2. Derivation of relativistic HRMHD

Consider quasineutral relativistic ion and electron fluids with infinitely small electron-to-ion mass ratio. To describe the time evolution of such plasmas, we use relativistic HMHD (Kawazura Reference Kawazura2017), which consists of the mass conservation law

(2.1a)\begin{equation} \frac{\partial }{\partial t}\left( n\gamma \right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\left( n\gamma\boldsymbol{u} \right) = 0, \end{equation}

the momentum equation

(2.1b)\begin{equation} \frac{\partial }{\partial t}\left( nh\gamma^2\boldsymbol{u} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\left( nh\gamma^2\boldsymbol{u}\boldsymbol{u} \right) ={-}c^2\boldsymbol{\nabla} p + c^2\rho_\mathrm{q}\boldsymbol{E} + c\boldsymbol{J}\times\boldsymbol{B}, \end{equation}

the generalized Ohm's law

(2.1c)\begin{equation} \boldsymbol{E} + \frac{\boldsymbol{u}}{c}\times\boldsymbol{B} = \frac{1}{\gamma e n}\left( \rho_\mathrm{q} \boldsymbol{E} + \frac{\boldsymbol{J}}{c}\times\boldsymbol{B} - \boldsymbol{\nabla} p_\mathrm{e} \right) \end{equation}

and Maxwell's equations

(2.1d)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{E} = 4{\rm \pi}\rho_\mathrm{q}, \end{gather}

(2.1e)\begin{gather}-\frac{\partial \boldsymbol{E}}{\partial t} + c\boldsymbol{\nabla}\times\boldsymbol{B} = 4{\rm \pi}\boldsymbol{J}, \end{gather}

(2.1f)\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{B} = 0, \end{gather}

(2.1g)\begin{gather}\frac{\partial \boldsymbol{B}}{\partial t} + c\boldsymbol{\nabla}\times\boldsymbol{E} = 0, \end{gather}

where $e$ is the elementary charge, $c$ is the speed of light, $n$ is the rest frame number density, $h$ is the total thermal enthalpy, $p$ is the total thermal pressure, $p_\mathrm{e}$ is the thermal pressure of electrons, $\boldsymbol {u}$ is the flow velocity, $\gamma = 1/\sqrt {1 - |\boldsymbol {u}|^2/c^2}$ is the Lorentz factor, $\boldsymbol {J}$ is the electric current, $\rho _\mathrm{q}$ is the charge density, $\boldsymbol {E}$ is the electric field and $\boldsymbol {B}$ is the magnetic field. The relativistic ideal MHD is recovered when the right-hand side of (2.1c) is neglected.

In what follows, we assume that all fields are separable into spatiotemporally constant background (symbols with a subscript 0) and fluctuations (symbols with $\delta$ in front), viz. $n = n_0 + \delta n, \boldsymbol {B} = (B_0 + \delta B_{\|})\hat {\boldsymbol {z}} + \delta \boldsymbol {B}_{\bot }$, and so on. Electrons are assumed to be isothermal, i.e. $\delta p_{{\rm e}} = T_{{{\rm e}}0}\delta n$, where $T_{{{\rm e}}0}$ is the background electron temperature. Here, $\hat {\boldsymbol {z}} = \boldsymbol {B}_0/|\boldsymbol {B}_0|$, and $\| (\bot )$ denotes the parallel (perpendicular) component to $\boldsymbol {B}_0$. We also assume that the mean flow is absent, i.e. $\boldsymbol {u}_0 = 0$. Plugging the constant background fields into (2.1c), (2.1d) and (2.1e), one finds $\boldsymbol {E}_0 = \boldsymbol {J}_0 = \rho _{\mathrm{q}0} = 0$. Then, we impose the reduced MHD ordering

(2.2)\begin{equation} \frac{\delta n}{n_0} \sim \frac{\delta \boldsymbol{B}}{B_0} \sim \frac{\boldsymbol{u}}{v_\mathrm{A}} \sim \frac{\delta p_\mathrm{e}}{p_{\mathrm{e}0}} \sim \frac{k_{\|}}{k_{\bot}} \sim \epsilon \ll 1, \quad \frac{\partial }{\partial t} \sim \omega \sim k_{\|} v_\mathrm{A}, \end{equation}

where $\omega$ and $\boldsymbol {k}$ are the frequency and wavenumber of the fluctuations, respectively. Here, we have defined the relativistic Alfvén speed,

(2.3)\begin{equation} v_\mathrm{A} = \frac{cB_0}{\sqrt{4{\rm \pi} n_0 h_0 + B_0^2}} = \sqrt{\frac{\sigma}{1 + \sigma}}\,c, \end{equation}

where $\sigma = B_0^2/4{\rm \pi} n_0 h_0$ is the magnetization parameter. When $\sigma \gtrsim 1$ (equivalently $v_\mathrm{A} \approx c$), the plasma is relativistically magnetized. Since $\boldsymbol {u}$ is small, the relativistic effect of bulk flow is absent, i.e. $\gamma \approx 1$, but this is acceptable because we are interested in the microscopic scales where bulk flow is generally small while the thermal energy and/or magnetic energy can be relativistic. We also assume that the ions are cold while the electron can be relativistically hot,

(2.4)\begin{equation} T_{\mathrm{i}0} \ll T_{\mathrm{e}0} \ll \sqrt{\frac{m_\mathrm{i}}{m_\mathrm{e}}}\,m_\mathrm{e} c^2 \approx 20\,\mathrm{MeV}, \end{equation}

where $T_{s0}$ and $m_s$ are temperature and mass of the species $s$. The upper bound for the electron temperature is required so that the relativistic thermal inertia of electrons is negligible at the ion inertial scale. Since the ions are cold, the thermal inertia of the ions is also negligible, i.e. $h_0 \approx m_\mathrm{i} c^2$. These conditions enforce the restriction,

(2.5)\begin{equation} \beta_\mathrm{e} = \left(\frac{2T_{\mathrm{e}0}}{m_\mathrm{i} c^2}\right)\frac{1}{\sigma} \ll \sqrt{\frac{m_\mathrm{e}}{m_\mathrm{i}}}, \end{equation}

where $\beta _\mathrm{e} = 8{\rm \pi} p_{\mathrm{e}0}/B_0^2$ is the electron beta. This is different from the assumptions $T_{\mathrm{i}0} \ll T_{\mathrm{e}0}$ and $\beta _\mathrm{e} \sim 1$ that are used in the non-relativistic HRMHD (S19) because they cannot be satisfied when the magnetization is relativistic, i.e. $\sigma \gtrsim 1$. In this work, we consider the electron beta to be lower than that of non-relativistic HMHD, which is formally allowed as long as $\epsilon \ll \beta _\mathrm{e}$. In other words, we treat $\beta _\mathrm{e}$ as order unity, although it is much smaller than $\sqrt {m_\mathrm{e}/m_\mathrm{i}}$, because $\epsilon$ is assumed to be even smaller than $\beta _\mathrm{e}$. However, $\beta _\mathrm{e}$ may not be too small because when $\beta _\mathrm{e} \sim m_\mathrm{e}/m_\mathrm{i}$, electron rest mass inertia becomes non-negligible, and thus the HMHD approximation breaks down (Zocco & Schekochihin Reference Zocco and Schekochihin2011). To summarize, we assume the range of $\beta _\mathrm{e}$ and $\epsilon$ as

(2.6)\begin{equation} \epsilon \sim \frac{m_\mathrm{e}}{m_\mathrm{i}} \ll \beta_\mathrm{e} \ll \sqrt{\frac{m_\mathrm{e}}{m_\mathrm{i}}}. \end{equation}

We, then, follow the derivation of non-relativistic RMHD by Schekochihin et al. (Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). We explicitly keep $v_\mathrm{A}/c$ and $k_{\bot } d_\mathrm{i}$ in the ordering so that one can take the non-relativistic and/or long-wavelength limit simply by neglecting the corresponding terms. First, we adopt the expansion $\boldsymbol {u} = \boldsymbol {u}^{(1)} + \boldsymbol {u}^{(2)} + O(\epsilon ^3 v_\mathrm{A})$ and substitute it into (2.1a). The $O(\epsilon ^0 n_0 \omega )$ terms yield

(2.7)\begin{equation} \boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{u}_{\bot}^{(1)} = 0. \end{equation}

This allows us to use a stream function leading to $\boldsymbol {u}_{\bot }^{(1)} = \hat {\boldsymbol {z}}\times \boldsymbol {\nabla }_{\bot }\varPhi$, where $\varPhi = (c/B_0)\phi$, and $\phi$ is the electrostatic potential. From the $O(\epsilon ^1 n_0 \omega )$ terms, one obtains

(2.8)\begin{equation} \left( \frac{\partial }{\partial t} + \boldsymbol{u}_{\bot}^{(1)}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\bot} \right)\frac{\delta n}{n_0} ={-}\left( \boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{u}_{\bot}^{(2)} + \frac{\partial u_{\|}^{(1)}}{\partial z} \right). \end{equation}

From the lowest-order terms in (2.1f), one can use a magnetic flux function leading to $v_\mathrm{A}(\delta \boldsymbol {B}_{\bot }^{(1)}/B_0) = \hat {\boldsymbol {z}}\times \boldsymbol {\nabla }_{\bot }\varPsi$, where $\varPsi = -(v_\mathrm{A}/B_0)A_{\|}$, and $A_{\|}$ is the parallel component of the vector potential. The $O(\epsilon ^0\omega )$ terms in (2.1b) give the pressure balance

(2.9)\begin{equation} \frac{B_0}{4{\rm \pi}}\delta B_{\|} ={-}\delta p_\mathrm{e} ={-}T_{\mathrm{e}0}\delta n. \end{equation}

Up to this point, the derivation is the same as the non-relativistic RMHD.

Next, we expand $\boldsymbol {E} = \boldsymbol {E}^{(1)} + \boldsymbol {E}^{(2)} + O(\epsilon ^3 v_{{\rm A}} B_0/c)$. Using electromagnetic potentials, the first- and second-order terms become

(2.10a)\begin{equation} \boldsymbol{E}^{(1)} ={-}\boldsymbol{\nabla}_{\bot}\phi, \quad \boldsymbol{E}^{(2)} ={-}\frac{1}{c}\frac{\partial \boldsymbol{A}}{\partial t} - \hat{\boldsymbol{z}}\frac{\partial \phi}{\partial z}. \end{equation}

We also expand $\boldsymbol {J} = \boldsymbol {J}^{(1)} + \boldsymbol {J}^{(2)} + O(\epsilon ^3 ck_{\bot }B_0)$, plug it into (2.1c) and remove $\rho _{\rm q}$ using (2.1d) to yield

(2.11) \begin{gather} \frac{c}{v_\mathrm{A} B_0}\left( \underbrace{\boldsymbol{E}^{(1)}}_{{\sim}\epsilon} + \underbrace{\boldsymbol{E}^{(2)}}_{{\sim}\epsilon^2} \right) + \frac{c}{v_\mathrm{A}}\left( \underbrace{\frac{\delta n}{n_0}}_{{\sim}\epsilon^2} - \underbrace{\frac{\boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{E}_{\bot}^{(1)}}{4{\rm \pi} e n_0}}_{{\sim}\epsilon^2(k_{\bot}d_{{\rm i}})(v_{{\rm A}}/c)^2} \right)\frac{\boldsymbol{E}^{(1)}}{B_0} + \frac{1}{v_{{\rm A}}}\left( \underbrace{\boldsymbol{u}^{(1)}}_{{\sim}\epsilon} - \underbrace{\frac{\boldsymbol{J}^{(1)}}{en_0}}_{{\sim}\epsilon (k_{\bot} d_{{\rm i}})} \right)\times\hat{\boldsymbol{z}} \nonumber\\ +\, \frac{1}{v_{{\rm A}}}\left( \underbrace{\boldsymbol{u}^{(1)}}_{{\sim}\epsilon^2} - \underbrace{\frac{\boldsymbol{J}^{(1)}}{en_0}}_{{\sim}\epsilon^2 (k_{\bot} d_{{\rm i}})} \right)\times\frac{\delta \boldsymbol{B}}{B_0} + \frac{1}{v_{{\rm A}}}\left[ \underbrace{\left( \frac{\delta n}{n_0} \right)\boldsymbol{u}^{(1)} + \boldsymbol{u}^{(2)}}_{{\sim}\epsilon^2} + \underbrace{\left( \frac{\delta n}{n_0} \right)\frac{\boldsymbol{J}^{(1)}}{en_0} - \frac{\boldsymbol{J}^{(2)}}{en_0}}_{{\sim}\epsilon^2 (k_{\bot} d_{{\rm i}})} \right]\times\hat{\boldsymbol{z}} \nonumber\\ +\, \frac{c}{v_{{\rm A}} en_0 B_0}\left[\underbrace{\boldsymbol{\nabla}_{\bot}\delta p_{{\rm e}}}_{{\sim}\epsilon\beta_{{\rm e}} (k_{\bot}d_{{\rm i}})} + \underbrace{\left( \frac{\delta n}{n_0} \right)\boldsymbol{\nabla}_{\bot}\delta p_{{\rm e}} + \hat{\boldsymbol{z}}\frac{\partial }{\partial z}\delta p_{{\rm e}}}_{{\sim}\epsilon^2\beta_{{\rm e}} (k_{\bot}d_{{\rm i}})}\right] = 0, \end{gather}

where $d_{{\rm i}} = \sqrt {m_{{\rm i}} c^2/4{\rm \pi} n_0 e^2}$ is the ion inertial length. As we mentioned above, $\beta _{{\rm e}}$ is assumed to be $O(\epsilon ^0)$ because $\epsilon \ll \beta _{{\rm e}}$. We then manipulate (2.1e) to get

(2.12)\begin{align} \frac{4{\rm \pi}\boldsymbol{J}}{cB_0} & =-\frac{1}{c^2}\hat{\boldsymbol{z}}\times\frac{\partial }{\partial t}\left( \underbrace{\boldsymbol{u}_{\bot}^{(1)}}_{{\sim} \epsilon^2 k_{\bot}(v_{{\rm A}}/c)^2} - \underbrace{\frac{\boldsymbol{J}_{\bot}^{(1)}}{en_0}}_{{\sim} \epsilon^2 k_{\bot}(k_{\bot} d_{{\rm i}})(v_{{\rm A}}/c)^2} \right) + \underbrace{\frac{1}{ce n_0 B_0}\frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}\delta p_{{\rm e}}}_{{\sim} \epsilon^2 \beta_{{\rm e}} k_{\bot}(k_{\bot} d_{{\rm i}})(v_{{\rm A}}/c)^2} \nonumber\\ & \quad +\, \underbrace{\hat{\boldsymbol{z}}\times\frac{\partial }{\partial z}\frac{\delta \boldsymbol{B}_{\bot}}{B_0}}_{{\sim} \epsilon^2 k_{\bot}} + \underbrace{\boldsymbol{\nabla}_{\bot}\left( \frac{\delta B_{\|}}{B_0} \right)\times\hat{\boldsymbol{z}}}_{{\sim} \epsilon k_{\bot}} + \underbrace{\boldsymbol{\nabla}_{\bot} \times\frac{\delta\boldsymbol{B}_{\bot}}{B_0}}_{{\sim} \epsilon k_{\bot}} + O(\epsilon^3 k_{\bot}). \end{align}

Collecting the terms order-by-order, one obtains

(2.13) \begin{equation} \begin{gathered} \boldsymbol{J}_{\bot}^{(1)} = \frac{c}{4{\rm \pi}}\boldsymbol{\nabla}_{\bot} \delta B_{\|} \times \hat{\boldsymbol{z}},\quad J_{\|}^{(1)} = \frac{c}{4{\rm \pi}}\hat{\boldsymbol{z}}\boldsymbol{\cdot}(\boldsymbol{\nabla}_{\bot}\times\delta\boldsymbol{B}_{\bot}),\\ \boldsymbol{J}_{\bot}^{(2)} = \frac{c}{4{\rm \pi}}\left[ -\frac{B_0}{c^2}\hat{\boldsymbol{z}}\times\frac{\partial }{\partial t}\left( \boldsymbol{u}_{\bot}^{(1)} - \frac{\boldsymbol{J}_{\bot}^{(1)}}{en_0} \right) + \frac{1}{cen_0}\frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}\delta p_{{\rm e}} + \hat{\boldsymbol{z}}\times\frac{\partial }{\partial z}\delta\boldsymbol{B}_{\bot} \right], \quad J_{\|}^{(2)} = 0. \end{gathered} \end{equation}

The terms including $\partial /\partial t$ in $\boldsymbol {J}_{\bot }^{(2)}$ originated from the displacement current, which disappears in the non-relativistic limit, and all the other terms are the same as the non-relativistic case. Plugging $\boldsymbol {J}_{\bot }^{(1)}$ into the $O(\epsilon )$ terms in (2.11), one obtains the pressure balance (2.9) again, where the isothermal electrons are assumed. Then, we can further manipulate $\boldsymbol {J}_{\bot }^{(2)}$ as

(2.14)\begin{equation} \boldsymbol{J}_{\bot}^{(2)} = \frac{c}{4{\rm \pi}}\left( \frac{B_0}{c^2}\frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}\varPhi + \hat{\boldsymbol{z}}\times\frac{\partial }{\partial z}\delta\boldsymbol{B}_{\bot} \right). \end{equation}

This is notable because the Hall term and the electron pressure gradient term, both of which becomes finite at the $d_{{\rm i}}$ scale, are cancelled, and therefore, the electric current (up to the second order) is identical to that of relativistic RMHD. The reason for this cancellation is rather straightforward. We evaluate the electric field $\boldsymbol {E}$ in the displacement current by using Ohm's law, which is the momentum equation of electrons. Therefore, the pressure balance cancels some of the terms of the electric current exactly in the same way as the first order of the momentum equation in the relativistic RMHD (which is identical to (2.16) shown below).

Next, we substitute (2.10a,b) and (2.13) into (2.1b) to get the equations for $\varPhi$ and $u_{\|}$. As we found in the previous paragraph, both $\boldsymbol {E}$ and $\boldsymbol {J}$ do not contain corrections due to the Hall effect, meaning that (2.1b) ends up with the same equations as the relativistic RMHD. The $z$ component of $O(\epsilon ^1\omega )$ terms give

(2.15)\begin{equation} \frac{\partial u_{\|}}{\partial t} + \left\{ \varPhi, u_{\|} \right\} = (1 + \sigma)v_{{\rm A}}^2\left[ \frac{\partial }{\partial z}\left( \frac{\delta B_{\|}}{B_0} \right) + \frac{1}{v_{{\rm A}}}\left\{ \varPsi, \frac{\delta B_{\|}}{B_0}\right\} \right], \end{equation}

while multiplying $\hat {\boldsymbol {z}}\boldsymbol {\cdot }\boldsymbol {\nabla }_{\bot }\times$ to $O(\epsilon ^1\omega )$ terms gives

(2.16)\begin{equation} \frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}^2\varPhi + \left\{ \varPhi, \boldsymbol{\nabla}_{\bot}^2\varPhi \right\} = v_{{\rm A}}\left( \frac{\partial }{\partial z}\boldsymbol{\nabla}_{\bot}^2\varPsi + \frac{1}{v_{{\rm A}}}\left\{ \varPsi, \boldsymbol{\nabla}_{\bot}^2\varPsi \right\} \right), \end{equation}

where $\left \{ f, g \right \} = (\partial _x f)(\partial _y g) - (\partial _x g)(\partial _y f)$.

Next, we derive the equations for $\varPsi$ and $\delta B_{\|}$. This time, unlike the momentum equation, there will be the corrections due to the Hall effect and electron pressure gradient. The $O(\epsilon ^2)$ terms in (2.11) are

(2.17)\begin{gather} \frac{\boldsymbol{E}^{(2)}}{B_0} + \left(\frac{\delta n}{n_0} - \frac{\boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{E}_{\bot}^{(1)}}{4{\rm \pi} e n_0} \right)\frac{\boldsymbol{E}^{(1)}}{B_0} + \frac{1}{c}\left( \boldsymbol{u}^{(1)} - \frac{\boldsymbol{J}^{(1)}}{en_0} \right)\times\frac{\delta \boldsymbol{B}}{B_0} \nonumber\\ +\, \frac{1}{c}\left[ \left( \frac{\delta n}{n_0} \right)\boldsymbol{u}^{(1)} + \boldsymbol{u}^{(2)} - \frac{\boldsymbol{J}^{(2)}}{en_0} \right]\times\hat{\boldsymbol{z}} + \frac{1}{en_0 B_0}\left(\hat{\boldsymbol{z}}\frac{\partial }{\partial z}\delta p_{{\rm e}}\right) = 0. \end{gather}

Note that a few terms have been cancelled out using the pressure balance (2.9). The $z$-component of (2.17) yields

(2.18)\begin{equation} \frac{\partial \varPsi}{\partial t} = v_{{\rm A}}\left[ \frac{\partial }{\partial z}\left( \varPhi + \sqrt{(1 + \sigma)}\, d_{{\rm i}} v_{{\rm A}}\frac{\delta B_{\|}}{B_0} \right) + \frac{1}{v_{{\rm A}}}\left\{ \varPsi,\, \varPhi + \sqrt{(1 + \sigma)}\, d_{{\rm i}} v_{{\rm A}}\frac{\delta B_{\|}}{B_0} \right\} \right]. \end{equation}

Meanwhile, (2.8) and (2.9) are rearranged as

(2.19)\begin{equation} \boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{u}_{\bot}^{(2)} + \frac{\partial u_{\|}^{(1)}}{\partial z} = \frac{\sigma c^2}{c_{{\rm S}}^2}\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\delta B_{\|}}{B_0} \right), \end{equation}

where $c_{{\rm S}} = \sqrt {(n_{{{\rm i}}0}T_{{{\rm e}}0})/(n_{{{\rm e}}0}m_{\rm i})} \ll (m_{{\rm e}}/m_{{\rm i}})^{1/4}c$ is the sound speed, and (2.13) and (2.14) are rearranged as

(2.20)\begin{equation} \frac{1}{en_0}\left( \boldsymbol{\nabla}_{\bot}\boldsymbol{\cdot}\boldsymbol{J}_{\bot}^{(2)} + \frac{\partial J_{\|}^{(1)}}{\partial z} \right) = \frac{\sqrt{(1 + \sigma)}\, d_{{\rm i}} v_{{\rm A}}}{c^2}\frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}^2\varPhi. \end{equation}

Note that the right-hand side of (2.20) is equal to zero in the non-relativistic limit (i.e. $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {J} = 0$). Combining (2.19) and (2.20) with $\hat {\boldsymbol {z}}\boldsymbol {\cdot }\boldsymbol {\nabla }_{\bot }\times$ to (2.17) yields

(2.21)\begin{gather} -\left( 1 + \frac{\sigma c^2}{c_{{\rm S}}^2} \right)\left[ \frac{\partial }{\partial t}\left( \frac{\delta B_{\|}}{B_0} \right) + \left\{ \varPhi,\, \frac{\delta B_{\|}}{B_0} \right\} \right] + \frac{\sqrt{(1 + \sigma)}\, d_{{\rm i}} v_{{\rm A}}}{c^2}\left( \frac{\partial }{\partial t}\boldsymbol{\nabla}_{\bot}^2\varPhi + \left\{ \varPhi,\, \boldsymbol{\nabla}_{\bot}^2\varPhi \right\} \right) \nonumber\\ +\, \frac{\partial u_{\|}}{\partial z} + \frac{1}{v_{{\rm A}}}\left\{ \varPsi,\, u_{\|} \right\} - \sqrt{(1 + \sigma)}\, d_{{\rm i}} \left( \frac{\partial }{\partial z}\boldsymbol{\nabla}_{\bot}^2\varPsi + \frac{1}{v_{{\rm A}}}\left\{ \varPsi,\, \boldsymbol{\nabla}_{\bot}^2\varPsi \right\} \right) = 0. \end{gather}

The time derivative of $\boldsymbol {\nabla }_{\bot }^2\varPhi$ can be eliminated by substituting (2.16), which results in

(2.22)\begin{gather} \left( 1 + \frac{\sigma c^2}{c_{{\rm S}}^2} \right)\left[ \frac{\partial }{\partial t}\left( \frac{\delta B_{\|}}{B_0} \right) + \left\{ \varPhi,\, \frac{\delta B_{\|}}{B_0} \right\} \right] \nonumber\\ = \frac{\partial u_{\|}}{\partial z} + \frac{1}{v_{{\rm A}}}\left\{ \varPsi,\, u_{\|} \right\} - \sqrt{(1 + \sigma)}\, d_{{\rm i}}\left( 1 - \frac{v_{{\rm A}}^2}{c^2} \right)\left( \frac{\partial }{\partial z} + \frac{1}{v_{{\rm A}}^2}\left\{ \varPsi,\, \boldsymbol{\nabla}_{\bot}^2\varPsi \right\} \right). \end{gather}

Finally, (2.15), (2.16), (2.18) and (2.22) are reorganized into

(2.23a)\begin{gather} \frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{\nabla}_{\bot}^2\varPhi = v_{{\rm A}}\boldsymbol{\nabla}_{\|}\boldsymbol{\nabla}_{\bot}^2\varPsi, \end{gather}

(2.23b)\begin{gather}\frac{\partial \varPsi}{\partial t} = v_{{\rm A}}\boldsymbol{\nabla}_{\|}\left( \varPhi + v_{{\rm A}}\rho_{{\rm H}}\mathcal{B} \right), \end{gather}

(2.23c)\begin{gather}\frac{\mathrm{d}\mathcal{U}}{\mathrm{d}t} = v_{{\rm S}}\boldsymbol{\nabla}_{\|}\mathcal{B}, \end{gather}

(2.23d)\begin{gather}\frac{\mathrm{d}\mathcal{B}}{\mathrm{d}t} = \boldsymbol{\nabla}_{\|}\left( v_{{\rm S}}\mathcal{U} - \rho_{{\rm H}} \boldsymbol{\nabla}_{\bot}^2\varPsi \right), \end{gather}

where

(2.24a)\begin{gather} \frac{\mathrm{d}}{\mathrm{d}t} = \frac{\partial }{\partial t} + \boldsymbol{u}_{\bot}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\bot} = \frac{\partial }{\partial t} + \left\{ \varPhi, \ldots \right\},\quad \boldsymbol{\nabla}_{\|} = \frac{\partial }{\partial z} + \frac{\delta\boldsymbol{B}_{\bot}}{B_0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{\bot} = \frac{\partial }{\partial z} + \frac{1}{v_{{\rm A}}} \left\{ \varPsi, \ldots \right\}, \end{gather}

(2.24b)\begin{gather}\mathcal{U} = \frac{u_{\|}}{v_{{\rm A}}}, \quad \mathcal{B} = \sqrt{(1 + \sigma)\left[ 1 + (1 + \sigma)v_{{\rm A}}^2/c_{{\rm S}}^2 \right]}\,\frac{\delta B_{\|}}{B_0}, \end{gather}

(2.24c) \begin{gather}v_{{\rm A}} = \sqrt{\frac{\sigma}{1 + \sigma}}\,c, \quad v_{{\rm S}} = \sqrt{\frac{\sigma}{1 + (1 + \sigma)v_{{\rm A}}^2/c_{{\rm S}}^2}}\,c, \quad \rho_{H} = \sqrt{\frac{1}{1 + (1 + \sigma)v_{{\rm A}}^2/c_{{\rm S}}^2}}\,d_{{\rm i}}. \end{gather}

Note that $v_{{\rm A}}^2/c_{{\rm S}}^2 \approx 2/\beta _{{\rm e}}$ when $T_{{{\rm i}}0}/T_{{{\rm e}}0} \ll 1$. We find that (2.23a)–(2.23d) are identical to non-relativistic HRMHD ((5.14)–(5.17) of S19), except for the definition of constants (2.24c) which become those of S19 in the limit of $\sigma \to 0$. Thereby, all the physical properties of the non-relativistic HRMHD (e.g. the conservation of energy and helicity ((5.24) and (5.68) of S19) and the linear dispersion relation ((5.26) of S19) are valid even in a magnetically dominated regime $\sigma \gtrsim 1$. Meanwhile, one retrieves the relativistic RMHD by taking the limit of $\rho _{{\rm H}} \to 0$. Note that RMHD of Chandran et al. (Reference Chandran, Foucart and Tchekhovskoy2018) and TenBarge et al. (Reference TenBarge, Ripperda, Chernoglazov, Bhattacharjee, Mahlmann, Most, Juno, Yuan and Philippov2021) are written in the Elsässer variables while (2.23a)–(2.23d) are not; as is shown by Galtier (Reference Galtier2006) that introducing the Elsässer for HMHD is complicated.

There are three important facts that make relativistic HRMHD and non-relativistic HRMHD formally identical. The first is that the electric current of relativistic HRMHD is identical to that of relativistic RMHD due to the electron pressure balance (see (2.13)). The second is that the displacement current only appears in the second order of the electric current. The third is that the Lorentz force due to the displacement current (i.e. $\boldsymbol {J}^{(2)}\times \boldsymbol {B}_0$) is the time derivative of $\boldsymbol {E}^{(1)}\times \boldsymbol {B}_0$ drift, which happens to be the left-hand side of the momentum equation (2.1b) and balances the non-relativistic Lorentz force $\boldsymbol {J}^{(1)}\times \delta \boldsymbol {B}$ in the right-hand side (given the pressure balance). Thus, the relativistic Lorentz force ends up with merely proportional to the non-relativistic Lorentz force, and its prefactor does not include $d_{{\rm i}}$.

Lastly, one finds that the Hall transition scale $\rho _{{\rm H}}$ becomes smaller and vanishes eventually as the magnetization $\sigma$ increases, meaning that the fluctuations behave like those of RMHD at $d_{{\rm i}}$ scale when the magnetization is strong. This behaviour was first discovered by Kawazura (Reference Kawazura2017), and the reason for it is that the Hall transition happens at the scale of $\sim v_{{\rm A}}/\varOmega _{{\rm i}}$ (which is equal to $d_{{\rm i}}$ in the non-relativistic limit) where $\varOmega _{{\rm i}}$ is ion cyclotron frequency, and this scale becomes smaller as the magnetization becomes larger because $v_{{\rm A}} \to c$ and $\varOmega _{{\rm i}} \to \infty$ as $\sigma \to \infty$.