Applicability criteria of proper charge neutrality and special relativistic MHD models extended by two-fluid effects

Yoshino, Shuntaro; Hirota, Makoto; Hattori, Yuji

doi:10.1063/5.0243293

The applicability of relativistic magnetohydrodynamics (RMHD) and its generalization to two-fluid models (including the Hall and inertial effects) is systematically investigated by using the method of dominant balance in the two-fluid equations. Although proper charge neutrality or quasi-neutrality is the key assumption for all MHD models, this condition is difficult to be met when both relativistic and inertial effects are taken into account. The range of application for each MHD model is illustrated in the space of dimensionless scale parameters. Moreover, the number of field variables of relativistic Hall MHD (RHMHD) is shown to be greater than that of RMHD and Hall MHD. Nevertheless, the RHMHD equations may be solved at a lower computational cost than RMHD in the limit of cold plasma, since root-finding algorithm, which is the most time-consuming part of the RMHD code, is no longer required to compute the primitive variables.

I. INTRODUCTION

Magnetized plasmas subject to relativistic effects are common in various high-energy celestial bodies, such as pulsar, black hole magnetosphere, corona of accretion disk, jet from active galactic nucleus, and gamma-ray burst. Relativistic magnetohydrodynamics (RMHD) has been used in theoretical and numerical studies as a model to analyze the macroscopic motion of relativistic, magnetized plasmas. Since RMHD ignores the microscopic scales of such as inertial length and gyro-radius, it is well known that the magnetic field is frozen in the plasma motion in the zero resistivity limit. Therefore, RMHD is inappropriate for dealing with magnetic reconnection^1,2 at least in the microscopic region where magnetic field lines reconnect. In many cases, magnetic reconnection is a key process in which magnetic energy is efficiently converted into kinetic and thermal energy. In addition, RMHD becomes invalid in the limit of low plasma density or weak magnetic field. The Vlasov–Maxwell equations, on the other hand, are based on first principles and have been solved by Particle-In-Cell (PIC) simulations in recent years. However, this direct approach is the most computationally expensive and these kinetic models are difficult to solve analytically. Thus, there is a demand for intermediate models which bridge the gap between RMHD and kinetic ones. In this study, we focus on extended RMHD models that include the two-fluid effects, which are expected to be more widely applicable than RMHD while maintaining a moderate computational cost.

In non-relativistic electron-ion plasmas, a model^3,4 including the two-fluid effects (i.e., the Hall effect and the electron inertia effect) is called extended MHD (XMHD) in the recent literature.⁵ The XMHD equations are derived from the two-fluid equations by imposing the quasi-neutrality (QN) condition, which approximately eliminates microscopic motions such as plasma oscillation and cyclotron oscillation. XMHD is also shown to have a Hamiltonian structure that conserves canonical vorticities (instead of magnetic flux).^6–9 Due to the electron-inertia effect, magnetic reconnection can occur even in the zero resistivity limit.^10–12

Furthermore, the Hall effect is well-known for significantly enhancing the reconnection speed, according to the Global Environment Modeling (GEM) Reconnection Challenge.¹³ Since the electron-inertia effect manifests itself on an even smaller scale than the Hall effect, Hall MHD¹⁴ is often used as well, neglecting only the electron-inertia effect. In the case of electron-positron plasmas, the Hall effect vanishes and only the inertial effect remains, so this model is called inertial MHD (IMHD).⁵

It is natural to assume that there are also some MHD models that include the relativistic effects alongside the two-fluid effects. Such an extension of RMHD in electron-positron plasma was explored early on the literature.^15,16 Additionally, the extension of generalized Ohm's law was attempted and applied to pulsar magnetosphere¹⁷ (generalized relativistic Ohm's law is also proposed by Ref. 18 in a different way). Koide¹⁹ derived a generalized RMHD model from the relativistic two-fluid equations by imposing the proper charge neutrality (PCN) condition, which will be referred to as relativistic extended MHD (RXMHD) in this paper. A variational principle of RXMHD was later proposed by Kawazura et al.²⁰ The general relativistic version of RXMHD was presented by Koide²¹ and by Comisso and Asenjo²² using a covariant form. RXMHD was applied to relativistic collisionless magnetic reconnection.²³ Relativistic Hall MHD (RHMHD) is similarly obtained by neglecting electron-inertia, and its properties have been studied by Kawazura.^24,25 However, for the QN condition to hold in non-relativistic MHD, the flow velocity must be sufficiently slower than the speed of light. Moreover, as will be clarified in this paper, the PCN condition in RMHD actually holds in the limit of neglecting the two-fluid effects. Therefore, hybrid models, which include both the relativistic and two-fluid effects, may violate the charge neutrality condition, requiring careful consideration of the applicability of RXMHD (and RHMHD). In fact, all models bearing the name “MHD” assume either QN or PCN a priori. Once these neutrality conditions (i.e., single-fluid approximation) fail, we should solve the two-fluid equations or kinetic models directly. For example, the two-fluid model for relativistic electron-positron plasmas²⁶ has been solved numerically and applied to the simulations of magnetic reconnection.²⁷

In this study, starting from the relativistic two-fluid equations, we systematically reproduce various MHD models (including RXMHD) using the method of dominant balance²⁸ and theoretically illustrate their scopes of application. Since there are too many dimensionless parameters in the original two-fluid equations, we will not explore all cases but focus only on the realm of MHD where the MHD balances hold; the MHD terms are not negligible but dominant. Specifically, we consider a situation in which the Lorentz force (⁠ $J \times B$ term) is dominant in the equation of motion for the center-of-mass velocity of the two fluids. If the pressure term or the electric force is far more dominant than the Lorentz force, the MHD model is unlikely to be applicable.²⁹ Therefore, in order to make nonessential parameters invisible and highlight only the dominant terms, the plasma pressure and the external electric field will be ignored from the beginning. The MHD models are finally classified in terms of three dimensionless parameters corresponding to the scales of the plasma density, the flow velocity, and the external magnetic field. Furthermore, the dimensionless parameters can be reduced to two if the flow velocity is assumed to be on the same order of the Alfvén velocity. The applicability of the various MHD models will be visualized in this parameter space, supposing that a dimensionless coefficient before each term is considered negligible if it is less than, say, $10^{- 4}$ ⁠. For these relativistic and two-fluidic MHD models, we will write them in the form of a dynamical system $\partial_{t} u = F (u)$ and identify the number of the time-evolving field variables u. We will show that RHMHD has more variables than HMHD and RMHD. In the case of RMHD, the right-hand side F(u) is notorious for being an implicit function of u, which requires extra computational cost.³⁰ RHMHD will be shown to resolve this problem of RMHD, although the number of variables increases.

II. BASIC EQUATIONS

A. Two-fluid equations

We denote the Minkowski spacetime of the reference frame by

x^{μ} : = (c t, x, y, z) = (x^{0}, x^{i}), x_{μ} : = (- c t, x, y, z) = (x_{0}, x_{i}),

(1)

where c is the speed of light and the Minkowski metric tensor is diag(−1,1,1,1). The partial derivatives will be shortly denoted by

\partial_{μ} = \partial / \partial x^{μ}

and

\partial^{μ} = \partial / \partial x_{μ}

⁠. The proper four-velocity is defined as

U^{μ} : = (γ, γ v / c), U_{μ} : = (- γ, γ v / c),

(2)

where v is the reference-frame three-velocity (called simply “velocity”), and

γ : = 1 / \sqrt{1 - | v |^{2} / c^{2}}

is the Lorentz factor. In this paper, Greek indices (⁠

μ = 0, 1, 2, 3

⁠) denote the time-space (4D) components, while Roman indices (⁠

v^{i} = v_{i}

⁠, i = 1, 2, 3) or bold faces (v) denote the spatial (3D) components. The Einstein summation convention will be used in what follows.

Momentarily Co-moving Reference Frame (MCRF) refers to the inertial frame co-moving with particles. Physical quantities of relativistic fluid are said to be “proper” when they are observed in the frame co-moving with the velocity v. Therefore, the proper number density is given by $N = n / γ$ ⁠, when n is the number density in the reference frame.

In this study, we start with the special-relativistic fluid equations for both positively and negatively charged gases, where dissipation due to collision is neglected for simplicity. Namely, we ignore resistivity and viscosity. The equations of motion, the continuity equations and Maxwell's equations are written as

\partial_{ν} (h_{\pm} N_{\pm} U_{\pm}^{μ} U_{\pm}^{ν}) = - \partial^{μ} p_{\pm} \pm e c N_{\pm} U_{\pm}^{ν} F^{μ}_{ν},

(3)

\partial_{ν} (N_{\pm} U_{\pm}^{ν}) = 0,

(4)

\partial_{ν}^{*} F^{μ ν} = 0,

(5)

\partial_{ν} F^{μ ν} = μ_{0} J^{μ},

(6)

where the subscripts plus

(+)

and minus

(-)

indicate that they are the quantities for positively and negatively charged particles, respectively. Moreover,

h_{\pm}

is the entropy per unit particle,

p_{\pm}

is the pressure,

F^{μ ν}

is the electromagnetic field tensor,

^{*} F^{μ ν}

is the Hodge dual tensor of

F^{μ ν}

⁠, and

J^{μ} : = e c (N_{+} U_{+}^{μ} - N_{-} U_{-}^{μ})

is the four-current. The governing equations, to , are called the two-fluid equations. We use the SI unit system; μ₀ is the vacuum magnetic permeability, and e is the elementary charge.

Maxwell's equations are also expressed in 3 + 1 form as

\partial_{j} E^{j} / c = μ_{0} e c \tilde{n},

(7)

\partial_{j} B^{j} / c = 0,

(8)

ε^{i}_{j k} \partial^{j} E^{k} / c = - \partial_{0} B^{i},

(9)

ε^{i}_{j k} \partial^{j} B^{k} = μ_{0} J^{i} - \partial_{0} E^{i} / c,

(10)

where Eⁱ is the electric field, Bⁱ is the magnetic field,

e \tilde{n} = e (n_{+} - n_{-})

is the charge density, and ε_ijk is the Levi-Civita symbol.

In this paper, the four potential

A^{μ} = (ϕ / c, A^{i})

is also introduced to express the electromagnetic field and we employ the Coulomb gauge

\partial_{j} A^{j} = 0

⁠. Maxwell's equations are then transformed into

- \partial_{j} \partial_{j} ϕ = μ_{0} e c^{2} \tilde{n},

(11)

\frac{1}{c^{2}} \partial_{t} \partial_{t} A^{i} + \partial_{i} \partial_{j} A^{j} - \partial_{j} \partial_{j} A^{i} = - \frac{1}{c^{2}} \partial_{i} (\partial_{t} ϕ) + μ_{0} J^{i} .

(12)

B. Transformation into MHD variables

Let us rewrite Eqs. and in terms of MHD variables without any approximation. For this purpose, we define the mass-weighted average of number density n, the center-of-mass velocity vⁱ, the number density difference

\tilde{n}

⁠, and the velocity uⁱ for the electric current as follows:

\begin{matrix} n : = \frac{m_{+} n_{+} + m_{-} n_{-}}{m_{+} + m_{-}}, v^{i} : = \frac{m_{+} n_{+} v_{+}^{i} + m_{-} n_{-} v_{-}^{i}}{(m_{+} + m_{-}) n}, \\ \tilde{n} : = n_{+} - n_{-}, u^{i} : = \frac{n_{+} v_{+}^{i} - n_{-} v_{-}^{i}}{n}, \end{matrix}

(13)

where

m_{\pm}

is the mass of the particle. Moreover, these variables are associated with four-dimensional center-of-mass flux (divided by c)

Q^{μ} = (n, n \frac{v^{i}}{c}) : = \frac{m_{+} N_{+} U_{+}^{μ} + m_{-} N_{-} U_{-}^{μ}}{m_{+} + m_{-}},

(14)

and four-dimensional current (with the same dimension as

Q^{μ}

⁠)

K^{μ} = \frac{J^{μ}}{e c} = (\tilde{n}, n \frac{u^{i}}{c}) : = N_{+} U_{+}^{μ} - N_{-} U_{-}^{μ} .

(15)

The MHD equations are sometimes called the single-fluid model, assuming that the two species of charged fluid move together approximately; $\tilde{n} ≪ n$ and $| u | ≪ | v |$ ⁠.

By denoting for the positive and negative species by (3)₊ and (3)₋, respectively, the equation of center-of-mass motion is obtained from the sum (3)₊ + (3)₋ as follows:

\begin{matrix} m c^{2} \partial_{ν} [f Q^{μ} Q^{ν} + μ^{2} \tilde{f} (Q^{μ} K^{ν} + K^{μ} Q^{ν}) + μ^{2} f' K^{μ} K^{ν}] \\ = - \partial^{μ} p + e c K^{ν} F^{μ}_{ν} . \end{matrix}

(16)

On the other hand, generalized Ohm's law is obtained from

[m_{-}

(3)

_{+} - m_{+}

(3)

] / (m_{+} + m_{-})

⁠,

\begin{matrix} m c^{2} \partial_{ν} [μ^{2} \tilde{f} Q^{μ} Q^{ν} + μ^{2} f' (Q^{μ} K^{ν} + K^{μ} Q^{ν}) \\ + μ^{2} (- \tilde{μ} f' + μ^{2} \tilde{f}) K^{μ} K^{ν}] \\ = \frac{μ}{2} \partial^{μ} p + \frac{1}{2} \partial^{μ} \tilde{p} + e c (Q^{ν} - \tilde{μ} K^{ν}) F^{μ}_{ν}, \end{matrix}

(17)

where the following abbreviations are used:

m : = m_{+} + m_{-}, μ^{2} : = \frac{m_{+} m_{-}}{m^{2}}, \tilde{μ} : = \frac{m_{+} - m_{-}}{m},

(18)

p : = p_{+} + p_{-}, \tilde{p} : = p_{+} - p_{-},

(19)

f_{\pm} : = \frac{h_{\pm}}{N_{\pm} m_{\pm} c^{2}},

(20)

\tilde{f} : = f_{+} - f_{-},

(21)

f : = \frac{1}{m} (m_{+} f_{+} + m_{-} f_{-}),

(22)

f' : = \frac{1}{m} (m_{-} f_{+} + m_{+} f_{-}) = f - \tilde{μ} \tilde{f} .

(23)

Both the classical and relativistic MHD equations are derived by neglecting

\tilde{f}

owing to

\tilde{f} ≪ f

(which then leads to

f' = f

⁠). Therefore, the orders of f and

\tilde{f}

are important for the validity of the MHD approximation.

Similarly, we obtain the conservation law of mass density

\partial_{ν} Q^{ν} = 0

(24)

from

[m_{+}

(4)

_{+} + m_{-}

(4)

_{-}] / m

⁠, and the conservation law of charge density

\partial_{ν} K^{ν} = 0

(25)

from (4)₊ to (4)_–.

C. Assumption of cold plasma

The validity of the MHD approximation primarily relies on

\tilde{n} ≪ n, | u | ≪ | v |

and

\tilde{f} ≪ f

being sufficiently fulfilled. To focus on this topic, we neglect the pressure terms (i.e., the cold plasma approximation) in what follows because they simply appear as additional terms and make the governing equations lengthy. Therefore,

p = \tilde{p} = 0

and

h_{\pm} = m_{\pm} c^{2}

are assumed. Then, Eqs. and are reduced to

f = \frac{1}{m} (\frac{m_{+}}{N_{+}} + \frac{m_{-}}{N_{-}}), \tilde{f} = \frac{1}{N_{+}} - \frac{1}{N_{-}} .

(26)

Using the relations,

\begin{matrix} N_{\pm} = \sqrt{- N_{\pm} N_{\pm} U_{\pm}^{α} U_{\pm α}} \\ = \sqrt{- (Q^{α} Q_{α} + \frac{m_{\mp}^{2}}{m^{2}} K^{α} K_{α}) \mp \frac{m_{\mp}}{m} (Q^{α} K_{α} + K^{α} Q_{α})}, \end{matrix}

(27)

we can express f and

\tilde{f}

in terms of

Q^{μ}

and

K^{μ}

(in a very complicated way). In Maxwell's equations, the electromagnetic field

F^{μ ν}

is generated by

J^{μ}

⁠, which is

e c K^{μ}

⁠. Therefore, the two-fluid equations are fully expressed by the MHD variables,

Q^{μ}, K^{μ}

⁠, and

F^{μ ν}

⁠.

Let us clarify the number of field variables in the two-fluid equations. From the definitions given above, Eqs. (16), (17), (24), and (25) clearly describe the time evolution of the eight variables $Q^{μ}$ and $K^{μ}$ (which correspond to n, vⁱ, $\tilde{n}$ ⁠, and uⁱ). Maxwell's equations provide the time evolution of the six variables Eⁱ and Bⁱ (which is $F^{μ ν}$ ⁠), but they must be solved under the two constraints (7) and (8) (which include no time derivative). In fact, we can eliminate the variable $\tilde{n}$ because $\tilde{n}$ is uniquely determined by Eⁱ via Eq. (7), and the charge conservation law (25) is automatically satisfied by Eqs. (7) and (10). Therefore, in the cold plasma approximation, the two-fluid equations constitute a dynamical system of 13 field variables under 1 constraint in total. In a sense, the degree of freedom is $13 - 1 = 12$ ⁠. Even when Aⁱ is used instead of Bⁱ, the Coulomb gauge $\partial_{i} A^{i} = 0$ is imposed instead of $\partial_{i} B^{i} = 0$ and the degree of freedom is the same. Reducing the number of field variables is one of the major purpose of the following MHD approximation.

III. DOMINANT BALANCE

To derive reduced models from the two-fluid equations systematically, we first normalize all terms in the equations and consider the dominant balances that are suitable for magnetized plasma.

A. Normalization

We normalize all the equations by introducing eight representative scales (with subscript

⋆

⁠) as follows:

\begin{matrix} \hat{n} = \frac{n}{n_{⋆}}, {\hat{v}}^{i} = \frac{v^{i}}{v_{⋆}}, \hat{\tilde{n}} = \frac{\tilde{n}}{{\tilde{n}}_{⋆}}, {\hat{u}}^{i} = \frac{u^{i}}{u_{⋆}}, \\ {\hat{B}}^{i} = \frac{B^{i}}{B_{⋆}}, {\hat{ϕ}}^{i} = \frac{ϕ}{ϕ_{⋆}}, {\hat{x}}^{i} = \frac{x^{i}}{L_{⋆}}, \hat{t} = \frac{t}{T_{⋆}}, \end{matrix}

(28)

where we have introduced the common scale for all three-dimensional components of vector fields (i.e.,

v^{1} \sim v^{2} \sim v^{3}

⁠) for simplicity. Note that

L_{⋆}

and

T_{⋆}

are the representative spatial and temporal scales, respectively, of plasma dynamics that we are interested in.

The conservation law of mass is written in terms of the normalized quantities (with the hat symbol) as

\frac{L_{⋆}}{v_{⋆} T_{⋆}} \hat{\partial_{0}} \hat{Q^{0}} + \hat{\partial_{i}} \hat{Q^{i}} = 0.

(29)

Except when we consider the special cases (such as steady solution or incompressible limit), the two terms on the left hand side balance each other. First of all, we assume this balance as usual:

Balance 1 : L_{⋆} = v_{⋆} T_{⋆} .

(30)

Since this balance is merely a relation among scale parameters, it should be actually interpreted as

L_{⋆} \sim v_{⋆} T_{⋆}

or

O (L_{⋆}) = O (v_{⋆} T_{⋆})

⁠. However, in this paper, the equality “

=

” will be used to reduce the number of the scale parameters by imposing this balance.

Next, we consider the Poisson equation that is normalized to

- \frac{ϕ_{⋆}}{L_{⋆}^{2}} {\hat{\partial}}_{j} {\hat{\partial}}_{j} \hat{ϕ} = μ_{0} e c^{2} {\tilde{n}}_{⋆} \hat{\tilde{n}}

(31)

and assume that there is no externally applied electrostatic potential (e.g.,

ϕ \to 0

at infinity). Then,

\hat{ϕ}

is generated only by the charge density of plasma itself via this equation, and it is natural to assume the balance between the left and right hand sides,

Balance 2 : ϕ_{⋆} = μ_{0} e c^{2} {\tilde{n}}_{⋆} L_{⋆}^{2} .

(32)

Since the two balances 1 and 2 are assumed among the eight representative scales, let us define five dimensionless parameters for later use as follows:

ε : = \frac{1}{L_{⋆}} \sqrt{\frac{m}{μ_{0} n_{⋆} e^{2}}},

(33)

σ : = B_{⋆}^{2} / (μ_{0} m n_{⋆} c^{2}),

(34)

β_{⋆} : = v_{⋆} / c,

(35)

α : = {\tilde{n}}_{⋆} / n_{⋆},

(36)

ε_{m} : = u_{⋆} / v_{⋆},

(37)

where ε denotes the normalized inertial length and σ is called the magnetization parameter. As we have mentioned earlier, the smallness of α and ε_m will be essential for the MHD approximation.

Using

{\hat{A}}^{i} = A^{i} / (B_{⋆} L_{⋆})

⁠, Ampère–Maxwell's law is normalized as

β_{⋆}^{2} {\hat{\partial}}_{0} {\hat{\partial}}_{0} {\hat{A}}^{i} + {\hat{\partial}}_{i} {\hat{\partial}}_{j} {\hat{A}}^{j} - {\hat{\partial}}_{j} {\hat{\partial}}_{j} {\hat{A}}^{i} = \frac{β_{⋆}}{ε \sqrt{σ}} (- α {\hat{\partial}}_{i} {\hat{\partial}}_{0} \hat{ϕ} + ε_{m} \hat{n} {\hat{u}}^{i}) .

(38)

The right hand side is regarded as the source terms which generate magnetic field and, hence, cannot be much larger than the left hand side. In contrast to the Poisson equation , we allow for externally applied magnetic field, which can exist (⁠

A^{i} \neq 0

⁠) even when the right hand side is small or zero (that is vacuum magnetic field). Thus, we should consider only the following regime:

Balance 3 : \frac{β_{⋆}}{ε \sqrt{σ}} max (α, ε_{m}) \leq 1.

(39)

Again, this inequality “

\leq

” actually means “

≲

” because this is a relation among the scale parameters.

Next, to estimate the orders of f and

\tilde{f}

⁠, let us normalize

N_{+}

and

N_{-}

as follows:

\begin{matrix} {\hat{N}}_{\pm}^{2} = \frac{N_{\pm}^{2}}{n_{⋆}^{2}} = {\hat{n}}^{2} (1 - β_{⋆}^{2} | \hat{v} |^{2}) \pm 2 \frac{m_{\mp}}{m} \hat{n} (α \hat{\tilde{n}} - β_{⋆}^{2} ε_{m} \hat{v} \cdot \hat{n} \hat{u}) \\ + \frac{m_{\mp}^{2}}{m^{2}} (α^{2} {\hat{\tilde{n}}}^{2} - β_{⋆}^{2} ε_{m}^{2} | \hat{n} \hat{u} |^{2}) . \end{matrix}

(40)

The first term on the right hand side is O(1). Since we are interested in the case of

α, ε_{m} ≪ 1

and the inequalities

m_{\mp} / m \leq 1

and

β_{⋆} < 1

always hold, the second and third terms on the right hand side are of small order;

max (α, β_{⋆}^{2} ε_{m}) ≪ 1

⁠. As a loose assumption, we consider the situation where

Balance 4 : η : = max (α, β_{⋆}^{2} ε_{m}) \leq 1,

(41)

holds. Namely, we give up applying the MHD approximation when

η ≫ 1

⁠. Assuming Eq. , we obtain the estimates,

n_{⋆} f = O (1)

and

n_{⋆} \tilde{f} = O (η)

⁠, and hence normalize them by

\hat{f} = n_{⋆} f

and

\hat{\tilde{f}} = n_{⋆} \tilde{f} / η

⁠. More explicitly, when

η ≪ 1

⁠, the leading-order terms are calculated by series expansion as follows:

\hat{f} = \frac{γ}{\hat{n}} + \frac{γ^{3}}{2 \hat{n}} μ^{2} (3 γ^{2} Λ_{a}^{2} - Λ_{b}) + O (α^{3}, β_{⋆}^{2} ε_{m}^{3}),

(42)

η \hat{\tilde{f}} = - \frac{γ^{2}}{\hat{n}} Λ_{a} + O (α^{2}, β_{⋆}^{2} ε_{m}^{2}),

(43)

where

γ = \frac{1}{\sqrt{1 - β_{⋆}^{2} | \hat{v} |^{2}}},

(44)

Λ_{a} : = α \frac{\hat{\tilde{n}}}{\hat{n}} - β_{⋆}^{2} ε_{m} \hat{v} \cdot \hat{u} = O (α, β_{⋆}^{2} ε_{m}),

(45)

Λ_{b} : = α^{2} \frac{{\hat{\tilde{n}}}^{2}}{{\hat{n}}^{2}} - β_{⋆}^{2} ε_{m}^{2} | \hat{u} |^{2} = O (α^{2}, β_{⋆}^{2} ε_{m}^{2}) .

(46)

Here, we emphasize that the first order terms in α and ε_m are vacant in the series expansion of

\hat{f}

⁠, which turns out to be important later.

It should be also remarked that we exclude the strongly-relativistic situation such as $β_{⋆} | \hat{v} | = | v | / c = 0.9999$ ⁠, in which the Lorenz factor γ becomes much greater than 1 and our estimate $\hat{f} = O (1)$ is no longer valid. This means a breakdown of the assumed balance³² and strongly relativistic flow regions must be treated separately using a different normalization. For example, we suggest that all the equations should be Lorenz-transformed to the inertia frame moving with the flow speed $0.9999 c$ so that the Lorenz factor becomes $γ = O (1)$ ⁠.

Now, Eqs. , , , and are normalized as follows:

\begin{matrix} \hat{\partial_{0}} [\hat{f} \hat{Q^{i}} \hat{Q^{0}} + μ^{2} η \hat{\tilde{f}} (α \hat{Q^{i}} \hat{K^{0}} + ε_{m} \hat{K^{i}} \hat{Q^{0}}) + μ^{2} α ε_{m} \hat{f'} \hat{K^{i}} \hat{K^{0}}] \\ + \hat{\partial_{j}} [\hat{f} \hat{Q^{i}} \hat{Q^{j}} + μ^{2} η ε_{m} \hat{\tilde{f}} (\hat{Q^{i}} \hat{K^{j}} + \hat{Q^{j}} \hat{K^{i}}) + μ^{2} ε_{m}^{2} \hat{f'} \hat{K^{i}} \hat{K^{j}}] \\ - \frac{α^{2}}{β_{⋆}^{2} ε^{2}} \hat{K^{0}} {\hat{F}}^{i}_{0} - \frac{ε_{m}}{β_{⋆} ε} \sqrt{σ} \hat{K^{j}} {\hat{F}}^{i}_{j} = 0, \end{matrix}

(47)

\begin{matrix} \hat{\partial_{0}} [μ^{2} η ε_{m} \hat{\tilde{f}} \hat{Q^{i}} \hat{Q^{0}} + μ^{2} ε_{m} \hat{f'} (α \hat{Q^{i}} \hat{K^{0}} + ε_{m} \hat{K^{i}} \hat{Q^{0}}) \\ - μ^{2} ε_{m}^{2} α (\tilde{μ} \hat{f'} - μ^{2} η \hat{\tilde{f}}) \hat{K^{i}} \hat{K^{0}}] + \hat{\partial_{j}} [μ^{2} η ε_{m} \hat{\tilde{f}} \hat{Q^{i}} \hat{Q^{j}} \\ + μ^{2} ε_{m}^{2} \hat{f'} (\hat{Q^{i}} \hat{K^{j}} + \hat{Q^{j}} \hat{K^{i}}) \\ - μ^{2} ε_{m}^{3} (\tilde{μ} \hat{f'} - μ^{2} η \hat{\tilde{f}}) \hat{K^{i}} \hat{K^{j}}] \\ - \frac{α ε_{m}}{β_{⋆}^{2} ε^{2}} (\hat{Q^{0}} - \tilde{μ} α \hat{K^{0}}) {\hat{F}}^{i}_{0} - \frac{ε_{m}}{β_{⋆} ε} \sqrt{σ} (\hat{Q^{j}} - \tilde{μ} ε_{m} \hat{K^{j}}) {\hat{F}}^{i}_{j} = 0, \end{matrix}

(48)

\hat{\partial_{0}} \hat{Q^{0}} + \hat{\partial_{i}} \hat{Q^{i}} = 0,

(49)

\frac{α}{ε_{m}} \hat{\partial_{0}} \hat{K^{0}} + \hat{\partial_{i}} \hat{K^{i}} = 0,

(50)

where the normalized electric field

{\hat{E}}^{i}

is given by

{\hat{E}}^{i} = {\hat{F}}^{i}_{0} = \frac{c L_{⋆}}{ϕ_{⋆}} F^{i}_{0} = - {\hat{\partial}}_{i} \hat{ϕ} - \frac{β_{⋆} ε \sqrt{σ}}{α} {\hat{\partial}}_{0} {\hat{A}}^{i} .

(51)

B. Imposition of MHD balance

The MHD approximation is understood as the reduction to a single-fluid model, satisfying

α ≪ 1 and ε_{m} ≪ 1.

(52)

If they were not satisfied, we would have to solve the two-fluid equations as they are. However, in the limit of

α, ε_{m} \to 0

(then

η \to 0

⁠), many terms in Eq. are negligible and ultimately Eq. becomes the equation of motion for neutral fluid. Since this simple limit is not interesting, we assume that the electromagnetic (⁠

J \times B

⁠) force, which is

\hat{K^{j}} {\hat{F}}^{i}_{j}

in , is not negligible but dominant. Namely, the flow v is dominantly accelerated by this term due to

Balance 5 : \frac{ε_{m}}{β_{⋆} ε} \sqrt{σ} = 1.

(53)

The another meaning of this balance can be understood by defining a representative cyclotron frequency as

ω_{c ⋆} = \frac{e B_{⋆}}{m} = \frac{c}{L_{⋆}} \frac{\sqrt{σ}}{ε} .

(54)

Then, balance 5 indicates that

ε_{m} = \frac{1 / ω_{c ⋆}}{T_{⋆}}

(55)

is small if this frequency is faster than the timescale

T_{⋆}

of flow dynamics. It is well known that the two-fluid equations generally encompass ion's and electron's cyclotron motions. By taking the limit of

ε_{m} \to 0

while keeping balance 5, we can eliminate these fast motions from the flow dynamics. We also remark that the

v \times B

term (which is

{\hat{Q}}^{j} {\hat{F}}^{i}_{j}

⁠) in Ohm's law becomes of order 1 due to Eq. .

Although balance 5 is assumed in this work, we do not claim here that Hall or extended MHD must always satisfy $ε_{m} = ω_{c ⋆}^{- 1} / T_{⋆} ≪ 1$ ⁠. In electron-ion plasmas, $ω_{c ⋆}$ is approximately the ion cyclotron frequency ω_ci, and the frequency of the whistler wave is known to be higher than ω_ci, i.e., $ε_{m} > 1$ ⁠. As will be shown later in Secs. IV and V, the electron inertia effect (⁠ $ε_{I}^{2}$ ⁠) can be neglected due to the smallness of the mass ratio $m_{-} / m_{+}$ ⁠. If we take the limit of $m_{-} / m_{+} \to 0$ ⁠, the Hall and extended MHD would be applicable even if $ε_{m} > 1$ ⁠.

On the other hand, the terms involving the electric field

{\hat{F}}^{i}_{0}

in Eqs. and are, respectively, written as

\frac{α^{2}}{β_{⋆}^{2} ε^{2}} \hat{K^{0}} {\hat{F}}^{i}_{0} = \frac{1}{σ} \frac{α^{2}}{ε_{m}^{2}} \hat{K^{0}} [- {\hat{\partial}}_{i} \hat{ϕ} - σ \frac{ε_{m}}{α} {\hat{\partial}}_{0} {\hat{A}}^{i}],

(56)

and

\begin{matrix} \frac{α ε_{m}}{β_{⋆}^{2} ε^{2}} (\hat{Q^{0}} - \tilde{μ} α \hat{K^{0}}) {\hat{F}}^{i}_{0} \\ = \frac{1}{σ} \frac{α}{ε_{m}} (\hat{Q^{0}} - \tilde{μ} α \hat{K^{0}}) [- {\hat{\partial}}_{i} \hat{ϕ} - σ \frac{ε_{m}}{α} {\hat{\partial}}_{0} {\hat{A}}^{i}], \end{matrix}

(57)

using balance 5. In the limit of

ε_{m} \to 0

or

σ \to 0

⁠, only the electrostatic force term (⁠

{\hat{\partial}}_{i} \hat{ϕ}

⁠) gets too large

1 / (σ ε_{m}) \to \infty

to balance with other terms. This implies that the existence of very fast plasma oscillation breaks down the assumed balance totally. To maintain the balance consistently, the charge separation α must be small enough that all terms in Eqs. and are equal or less than the order 1, which requires

α \leq \sqrt{σ} ε_{m}, α \leq σ ε_{m}

and

α \leq ε_{m}

⁠. To consider the most general situation satisfying all of them, we assume

Balance 6 : α = ε_{m} ε_{σ},

(58)

where

ε_{σ}

is the abbreviation of

ε_{σ} : = min (σ, 1) .

(59)

This is the last balance that we impose to derive MHD models. The magnitude of α is now determined by other scale parameters. The meaning of this balance is again understood by introducing a representative plasma frequency as

ω_{p ⋆} : = c \sqrt{\frac{e^{2} μ_{0} n_{⋆}}{m}} = \frac{1}{L_{⋆}} \frac{c}{ε} .

(60)

Since

ε_{σ} \leq \sqrt{σ}

holds mathematically (see Fig. 1), balance 6 leads to

α \leq \frac{1 / ω_{p ⋆}}{T_{⋆}} = β_{⋆} ε = ε_{m} \sqrt{σ} .

(61)

Therefore, when the plasma frequency is much faster than the timescale of the flow dynamics (⁠

ω_{p ⋆}^{- 1} / T_{⋆} \to 0

⁠), balance 6 requires α to be small (⁠

α \to 0

⁠), which diminishes the fast plasma oscillation. Note that

ω_{p ⋆}^{- 1} / T_{⋆}

is not always a small number when σ is much greater than 1. Balance 6 requires smallness of α more strictly than the condition

α \leq ω_{p ⋆}^{- 1} / T_{⋆}

when

σ \geq 1

⁠.

FIG. 1.

View large Download slide

Plots of $ε_{σ}, \sqrt{σ}$ and $σ / (1 + σ)$ ⁠.

At this point, we summarize the situation where all the balances

1, 2, \dots, 6

are imposed together. Given the balances 5 and 6, balance 3 can be reduced to

Balance 3' : \frac{β_{⋆}^{2}}{σ} \leq 1 .

(62)

Therefore, the situation can be divided into the two cases,

β_{⋆}^{2} \leq σ \leq 1

or

1 \leq σ

⁠. In either case, balance 4 is simply rewritten as

Balance 4' : η = ε_{m} ε_{σ} \leq 1.

(63)

By omitting the hat symbol

\hat{}

in what follows, the normalized equations are summarized as follows:

\begin{matrix} \partial_{0} [f Q^{i} Q^{0} + ε_{I}^{2} ε_{σ} \tilde{f} (ε_{σ} Q^{i} K^{0} + K^{i} Q^{0}) + ε_{σ} ε_{I}^{2} f' K^{i} K^{0}] \\ + \partial_{j} [f Q^{i} Q^{j} + ε_{I}^{2} ε_{σ} \tilde{f} (Q^{i} K^{j} + Q^{j} K^{i}) + ε_{I}^{2} f' K^{i} K^{j}] \\ - ε_{σ} K^{0} [- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}] - K^{j} F^{i}_{j} = 0, \end{matrix}

(64)

\begin{matrix} ε_{I}^{2} \partial_{0} [ε_{σ} \tilde{f} Q^{i} Q^{0} + f' (ε_{σ} Q^{i} K^{0} + K^{i} Q^{0}) \\ - ε_{σ} (ε_{H} f' - ε_{σ} ε_{I}^{2} \tilde{f}) K^{i} K^{0}] \\ + ε_{I}^{2} \partial_{j} [ε_{σ} \tilde{f} Q^{i} Q^{j} + f' (Q^{i} K^{j} + Q^{j} K^{i}) \\ - (ε_{H} f' - ε_{σ} ε_{I}^{2} \tilde{f}) K^{i} K^{j}] \\ - (Q^{0} - ε_{H} ε_{σ} K^{0}) [- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}] \\ - (Q^{j} - ε_{H} K^{j}) F^{i}_{j} = 0, \end{matrix}

(65)

\partial_{0} Q^{0} + \partial_{i} Q^{i} = 0,

(66)

ε_{σ} \partial_{0} K^{0} + \partial_{i} K^{i} = 0,

(67)

- \partial_{j} \partial_{j} ϕ = \tilde{n},

(68)

β_{⋆}^{2} \partial_{0} \partial_{0} A^{i} + \partial_{i} \partial_{j} A^{j} - \partial_{j} \partial_{j} A^{i} = \frac{β_{⋆}^{2}}{σ} (- ε_{σ} \partial_{i} \partial_{0} ϕ + n u^{i}),

(69)

where

f' = f - ε_{H} ε_{σ} \tilde{f} .

(70)

We have introduced

ε_{H} : = \tilde{μ} ε_{m},

(71)

ε_{I} : = μ ε_{m},

(72)

because ε_m appears only in these forms. By noting

Λ_{a} = O (ε_{m} ε_{σ})

and

Λ_{b} = O (ε_{m}^{2} ε_{σ})

⁠, the estimate becomes

f = \frac{1}{n \sqrt{1 - β_{⋆}^{2} | v |^{2}}} + O (ε_{σ} ε_{I}^{2}) .

(73)

Due to balances 5 and 6, the number of non-dimensional parameters (namely, the dimension of the parameter space) has been reduced to three; $ε_{m}, σ, β_{⋆}$ ⁠. However, the governing equations are still equivalent to the two-fluid equations. In Sec. IV, we derive reduced models by taking the specific limit of $ε_{m}, σ, β_{⋆}$ ⁠.

IV. REDUCTION TO VARIOUS MHD MODELS

A. Vacuum limit $β_{⋆}^{2} / σ \to 0$

The limit $β_{⋆}^{2} / σ \to 0$ corresponds to the vacuum state since the right hand side of the Ampère–Maxwell equation (69) vanishes. This limit may be uninteresting because the plasma current is too small to disturb the vacuum magnetic field. For example, the limit of large magnetization parameter $σ \to \infty$ inevitably results in this vacuum state (due to $β_{⋆} \leq 1$ ⁠).

As another high magnetization limit $σ \to \infty$ ⁠, the force-free approximation is widely used.³¹ When the current is divided into parallel (⁠ $J_{∥}$ ⁠) and perpendicular (⁠ $J_{⊥}$ ⁠) components to the magnetic field, $J_{∥}$ remains finite while $J_{⊥} = 0$ (i.e., $J \times B = 0$ ⁠) in the force-free state. In this study, we set $n_{⋆} u_{⋆}$ as the single representative scale of the current, and no distinction is made between the parallel and perpendicular components. To obtain the force-free state, only the parallel component should diverge (⁠ $u_{∥ ⋆} \to \infty$ ⁠) to keep the parallel current finite in the vacuum limit $n_{⋆} \to 0$ ⁠. Although the force-free model will be reproduced later in Sec. VI, the existence of the “extended” force-free model is nontrivial and will be the subject of future work.

According to the dispersion relation of RMHD, the relativistic Alfvén velocity

v_{A ⋆}

is well-known as

v_{A *} = \frac{B_{⋆} c}{\sqrt{μ_{0} m n_{⋆} c^{2} + B_{⋆}^{2}}} = \sqrt{\frac{σ}{1 + σ}} c .

(74)

When

σ \to \infty

(i.e., strong magnetic field or low density limit), the displacement current becomes dominant and the Alfvén wave turns into the electromagnetic wave in vacuum.

The interaction between plasma motion and electromagnetic field is most active in the situation $β_{⋆}^{2} / σ ≃ 1$ ⁠. Therefore, the Alfvén ordering $β_{⋆}^{2} / σ = 1$ is conventionally employed in (non-relativistic) MHD focusing on only this situation, which is admissible as far as $σ \leq 1$ ⁠. It is also interesting to note that the scale of $ε_{σ}$ is similar to $v_{A ⋆}^{2} / c^{2} = σ / (1 + σ)$ ⁠.

B. Single-fluid limit $ε_{m} \to 0$

In the limit of

ε_{m} \to 0

(then

ε_{H}, ε_{I} \to 0

⁠), many terms can be neglected in Eqs. and as follows:

\begin{matrix} \partial_{0} (γ n v^{i}) + \partial_{j} (γ n v^{i} v^{j}) \\ - ε_{σ} \tilde{n} (- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}) - n u^{j} F^{i}_{j} = 0, \end{matrix}

(75)

- n (- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}) - n v^{j} F^{i}_{j} = 0.

(76)

These are the well-known RMHD equations, where the expression of f has been simplified into

f = \frac{γ}{n} = \frac{1}{n \sqrt{1 - β_{⋆}^{2} | v |^{2}}},

(77)

and

\tilde{f}

is completely neglected as if the “proper charge neutrality”

N_{+} = N_{-}

holds. In fact, there exists small-order charge separation

\tilde{n} \neq 0

(with

\nabla \cdot J \neq 0

⁠) and the associated electrostatic force takes part in the dominant balance. We will discuss more about the RMHD equations in Sec. VI.

C. Nonrelativistic limit $σ \to 0$

Here, we consider the limit of small σ. Because of balance 3′, the limit of

σ \to 0

forces

β_{⋆}^{2} \to 0

as well (and

ε_{σ} \to 0

⁠). Therefore, let us take these nonrelativistic limits while keeping

\frac{β_{⋆}^{2}}{σ} \to \frac{v_{⋆}^{2}}{v_{A ⋆}^{2}} = const . (\leq 1) .

(78)

Then, Eqs. , , and are reduced to the extended MHD (XMHD) equations

\partial_{0} (n v^{i}) + \partial_{j} (n v^{i} v^{j} + ε_{I}^{2} n u^{i} u^{j}) - n u^{j} F^{i}_{j} = 0,

(79)

\begin{matrix} ε_{I}^{2} \partial_{0} (n u^{i}) + ε_{I}^{2} \partial_{j} [n (v^{i} u^{j} + v^{j} u^{i}) - ε_{H} n u^{i} u^{j}] \\ - n (- \partial_{i} ϕ - \partial_{0} A^{i}) - n (v^{j} - ε_{H} u^{j}) F^{i}_{j} = 0, \end{matrix}

(80)

\partial_{i} \partial_{j} A^{j} - \partial_{j} \partial_{j} A^{i} = \frac{v_{⋆}^{2}}{v_{A ⋆}^{2}} n u^{i} .

(81)

where

f = 1 / n

has been substituted. Since

v_{⋆}^{2} / v_{A ⋆}^{2} \to 0

is again the uninteresting vacuum limit, it is conventional to use the Alfvén ordering

v_{⋆} = v_{A ⋆}

⁠.

Equation (80) is the generalized Ohm's law. Recall that we neglected resistivity and viscosity at the beginning, assuming that they are not dominant. The inclusion of these terms with large Lundquist and Reynolds numbers does not affect the balances we have discussed so far.

Note that the displacement current in Eq. (69) has been neglected and hence $\nabla \times B = (v_{⋆}^{2} / v_{A ⋆}^{2}) J$ is a constraint; we can eliminate nuⁱ (or J) using this relation. The generalized Ohm's law (80) is regarded as the evolution equation for Aⁱ, where $ϕ$ is determined such that the Coulomb gauge $\partial_{i} A^{i} = 0$ holds. Therefore, the XMHD equations constitute a dynamical system of seven fields $(n, v^{i}, A^{i})$ with one constraint $\partial_{i} A^{i} = 0$ ⁠. Since $\nabla \cdot J = \partial_{i} (n u^{i}) = 0$ ⁠, the so-called “quasi-neutrality condition” holds as if $\tilde{n} = 0$ ⁠. In fact, small charge separation $\tilde{n} = - \partial_{j} \partial_{j} ϕ \neq 0$ exists although it no longer appears explicitly in the XMHD equations. For small $σ ≪ 1$ ⁠, balance 6 (⁠ $α = ε_{m} σ$ ⁠) requests α to be further smaller than ε_m. The charge conservation law is indeed reduced to $\nabla \cdot J = 0$ in the limit $σ \to 0$ ⁠.

The electron-inertia effect is manifested by the terms with $ε_{I}^{2}$ ⁠, which is the second-order of ε_m. If we neglect only $ε_{I}^{2}$ ⁠, the Hall MHD (HMHD) equations are reproduced. If we neglect ε_H too or simply take the limit of $ε_{m} \to 0$ ⁠, the MHD equations are finally obtained.

D. Relativistic Hall MHD model

Now, we are positioned to search for the other MHD models, which include both the relativistic and two-fluid effects. To derive a reduced model, we still need to assume the smallness of ε_m but should not neglect it completely. An approximation that comes to mind immediately is to neglect $O (ε_{m}^{2})$ ⁠, namely, the electron-inertia effect $ε_{I}^{2}$ only. The resultant equations deserve to be called relativistic Hall MHD (RHMHD). It is remarkable that f in Eq. (73) includes no additional term due to the Hall effect, $O (ε_{H})$ or $O (ε_{m})$ ⁠. By neglecting $ε_{I}^{2}$ ⁠, the proper charge neutrality $f = γ / n$ still holds approximately and $\tilde{f}$ vanishes.

Therefore, in RHMHD, Eqs. and are reduced to

\begin{matrix} \partial_{0} (γ n v^{i}) + \partial_{j} (γ n v^{i} v^{j}) \\ - ε_{σ} \tilde{n} (- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}) - n u^{j} F^{i}_{j} = 0, \end{matrix}

(82)

\begin{array}{c} - (n - ε_{H} ε_{σ} \tilde{n}) (- \frac{ε_{σ}}{σ} \partial_{i} ϕ - \partial_{0} A^{i}) \\ - n (v^{j} - ε_{H} u^{j}) F^{i}_{j} = 0. \end{array}

(83)

The terms including ε_H are the difference from RMHD.

If we further assume the smallness of $σ ≪ 1$ additionally, we can also neglect the term of $O (ε_{H} ε_{σ})$ in Ohm's law. Although it is just a minor reduction, let us call it weakly-relativistic Hall MHD (W-RHMHD).

E. Weakly-relativistic XMHD model

If one wants to allow for both the electron-inertia and relativistic effects, it is difficult to derive a reduced model from the two-fluid equations. One option is to neglect

O (ε_{I}^{2} σ)

by assuming the sufficient smallness of both ε_m and

σ (≪ 1)

⁠. Then, the proper charge neutrality holds again and we can derive similar equations to XMHD.

\begin{matrix} \partial_{0} (γ n v^{i}) + \partial_{j} (γ n v^{i} v^{j} + ε_{I}^{2} γ n u^{i} u^{j}) \\ - σ \tilde{n} (- \partial_{i} ϕ - \partial_{0} A^{i}) - n u^{j} F^{i}_{j} = 0, \end{matrix}

(84)

\begin{matrix} ε_{I}^{2} \partial_{0} (γ n u^{i}) + ε_{I}^{2} \partial_{j} [γ n (v^{i} u^{j} + v^{j} u^{i}) - ε_{H} γ n u^{i} u^{j}] \\ - (n - ε_{H} σ \tilde{n}) (- \partial_{i} ϕ - \partial_{0} A^{i}) - n (v^{j} - ε_{H} u^{j}) F^{i}_{j} = 0. \end{matrix}

(85)

We call this model weakly-relativistic XMHD because it is valid only when σ and

β_{⋆}^{2}

are sufficiently smaller than 1.

V. RANGE OF APPLICATION

Under balances 1–6, there remain three nondimensional parameters $(ε_{m}, σ, β_{⋆})$ ⁠, which are related to the three physical scales $(n_{⋆}, v_{⋆}, B_{⋆})$ of plasma (for fixed length scale $L_{⋆}$ ⁠). More rigorously speaking, $\tilde{μ}$ and μ are two additional parameters which appear only in the forms, $ε_{H} = \tilde{μ} ε_{m}$ and $ε_{I} = μ ε_{m}$ ⁠. Because of the inequalities $\tilde{μ} \leq 1$ and $μ \leq 1$ ⁠, they do not alter balances 1–6 but possibly make ε_H and ε_I further smaller than ε_m. As the two typical examples, we will consider electron-ion (hydrogen) plasma $m_{-} / m_{+} ≃ 10^{- 3}$ (for which $μ ≃ 10^{- 3 / 2} = 0.0316$ and $\tilde{μ} ≃ 1$ ⁠) and electron-positron plasma $m_{-} / m_{+} = 1$ (for which $μ = 0.5$ and $\tilde{μ} = 0$ ⁠).

Now, we carefully consider the magnitude of ε_m, which obviously measures the impact of the two-fluid effect as we have seen in Sec. IV. According to balances 5 and 3′, it depends on the other parameters as follows:

ε_{m} = \frac{β_{⋆}}{\sqrt{σ}} ε = \frac{v_{⋆}}{L_{⋆} ω_{c ⋆}} with \frac{β_{⋆}}{\sqrt{σ}} \leq 1.

(86)

Here, we obtain

ε = d_{⋆} / L_{⋆}

by newly introducing a representative inertial length (or skin depth)

d_{⋆} = c / ω_{p ⋆}

⁠. More specifically, the inertial lengths for positively (+) and negatively (–) charged gases are given by

d_{\pm} = \sqrt{\frac{m_{\pm}}{m_{+} + m_{-}}} d_{⋆} .

(87)

Indeed,

d_{+}

and

d_{-}

respectively correspond to ion's and electron's inertial lengths for electron-ion plasma.

In the case of non-relativistic limit (⁠

σ \to 0

and

β_{⋆} \to 0

⁠) with application of the Alfvén ordering

β_{⋆}^{2} / σ \to 1

(⁠

v_{⋆} = v_{A ⋆}

⁠), we simply obtain

ε_{m} = ε

⁠. Therefore,

ε_{H} ≃ d_{+} / L_{⋆} and ε_{I} ≃ d_{-} / L_{⋆} (≃ \sqrt{m_{-} / m_{+}} ε_{H}),

(88)

for electron-ion plasma, and

ε_{H} = 0 and ε_{I} = \sqrt{2} d_{+} / L_{⋆} = \sqrt{2} d_{-} / L_{⋆},

(89)

for electron-positron plasma. In this way, the Hall and electron-inertia effects are associated with the small-scales

d_{+}

and

d_{-}

⁠, where the density

n_{⋆}

is important because

d_{⋆}

is proportional to

1 / \sqrt{n_{⋆}}

only. For sufficiently dense plasma

n_{⋆} \to \infty

⁠, the two-fluid effect becomes negligible

ε \to 0

⁠. However, this is a consequence of applying the Alfvén ordering. Namely,

v_{⋆}

and

B_{⋆}

are not fixed independently but varied along with

n_{⋆}

⁠.

In general, it is interesting to note that ε_m does not originally depend on the density $n_{⋆}$ but on the ratio $v_{⋆} / B_{⋆}$ in Eq. (86). When σ gets larger than 1, the two-fluid effect ε_m gets smaller than ε by the factor $β_{⋆} / \sqrt{σ}$ ⁠. This tendency agrees with Kawazura²⁴ in which the ion skin depth is modified to shrink as the magnetic field strength increases relativistically. In the limit of $B_{⋆} \to \infty$ with fixed $n_{⋆}$ and $v_{⋆}$ ⁠, we find that the two-fluid effect becomes negligible and the use of RMHD is justified (although it tends to be almost vacuum plasma, $β_{⋆}^{2} / σ \to 0$ ⁠).

In the case of electron-positron plasma, the Hall effect vanishes identically, $ε_{H} = 0$ ⁠. Then, the XMHD model includes only the electron-inertia effect, which is especially called Inertial MHD (IMHD). Similarly, we can obtain weakly relativistic IMHD from weakly relativistic XMHD when $ε_{H} = 0$ ⁠.

All the models which we have been presented so far are summarized in Table I. All these reduced models need to neglect the order of

ε_{σ} ε_{I}^{2} = \frac{m_{+} m_{-}}{m^{2}} β_{⋆}^{2} \frac{ε_{σ}}{σ} ε^{2},

(90)

in common, which is necessary for

f = γ / n

to hold approximately and to get rid of

\tilde{f}

⁠. Then, either proper charge neutrality or quasi-neutrality holds. In other words, we have to solve the two-fluid equations directly if this

ε_{σ} ε_{I}^{2}

is not sufficiently smaller than 1.

TABLE I.

Classification of models (H = Hall, I = Inertial, X = eXtended, R = Relativistic, W-R = Weakly-Relativistic).

Model	Included order	Neglected order
MHD		$O (σ), O (ε_{H}), O (ε_{I}^{2})$
HMHD	$O (ε_{H})$	$O (σ), O (ε_{I}^{2})$
IMHD	$O (ε_{I}^{2})$	$O (σ), O (ε_{H})$
XMHD	$O (ε_{H}), O (ε_{I}^{2})$ ⁠,	$O (σ)$
RMHD	$O (σ)$	$O (ε_{H}), O (ε_{I}^{2})$
RHMHD	$O (σ), O (ε_{H})$	$O (ε_{I}^{2})$
W-RHMHD	$O (σ), O (ε_{H})$	$O (ε_{I}^{2}), O (ε_{H} σ)$
W-RIMHD	$O (σ), O (ε_{I}^{2})$	$O (ε_{H}), O (ε_{I}^{2} σ)$
W-RXMHD	$O (σ), O (ε_{H}), O (ε_{I}^{2})$	$O (ε_{I}^{2} σ)$

Since it is still difficult to imagine the applicable scope of each model, let us assume $10^{- 4}$ as a clear threshold for example. Namely, the non-dimensional parameters (such as σ and ε_H) can be considered negligible if they are below $10^{- 4}$ ⁠. Otherwise, they are not neglected. Then, the lines such as $σ = 10^{- 4}$ and $ε_{H} = 10^{- 4}$ divide the parameter space $(ε_{m}, σ, β_{⋆})$ into subspaces, in which a certain group of MHD models is applicable. Recall that $β_{⋆}$ should satisfy $β_{⋆}^{2} \leq σ$ and it appears only in Ampère-Maxwell's equation (69). In Table I, the models are classified in terms of σ and ε_m (regardless of $β_{⋆}$ ⁠), which are illustrated in Fig. 2 for electron-ion plasma and in Fig. 3 for electron-positron plasmas. Due to the smallness of mass ratio $m_{-} / m_{+} ≃ 10^{- 3}$ ⁠, the electron-inertia effect is readily neglected (⁠ $ε_{I}^{2} < 10^{- 4}$ ⁠) in the majority of cases for electron-ion plasma. However, we have to keep in mind that electron inertia may be important locally at singular point or layer, where the small-scale structure $L_{⋆} \sim d_{-}$ emerges (as in the location where magnetic reconnection occurs).

FIG. 2.

View large Download slide

Electron-ion plasma (σ vs ε_m).

FIG. 3.

View large Download slide

Electron-positron plasma (σ vs ε_m).

Although Figs. 2 and 3 look simple enough, let us further illustrate the application ranges in terms of

(n_{⋆}, v_{⋆}, B_{⋆})

⁠. Since the vacuum state

β_{⋆}^{2} / σ ≪ 1

is uninteresting, it is reasonable to fix

β_{⋆}

to the maximal value

β_{⋆} = \sqrt{ε_{σ}}

⁠. As shown in the plots of Fig. 1, there is in fact no order difference between

\sqrt{ε_{σ}}

and

v_{A ⋆} / c = \sqrt{σ / (1 + σ)}

in the scale analysis;

\sqrt{ε_{σ}} ≃ v_{A ⋆} / c

⁠. Therefore, we refer to

β_{⋆} = \sqrt{ε_{σ}} ≃ v_{A ⋆} / c

(91)

as relativistic Alfvén ordering. By imposing this relativistic Alfvén ordering on

β_{⋆}

⁠, we obtain

ε_{m} = ε \sqrt{ε_{σ} / σ}

and the two remaining parameters are chosen as

ε^{- 2} = \frac{μ_{0} n_{⋆} e^{2} L_{⋆}^{2}}{m} \propto n_{⋆},

(92)

\sqrt{σ} / ε = \frac{e B_{⋆} L_{⋆}}{m c} \propto B_{⋆},

(93)

representing the scales of

(n_{⋆}, B_{⋆})

directly. Therefore, we can remap Fig. 2 into Figs. 4 and 3 into Fig. 5 on the 2D plane

(ε^{- 2}, \sqrt{σ} / ε)

⁠. The region of

σ > 10^{4}

is filled in gray because it is considered vacuum (⁠

β_{⋆}^{2} / σ < 10^{- 4}

⁠). In the strong magnetic field limit

B_{⋆} \to \infty

⁠, we inevitably enter this vacuum regime but RMHD is still valid and no problem to keep using it. In the dense plasma limit

n_{⋆} \to \infty

⁠, we enter the conventional MHD regime. In this figure, the limit of large scale

L_{⋆} \to \infty

corresponds to the movement in the direction indicated by the fat arrow, which is parallel to the

σ =

const. line. In the triangle region indicated by “2-fluid”, the charge neutrality approximation

ε_{σ} ε_{I}^{2} < 10^{- 4}

is not satisfied. This region exists on the low density side

ε^{- 2} < 10^{4} μ^{2}

and for intermediate strength of magnetic field. In the weak magnetic field limit

B_{⋆} \to 0

⁠, we can apply the non-relativistic MHD models (such as XMHD, HMHD, IMHD, and MHD). But, when magnetic field is weak such that

\sqrt{σ} / ε < \sqrt{10^{4} μ^{2}}

holds and in the low density limit

n_{⋆} \to 0

⁠, we have to solve the two-fluid equations without assuming charge neutrality. The limit of small scale

L_{⋆} \to 0

also enters the “2-fluid” region eventually.

FIG. 4.

View large Download slide

Electron-ion plasma (⁠ $n_{⋆}$ vs $B_{⋆}$ vs $L_{⋆}$ ⁠).

FIG. 5.

View large Download slide

Electron-positron plasma (⁠ $n_{⋆}$ vs $B_{⋆}$ vs $L_{⋆}$ ⁠).

VI. REMARKS ON RHMHD

In comparison to RMHD, the RHMHD equations just have a few additional terms in Ohm's law due to the Hall effect. However, this difference is quite influential when solving these equations theoretically and numerically.

In general, when a dynamical system $\partial_{t} u = F (u)$ for u is solved numerically, the recurrence formula such as $u^{n + 1} = u^{n} + F (u^{n}) Δ t$ is iterated for time marching $t \to t + Δ t$ ⁠. To execute this iteration, the right hand side F(u) must be calculated uniquely using the dynamical variable u. This is a fundamental requirement for the well-posedness of the time-evolving system.

In 3D vector format, the RHMHD equations are composed of the evolution equations (that include time derivative

\partial_{t}

of some quantity),

\partial_{t} n = - \nabla \cdot (n v),

(94)

\partial_{t} (n γ v) = - \nabla \cdot (n γ vv) + ε_{σ} \tilde{n} E + J \times B,

(95)

β_{⋆}^{2} \partial_{t} E = \nabla \times B - \frac{β_{⋆}^{2}}{σ} J,

(96)

\partial_{t} B = - \nabla \times E,

(97)

and the constraints,

n (E + v \times B) - ε_{H} (ε_{σ} \tilde{n} E + J \times B) = 0,

(98)

\nabla \cdot E = \frac{ε_{σ}}{σ} \tilde{n},

(99)

\nabla \cdot B = 0.

(100)

A drastic change from the two-fluid equations is that the time derivative of the current J no longer exists in Eq. due to neglect of electron-inertia

ε_{I}^{2} \to 0

⁠. Therefore, to calculate the right hand sides of the evolution equations, we have to determine (or eliminate) J using the other variables.

From Ohm's law , the electric field of the component parallel to the magnetic field must be zero (⁠

E \cdot B = 0

⁠). By combining and , we obtain

\begin{matrix} \partial_{t} (E \times B) = \nabla \cdot (\frac{BB}{β_{⋆}^{2}} + EE) - \nabla (\frac{| B |^{2}}{2 β_{⋆}^{2}} + \frac{| E |^{2}}{2}) \\ - \frac{1}{σ} (ε_{σ} \tilde{n} E + J \times B), \end{matrix}

(101)

\partial_{t} (E \cdot B) = \frac{1}{β_{⋆}^{2}} B \cdot (\nabla \times B) - E \cdot (\nabla \times E) - \frac{J \cdot B}{σ} .

(102)

To preserve the constraint

E \cdot B = 0

⁠, the current parallel to the magnetic field (⁠

J \cdot B

⁠) is determined such that the right hand side of Eq. becomes always zero.

\frac{J \cdot B}{σ} = \frac{1}{β_{⋆}^{2}} B \cdot (\nabla \times B) - E \cdot (\nabla \times E) .

(103)

On the other hand, the evolution equation for

E \times B

can be solved instead of Eq. because the electric field has only the perpendicular component and can be reproduced by

E = B \times (E \times B) / | B |^{2}

⁠. Moreover, we can easily eliminate

ε_{σ} \tilde{n} E + J \times B

in Eqs. and by using Ohm's law . Then, the right hand sides no longer include J. Therefore, the cold RHMHD equations are regarded as a dynamical system of 10 fields

(n, M_{P}, M_{EM}, B)

⁠, where

M_{P} = n γ v

and

M_{EM} = E \times B

⁠, satisfying two constraints

M_{EM} \cdot B = 0

and

\nabla \cdot B = 0

⁠. The right hand sides of Eqs. , , , and are explicitly written in terms of

(n, M_{P}, M_{EM}, B)

⁠, using

v = \frac{M_{P} / n}{\sqrt{1 + β_{⋆}^{2} {(M_{P} / n)}^{2}}},

(104)

E = B \times M_{EM} / | B |^{2} .

(105)

In this way, the RHMHD equations are numerically solvable in the cold plasma limit.

We also note that the force-free state can be obtained in the high magnetization limit $σ \to \infty$ ⁠, where only Eqs. (97) and (101) are solved by neglecting the last term of Eq. (101). Although J is already eliminated from these equations, it must exist at orders $J \times B = O (1)$ and $J \cdot B = O (σ)$ ⁠. Our normalization $J = O (1)$ is therefore violated in the parallel component and needs to be modified to treat this model.

On the other hand, in the case of RMHD which neglects the Hall effect

ε_{H} = 0

⁠, Ohm's law becomes

E + v \times B = 0

that does not include J. Therefore, we are forced to eliminate J by combining Eqs. and as follows:

\begin{matrix} \partial_{t} (n γ v + σ E \times B) = - \nabla \cdot (n γ vv) + σ \nabla \cdot (\frac{BB}{β_{⋆}^{2}} + EE) \\ - σ \nabla (\frac{| B |^{2}}{2 β_{⋆}^{2}} + \frac{| E |^{2}}{2}), \end{matrix}

(106)

where

M_{tot} : = n γ v + σ E \times B

is the total momentum of plasma and electromagnetic field. According to Ohm's law, E is always replaced by

- v \times B

⁠. Thus, the RMHD equations are a dynamical system of seven fields

(n, M_{tot}, B)

with a constraint

\nabla \cdot B = 0

⁠. However, to calculate the right hand sides of Eqs. , , and , we need to write v in terms of

(n, M_{tot}, B)

⁠. It is well known that this is not analytically feasible and requires the use of a root-finding algorithm (such as the Newton–Raphson method), which is one of the most computationally expensive part of RMHD simulation.

For the case of hot plasma, the momentum equation is rewritten as

\partial_{t} (h γ^{2} v) = - \nabla p - \nabla \cdot (h γ^{2} v v) + ε_{σ} \tilde{n} E + J \times B,

(107)

including the enthalpy density h and pressure p. The evolution equation for the total energy

E_{tot}

should be solved additionally when the barotropic condition is not satisfied, and the primitive variables

(N, v, p, B)

should be reconstructed from the time-evolving variables. Since

M_{p} = h γ^{2} v

depends on

N = n / γ

even in the barotropic case h(N), Eq. no longer determines v explicitly. Unfortunately, a root-finding algorithm is again necessary for RHMHD.

Therefore, in the presence of the Hall effect and the cold plasma limit, we can avoid using root-finding algorithm and the time-marching algorithm becomes straightforward while the number of field variables increases from 7 to 10. The cold RHMHD equations are possibly solved at a lower cost than the RMHD equations.

Finally, in the presence of the electron-inertia effect $ε_{I}^{2} \neq 0$ ⁠, Ohm's law is regarded as the evolution equation for $γ J$ ⁠. The number of field variables is 13 under one constraint $\nabla \cdot B = 0$ ⁠, which is essentially the same as the original two-fluid equations. The numbers of field variables and constraints are summarized in Table II. As we have remarked before, it is more natural in XMHD (and IMHD) to solve A instead of B under the constraint $\nabla \cdot A = 0$ ⁠. Since cold plasma is assumed in this work for simplicity, one more field variable (such as pressure or temperature) would be added when temperature is not negligible.

TABLE II.

Number of field variables for cold plasma (H = Hall, I = Inertial, X = eXtended, R = Relativistic, W-R = Weakly-Relativistic).

Model	Field variables	Constraints
MHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
HMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
IMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
XMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
RMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
RHMHD	10 $(n, v, B, E)$	2 (⁠ $\nabla \cdot B = 0, E \cdot B = 0$ ⁠)
W-RHMHD	10 $(n, v, B, E)$	2 (⁠ $\nabla \cdot B = 0, E \cdot B = 0$ ⁠)
W-RIMHD	13 $(n, v, J, B, E)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
W-RXMHD	13 $(n, v, J, B, E)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)

Model	Field variables	Constraints
MHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
HMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
IMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
XMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
RMHD	7 $(n, v, B)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
RHMHD	10 $(n, v, B, E)$	2 (⁠ $\nabla \cdot B = 0, E \cdot B = 0$ ⁠)
W-RHMHD	10 $(n, v, B, E)$	2 (⁠ $\nabla \cdot B = 0, E \cdot B = 0$ ⁠)
W-RIMHD	13 $(n, v, J, B, E)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)
W-RXMHD	13 $(n, v, J, B, E)$	1 (⁠ $\nabla \cdot B = 0$ ⁠)

VII. CONCLUSION

In this paper, we have investigated the applicability of various MHD models to special relativistic plasmas, using the method of dominant balance in the two-fluid equations. To simplify the formulation and consideration, we have assumed cold plasma (the limit of zero temperature and pressure) because electromagnetic force, not pressure, is a dominant force in the MHD balance. Although there is no problem in including nondominant pressure effect, the case of relativistic pressure should be investigated as a future topic which might also breaks down the MHD balance (or the charge neutrality approximation). Similarly, externally-applied electric field is assumed to be absent because it is rarely dominant.

Under these assumptions, the relativistic two-fluid equations are nondimensionalized by eight representative scales, resulting in seven nondimensional parameters. For the electromagnetic force to be a dominant term, the six balances (1–6) are imposed as constraints among these parameters. Since balances 3 and 4 are inequalities, the number of the nondimensional parameters is reduced to three $(ε_{m}, σ, β_{⋆})$ satisfying an inequality $β_{⋆}^{2} / σ \leq 1$ ⁠. The parameter $ε_{m} = v_{⋆} / (L_{⋆} ω_{c ⋆})$ is smaller than 1 if we focus on the flow dynamics slower than the cyclotron frequency $ω_{c ⋆}$ ⁠. The RMHD equations are obtained in the limit $ε_{m} \to 0$ ⁠. By taking the mass ratio as an additional parameter, this parameter ε_m appears only through either $ε_{H} = \tilde{μ} ε_{m}$ or $ε_{I}^{2} = μ^{2} ε_{m}^{2}$ in the two-fluid equations. We have shown that the approximation of proper charge neutrality can be justified by neglecting the order of $ε_{σ} ε_{I}^{2}$ where $ε_{σ} = min (σ, 1)$ ⁠. When $σ ≪ 1$ ⁠, this approximation naturally corresponds to the quasi-neutrality condition of non-relativistic MHD. All the reduced models, or the generalized MHD models, are derived by neglecting $O (ε_{σ} ε_{I}^{2})$ while allowing for the Hall effect $O (ε_{H})$ ⁠, electron-inertia effect $O (ε_{I}^{2})$ and relativistic effects $O (β_{⋆}^{2})$ and $O (σ)$ ⁠. A special care is therefore needed when both the electron-inertia and relativistic effects are taken into account simultaneously, because their multiplication $O (ε_{σ} ε_{I}^{2})$ is not negligible unless both σ and $ε_{I}^{2}$ are much smaller than 1. Only for the weakly relativistic case $σ ≪ 1$ ⁠, we can use the W-RXMHD and W-RIMHD models, where proper charge neutrality is still valid. If $ε_{σ} ε_{I}^{2} ≪ 1$ is not fulfilled, the two-fluid equations should be solved without any approximation.

The case of $β_{⋆}^{2} / σ ≪ 1$ is often uninteresting because it is almost vacuum (i.e., the kinetic energy is much smaller than the energy of externally applied magnetic field). In this paper, we set aside the force-free state for simplicity, which allows for only the existence of finite parallel current even in the vacuum limit. On the other hand, the inequality $β_{⋆}^{2} / σ \leq 1$ indicates that the maximum velocity scale should be $β_{⋆} = \sqrt{ε_{σ}}$ which is understood as relativistic Alfvén ordering (⁠ $v_{⋆} ≃ v_{A ⋆}$ ⁠). Interesting MHD phenomena are expected in this velocity scale. By focusing on this velocity scale, the number of the nondimensional parameters is further reduced to two $(ε, σ)$ which are related to the scales of number density $n_{⋆}$ ⁠, magnetic field $B_{⋆}$ and length $L_{⋆}$ ⁠. We have illustrated the applicable ranges of the various MHD models in terms of these scales. For a low density case or in a small scale, it is shown that the charge neutrality condition is violated at an intermediate strength of magnetic field around $σ \sim 1$ ⁠.

We have also summarized the number of field variables for each generalized MHD model. The RHMHD model is shown to be a dynamical system of 10 fields $(n, v, B, E)$ satisfying two constraints (⁠ $\nabla \cdot B = 0$ and $E \cdot B = 0$ ⁠). This number 10 is different from seven of the other non-relativistic MHD models and 13 of the original two-fluid model. Moreover, the cold RHMHD equations describe the time marching of variables $(n, n γ v, E \times B, B)$ and the primitive variables $(n, v, B, E)$ can be written explicitly by them. Since the cold RMHD equations requires the root-finding algorithm to calculate $(n, v, B)$ from $(n, n γ v + σ E \times B, B)$ ⁠, the cold RHMHD model has an advantage in that the time-marching algorithm is simpler and less expensive than RMHD at the expense of increasing the field variables from 7 to 10. Naturally, RHMHD is a higher fidelity model than RMHD since it includes the Hall effect, which is known to be important for magnetic reconnection process in electron-ion plasma. The application of RHMHD is therefore expected to be beneficial both theoretically and numerically for analyzing relativistic plasma phenomena.

ACKNOWLEDGMENTS

We thank Y. Kawazura and K. Toma for helpful discussion. This work was supported by JST SPRING (Grant No. JPMJSP2114), a Scholarship of Tohoku University, Division for Interdisciplinary Advanced Research and Education, and IFS Graduate Student Overseas Presentation Award.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Shuntaro Yoshino: Conceptualization (equal); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (equal); Validation (supporting); Visualization (equal); Writing – original draft (lead); Writing – review & editing (supporting). Makoto Hirota: Conceptualization (equal); Formal analysis (supporting); Investigation (supporting); Methodology (equal); Project administration (lead); Supervision (lead); Validation (lead); Writing – review & editing (lead). Yuji Hattori: Project administration (supporting); Supervision (supporting); Writing – review & editing (supporting).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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This work is licensed under a Creative Commons Attribution 4.0 International License .

Applicability criteria of proper charge neutrality and special relativistic MHD models extended by two-fluid effects

I. INTRODUCTION

II. BASIC EQUATIONS

A. Two-fluid equations

B. Transformation into MHD variables

C. Assumption of cold plasma

III. DOMINANT BALANCE

A. Normalization

B. Imposition of MHD balance

IV. REDUCTION TO VARIOUS MHD MODELS

A. Vacuum limit $β_{⋆}^{2} / σ \to 0$

B. Single-fluid limit $ε_{m} \to 0$

C. Nonrelativistic limit $σ \to 0$

D. Relativistic Hall MHD model

E. Weakly-relativistic XMHD model

V. RANGE OF APPLICATION

VI. REMARKS ON RHMHD

VII. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Applicability criteria of proper charge neutrality and special relativistic MHD models extended by two-fluid effects

I. INTRODUCTION

II. BASIC EQUATIONS

A. Two-fluid equations

B. Transformation into MHD variables

C. Assumption of cold plasma

III. DOMINANT BALANCE

A. Normalization

B. Imposition of MHD balance

IV. REDUCTION TO VARIOUS MHD MODELS

A. Vacuum limit β⋆2/σ→0

B. Single-fluid limit εm→0

C. Nonrelativistic limit σ→0

D. Relativistic Hall MHD model

E. Weakly-relativistic XMHD model

V. RANGE OF APPLICATION

VI. REMARKS ON RHMHD

VII. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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A. Vacuum limit $β_{⋆}^{2} / σ \to 0$

B. Single-fluid limit $ε_{m} \to 0$

C. Nonrelativistic limit $σ \to 0$