol A}}\times G_{{\boldsymbol A}})-e^{{-}1} \sum_p \frac{e_p}{m_p}f_p[\boldsymbol{\nabla} G_{n_e}\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} F_{f_p}-\boldsymbol{\nabla} F_{n_e} \boldsymbol{\cdot} \nabla_{\boldsymbol{v}} G_{f_p} ] \nonumber\\ & \quad +\frac{1}{en_e}\sum_p\frac{e_p}{m_p}f_p (\boldsymbol{\nabla}\times{\boldsymbol A})\boldsymbol{\cdot} (\nabla_{\boldsymbol{v}} F_{f_p}\times G_{{\boldsymbol A}}-\nabla_{\boldsymbol{v}} G_{f_p}\times F_{{\boldsymbol A}})\nonumber\\ & \quad -\frac{1}{e n_e}(\boldsymbol{\nabla}\times {\boldsymbol A})\boldsymbol{\cdot} \int \int \, \textrm{d}^{3}v\, \textrm{d}^{3}v' \left(\sum_p \frac{e_p}{m_p}f_p\nabla_{\boldsymbol{v}} F_{f_p}\right)\times\left(\sum_{p'} \frac{e_{p'}}{m_{p'}}f_{p'}\boldsymbol{\nabla}_{\boldsymbol{v}'} F_{f_{p'}}\right)\nonumber\\ & \quad +\left.\sum_p \int \, \textrm{d}^{3}v \frac{f_p}{m_p} \left[[\![F_{f_p},G_{f_p}]\!]+ \frac{e_p}{m_p}(\boldsymbol{\nabla}\times {\boldsymbol A})\boldsymbol{\cdot} \left(\nabla_{\boldsymbol{v}} F_{f_p}\times\nabla_{\boldsymbol{v}} G_{f_p} \right) \right]\right\}. \end{align}

Now, we can proceed by expressing this bracket in terms of ${\boldsymbol V}$ and ${\boldsymbol B}=\boldsymbol {\nabla }\times {\boldsymbol A}$, instead of $\bar {{\boldsymbol M}}$ and ${\boldsymbol A}$. Let us note first that from (2.33), an arbitrary variation of ${\boldsymbol V}$ can be written as

(2.38)\begin{equation} \delta{\boldsymbol V}=\rho^{{-}1}\delta \bar{{\boldsymbol M}}-\rho^{{-}2}\bar{{\boldsymbol M}}\delta\rho -\frac{e}{m}\delta{\boldsymbol A}. \end{equation}

Considering a functional $\bar {F}$ of $\bar {{\boldsymbol M}}, {\boldsymbol A}$ and $\rho$, and then expressing it in terms of ${\boldsymbol V}, {\boldsymbol B}$ and $\rho$, the equality $\bar {F}[\bar {{\boldsymbol M}},{\boldsymbol A},\rho ]=F[{\boldsymbol V},{\boldsymbol B},\rho ]$ must hold. Upon taking the first variation of this relation and using the chain rule for the functional derivatives, the following relations can be deduced:

(2.39)\begin{gather} \bar{F}_{\bar{{\boldsymbol M}}}=\rho^{{-}1}F_{{\boldsymbol V}}, \end{gather}

(2.40)\begin{gather}\bar{F}_\rho= F_\rho-\rho^{{-}1} \left({\boldsymbol V}+\frac{e}{m}{\boldsymbol A}\right)\boldsymbol{\cdot} F_{{\boldsymbol V}}, \end{gather}

(2.41)\begin{gather}\bar{F}_{{\boldsymbol A}}= \boldsymbol{\nabla}\times F_{{\boldsymbol B}}-\frac{e}{m}F_{{\boldsymbol V}}. \end{gather}

Substituting (2.33) and (2.39)–(2.41) to (2.37), we find the final bracket, which readsFootnote ¹

(2.42)\begin{align} \{F,G\}& = \int \, \textrm{d}^{3}x\left\{\vphantom{\sum_p} (G_{\boldsymbol V}\boldsymbol{\cdot} \nabla F_\rho -F_{\boldsymbol V}\boldsymbol{\cdot} \nabla G_\rho)+ \rho^{{-}1} (\boldsymbol{\nabla}\times{\boldsymbol V})\boldsymbol{\cdot} (F_{\boldsymbol V}\times G_{\boldsymbol V})\right.\nonumber\\ & \quad +\frac{e}{m^{2}}\left(\frac{mn_e-\rho}{\rho n_e}\right){\boldsymbol B}\boldsymbol{\cdot} (F_{\boldsymbol V}\times G_{\boldsymbol V})+m^{{-}1} (G_{\boldsymbol V}\boldsymbol{\cdot} \nabla F_{n_e}-F_{\boldsymbol V}\boldsymbol{\cdot} \nabla G_{n_e})\nonumber\\ & \quad + \frac{1}{m n_e} {\boldsymbol B} \boldsymbol{\cdot} [ F_{{\boldsymbol V}}\,{\times}\,(\boldsymbol{\nabla}\,{\times}\, G_{\boldsymbol B})- G_{{\boldsymbol V}}\,{\times}\,(\boldsymbol{\nabla}\,{\times}\, F_{\boldsymbol B}) ]-\frac{1}{en_e}{\boldsymbol B}\boldsymbol{\cdot} [(\boldsymbol{\nabla}\,{\times}\, F_{\boldsymbol B})\,{\times}\,(\boldsymbol{\nabla}\,{\times}\, G_{\boldsymbol B})]\nonumber\\ & \quad +\frac{1}{en_e}\sum_p\frac{e_p}{m_p}\int \, \textrm{d}^{3}v f_p {\boldsymbol B} \boldsymbol{\cdot} \left[\vphantom{\frac{e}{m}}\nabla_{\boldsymbol{v}} F_{f_p}\times(\boldsymbol{\nabla}\times G_{\boldsymbol B})- \nabla_{\boldsymbol{v}} G_{f_p}\times(\boldsymbol{\nabla}\times F_{\boldsymbol B})\right.\nonumber\\ & \quad +\left.\frac{e}{m}(\nabla_{\boldsymbol{v}} G_{f_p}\times F_{{\boldsymbol V}}-\nabla_{\boldsymbol{v}} F_{f_p} \times G_{{\boldsymbol V}})\right]\nonumber\\ & \quad -\frac{1}{e}\sum_p \frac{e_p}{m_p}\int \, \textrm{d}^{3}v f_p (\boldsymbol{\nabla} G_{n_e}\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} F_{f_p}-\boldsymbol{\nabla} F_{n_e}\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} G_{f_p})\nonumber\\ & \quad -\frac{1}{en_e}{\boldsymbol B} \boldsymbol{\cdot} \left( \sum_{p,p'}\frac{e_p}{m_p} \frac{e_{p'}}{m_{p'}}\int\int \, \textrm{d}^{3}v \, \textrm{d}^{3}v' f_p f_{p'} \nabla_{\boldsymbol{v}} F_{f_p}\times \nabla_{\boldsymbol{v}} G_{f_{p'}}\right) \nonumber\\ & \quad +\left.\sum_p \int \, \textrm{d}^{3}v \frac{f_p}{m_p}\left[[\![F_{f_p},G_{f_p}]\!]+ \frac{e_p}{m_p}{\boldsymbol B}\boldsymbol{\cdot} (\nabla_{\boldsymbol{v}} F_{f_p}\times \nabla_{\boldsymbol{v}} G_{f_p})\right] \right\}. \end{align}

Observe, unlike the bracket (2.30), (2.42) is gauge invariant. Now, having computed the bracket of our model, we write down the Hamiltonian functional, which is the direct sum of the fluid and particle Hamiltonians, i.e.

(2.43)\begin{equation} {\mathcal{H}}=\int \, \textrm{d}^{3}x \left[\rho \frac{|{\boldsymbol V}|^{2}}{2}+\rho U(\rho)+n_e {\mathcal{U}} _e+\frac{|{\boldsymbol B}|^{2}}{2\mu_0}\right]+\sum_p\int\int \, \textrm{d}^{3}x\,\textrm{d}^{3}v m_p \frac{v^{2}}{2} f_p. \end{equation}

Here, $U(\rho )$ is the specific internal energy of the thermal ion fluid and ${\mathcal {U}} _e$ is some electron internal energy function. The dependence of this function is dictated by the nature of the electron pressure. In our case, it was first shown by Morrison (Reference Morrison1982) (and subsequent work, e.g. Hazeltine, Mahajan & Morrison Reference Hazeltine, Mahajan and Morrison2013) that gyrotropic pressure tensors of the form (2.15) follow from an internal energy that depends explicitly on $n_e$ and $|{\boldsymbol B}|$. If an electron internal energy has the form

(2.44)\begin{equation} {\mathcal{U}} _e= {\mathcal{U}} _e(n_e,|{\boldsymbol B}|), \end{equation}

then the following equations give the pressure tensor components $P_{e\parallel }$ and $P_{e\perp }$:

(2.45)\begin{gather} \frac{\partial {\mathcal{U}} _e}{\partial n_e}=\frac{P_{e\parallel}}{n_e^{2}}, \end{gather}

(2.46)\begin{gather}\frac{\partial {\mathcal{U}} _e}{\partial |{\boldsymbol B}|}=\frac{P_{e\perp}-P_{e\parallel}}{n_e|{\boldsymbol B}|}; \end{gather}

and hence

(2.47)\begin{gather} \frac{\delta {\mathcal{H}}}{\delta n_e}= {\mathcal{U}} _e+\frac{P_{e\parallel}}{n_e}, \end{gather}

(2.48)\begin{gather}\frac{\delta {\mathcal{H}}}{\delta {\boldsymbol B}}=\mu_0^{{-}1}(1-\gamma){\boldsymbol B}, \end{gather}

where $\gamma := ({P_{e\parallel }-P_{e\perp }})/(B^2/ \mu_0)$ is a function that measures the electron pressure anisotropy. Note that the Poisson structure (2.42) is valid for barotropic (${\mathcal {U}} _e= {\mathcal {U}} _e(n_e$)) and gyrotropic [(2.45) and (2.46)] electron pressure. The identification of an appropriate Hamiltonian structure for a generally non-gyrotropic electron pressure tensor will be pursued in a future work.

The equations of motion follow from (2.29) with the Poisson bracket (2.42), the Hamiltonian (2.43) and also using (2.47), (2.48). It can be readily seen that $\partial _t n_e=\{n_e,{\mathcal {H}}\}$ and $\partial _t\rho =\{\rho,{\mathcal {H}}\}$ are indeed (2.22) and the continuity equation for the fluid ions, respectively. The latter is

(2.49)\begin{equation} \partial_t\rho={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (\rho{\boldsymbol V}). \end{equation}

The remaining equations, i.e. the momentum, the induction and the Vlasov equation that stem from (2.29), are

(2.50)\begin{align} \partial_t{\boldsymbol V} & ={\boldsymbol V}\times\boldsymbol{\nabla}\times{\boldsymbol V} -\boldsymbol{\nabla}\left( h+\frac{{V}^{2}}{2} \right)+\frac{e}{m}\left(1-\frac{n_i}{n_e} \right){\boldsymbol V}\times{\boldsymbol B}\nonumber\\ & \quad +\frac{1}{mn_e}(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B} -{\boldsymbol J}_k)\times{\boldsymbol B}-m^{{-}1}\boldsymbol{\nabla}\left( {\mathcal{U}} _e+\frac{P_{e\parallel}}{n_e}\right)-\frac{\boldsymbol{\nabla}\times(\gamma {\boldsymbol B})}{\mu_0mn_e}\times{\boldsymbol B} , \end{align}

(2.51)\begin{align} \partial_t {\boldsymbol B} & =\boldsymbol{\nabla}\times\left[\frac{n_i}{n_e}{\boldsymbol V}\times{\boldsymbol B}- \frac{1}{en_e}(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-{\boldsymbol J}_k)\times{\boldsymbol B}\right]\nonumber\\ & \quad +\mu_0^{{-}1}\boldsymbol{\nabla}\times\left[\frac{\gamma}{en_e} ({\boldsymbol B}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol B}-\boldsymbol{\nabla}{\boldsymbol B}\boldsymbol{\cdot} {\boldsymbol B})+\frac{\boldsymbol{\nabla}\gamma\times{\boldsymbol B}}{en_e} \times{\boldsymbol B}\right], \end{align}

(2.52)\begin{align} \partial_t f_p& ={-}\boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla} f_p-\frac{e_p}{m_p}\left[ \left(\boldsymbol{v}-\frac{n_i}{n_e}{\boldsymbol V}\right)\times{\boldsymbol B}\right.\nonumber\\ & \quad \left.+\frac{1}{en_e}\left(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-{\boldsymbol J}_k\right)\times{\boldsymbol B} -\frac{1}{e}\boldsymbol{\nabla}\left( {\mathcal{U}} _e+\frac{P_{e\parallel}}{n_e}\right)- \frac{\boldsymbol{\nabla}\times(\gamma{\boldsymbol B})}{\mu_0en_e}\times{\boldsymbol B}\right]\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} f_p. \end{align}

It is a matter of vector calculus manipulations to prove that

(2.53)\begin{align} & -\frac{1}{m}\boldsymbol{\nabla}\left( {\mathcal{U}} _e+\frac{P_{e\parallel}}{n_e}\right)- \frac{\boldsymbol{\nabla}\times(\gamma{\boldsymbol B})}{\mu_0mn_e}\times{\boldsymbol B}\nonumber\\ & \quad ={-}\frac{1}{mn_e}[\boldsymbol{\nabla} P_{e\perp}+\mu_0^{{-}1}(\gamma {\boldsymbol B}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol B} +{\boldsymbol B}{\boldsymbol B}\boldsymbol{\cdot} \boldsymbol{\nabla}\gamma) ]={-}\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{mn_e}. \end{align}

The last equality can be easily proven by taking the divergence of the gyrotropic pressure equation (2.15). In light of (2.53), the momentum and the Vlasov equations, (2.50) and (2.52), respectively, which arise from the Hamiltonian formulation, are identical to (2.20) and (2.21). In addition, performing some manipulations on the last term of (2.51), it can be seen that

(2.54)\begin{align} & \mu_0^{{-}1}\boldsymbol{\nabla}\times\left[\frac{\gamma}{en_e} ({\boldsymbol B}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol B}-\boldsymbol{\nabla}{\boldsymbol B}\boldsymbol{\cdot} {\boldsymbol B})+ \frac{\boldsymbol{\nabla}\gamma\times{\boldsymbol B}}{en_e}\times{\boldsymbol B}\right]\nonumber\\ & \quad =\boldsymbol{\nabla}\times\left[\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e}- \frac{1}{e}\boldsymbol{\nabla} {\mathcal{U}} _e-\frac{1}{e}\boldsymbol{\nabla}\left(n_e \frac{\partial {\mathcal{U}} _e}{\partial n_e}\right)\right]= \boldsymbol{\nabla}\times\frac{\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e}{en_e}. \end{align}

Hence, (2.51) is indeed the induction equation (2.19).

2.3. Casimir invariants

A typical procedure to identify the Casimirs of a noncanonical Poisson bracket, e.g. (2.42), is to rearrange it in the following form:

(2.55)\begin{equation} \{F,G\}=\int \, \textrm{d}^{3}x F_{u_i} \mathcal{J}_{ij} G_{u_j}, \end{equation}

where ${\mathcal {J}}$ is the Poisson operator associated with this bracket, and then seek solutions to the following system of Casimir-determining equations:

(2.56)\begin{equation} \mathcal{J}_{ij} {\mathcal{C}} _{u_j}=0,\quad i=1,\ldots,5. \end{equation}

By this procedure, we find the following Casimir invariants:

(2.57a,b)\begin{gather} {\mathcal{C}} _1= \int \, \textrm{d}^{3}x \rho,\quad {\mathcal{C}} _2 = \int \, \textrm{d}^{3}x n_e, \end{gather}

(2.58)\begin{gather} {\mathcal{C}} _3= \frac{1}{2}\int \, \textrm{d}^{3}x {\boldsymbol A}\boldsymbol{\cdot} {\boldsymbol B}, \end{gather}

(2.59)\begin{gather}{\mathcal{C}} _4= \frac{1}{2}\int \, \textrm{d}^{3}x \left({\boldsymbol V}+\frac{e}{m}{\boldsymbol A}\right)\boldsymbol{\cdot} \left(\boldsymbol{\nabla}\times{\boldsymbol V}+\frac{e}{m}{\boldsymbol B}\right), \end{gather}

(2.60)\begin{gather}{\mathcal{C}} _p= \int\int \, \textrm{d}^{3}x\,\textrm{d}^{3}v \varLambda_p(f_p). \end{gather}

This set of Casimirs, typical of magnetofluid models (e.g. Lingam, Miloshevich & Morrison Reference Lingam, Miloshevich and Morrison2016), has two helicity invariants, but is augmented by an additional kinetic Casimir for each particle species $p$, which involves the corresponding distribution function, while there are no cross-fluid-kinetic Casimirs. Here, $\varLambda _p(f_p)$ are arbitrary functions of their respective distribution functions $f_p$.

3.1. Transformation of the Hamiltonian CCS

As it is well known, another method to couple the energetic and the fluid components is through the pressure tensor appearing in a center-of-mass momentum equation rather than through the current density. This can be done upon taking the first-order fluid moment of the Vlasov equations, which govern the dynamics of the energetic species, and adding it to the momentum equation of the thermal ions. The first-order fluid moment of (2.3) yields

(3.1)\begin{equation} m_p\partial_t (n_p{\boldsymbol V}_p)=e_pn_p({\boldsymbol E}+{\boldsymbol V}_p\times{\boldsymbol B})-\sum_p\boldsymbol{\nabla}\boldsymbol{\cdot} \tilde{\boldsymbol{\mathsf{P}}}_p, \end{equation}

where

(3.2a–c)\begin{equation} n_p=\int \, \textrm{d}^{3}v f_p,\quad {\boldsymbol V}_p=\frac{1}{n_p}\int \, \textrm{d}^{3}v \boldsymbol{v} f_p ,\quad \tilde{\boldsymbol{\mathsf{P}}}_p=m_p\int \, \textrm{d}^{3}v \boldsymbol{v}\boldsymbol{v} f_p. \end{equation}

Adding (3.1) and (2.2) for $s=i$ and using the Ohm's law (2.18), we obtain the following momentum equation:

(3.3)\begin{equation} \partial_t{\boldsymbol M}+m\boldsymbol{\nabla}\boldsymbol{\cdot} (n_i {\boldsymbol V}{\boldsymbol V})=\mu_0^{{-}1}(\boldsymbol{\nabla}\times{\boldsymbol B})\times{\boldsymbol B} -\boldsymbol{\nabla} P_i-\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e-\sum_p\boldsymbol{\nabla}\boldsymbol{\cdot} \tilde{\boldsymbol{\mathsf{P}}}_p, \end{equation}

where

(3.4)\begin{equation} {\boldsymbol M}:=mn_i{\boldsymbol V}+\sum_p m_pn_p {\boldsymbol V}_p. \end{equation}

If we replace (2.20) of the CCS model by (3.3), we obtain a totally equivalent model coupling though the fluid and the particle components through the particle species pressure tensor terms in (3.3). It is logical then to expect that this reformulation of the model would have a Hamiltonian structure resulting from some change of dynamical variables. Tronci (Reference Tronci2010) showed that the reformulation of the Hamiltonian structure is effected by considering ${\boldsymbol M}$ instead of ${\boldsymbol V}$ as an independent dynamical variable. Performing this change of variables, one can see how the functional derivatives with respect to the old and the new sets of dynamical variables should relate. Requiring

(3.5)\begin{equation} \delta F[n_i,n_e,{\boldsymbol V},{\boldsymbol B},f_p]=\delta \tilde{F}[n_i,n_e,{\boldsymbol M},{\boldsymbol B},f_p], \end{equation}

and employing the chain rule for functional derivatives, the following equations can be deduced

(3.6)\begin{equation} \left.\begin{array}{c@{}} F_{n_e}=\tilde{F}_{n_e}, \quad F_{{\boldsymbol B}}=\tilde{F}_{{\boldsymbol B}},\quad F_{{\boldsymbol V}}=m n_i \tilde{F}_{{\boldsymbol M}},\\ F_{n_i}=\tilde{F}_{n_i}+m {\boldsymbol V}\boldsymbol{\cdot} \tilde{F}_{{\boldsymbol M}},\quad F_{f_p}=\tilde{F}_{f_p}+m_p\boldsymbol{v}\boldsymbol{\cdot} \tilde{F}_{{\boldsymbol M}}. \end{array}\right\} \end{equation}

Substituting equations (3.6) into the Poisson bracket (2.42) and using the definition (3.4) of ${\boldsymbol M}$, we find, after some algebra, the following bracket:

(3.7)\begin{align} \{F,G\}& = \int \, \textrm{d}^{3}x\,\left\{\vphantom{\frac{1}{en_e}}{\boldsymbol M}\boldsymbol{\cdot} (G_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{{\boldsymbol M}}-F_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{{\boldsymbol M}}) +n_i (G_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_i}-F_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_i})\right.\nonumber\\ & \quad + n_e(G_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_e}-F_{{\boldsymbol M}}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_e}) +{\boldsymbol B}\boldsymbol{\cdot} [F_{{\boldsymbol M}}\times(\boldsymbol{\nabla}\times{\boldsymbol B})-G_{{\boldsymbol M}}\times(\boldsymbol{\nabla}\times F_{{\boldsymbol B}})] \nonumber\\ & \quad - \sum_p\frac{e_p}{em_p}\int \, \textrm{d}^{3}v f_p(\boldsymbol{\nabla}_{\boldsymbol{v}}F_{f_p}\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_e}-\boldsymbol{\nabla}_{\boldsymbol{v}}G_{f_p}\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_e})\nonumber\\ & \quad + \frac{1}{en_e}\sum_p \frac{e_p}{m_p}\int \, \textrm{d}^{3}v f_p{\boldsymbol B}\boldsymbol{\cdot} [\boldsymbol{\nabla}_{\boldsymbol{v}}F_{f_p}\times(\boldsymbol{\nabla}\times G_{{\boldsymbol B}})-\boldsymbol{\nabla}_{\boldsymbol{v}}G_{f_p} \times(\boldsymbol{\nabla}\times F_{{\boldsymbol B}})]\nonumber\\ & \quad - \frac{1}{en_e}{\boldsymbol B}\boldsymbol{\cdot} \int \int \, \textrm{d}^{3}v\,\textrm{d}^{3}v'\, \sum_{p,p'}\frac{e_pe_{p'}}{m_pm_{p'}}f_p(\boldsymbol{v})f_{p'}(\boldsymbol{v}')\nabla_{\boldsymbol{v}} F_{f_p} \times\boldsymbol{\nabla}_{\boldsymbol{v}'}G_{f_{p'}}\nonumber\\ & \quad + \sum_p m_p^{{-}1}\int \, \textrm{d}^{3}v \, f_p \left[[\![F_{f_p},G_{f_p}]\!]+\frac{e_p}{m_p}{\boldsymbol B}\boldsymbol{\cdot} (\nabla_{\boldsymbol{v}} F_{f_p}\times \nabla_{\boldsymbol{v}} G_{f_p})\right.\nonumber\\ & \quad \left.\left.+\vphantom{\frac{e_p}{m_p}} m_p([\![F_{f_p}, \boldsymbol{v}\boldsymbol{\cdot} G_{{\boldsymbol M}}]\!]-[\![G_{f_p},\boldsymbol{v}\boldsymbol{\cdot} F_{{\boldsymbol M}}]\!])\right]-\frac{1}{en_e}{\boldsymbol B}\boldsymbol{\cdot} [(\boldsymbol{\nabla}\times F_{{\boldsymbol B}})\times(\boldsymbol{\nabla}\times G_{{\boldsymbol B}})]\right\} . \end{align}

We should also write the Hamiltonian in terms of ${\boldsymbol M}$ and then invoke Hamilton's equations to retrieve the dynamical system in this PCS formulation. The transformed Hamiltonian reads

(3.8)\begin{align} {\mathcal{H}}& = \int \, \textrm{d}^{3}x \, \left[\frac{|{\boldsymbol M}-\boldsymbol{{\mathcal{M}}}|^{2}}{2m n_i}+ n_i {\mathcal{U}} _i(n_i)+n_e {\mathcal{U}} _e(n_e,|{\boldsymbol B}|)+\frac{|{\boldsymbol B}|^{2}}{2\mu_0} \right] \nonumber\\ & \quad + \sum_p \frac{m_p}{2}\int \int \, \textrm{d}^{3}x\,\textrm{d}^{3}v f_pv^{2}, \end{align}

where

(3.9)\begin{equation} \boldsymbol{{\mathcal{M}}}:=\sum_p \int \, \textrm{d}^{3}v m_pf_p \boldsymbol{v}. \end{equation}

The functional derivative of ${\mathcal {H}}$ with respect to the new variable ${\boldsymbol M}$ is

(3.10)\begin{equation} \frac{\delta {\mathcal{H}}}{\delta {\boldsymbol M}}={\boldsymbol V}. \end{equation}

Also, one should notice that the functional derivative with respect to $f_p$ is not the same as in the previous case of the CCS, but

(3.11)\begin{equation} \frac{\delta {\mathcal{H}}}{\delta f_p}=m_p\frac{v^{2}}{2}-m_p\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol V}. \end{equation}

We can verify that Hamilton's equations are indeed the dynamical equations of the CCS model with the momentum equation being replaced by (3.3).

3.2. Conventional construction of a PCS

The Hamiltonian system described in the previous subsection is equivalent to the CCS model of § 2 because it is obtained from CCS by a mere change of variables. However, it is a commonality in the hybrid fluid–kinetic models to employ pressure coupling schemes which involve a simpler coupling of the bulk plasma with the energetic particles assuming that $n_p\ll n_i$ and ${\boldsymbol M}/\rho \sim {\boldsymbol V}$ in the dynamical equations. These assumptions result in a momentum equation governing the ion momentum density, instead of the total momentum density ${\boldsymbol M}$, and containing the divergence of a pressure tensor associated with the energetic particles (e.g. see Park et al. Reference Park, Parker, Biglari, Chance, Chen, Cheng, Hahm, Lee, Kulsrud, Monticello, Sugiyama and White1992; Kim, Sovinec & Parker Reference Kim, Sovinec and Parker2004; Fu et al. Reference Fu, Park, Strauss, Breslau, Chen, Jardin and Sugiyama2006; Takahashi, Brennan & Kim Reference Takahashi, Brennan and Kim2009). A similar treatment in our case with a two-fluid, Hall-MHD bulk, results in the following set of equations:

(3.12)\begin{gather} \partial_t n +\boldsymbol{\nabla}\boldsymbol{\cdot} (n{\boldsymbol V})=0, \end{gather}

(3.13)\begin{gather}\rho (\partial_t{\boldsymbol V}+{\boldsymbol V}\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol V})={\boldsymbol J}\times{\boldsymbol B} -\sum_p\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_p-\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e-\boldsymbol{\nabla} P_i, \end{gather}

(3.14)\begin{gather}\partial_t {\boldsymbol B}=\boldsymbol{\nabla}\times\left[ {\boldsymbol V}\times {\boldsymbol B} -\frac{1}{en}{\boldsymbol J}\times {\boldsymbol B}+\frac{1}{en}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e\right], \end{gather}

(3.15)\begin{gather}\partial_t f_p+\boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla} f_p+\frac{e_p}{m_p} \left[ (\boldsymbol{v}-{\boldsymbol V})\times {\boldsymbol B} +\frac{1}{en}{\boldsymbol J}\times{\boldsymbol B}- \frac{1}{en}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e\right]\boldsymbol{\cdot} \nabla_{\boldsymbol{v}} f_p=0. \end{gather}

Here, $n=n_i=n_e$, which is the zeroth order quasineutrality condition for $n_p/n_i \ll 1$.

3.3. Hamiltonian construction of a PCS

It has been stressed in several works (Tronci Reference Tronci2010; Tronci et al. Reference Tronci, Tassi, Camporeale and Morrison2014; Burby & Tronci Reference Burby and Tronci2017; Close et al. Reference Close, Burby and Tronci2018) that such PC models, which are widely employed in the study of energetic particle effects on plasma stability, do not conserve some energy functional as all consistent ideal plasma models do (e.g. Morrison & Greene Reference Morrison and Greene1980; Marsden & Weinstein Reference Marsden and Weinstein1982; Lingam et al. Reference Lingam, Morrison and Miloshevich2015). In addition, it has been shown that a Vlasov–MHD model in the pressure-coupling scheme exhibits spurious instabilities attributed to the lack of energy conservation (Tronci et al. Reference Tronci, Tassi, Camporeale and Morrison2014). In contrast, a Hamiltonian variant of the model, derived by a procedure that ensures energy conservation (Tronci Reference Tronci2010), does not contain these unphysical modes. Prompted by this observation, we employ the method of Tronci (Reference Tronci2010) in our case so as to derive a simplified, yet Hamiltonian, PCS model. The idea is, instead of assuming $n_p\ll n_i$ to simplify the equations of motion, to replace the ion momentum density ${\boldsymbol M}-\boldsymbol {{\mathcal {M}}}$ by the total momentum density ${\boldsymbol M}$ in the Hamiltonian functional, implicitly assuming that ${\boldsymbol M}\approx mn_i{\boldsymbol V}$ on the Hamiltonian level. Hence, the term $|{\boldsymbol M}-\boldsymbol {{\mathcal {M}}}|^{2}/(2m n_i)$ in (3.8) becomes $|{\boldsymbol M}|^{2}/(2mn_i)$, which leads to

(3.16)\begin{align} {\mathcal{H}}& = \int \, \textrm{d}^{3}x \left[\frac{|{\boldsymbol M}|^{2}}{2mn_i}+n_i {\mathcal{U}} _i(n_i)+n_e {\mathcal{U}} _{e}(n_e,|{\boldsymbol B}|)+\frac{|{\boldsymbol B}|^{2}}{2\mu_0}\right]\nonumber\\ & \quad +\sum_p \frac{m_p}{2}\int \int \, \textrm{d}^{3}x\, \textrm{d}^{3}v f_p v^{2}\nonumber\\ & =\int \, \textrm{d}^{3}x \left[\frac{mn_i}{2}|{\boldsymbol u}|^{2}+n_i {\mathcal{U}} _i(n_i) +n_e {\mathcal{U}} _{e}(n_e,|{\boldsymbol B}|)+\frac{|{\boldsymbol B}|^{2}}{2\mu_0}\right]\nonumber\\ & \quad +\sum_p \frac{m_p}{2}\int \int \, \textrm{d}^{3}x\, \textrm{d}^{3}v f_p v^{2}, \end{align}

where ${\boldsymbol u}:={\boldsymbol M}/(m n_i)$ is a weighted sum of the ion velocities. Note that for $n_p\ll n_i$, and if ${\boldsymbol V}_p$ are comparable to ${\boldsymbol V}$, then ${\boldsymbol u}={\boldsymbol V}$ up to zeroth order in $n_p/n_i$. It is convenient for us, in terms of comparing with the results of the previous section and other studies, to write the Poisson bracket (3.7) in terms of the velocity variable ${\boldsymbol u}$. This is done in Appendix A. The Casimirs of the bracket given in Appendix A are

(3.17a,b)\begin{align} {\mathcal{C}} _1 & = \int \, \textrm{d}^{3}x n_i, \quad {\mathcal{C}} _2 = \int \, \textrm{d}^{3}x n_e, \end{align}

(3.18)\begin{align} {\mathcal{C}} _3 & = \frac{1}{2}\int \, \textrm{d}^{3}x {\boldsymbol A}\boldsymbol{\cdot} {\boldsymbol B}, \end{align}

(3.19)\begin{align} {\mathcal{C}} _4 & = \int \, \textrm{d}^{3}x\,{\boldsymbol u}\boldsymbol{\cdot} \left(\frac{1}{2}\boldsymbol{\nabla}\times{\boldsymbol u}+ \frac{e}{m}{\boldsymbol B}\right)\nonumber\\ & \quad -\sum_p\int \, \textrm{d}^{3}x\frac{m_p}{m n_i}\int \, \textrm{d}^{3}v f_p\boldsymbol{v}\boldsymbol{\cdot} \left[\boldsymbol{\nabla}\times\left( {\boldsymbol u}-\frac{1}{2m n_i} \sum_{p'}m_{p'}\int \, \textrm{d}^{3}v' f_{p'}\boldsymbol{v}'\right)+\frac{e}{m}{\boldsymbol B}\right], \end{align}

(3.20)\begin{align} {\mathcal{C}} _p & = \int \, \textrm{d}^{3}x\,\textrm{d}^{3}v \varLambda_p(f_p). \end{align}

The equations of motion that arise from Hamilton's equations with the Poisson bracket (A2) (Appendix A) and the Hamiltonian (3.16) are

(3.21)\begin{align} \partial_t n_i& ={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (n_i{\boldsymbol u}), \end{align}

(3.22)\begin{align} \partial_t n_e& ={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (n_e {\boldsymbol u})- \frac{1}{e}\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_k, \end{align}

(3.23)\begin{align} \partial_t {\boldsymbol B}& =\boldsymbol{\nabla}\times\left[{\boldsymbol u}\times{\boldsymbol B} -\frac{1}{en_e} (\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-{\boldsymbol J}_k)\times{\boldsymbol B}+ \frac{1}{en_e}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e\right], \end{align}

(3.24)\begin{align} \partial_t {\boldsymbol u}& ={\boldsymbol u}\times\boldsymbol{\nabla}\times{\boldsymbol u}-\boldsymbol{\nabla}( h_i+u^{2}/2) - \rho^{{-}1}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e\nonumber\\ & \quad -\rho^{{-}1}\sum_p \boldsymbol{\nabla}\boldsymbol{\cdot} \tilde{\boldsymbol{\mathsf{P}}}_p+\mu_0^{{-}1} \rho^{{-}1}(\boldsymbol{\nabla}\times{\boldsymbol B})\times{\boldsymbol B}, \end{align}

(3.25)\begin{align} \partial_t f_p& ={-}(\boldsymbol{v}+{\boldsymbol u})\boldsymbol{\cdot} \boldsymbol{\nabla} f_p-\frac{e_p}{m_p} \left[\boldsymbol{v}\times{\boldsymbol B}+\frac{1}{en_e}(\mu_0^{{-}1}\boldsymbol{\nabla}\times{\boldsymbol B}-{\boldsymbol J}_k)\times{\boldsymbol B}\right.\nonumber\\ & \quad -\left.\frac{1}{en_e}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\mathsf{P}}_e\right] \boldsymbol{\cdot} \boldsymbol{\nabla}_{v}f_p+\nabla_{\boldsymbol{v}} f_p\boldsymbol{\cdot} \boldsymbol{\nabla}{\boldsymbol u}\boldsymbol{\cdot} \boldsymbol{v}, \end{align}

where $\rho =m n_i$ and $h_i=(1/m) \textrm{d}[n_i \mathcal{U}_i(n_i)]/\textrm{d}n_i$ is the specific enthalpy of the thermal ion fluid. Computing the first-order velocity moment of the Vlasov equation (3.25), we find

(3.26)\begin{equation} \partial_t\sigma_k={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (\sigma_k{\boldsymbol u})-\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol J}_k, \end{equation}

which, combined with (3.21) and (3.22), yields

(3.27)\begin{equation} \partial_t \sigma={-}\boldsymbol{\nabla}\boldsymbol{\cdot} (\sigma{\boldsymbol u}), \end{equation}

which is a continuity equation implying global but not local charge conservation. However, if the plasma is initially quasineutral, i.e. $\sigma =0$ at $t=0$, then quasineutrality is ensured by (3.27) $\forall t>0$.

4.1. Translationally symmetric formulation

In this section, we consider simplified dynamics with all dynamical variables being invariant along a fixed straight axis in physical space. In this translationally symmetric case, the magnetic field and macroscopic velocity can be expressed in terms of five scalar Clebsch potentials. Note that the distribution functions, although translationally symmetric in space, still depend on all three microscopic velocity coordinates. Translationally symmetric models facilitate computer simulations, because the dependency on one spatial coordinate is dropped and can be considered good approximations whenever a strong guiding magnetic field directed in a fixed direction is present. Using Cartesian coordinates $(x,y,z)$ in physical space and assuming invariance along the $z$ axis, the velocity and the magnetic field can be written as

(4.1)\begin{gather} {\boldsymbol u} = u_z(x,y,t)\hat{z} +\boldsymbol{\nabla}\chi(x,y,t)\times \hat{z}+\boldsymbol{\nabla}\varUpsilon(x,y,t), \end{gather}

(4.2)\begin{gather}{\boldsymbol B} = B_z(x,y,t)\hat{z}+\boldsymbol{\nabla}\psi(x,y,t)\times\hat{z}. \end{gather}

To obtain a Hamiltonian formulation in this symmetric case, we need to translate this field decomposition in a decomposition of the vector functional derivatives in terms of functional derivatives with respect to the scalar fields $u_z,B_z,\chi,\psi,\varUpsilon$. The process of deriving these relations has been described several times before, e.g. by Andreussi, Morrison & Pegoraro (Reference Andreussi, Morrison and Pegoraro2010), Kaltsas, Throumoulopoulos & Morrison (Reference Kaltsas, Throumoulopoulos and Morrison2017), Grasso et al. (Reference Grasso, Tassi, Abdelhamid and Morrison2017) and Kaltsas, Throumoulopoulos & Morrison (Reference Kaltsas, Throumoulopoulos and Morrison2018). Following the same procedure here, we find

(4.3)\begin{gather} F_{{\boldsymbol u}}=F_{u_z}\hat{z}+\boldsymbol{\nabla} F_\varOmega \times\hat{z}-\boldsymbol{\nabla} F_{w}, \end{gather}

(4.4)\begin{gather}F_{{\boldsymbol B}}=F_{B_z}\hat{z}-\boldsymbol{\nabla}(\varDelta^{{-}1}F_\psi)\times\hat{z}, \end{gather}

where $\varDelta ^{-1}$ is the inverse Laplacian operator, $\varOmega :=-\Delta \chi \hat {z}$ and $w:=\Delta \varUpsilon$. The translationally symmetric Hall MHD bracket is known from Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017), Grasso et al. (Reference Grasso, Tassi, Abdelhamid and Morrison2017), and hence we have to compute the translationally symmetric cross-kinetic-Hall MHD terms and the term accounting for electron fluid thermodynamics. This is done in Appendix B upon substituting (4.3) and (4.4) in the bracket (A2) of Appendix A. The resulting translationally symmetric bracket is

(4.5)\begin{align} \{F,G\}_{TS}& = \int \, \textrm{d}^{2}x \left\{ \vphantom{\frac{e_p}{m_p}} F_\rho \Delta G_w-G_\rho\Delta F_w\right.\nonumber\\ & \quad +\frac{n_e}{\rho}([F_\varOmega,G_{n_e}]- [G_\varOmega,F_{n_e}]+\boldsymbol{\nabla} F_w\boldsymbol{\cdot} \boldsymbol{\nabla} G_{n_e}-\boldsymbol{\nabla} G_w\boldsymbol{\cdot} \boldsymbol{\nabla} F_{n_e})\nonumber\\ & \quad +\rho^{{-}1} \varOmega ([F_\varOmega,G_\varOmega]+[F_w,G_w]+\boldsymbol{\nabla} F_w \boldsymbol{\cdot} \boldsymbol{\nabla} G_\varOmega -\boldsymbol{\nabla} F_\varOmega\boldsymbol{\cdot} \boldsymbol{\nabla} G_w)\nonumber\\ & \quad +u_z([F_\varOmega,\rho^{{-}1}G_{u_z}]-[G_\varOmega,\rho^{{-}1}F_{u_z}]+ \boldsymbol{\nabla}(\rho^{{-}1}G_{u_z})\boldsymbol{\cdot} \boldsymbol{\nabla} F_w\nonumber\\ & \quad-\boldsymbol{\nabla}(\rho^{{-}1}F_{u_z})\boldsymbol{\cdot} \boldsymbol{\nabla} G_w+\rho^{{-}1}F_\varUpsilon G_{u_z}- \rho^{{-}1}G_\varUpsilon F_{u_z}) \nonumber\\ & \quad +\psi ([F_\varOmega,\rho^{{-}1}G_\psi]-[G_\varOmega, \rho^{{-}1} F_\psi]+[F_{B_z},\rho^{{-}1}G_{u_z}]-[G_{B_z},\rho^{{-}1}F_{u_z}]\nonumber\\ & \quad +\boldsymbol{\nabla} F_w \boldsymbol{\cdot} \boldsymbol{\nabla} (\rho^{{-}1} G_\psi)-\boldsymbol{\nabla} G_w \boldsymbol{\cdot} \boldsymbol{\nabla} (\rho^{{-}1} F_\psi)+\rho^{{-}1}F_\varUpsilon G_\psi -\rho^{{-}1} G_\varUpsilon F_\psi ) \nonumber\\ & \quad +\rho^{{-}1}B_z([F_\varOmega,G_{B_z}]-[G_\varOmega,F_{B_z}]+ \boldsymbol{\nabla} F_w\boldsymbol{\cdot} \boldsymbol{\nabla} G_{B_z}-\boldsymbol{\nabla} G_w\boldsymbol{\cdot} \boldsymbol{\nabla} F_{B_z} )\nonumber\\ & \quad +\frac{1}{e}\psi([G_{B_z},n_{e}^{{-}1}F_\psi]-[F_{B_z},n_e^{{-}1}G_\psi]) -\frac{1}{en_e}B_z[F_{B_z},G_{B_z}]\nonumber\\ & \quad -\frac{1}{e}\sum_p\frac{e_p}{m_p}\int \, \textrm{d}^{3}v [n_e^{{-}1}B_z\nabla_{v_\perp} f_p \boldsymbol{\cdot} (G_{f_p} \boldsymbol{\nabla} F_{B_z}-F_{f_p}\boldsymbol{\nabla} G_{B_z})\nonumber\\ & \quad +\psi([n_e^{{-}1}F_{f_p}\partial_{v_z}f_p,G_{B_z}]-[n_e^{{-}1}G_{f_p} \partial_{v_z}f_p,F_{B_z}]\nonumber\\ & \quad +\boldsymbol{\nabla}\boldsymbol{\cdot} (n_e^{{-}1}F_\psi G_{f_p}\nabla_{v_\perp} f_p)-\boldsymbol{\nabla}\boldsymbol{\cdot} (n_e^{{-}1}G_\psi F_{f_p}\nabla_{v_\perp} f_p))]\nonumber\\ & \quad +\frac{1}{e}\sum_p\frac{e_p}{m_p}\int \, \textrm{d}^{3}v (\nabla_{v_\perp} f_p)\boldsymbol{\cdot} (F_{f_p}\boldsymbol{\nabla} G_{n_e}-G_{f_p}\boldsymbol{\nabla} F_{n_e})\nonumber\\ & \quad -\frac{1}{en_e}\sum_{p,p'} \frac{e_p}{m_p}\frac{e_{p'}}{m_{p'}}\int \int \, \textrm{d}^{3}v \, \textrm{d}^{3}v' f_p f_{p'}[B_z\langle F_{f_p},G_{f_{p'}}\rangle\nonumber\\ & \quad -\boldsymbol{\nabla}\psi\boldsymbol{\cdot} (\partial_{v_z}F_{f_p}\boldsymbol{\nabla}_{v_{{\perp}}'}G_{f_{p'}}- \partial_{v_z'}G_{f_p'}\boldsymbol{\nabla}_{v_{{\perp}}}F_{f_{p}})]\nonumber\\ & \quad +\sum_p\int \, \textrm{d}^{3}v \frac{f_p}{m_p}[[\![F_{f_p},G_{f_p}]\!]_{{\perp}}+ \frac{e_p}{m_p}(B_z\langle F_{f_p},G_{f_p} \rangle \nonumber\\ & \quad +\boldsymbol{\nabla}\psi \boldsymbol{\cdot} [\partial_{v_z}G_{f_p}\nabla_{v_\perp} F_{f_p}-\partial_{v_z}F_{f_p} \nabla_{v_\perp} G_{f_p}])\nonumber\\ & \quad +m_p([\![F_{f_p},\rho^{{-}1}(v_zG_{u_z}+\hat{z}\boldsymbol{\cdot} \boldsymbol{v}_\perp{\times}\boldsymbol{\nabla} G_{\varOmega}-\boldsymbol{v}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla} G_w )]\!]_{{\perp}}\nonumber\\ & \quad-\left.\vphantom{\frac{e_p}{m_p}} [\![G_{f_p},\rho^{{-}1}(v_zF_{u_z}+\hat{z}\boldsymbol{\cdot} \boldsymbol{v}_\perp{\times}\boldsymbol{\nabla} F_{\varOmega}-\boldsymbol{v}_\perp \boldsymbol{\cdot} \boldsymbol{\nabla} F_w )]\!]_{{\perp}})]\right\}. \end{align}

Note that we have introduced a new bracket notation, namely,

(4.6)\begin{gather} {[}a,b]:= (\boldsymbol{\nabla} a\times\boldsymbol{\nabla} b)\boldsymbol{\cdot} \hat{z}, \end{gather}

(4.7)\begin{gather}\langle a,b\rangle := (\nabla_{v_\perp} a\times \nabla_{v_\perp} b)\boldsymbol{\cdot} \hat{z}, \end{gather}

(4.8)\begin{gather}{[}\![a,b]\!]_{{\perp}}:= \boldsymbol{\nabla}_\perp a\boldsymbol{\cdot} \nabla_{v_\perp} b-\boldsymbol{\nabla}_\perp b\boldsymbol{\cdot} \nabla_{v_\perp} a. \end{gather}

The translationally symmetric Hamiltonian reads as follows:

(4.9)\begin{align} {\mathcal{H}} & = \int \, \textrm{d}^{3}x \left[\frac{1}{2}\rho (u_z^{2}+|\boldsymbol{\nabla}\chi|^{2}+2[\varUpsilon,\chi]+|\boldsymbol{\nabla} \varUpsilon|^{2})\right.\nonumber\\ & \quad +\left.\frac{B_z^{2}}{2\mu_0}+\frac{|\boldsymbol{\nabla}\psi|^{2}}{2\mu_0}+ \rho U_i(\rho)+n_e {\mathcal{U}} _e(n_e,B_z,|\boldsymbol{\nabla}\psi|)\right]\nonumber\\ & \quad +\sum_p\int \, \textrm{d}^{3}x\, \textrm{d}^{3}v \frac{1}{2}m_p f_p v^{2}, \end{align}

and the functional derivatives of ${\mathcal {H}}$ with respect to the two scalars associated with the magnetic field are

(4.10)\begin{gather} \frac{\delta{\mathcal{H}}}{\delta B_z} = \mu_0^{{-}1}(1-\gamma)B_z, \end{gather}

(4.11)\begin{gather}\frac{\delta {\mathcal{H}}}{\delta\psi} ={-}\boldsymbol{\nabla}\boldsymbol{\cdot} [\mu_0^{{-}1}(1-\gamma)\boldsymbol{\nabla}\psi]. \end{gather}

Having these expressions and also computing the functional derivatives with respect to the remaining scalars, one can derive the translationally symmetric dynamical equations, and in addition, equilibrium and stability conditions. For equilibrium and stability analysis, one has to identify the families of Casimir invariants ${\mathcal {C}}$ that span the non-trivial null space of the Poisson bracket (4.5). From the Casimir determining equations, stemming from the requirement $\mathfrak {C}_{\xi _i}=0$, where $\mathfrak {C}_{\xi _i}$ appear in the following rearrangement of (4.5):

(4.12)\begin{align} \{F,G\}=\int \, \textrm{d}^{3}x \left\{F_\rho \mathfrak{C}_\rho+F_{n_e} \mathfrak{C}_{n_e}+F_{B_z} \mathfrak{C}_{B_z}+F_\psi \mathfrak{C}_\psi+F_\chi \mathfrak{C}_\chi+F_\varUpsilon \mathfrak{C}_\varUpsilon+\sum_p\int \, \textrm{d}^{3}v\,F_{f_p} \mathfrak{C}_{f_p} \right\}, \end{align}

we were able to identify the following families of Casimirs

(4.13)\begin{gather} {\mathcal{C}} _1 = \int \, \textrm{d}^{3}x n_e N(\psi) , \end{gather}

(4.14)\begin{gather}{\mathcal{C}} _2 = \int \, \textrm{d}^{3}x \rho K(\varphi), \end{gather}

(4.15)\begin{gather}{\mathcal{C}} _3 = \int \, \textrm{d}^{3}x B_z F(\psi), \end{gather}

(4.16)\begin{gather}{\mathcal{C}} _4 = \int \, \textrm{d}^{3}x \left(\varOmega+\frac{e}{m}B_z-\sum_p m_p \int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}]\right)G(\varphi), \end{gather}

(4.17)\begin{gather}{\mathcal{C}} _{p} = \int \int \, \textrm{d}^{3}x\, \textrm{d}^{3}v\,\varLambda_p(f_p), \end{gather}

where $N,K,F,G$ and $\varLambda _p$ are arbitrary functions of their respective arguments, and

(4.18)\begin{equation} \varphi:=u_z+\frac{e}{m}\psi-\rho^{{-}1}\sum_pm_p\int \, \textrm{d}^{3}v f_p v_z, \end{equation}

is a generalized ion stream function, modified owing to the presence of kinetic particle species. For $f_p\rightarrow 0$, this stream function and the entire set of Casimir invariants (4.13)–(4.17) reduce to their translationally symmetric Hall MHD counterparts (Kaltsas et al. Reference Kaltsas, Throumoulopoulos and Morrison2017). In this fluid limit, quasineutrality implies $n_e=n_i$, because $n_p=\int \, \textrm {d}^{3}v\,f_p=0$.

4.2. Energy-Casimir equilibria

Owing to spatial symmetry, the Casimirs (4.13)–(4.17) constitute infinite families of invariants, thereby allowing for the derivation of sufficient stability conditions by the energy-Casimir method, which has been used for hybrid kinetic-MHD models by Tronci et al. (Reference Tronci, Tassi and Morrison2015) and for the extended and Hall MHD models by Kaltsas, Throumoulopoulos & Morrison (Reference Kaltsas, Throumoulopoulos and Morrison2020). A preliminary step though is the definition of the stationary state that serves as the initial condition for the dynamics. The set of equilibrium equations are derived in the first step of the energy-Casimir method by setting the first-order variation of the extended Hamiltonian equal to zero. In the MHD and extended MHD models, this procedure resulted in a system of Grad–Shafranov–Bernoulli (GSB) equations. Here, we derive the corresponding GSB system for translationally symmetric hybrid kinetic-Hall MHD in the PCS. The energy-Casimir functional, or extended Hamiltonian, is given by

(4.19)\begin{equation} \mathfrak{F}={\mathcal{H}}-\sum_{i=1}^{4} {\mathcal{C}} _i-\sum_p {\mathcal{C}} _p,\end{equation}

where ${\mathcal {H}}$ is given by (4.9) and ${\mathcal {C}}$ are the Casimirs (4.13)–(4.17). The first variation of (4.19) can be written as

(4.20)\begin{align} \delta\mathfrak{F} & = \int \, \textrm{d}^{3}x \left(\vphantom{\sum_p}\mathfrak{R}_1\delta n_e+\mathfrak{R}_2 \delta\rho+\mathfrak{R}_3 \delta u_z+\mathfrak{R}_4 \delta \chi+\mathfrak{R}_5 \delta \varUpsilon\right.\nonumber\\ & \quad+\left.\mathfrak{R}_6 \delta B_z+\mathfrak{R}_7 \delta\psi +\sum_p\int \, \textrm{d}^{3}v \mathfrak{R}_p\delta f_p \right). \end{align}

Assuming independent, arbitrary variations of the scalar dynamical variables, the requirement $\delta \mathfrak {F}=0$ is equivalent to $\mathfrak {R}_i=0, i=1,\ldots,7\,$ and $\mathfrak {R}_p=0, \forall p$. These equations lead to the following equilibrium conditions:

(4.21)\begin{align} & {\mathcal{U}} _e+n_{e}^{{-}1}P_{e_{{\parallel}}}-N(\psi)=0, \end{align}

(4.22)\begin{align} & h_i(\rho)+\frac{u^{2}}{2} -K(\varphi)-\rho^{{-}1}K'(\varphi)\sum_pm_p \int \, \textrm{d}^{3}v f_p v_z \nonumber\\ & \quad -\rho^{{-}2}\left(\varOmega+\frac{e}{m}B_z -\sum_p m_p \int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}] \right)G'(\varphi) \sum_{p'}m_{p'} \int \, \textrm{d}^{3}v f_{p'}v_z \nonumber\\ & \quad -\rho^{{-}2}\sum_p m_p \int \, \textrm{d}^{3}v f_{p}[\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x},G]=0, \end{align}

(4.23)\begin{align} & -\boldsymbol{\nabla}\boldsymbol{\cdot} (\rho \boldsymbol{\nabla}\varUpsilon )+[\chi,\rho]=0, \end{align}

(4.24)\begin{align} & -\boldsymbol{\nabla}\boldsymbol{\cdot} (\rho \boldsymbol{\nabla}\chi)+[\rho,\varUpsilon]+\Delta G(\varphi)=0, \end{align}

(4.25)\begin{align} & \rho u_z-\rho K'(\varphi)-\left(\varOmega+\frac{e}{m}B_z-\sum_p m_p \int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}]\right)G'(\varphi)=0, \end{align}

(4.26)\begin{align} & \mu_0^{{-}1}(1-\gamma)B_z-F(\psi)-\frac{e}{m}G(\varphi)=0, \end{align}

(4.27)\begin{align} & \mu_0^{{-}1}\boldsymbol{\nabla}\boldsymbol{\cdot} [(1-\gamma)\boldsymbol{\nabla}\psi]+n_e N'(\psi)+B_zF'(\psi) +\frac{e}{m}\rho K'(\varphi)\nonumber\\ & \quad +\frac{e}{m} \left(\varOmega+\frac{e}{m}B_z-\sum_p m_p \int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}]\right)G'(\varphi)=0, \end{align}

(4.28)\begin{align} & \frac{1}{2}m_pv^{2}-\varLambda_p'(f_p)+m_pv_z\left[K'(\varphi)\right.\nonumber\\ & \quad +\rho^{{-}1}\left.\left(\varOmega+\frac{e}{m}B_z-\sum_p m_p\int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}]\right)G'(\varphi)\right]\nonumber\\ & \quad +\rho^{{-}1}m_p[\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x},G(\varphi)]=0. \end{align}

Equation (4.21) provides a relation between $P_{e_\parallel }$ and the variables $n_e, \psi$ and $B_z$. Upon combining (4.21) with (2.46), we find an equation for $P_{e_\perp }$ also. This one reads as follows:

(4.29)\begin{equation} P_{e_\perp}=n_e\left[|{\boldsymbol B}|\frac{\partial {\mathcal{U}} _e}{\partial |{\boldsymbol B}|} +N(\psi)- {\mathcal{U}} _(n_e,|{\boldsymbol B}|)\right]. \end{equation}

In terms of $B_z$ and $|\boldsymbol {\nabla }\psi |$, (4.29) becomes

(4.30)\begin{equation} P_{e_\perp}=n_e\left[|{\boldsymbol B}|^{2}\left(\frac{1}{B_z}\frac{\partial {\mathcal{U}} _e}{\partial B_Z}+ \frac{1}{|\boldsymbol{\nabla}\psi|}\frac{\partial {\mathcal{U}} _e}{\partial |\boldsymbol{\nabla}\psi|}\right) +N(\psi)- {\mathcal{U}} _e(n_e,B_z,|\boldsymbol{\nabla}\psi|)\right]. \end{equation}

Therefore, to fully define an equilibrium state, we need an equation of state for the electron fluid, i.e. ${\mathcal {U}} _e= {\mathcal {U}} _e(n_e,B_z,|\boldsymbol {\nabla }\psi |)$. For example, such equations, which are associated with collisionless magnetic reconnection, are provided by Le et al. (Reference Le, Egedal, Daughton, Fox and Katz2009).

From (4.26), we readily obtain an equation that relates $B_z$ with $\psi, \varphi$ and $\gamma$,

(4.31)\begin{equation} B_z=\mu_0\frac{F(\psi)+(e/m)G(\varphi)}{1-\gamma}. \end{equation}

Equation (4.31) is in general an implicit equation for determining $B_z$ because the anisotropy function, $\gamma$, depends on $B_z$ as well, and hence $B_z$ appears nonlinearly in (4.26). We could possibly make $\gamma$ independent of $B_z$ under some special assumption, i.e. selecting the functional dependence of ${\mathcal {U}} _{e}$ on $B_z$ in a particular manner.

Now, by multiplying (4.28) with $f_p$, integrating over the velocity space and summing over the particle species, we obtain the following equation:

(4.32)\begin{align} & \sum_p\int \, \textrm{d}^{3}v \frac{m_p}{2}f_p v^{2}+\sum_p K'(\varphi) \int \, \textrm{d}^{3}v m_p f_p v_z-\sum_p\int \, \textrm{d}^{3}v f_p \varLambda_p'(f_p) \nonumber\\ & \quad +\rho^{{-}1}\left(\varOmega +\frac{e}{m} B_z -\sum_p m_p \int \, \textrm{d}^{3}v [\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}] \right)G'(\varphi)\sum_p \int \, \textrm{d}^{3}v m_p f_p v_z\nonumber\\ & \quad +\sum_pm_p \rho^{{-}1} \int \, \textrm{d}^{3}v f_p [\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x},G(\varphi)]=0. \end{align}

Combining equations (4.22) and (4.32), we find a Bernoulli equation of the form

(4.33)\begin{equation} \rho h(\rho)=\rho K(\varphi)-\frac{1}{2}\rho u^{2}-\sum_p \left[\int \, \textrm{d}^{3}v \, f_p \varLambda_p'(f_p)-\int \, \textrm{d}^{3}v \frac{m_p}{2}f_p v^{2}\right]. \end{equation}

The two Grad–Shafranov equations, one for the thermal ions and one for the electron fluid, stem from (4.25) and (4.27), respectively, using (4.23), (4.24) and (4.26), along with the definition of $\varphi$. These GS equations are

(4.34)\begin{align} & G'(\varphi)\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\frac{G'}{\rho}\boldsymbol{\nabla}\varphi\right)+ \rho\left(\varphi-\frac{e}{m}\psi\right)-\rho K'(\varphi)-\mu_0 \frac{e}{m}\frac{F(\psi)+\frac{e}{m}G(\varphi)}{1-\gamma}G'(\varphi)\nonumber\\ & \quad+\sum_p m_p \int \, \textrm{d}^{3}v (f_p v_z+G'[\rho^{{-}1}f_p,\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol x}])=0, \end{align}

(4.35)\begin{align} & \mu_0^{{-}1}\boldsymbol{\nabla}\boldsymbol{\cdot} [(1-\gamma)\boldsymbol{\nabla}\psi]+n_e N'(\psi)+\mu_0 \frac{F(\psi)+\frac{e}{m}G(\varphi)}{1-\gamma}F'(\psi)\nonumber\\ & \quad +\frac{e}{m}\rho\left(\varphi-\frac{e}{m}\psi+\rho^{{-}1} \sum_p m_p \int \, \textrm{d}^{3}v f_p v_z\right)=0. \end{align}

Note that in (4.35), the anisotropy parameter $\gamma$ appears inside the differential operator, and hence this partial differential equation might not be always elliptic (Ito et al. Reference Ito, Ramos and Nakajima2007) as is the case for an isotropic electron pressure. Finally, we need a set of equations for determining the equilibrium distribution functions $f_{p,e}$. Using (4.25), (4.28) is simplified to

(4.36)\begin{equation} \varLambda_p'(f_p)=\tfrac{1}{2}m_p v^{2}+m_p \boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol u}, \end{equation}

where we have used

(4.37)\begin{equation} {\boldsymbol u}_\perp = \rho^{{-}1}\boldsymbol{\nabla} G\times\hat{z}, \end{equation}

which can be deduced from (4.24) and from the fact that $[\boldsymbol {v}\boldsymbol {\cdot } {\boldsymbol x},G]=(\boldsymbol {\nabla } G\times \hat {z})\boldsymbol {\cdot } \boldsymbol {v}$. For invertible functions $\varLambda _p$, we take solutions to (4.36) of the form

(4.38)\begin{equation} f_{p,e}=f_{p,e}(\tfrac{1}{2} (|\boldsymbol{v}+{\boldsymbol u}|^{2}-|{\boldsymbol u}|^{2})), \end{equation}

as was the case in the paper by Morrison et al. (Reference Morrison, Tassi and Tronci2014) for the planar kinetic-MHD PCS model. Here, the subscript $e$ denotes equilibrium distribution functions. With (4.36), the Bernoulli equation becomes

(4.39)\begin{equation} \rho h(\rho)=\rho K(\varphi)-\frac{1}{2}\rho u^{2}-\sum_p m_p\int \, \textrm{d}^{3}v f_p\boldsymbol{v}\boldsymbol{\cdot} {\boldsymbol u}. \end{equation}

Assuming a special functional form for $f_{p,e}$, we can, in principle, compute the velocity space integrals appearing in (4.34), (4.35) and (4.39) to obtain a Hall-MHD GSB system, modified by the presence of energetic particles. As a final remark, note that the Hall-MHD GSB equations with anisotropic electron pressure are retrieved from (4.34), (4.35), (4.39), in the limit $f_p\rightarrow 0$, as can be seen by comparison with variants of this system obtained by Kaltsas et al. (Reference Kaltsas, Throumoulopoulos and Morrison2017,  Reference Kaltsas, Throumoulopoulos and Morrison2018) for isotropic electron pressure.