2.1 Objectives

In this section, the purpose is to unravel a series of asymptotic limits bridging the gap between the Vlasov–Maxwell system and a MHD model. The difficulty is therefore to identify parameters explaining the transition from one description to the other and to relate these parameters to specific characteristics of the system. The tools mobilized to achieve this aim are based on the asymptotic analysis of the Vlasov–Maxwell system. Since the low-frequency plasma modelling is related to a fluid plasma description, the kinetic model is upgraded with collision operators. Therefore, the most refined modelling consists of a Vlasov equation for the electrons and the ions, augmented with a collision operator and coupled to the Maxwell system. Even if the physical model is non-collisional or weakly collisional, the transition towards a fluid limit is accounted for by a collisional process, thanks to a Bhatnager–Gross–Krook (BGK) operator. This choice of collision operator is questionable from a strict modelling viewpoint; nonetheless, the purpose here is to easily derive the fluid limit at a limited computational cost. In this respect the BGK collision operator is a good candidate. First, the whole collisional processes are considered, including both intra- and inter-species collisions. Nonetheless, only the minimal collisional process will be accounted for to derive a MHD regime from the kinetic model. This point will be outlined in the following sections. The introduction of non-dimensional quantities will naturally reveal dimensionless parameters in the equations. Letting some of these parameters go to zero shapes the hierarchy of models derived for the Vlasov–Maxwell system and bridging the gap with MHD models.

2.2 The Vlasov–BGK–Maxwell system

The most refined description of the plasma is constituted by two Vlasov equations, $f_\textrm {i}$ and $f_\textrm {e}$ being the ion and electron distribution functions:

(2.1)\begin{gather} \partial_t\, f_\textrm{i} +v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{i} + \frac{q}{m_\textrm{i}} ( E + v \times B) \boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} = \mathcal{Q}_\textrm{i}, \end{gather}

(2.2)\begin{gather}\partial_t\, f_\textrm{e} +v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{e} - \frac{q}{m_\textrm{e}} ( E + v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} = \mathcal{Q}_\textrm{e}. \end{gather}

In these equations, $q$ is the elementary charge and $m_\alpha$ is the mass of the species $\alpha$ ($\alpha =\textrm {e}$ for the electrons and $\textrm {i}$ for the ions). The BGK collision operator $\mathcal {Q}_\alpha$ is given by (Huba Reference Huba2011)

(2.3)\begin{equation} \left.\begin{gathered} \mathcal{Q}_\alpha = \mathcal{Q}_{\alpha\alpha}+ \mathcal{Q}_{\alpha\beta},\\ \mathcal{Q}_{\alpha\alpha} = {\nu_{\alpha\alpha}} ( \mathcal{M}_{n_\alpha,u_\alpha,T_\alpha} - f_\alpha ),\quad \mathcal{Q}_{\alpha\beta}={\nu_{\alpha\beta}} ( \bar{\mathcal{M}}_{n_\alpha,\bar{u}_\beta,\bar{T}_\beta} - f_\alpha ). \end{gathered}\right\} \end{equation}

Here $\nu _{\alpha \alpha }$ and $\nu _{\alpha \beta }$ are the like-particle and inter-species collision frequencies which can be defined as (Degond Reference Degond2007; Spatschek Reference Spatschek2012)

(2.4a)\begin{gather} \nu_{\textrm{ii}} = K_0 \frac{n_\textrm{i}}{(k_\textrm{B} T_\textrm{i})^{{3}/{2}}}\frac{\sqrt{2}}{\sqrt{m_\textrm{i}}},\quad \nu_{\textrm{ie}} = K_0 \frac{n_\textrm{e}}{(k_\textrm{B} T_\textrm{i})^{3/2}}\frac{\sqrt{m_\textrm{e}}}{m_\textrm{i}}, \end{gather}

(2.4b)\begin{gather}\nu_{\textrm{ee}} = K_0 \frac{n_\textrm{e}}{(k_\textrm{B} T_\textrm{e})^{{3}/{2}}}\frac{\sqrt{2}}{\sqrt{m_\textrm{e}}},\quad \nu_{\textrm{ei}} = K_0 \frac{n_\textrm{i}}{(k_\textrm{B} T_\textrm{e})^{3/2}}\frac{1}{\sqrt{m_\textrm{e}}}, \end{gather}

where

(2.4c)\begin{equation} K_0 = C \left(\frac{q^2}{4 {\rm \pi} \epsilon_0}\right)^2 \ln(\varLambda), \end{equation}

$C$ denoting a constant with a magnitude equal to one, $\ln (\varLambda )$ the Coulomb logarithm and the ions being assumed mono-charged.

The Maxwellians $\mathcal {M}_{n_\alpha ,u_\alpha ,T_\alpha }$ and $\bar {\mathcal {M}}_{n_\alpha ,\bar {u}_\beta ,\bar {T}_\beta }$ are defined as

(2.5a)\begin{gather} \mathcal{M}_{n_\alpha,u_\alpha,T_\alpha} = n_\alpha(x,t) \left(\frac{ m_\alpha}{2{\rm \pi} k_\textrm{B} T_\alpha(x,t)}\right)^{{D_v}/{2}} \exp\left(-\frac{m_\alpha|u_\alpha(x,t)-v |^2}{2 k_\textrm{B} T_\alpha(x,t)} \right), \end{gather}

(2.5b)\begin{gather}\bar{\mathcal{M}}_{n_\alpha,\bar{u}_\beta,\bar{T}_\beta} = n_\alpha(x,t) \left(\frac{ m_\alpha}{2{\rm \pi} k_\textrm{B} \bar{T}_\beta(x,t)}\right)^{{D_v}/{2}} \exp\left(-\frac{m_\alpha|\bar{u}_\beta(x,t)-v |^2}{2 k_\textrm{B} \bar{T}_\beta(x,t)} \right), \end{gather}

$D_v$ denoting the dimension of the velocity space and $k_\textrm {B}$ is the Boltzmann constant. The Maxwellian parameters $n_\alpha$, $u_\alpha$ and $T_\alpha$ are the density, mean velocity and temperature associated with the distribution function $f_\alpha$ and defined as

(2.6a–c)\begin{align} n_\alpha = \int_{\varOmega_v} f_\alpha\,\textrm{d} v,\quad n_\alpha u_\alpha= \int_{\varOmega_v} v\, f_\alpha \,\textrm{d} v,\quad \frac{1}{\gamma-1}n_\alpha k_\textrm{B} T_\alpha = \frac{m_\alpha}{2}\int_{\varOmega_v} |v-u|^2 f_\alpha \,\textrm{d} v, \end{align}

with $\gamma$ the specific heat ratio whose value depends on the dimensionality of the velocity space $D_v$ through

(2.7)\begin{equation} \gamma - 1 = \frac{2}{D_v}. \end{equation}

The collision operators verify the following conservation properties:

(2.8a)\begin{gather} \int \mathcal{Q}_{\alpha\alpha} m_\alpha \left(\begin{matrix} 1 \\ v \\ |v|^2 \end{matrix}\right)\textrm{d} v = 0, \end{gather}

(2.8b)\begin{gather}\int \mathcal{Q}_{\alpha\beta} m_\alpha \left(\begin{matrix} 1 \\ v \\ |v|^2 \end{matrix}\right) \textrm{d} v + \int \mathcal{Q}_{\beta\alpha} m_\beta \left(\begin{matrix} 1 \\ v \\ |v|^2 \end{matrix}\right) \textrm{d} v = 0. \end{gather}

The temperature and the mean velocity $(\bar {u}_\beta ,\bar {T}_\beta )$ in the inter-species collision operator expression (2.5b) should be chosen with care in order to guarantee the total momentum and energy conservation. Indeed the identities

(2.9a)\begin{gather} \int \mathcal{Q}_{\alpha\beta} m_\alpha v \,\textrm{d} v = \nu_{\alpha\beta} m_\alpha n_\alpha ( \bar{u}_\beta - u_\alpha), \end{gather}

(2.9b)\begin{gather}\int \mathcal{Q}_{\alpha\beta} m_\alpha \frac{|v|^2}{2} \textrm{d} v = \nu_{\alpha\beta} \left( \frac{D_v}{2} n_\alpha k_\textrm{B} ( \bar{T}_\beta - T_\alpha ) + \frac{1}{2}m_\alpha n_\alpha (|\bar{u}_\beta|^2- |u_\alpha|^2) \right) \end{gather}

hold true for the operators defined by (2.5). The trivial choice $(\bar {u}_\beta ,\bar {T}_\beta )=({u}_\beta ,{T}_\beta )$ does ensure the plasma total momentum conservation, provided that $\nu _{\textrm {ei}} m_\textrm {e} n_\textrm {e} = \nu _{\textrm {ie}} m_\textrm {i} n_\textrm {i}$. However, in this case, the plasma total energy is not conserved. We refer to Greene (Reference Greene1973) for a seminal work, as well as to Klingenberg et al. (Reference Klingenberg, Pirner and Puppo2016) and references therein for recent advances, on the choice of these parameters compliant with the desired properties (2.8b) of the inter-species collision operators, therefore providing the conservation of the plasma momentum and energy.

The electromagnetic field $(E,B)$ evolution is driven by the Maxwell system:

(2.10)\begin{gather} \frac{1}{c^2}\partial_t E - \boldsymbol{\nabla}_x \times B= - \mu_0 J, \end{gather}

(2.11)\begin{gather}\partial_t B+ \boldsymbol{\nabla}_x \times E = 0, \end{gather}

(2.12)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} E= \frac{\rho}{\epsilon_0}, \end{gather}

(2.13)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0, \end{gather}

where $c$ is the speed of light, $\mu _0$ the vacuum permeability and $\epsilon _0$ the vacuum permittivity verifying $\mu _0\epsilon _0 c^2 = 1$. The Maxwell sources are the particle currents and densities

(2.14a)\begin{gather} \rho = q (n_\textrm{i} - n_\textrm{e}), \end{gather}

(2.14b)\begin{gather}J = q (n_\textrm{i} u_\textrm{i} - n_\textrm{e} u_\textrm{e}). \end{gather}

The definition of the collision frequencies as stated by (2.4) relates to different time scales. Indeed, because of their different masses, ions and electrons are not equally affected by collisions. These properties are more clearly emphasized working with dimensionless variables as proposed in the next section.

2.3 Scaling of the Vlasov–Maxwell system

The equations are written with dimensionless quantities in order to easily identify different regimes. The scaling is introduced under a priori assumptions that the electronic and ionic temperatures, densities and mean velocities are comparable with a magnitude denoted $T_0$, $n_0$ and $u_0$. These scales define the typical Debye length as well as the electron plasma period:

(2.15a,b)\begin{equation} \lambda_\textrm{D}=\sqrt{\frac{\epsilon_0 k_\textrm{B} T_0}{q^2 n_0}},\quad \tau_{\textrm{pe}}=\sqrt{\frac{m_\textrm{e} \epsilon_0}{q^2 n_0}}. \end{equation}

We denote by $x_0$ and $t_0$ the characteristic space and time scales of the phenomena observed, which yields the velocity of interest $\vartheta _0 = x_0/t_0$. The magnitude of the thermal velocity for the species $\alpha$ is denoted $v_{0,\alpha }$ with $v_{0,\alpha }=k_\textrm {B} T_0/m_\alpha$. Due to the different masses, the thermal velocity of the electrons is not that of the ions. The reference thermal velocity $v_0$ will be defined by the ion one $v_{0}^2=k_\textrm {B} T_0/m_\textrm {i}$, hence $v_{0,\textrm {e}}=v_0/\varepsilon$ and $v_{0,\textrm {i}}=v_0$, where $\varepsilon ^2=m_\textrm {e}/m_\textrm {i}$. Finally, the scale of the electromagnetic field is denoted $(E_0,B_0)$, the particle current scale being defined as $J_0 = q n_0 u_0$. The dimensionless variables are defined according to

(2.16a–f)\begin{gather} x^* = \frac{x}{x_0},\quad t^* = \frac{t}{t_0},\quad v^* = \frac{v}{v_{0,\alpha}},\quad f^* = \frac{f}{n_0/(v_{0,\alpha})^{D_v}},\quad n^* = \frac{n}{n_0},\quad J^* = \frac{J}{q n_0 u_0}, \end{gather}

(2.17a,b)\begin{gather}E^* = \frac{E}{E_0},\quad B^* = \frac{B}{B_0}, \end{gather}

the collision frequencies verifying

(2.18a–c)\begin{equation} \nu_{\textrm{ee},0} = \nu_{\textrm{ei},0} = \frac{1}{\varepsilon} \nu_{\textrm{ii},0},\quad \nu_{\textrm{ie},0} = \varepsilon \nu_{\textrm{ii},0},\quad \varepsilon = \sqrt{\frac{m_\textrm{e}}{m_\textrm{i}}}. \end{equation}

On the fastest time scales, the electron distribution function relaxes towards a Maxwellian. On the same time scale, the electron mean velocity and temperature relax towards those of the ions. The relaxation of the ionic distribution function towards the local equilibrium is slower, by a factor $\varepsilon ^{-1}=\sqrt {m_\textrm {i}/m_\textrm {e}}$. Finally, the ions are almost unaffected by the collisions with the electrons. The relaxation of the ionic distribution function towards that of the electrons defines the largest time scale, by a factor $\varepsilon ^{-1}$ compared to the relaxation towards the thermodynamical equilibrium.

The dimensionless ionic and electronic Vlasov equations can be rewritten as (keeping the same notations for dimensionless variables)

(2.19)\begin{align} & \xi \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} + \eta \left(E+ \frac{\beta}{\xi} v \times B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} =\frac{ \xi}{\kappa} ( \nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}) + \varepsilon \nu_{\textrm{ie}}( \bar{\mathcal{M}}_{n_\textrm{i},u_\textrm{e},T_\textrm{e}} - f_\textrm{i} )), \end{align}

(2.20)\begin{align} & \xi \varepsilon \partial_t\, f_\textrm{e} \!+\! v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} - \eta\left(E+\frac{\beta}{\varepsilon \xi} v \!\times\! B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} = \frac{\xi}{\kappa} ( \nu_{\textrm{ee}} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}}-f_\textrm{e}) \!+\! \nu_{\textrm{ei}} (\bar{\mathcal{M}}_{n_\textrm{e},u_\textrm{i},T_\textrm{i}} - f_\textrm{e} )), \end{align}

together with the dimensionless Maxwell system written

(2.21a)\begin{gather} \lambda^2 \eta \left(\alpha^2\frac{\partial {E}}{\partial t} - \beta \boldsymbol{\nabla}_x \times B\right)=-\alpha^2 \frac{M}{\xi} J, \end{gather}

(2.21b)\begin{gather}\beta \partial_t B + \boldsymbol{\nabla}_x \times E = 0, \end{gather}

(2.21c)\begin{gather}\lambda^2 \eta \boldsymbol{\nabla}_x \boldsymbol{\cdot} E= n_\textrm{i} - n_\textrm{e}, \end{gather}

(2.21d)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0, \end{gather}

(2.21e)\begin{gather}J = n_\textrm{i} u_\textrm{i} - n_\textrm{e} u_\textrm{e}. \end{gather}

This system is written using the following dimensionless parameters:

(2.22)\begin{align} \left.\begin{array}{l@{}} \displaystyle \varepsilon^2=\dfrac{m_\textrm{e}}{m_\textrm{i}} \text{ the ratio of the electronic and ionic masses},\\ \displaystyle \lambda= \dfrac{\lambda_\textrm{D}}{x_0} \text{ the scaled Debye length},\\ \displaystyle M = \dfrac{u_0}{v_{0}} \text{ the ionic Mach number, with } v_{0} = \sqrt{\dfrac{k_\textrm{B}T_0}{m_\textrm{i}}} \text{ the ionic speed of sound},\\ \displaystyle \xi=\dfrac{\vartheta_0}{v_0} \text{ the ratio of the typical velocity to the ionic speed of sound}, \\ \displaystyle \alpha=\dfrac{\vartheta_0}{c} \text{ the ratio of the typical velocity to the speed of light}, \\ \displaystyle \eta=\dfrac{q x_0 E_0}{k_\textrm{B} T_0} \text{ the ratio of the electric and plasma internal energies}, \\ \displaystyle \beta=\dfrac{\vartheta_0B_0}{E_0} \text{ the induced electric field relative to the total electric field},\\ \kappa^{-1} = \nu_{\textrm{ii},0} t_0 \text{ the number of ion--ion collisions during the typical time}.\end{array}\right\}\end{align}

The dimensionless Maxwellians are defined by

(2.23a)\begin{align} \mathcal{M}_{n_e,u_\alpha,T_\alpha} &= \bar{\mathcal{M}}_{n_e,u_\alpha,T_\alpha} \nonumber\\ &= n_\textrm{e}(x,t) \left(\frac{1}{2{\rm \pi} T_\alpha(x,t)}\right)^{{D_v}/{2}} \exp\left(-\frac{|M\varepsilon u_\alpha(x,t)-v |^2}{2T_\alpha(x,t)}\right), \end{align}

(2.23b)\begin{align} \mathcal{M}_{n_i,u_\alpha,T_\alpha} &= \bar{\mathcal{M}}_{n_i,u_\alpha,T_\alpha} \nonumber\\ &= n_\textrm{i}(x,t) \left(\frac{1}{2{\rm \pi} T_\alpha(x,t)}\right)^{{D_v}/{2}} \exp\left(-\frac{|M u_\alpha(x,t)-v |^2}{2T_\alpha(x,t)} \right). \end{align}

Some comments can be stated regarding the meaning of these parameters and the scaling relations.

The typical mean velocity and temperature are assumed to be the same for electrons and ions. Accordingly, the relaxation of the electron mean velocity and temperature towards those of the ions may be assumed to marginally contribute to the evolution of the system. This assumption is therefore consistent with the investigation of resistive-less plasma modellings and the neglect of the inter-species collisions.

The parameter $\xi$ is intended to provide a measure of how the electronic and ionic dynamics are resolved. The choice $\xi =1$ means that the system is assumed to evolve at a speed comparable to the ionic thermal velocity $v_0$, while $\xi \varepsilon =1$ performs a rescaling of this typical velocity to the electron microscopic velocity. Setting $\xi =M$ relates the typical speed of the system to the ionic mean velocity $u_0$. Actually, the Mach number measures the gap between the microscopic (thermal) and macroscopic velocity scales.

The scaling relation $\eta =1$ is generally assumed in single-fluid plasma representation. The plasma internal energy is then on a par with the electric energy. This equilibrium is fundamental in the derivation of the Boltzmann relation. The identity $\beta M = \xi$ is also common in single-fluid plasma models. This amounts to assuming that the induced electric field scales as the product of the plasma mean velocity and the typical magnetic field: $E_0= u_0 B_0$. In other words, the magnetic field is essentially transported with the plasma flow. This latter assumption is in line with the Alfvén frozen theorem (Moreau Reference Moreau1990; Davidson Reference Davidson2001; Schnack Reference Schnack2009; Freidberg Reference Freidberg2014) characteristic of ideal MHD models: the magnetic field is frozen into the plasma and transported by its flow.

The derivation of reduced models consists of identifying small dimensionless parameters and letting them go to zero. The smallness of the scaled Debye length refers to a typical space scale much larger than the physical Debye length. This means that the charge separations, occurring on space scales comparable to the Debye length, are assumed unimportant in explaining the evolution of the system. Sending the scaled Debye length to zero performs a low-frequency filtering into the equation deriving thus a quasi-neutral model. In the context of the derivation of numerical methods, the typical length relates to the mesh size. This outlines the advantage of reduced models: the low-frequency filtering operated by vanishingly small parameters permits the derivation of numerical methods with discretization parameters (mesh size and time step) unconstrained by the small scales filtered out from the original equations.

3.1 Handling the fluid and quasi-neutral limits

3.1.1 A hierarchy of fluid and kinetic quasi-neutral models

The aim here is to reduce the number of free dimensionless parameters, deriving by this means different reduced models well suited for the description of low-frequency phenomena. As depicted in figure 1, the starting point of this hierarchy of models implements the minimal upgrades of the Vlasov–Maxwell system to recover a MHD regime. Precisely, only the inter-species collisions are taken into account in the initial model in order for the distribution function to relax towards the local equilibrium. This yields

(3.1a)\begin{gather} \xi \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} + \eta \left(E+ \frac{\beta}{\xi} v \times B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} =\frac{ \xi}{\kappa}\nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}), \end{gather}

(3.1b)\begin{gather}\xi \varepsilon \partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} - \eta\left(E+ \frac{\beta}{\varepsilon \xi} v \times B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} =\frac{ \xi}{\kappa} \nu_{\textrm{ee}} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}} - f_\textrm{e}), \end{gather}

for the evolution of the ions and electrons coupled to the dimensionless Maxwell system defined by (2.21).

Figure 1.

Fluid and kinetic (quasi-neutral) model hierarchies derived from the Vlasov–BGK–Maxwell system.

From the scaling relations stated by (2.18a–c), discarding the inter-species collisions makes sense for the ions. Due to their large mass, the ions are almost unaffected by encounters with electrons. For the electrons, this assumption is not in line with the scaling of the intra- and inter-species collision frequencies. However, the purpose here is to propose a physically meaningful framework to clarify the foundation of a numerical method bridging the gap between a kinetic description of a weakly (or non-) collisional magnetized plasma with a MHD regime. The interspecies collisions give rise to the resistivity in the macroscopic system which is not the targeted class of modelling for this work.

3.1.2 Handling the fluid limit

To identify easily a fluid regime, the distribution function is decomposed into a Maxwellian $\mathcal {M}_{n_\alpha ,u_\alpha ,T_\alpha }$ and a deviation from this Maxwellian ${\kappa } g_\alpha$ according to

(3.2a)\begin{equation} f_\alpha = \mathcal{M}_{n_\alpha,u_\alpha,T_\alpha} + {\kappa} g_\alpha, \end{equation}

the deviation verifying

(3.2b)\begin{equation} \left\langle\left(\begin{matrix} 1 \\ v \\ |v|^2 \end{matrix}\right) g_\alpha \right\rangle = \int_{\varOmega_v} \left(\begin{matrix} 1 \\ v \\ |v|^2 \end{matrix}\right) g_\alpha \,\textrm{d} v =0. \end{equation}

With this decomposition, the Vlasov–Boltzmann equations (3.1) can be recast into a hydrodynamic set of equations with kinetic corrections, depending on the moment of the deviation $g_\alpha$, yielding

(3.3a)\begin{align} &\hspace{7.5pc}\frac{\xi}{M} \partial_t n_\textrm{i} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{i} u_\textrm{i}) = 0, \end{align}

(3.3b)\begin{align} & M^2\left(\frac{\xi}{M} \partial_t (n_\textrm{i} u_\textrm{i}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{i}u_\textrm{i} \otimes u_\textrm{i}) \right)+ \boldsymbol{\nabla}_x p_\textrm{i} - \eta n_\textrm{i} \left(E + \frac{\beta M}{\xi} u_\textrm{i} \times B \right) \nonumber\\ &\quad = - {\kappa}\boldsymbol{\nabla}_x \boldsymbol{\cdot} \left\langle v \otimes v g_\textrm{i} \right\rangle, \end{align}

(3.3c)\begin{align} &\hspace{-0.5pc}\frac{\xi}{M} \partial_t W_\textrm{i} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{i} + p_\textrm{i})u_\textrm{i} ) - \eta n_\textrm{i} E\cdot u_\textrm{i} = -\frac{\kappa}{M} \boldsymbol{\nabla}_x \boldsymbol{\cdot} \left\langle \frac{|v|^2}{2} v g_\textrm{i}\right\rangle, \end{align}

with

(3.3d)\begin{equation} W_\textrm{i} = (M)^2 \frac{1}{2} n_\textrm{i} {|u_\textrm{i}|}^2 + \frac{p_\textrm{i}}{\gamma-1},\quad p_\textrm{i} = n_\textrm{i} T_\textrm{i}, \end{equation}

for the ions, and an equivalent system for the electrons,

(3.4a)\begin{align} &\hspace{7.5pc}\frac{\xi}{M} \partial_t n_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e}) = 0, \end{align}

(3.4b)\begin{align} & \left(M\varepsilon\right)^2\left(\frac{\xi}{M} \partial_t (n_\textrm{e} u_\textrm{e}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e}u_\textrm{e} \otimes u_\textrm{e}) \right)+ \boldsymbol{\nabla}_x p_\textrm{e} + \eta n_\textrm{e} \left(E + \frac{\beta M}{\xi}u_\textrm{e} \times B\right) \nonumber\\ &\quad = -{\kappa}\boldsymbol{\nabla}_x \boldsymbol{\cdot} \left\langle v \otimes v g_\textrm{e} \right\rangle, \end{align}

(3.4c)\begin{align} &\hspace{0.5pc}\frac{\xi}{M} \partial_t W_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{e} + p_\textrm{e})u_\textrm{e} ) + \eta n_\textrm{e} E\cdot u_\textrm{e} = -\frac{\kappa}{M\varepsilon} \boldsymbol{\nabla}_x \boldsymbol{\cdot} \left\langle \frac{|v|^2}{2} v g_\textrm{e}\right\rangle, \end{align}

with

(3.4d)\begin{equation} W_\textrm{e} = (M\varepsilon)^2 \frac{1}{2} n_\textrm{e} {|u_\textrm{e}|}^2 + \frac{p_\textrm{e}}{\gamma-1},\quad p_\textrm{e} = n_\textrm{e} T_\textrm{e}. \end{equation}

These two systems are coupled to a set of equations (the Maxwell system (2.21)) driving the changes in the electromagnetic field, as well as an equation for the evolution of the deviations $g_\textrm {e}$ and $g_\textrm {i}$. The construction of these equations will be omitted in the present work; we refer for instance to Crouseilles & Lemou (Reference Crouseilles and Lemou2011), Crestetto et al. (Reference Crestetto, Crouseilles and Lemou2012) and De Cecco et al. (Reference De Cecco, Deluzet, Negulescu and Possanner2017) for examples of their derivation.

3.1.3 On the quasi-neutral limit

Omitting the collisions, the fastest velocity in this system is the propagation of waves at the speed of light described by the Maxwell system. The Debye length as well as the plasma period also define small space and time scales for large plasma densities. The quasi-neutral limit is defined by the following scaling relations:

(3.5)\begin{equation} (\alpha,\lambda) \to 0,\quad {\alpha}\sim{\lambda}. \end{equation}

This amounts to assuming that the scaled Debye length is small compared to the typical length and that the system evolves at a speed lower than the speed of light. By this means, the small scales related to these parameters are filtered out of the equations. The last hypothesis $\alpha \sim \lambda$ is essential to recover the low-frequency Ampère law, derived by neglecting the displacement current. This equation being common to MHD models, the quasi-neutral limit encompasses these two assumptions. With the vanishing of this generalized dimensionless Debye length $(\lambda ,\alpha )\to 0$, the Maxwell system degenerates into

(3.6a)\begin{gather} \beta \boldsymbol{\nabla}_x \times B = \frac{M}{\xi} J, \end{gather}

(3.6b)\begin{gather}\beta \partial_t B + \boldsymbol{\nabla}_x \times E = 0, \end{gather}

(3.6c)\begin{gather}n_\textrm{i} = n_\textrm{e}, \end{gather}

(3.6d)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0. \end{gather}

From Gauss's law, the property of the electronic density to match that of the ions is recovered, which genuinely enforces quasi-neutrality of the plasma. The electric field has no contribution in either of these degenerate Gauss and Ampère equations. The remaining occurrence of the electric field is limited to the Faraday equation (3.6b). Therefore, this set of equations is not well suited for the computation of the electric field. Indeed, the electrostatic component of the electric field can be arbitrarily chosen in (3.6): the electric field satisfying this system may be augmented by any gradient of a scalar potential (see also Degond & Deluzet (Reference Degond and Deluzet2017) for further details).

In the quasi-neutral limit, the electric field is provided by the particle current $J$ rather than the displacement current ($\partial E/\partial t$ originally present in Ampère's law). To close the system, the dependence of $J$ with respect to $E$ shall be explained to restore uniqueness of the electric field. This is related to the model describing the plasma.

3.2 A hierarchy of kinetic models for quasi-neutral plasmas

3.2.1 A kinetic formulation of the electron MHD

The aim here is to follow the microscopic dynamics of the electrons. The velocity of interest is the kinetic velocity of the electrons. This amounts to setting $\vartheta _0 = v_0/\varepsilon$ or equivalently $\xi \varepsilon = 1$, yielding

(3.7a)\begin{gather} \partial_t\, f_\textrm{i} + \varepsilon \left(v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} + \eta (E+ {\varepsilon\beta} v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i}\right) =\frac{1}{\kappa}\nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}), \end{gather}

(3.7b)\begin{gather}\partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} - \eta(E+ {\beta} v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} =\frac{1}{\varepsilon \kappa} \nu_{\textrm{ee}} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}} - f_\textrm{e}). \end{gather}

The collisions are assumed to be ineffective on the characteristic time scale $\kappa \varepsilon \gg {1}$, which amounts to neglecting the collision operator in (3.7), in particular for the ions, owing to $\varepsilon \ll 1$.

Performing the quasi-neutral limit $(\lambda =\alpha )\to 0$, the system at hand here is recast into (see Degond et al. Reference Degond, Deluzet and Doyen2015)

(3.8)\begin{gather} \partial_t\, f_\textrm{i} + \varepsilon (v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} +\eta (E+\varepsilon\beta v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i}) =0, \end{gather}

(3.9)\begin{gather}\partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} -\eta (E+\beta v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} =0, \end{gather}

(3.10)\begin{gather}\beta \boldsymbol{\nabla}_x \times B=(M\varepsilon)J, \end{gather}

(3.11)\begin{gather}\beta\partial_t B + \boldsymbol{\nabla}_x \times E = 0, \end{gather}

(3.12)\begin{gather}n_\textrm{e}= n_\textrm{i}=n, \end{gather}

(3.13)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0. \end{gather}

First, note that the formal time derivative of the Faraday equation (3.11) together with the curl of Ampère's law yields

(3.14)\begin{equation} \boldsymbol{\nabla}_x \times \boldsymbol{\nabla}_x \times E = - (M \varepsilon) \partial_t J, \end{equation}

which outlines that the electric field is known up to the gradient of a potential in this system. In Degond, Deluzet & Savelief (Reference Degond, Deluzet and Savelief2012a) and Degond et al. (Reference Degond, Deluzet and Doyen2015) the ill-posed nature of this equation is corrected by explaining the relation between the current density and the electric field. The conservation of the ionic and electronic momentum, as stated by (3.3b) and (3.4b), yields

(3.15a)\begin{align} (M \varepsilon) \frac{\partial J}{\partial t} &= - (M\varepsilon )^2 \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\mathbb{S}_\textrm{i} - \mathbb{S}_\textrm{e} ) - \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\varepsilon^2 P_\textrm{i} -P_\textrm{e} ) \nonumber\\ &\quad +\eta(\varepsilon^2 n_\textrm{i} +n_\textrm{e}) E + \eta \beta (M \varepsilon) ( \varepsilon^2 n_\textrm{i} u_\textrm{i} + n_\textrm{e} u_\textrm{e}) \times B, \end{align}

with, for any species $\alpha$ ($\alpha =\textrm {e}$ for the electrons and $\textrm {i}$ for the ions),

(3.15b)\begin{equation} \mathbb{S}_\alpha= n_\alpha u_\alpha \otimes u_\alpha,\quad P_\alpha = p_\alpha \mathbb{I}{\textrm{d}} + \kappa \left\langle v \otimes v g_\alpha \right\rangle. \end{equation}

Inserting the identity (3.15a), together with $n=n_\textrm {i}=n_\textrm {e}$, into (3.14) gives

(3.16)\begin{align} n (1+\varepsilon^2) E + \boldsymbol{\nabla}_x \times \boldsymbol{\nabla}_x \times E &= - (M\varepsilon)n ( \varepsilon^2 u_\textrm{i} + u_\textrm{e})\times B \nonumber\\ &\quad +( M\varepsilon)^2 \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\mathbb{S}_\textrm{i} - \mathbb{S}_\textrm{e})+ \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\varepsilon^2 P_\textrm{i} - P_\textrm{e}). \end{align}

This equation is well posed in the quasi-neutral limit ($n>0$) and can be used for the computation of the electric field. It is written under the assumption $\beta =1$ as well as $\eta =1$ which amounts to considering the thermal energy on a par with the electric one. This yields the following definition of the quasi-neutral model:

(3.17a)\begin{gather} \partial_t\, f_\textrm{i} + \varepsilon (v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} + (E+\varepsilon v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i}) =0, \end{gather}

(3.17b)\begin{gather}\partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} -(E+ v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} =0, \end{gather}

(3.17c)\begin{align} n (1+\varepsilon^2) E + \boldsymbol{\nabla}_x \times \boldsymbol{\nabla}_x \times E = &- (M\varepsilon) n ( \varepsilon^2 u_\textrm{i} + u_\textrm{e})\times B \nonumber\\ &+( M\varepsilon)^2 \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\mathbb{S}_\textrm{i}-\mathbb{S}_\textrm{e})+ \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\varepsilon^2 P_\textrm{i} - P_\textrm{e}), \end{align}

(3.17d)\begin{gather} \partial_t B + \boldsymbol{\nabla}_x \times E = 0, \end{gather}

(3.17e)\begin{gather}\boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0. \end{gather}

Note that the electric field provided by (3.17c) enforces a divergence-free particle current, or more precisely $\partial _t (\boldsymbol {\nabla } \boldsymbol {\cdot } J) = 0$. This yields, because of the continuity equation,

(3.18)\begin{equation} \frac{\partial^2 \rho}{\partial t^2} = 0. \end{equation}

This proves the consistency of this model with the quasi-neutrality assumption (matching of the electronic and ionic densities) as soon as the initial data are compliant with this regime. Note that, working a time semi-discretization, the vanishing of the charge density, rather than its time double derivative, may be recovered (see Degond et al. Reference Degond, Deluzet and Doyen2015). Note also that the evolution of the ions only brings a marginal correction, proportional to $\varepsilon ^2$, to the electron dynamics. On this time scale, the ions may be considered at rest.

The characteristics of this model are similar to the so-called electron MHD: the time scale of interest is that of the electrons, the ions merely creating a motionless background for the fast electron flows (Kingsep, Chukbar & Yank'kov Reference Kingsep, Chukbar and Yank'kov1990). In particular, this modelling accounts for the inertia of electrons. A noticeable difference from the electron MHD (see § 3.3.1) lies in the kinetic description of the plasma. An asymptotic-preserving method is proposed in Degond et al. (Reference Degond, Deluzet and Doyen2015) to bridge this quasi-neutral model and the Vlasov–Maxwell system. The properties of this quasi-neutral plasma description are investigated in Tronci & Camporeale (Reference Tronci and Camporeale2015) by means of a linear stability analysis.

3.2.2 A hybrid formulation of the Hall MHD

Hybrid modelling (Yin et al. Reference Yin, Winske, Gary and Birn2002; Winske et al. Reference Winske, Yin, Omidi, Karimabadi, Quest, Büchner, Scholer and Dum2003; Tronci et al. Reference Tronci, Tassi, Camporeale and Morrison2014) refers to a class of plasma models where the ions are described by a kinetic equation while the fluid limit is assumed for the electrons. This is in line with the scaling relations of the collision frequencies stated by (2.18a–c). The relaxation of the electronic distribution function towards the local equilibrium is indeed faster than for the ions. The aim of this modelling is to filter out of the equations the fast scales carried by the electron dynamics. Therefore, a zero-inertia regime is also assumed for the electrons together with the fluid limit and the quasi-neutrality of the plasma.

The typical velocity selected here is the microscopic (thermal) velocity of the ions. This translates into the identity $\xi = 1$ resulting in the following system for the plasma:

(3.19a)\begin{gather} \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{i} + \eta \left(E+ \frac{\beta}{\xi} v \times B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} =\frac{1}{\kappa}\nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}), \end{gather}

(3.19b)\begin{gather}\partial_t\, f_\textrm{e} + \frac{1}{\varepsilon}\left( v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{e} - \eta\left(E+ \frac{\beta}{\varepsilon \xi} v \times B\right)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} \right)= \frac{1}{\varepsilon \kappa} \nu_{\textrm{ee}} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}} - f_\textrm{e}). \end{gather}

The fluid limit for the electrons is selected assuming $(\varepsilon \kappa ) \ll 1$ meaning that the number of electron collisions during the typical time is large. The quasi-neutrality of the plasma amounts to setting $\lambda =\alpha \ll 1$. To overcome the degeneracy of the Maxwell system in the quasi-neutral limit, the electronic momentum is harnessed to provide the so-called generalized Ohm's law. The electronic system can be recast into

(3.20)\begin{align} & (M\varepsilon)^2\left(\frac{1}{M} \partial_t (n_\textrm{e} u_\textrm{e}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e}u_\textrm{e} \otimes u_\textrm{e}) \right)+ \boldsymbol{\nabla}_x p_\textrm{e} + \eta ( n_\textrm{e} E + (\beta M) u_\textrm{e}\times B) \nonumber\\ &\quad =-(\kappa (M\varepsilon)) \boldsymbol{\nabla}_x \boldsymbol{\cdot} \sigma_\textrm{e}, \end{align}

(3.21)\begin{align} & \frac{1}{M} \partial_t W_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{e} + p_\textrm{e})u_\textrm{e} ) + \eta n_\textrm{e} E\cdot u_\textrm{e} = -\frac{\kappa}{M\varepsilon}\boldsymbol{\nabla}_x \boldsymbol{\cdot} ((M\varepsilon)^2\sigma_\textrm{e} \cdot u_\textrm{e} + \mu_\textrm{e} \boldsymbol{\nabla}_xT_\textrm{e}), \end{align}

where the viscous stress tensor $\sigma _\textrm {e}$ as well as the thermal conductivity $\mu _\textrm {e}$ are defined in § 3 and, owing to the quasi-neutrality assumption, $n_\textrm {e} = n_\textrm {i} = n$.

The dynamics described by these equations is stiff. This is due to the smallness of $(\xi \varepsilon )$ in this regime: the thermal velocity of the ions (defined as the typical velocity) is small compared to that of the electrons. Therefore, the electrons are in a low-Mach regime. Assuming $(M\varepsilon )\ll 1$ gives rise to the following equilibria:

(3.22a)\begin{gather} \boldsymbol{\nabla}_x (n T_\textrm{e}) + \eta n ( E + (\beta M) u_\textrm{e}\times B) = 0, \end{gather}

(3.22b)\begin{gather}\boldsymbol{\nabla}_x T_\textrm{e} = 0. \end{gather}

The classical massless approximation for the electrons is recovered with the generalized Ohm's law and a homogeneous electronic temperature. The definition of the mean velocity $u_\textrm {e}$ is derived from the particle current density $J= n (u_\textrm {i} - u_\textrm {e})$ together with Ampère's law (3.10), yielding

(3.23)\begin{equation} u_\textrm{e} = u_\textrm{i} - \frac{\beta}{M}\frac{\boldsymbol{\nabla}_x \times B}{n}. \end{equation}

The hybrid plasma modelling is written (assuming $\eta =\beta =1$)

(3.24a)\begin{gather} \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} + (E+ v \times B)\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} =0, \end{gather}

(3.24b)\begin{gather}E = - M u \times B+ M \frac{\boldsymbol{\nabla}_x \times B}{n}\times B - T_\textrm{e} \frac{\boldsymbol{\nabla}_x n}{n}, \end{gather}

(3.24c)\begin{gather}\frac{\partial B}{\partial t} + \boldsymbol{\nabla}_x \times E = 0,\quad \boldsymbol{\nabla}_x \boldsymbol{\cdot} B = 0, \end{gather}

with

(3.24d)\begin{equation} n = \int f_\textrm{i} \,\textrm{d} v,\quad n u = \int v\, f_\textrm{i} \,\textrm{d} v. \end{equation}

The derivation of a similar model is proposed in Acheritogaray et al. (Reference Acheritogaray, Degond, Frouvelle and Liu2011) with numerical investigations in Degond et al. (Reference Degond, Deluzet, Dimarco, Gallice, Santagati and Tessieras2011a).

3.3 A fluid hierarchy of quasi-neutral models

3.3.1 The electron MHD system

This model is obtained by letting $\kappa \to 0$ in (3.1). This yields the following set of equations for the electrons:

(3.25a)\begin{gather} \frac{\xi}{M} \partial_t n_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e}) = 0, \end{gather}

(3.25b)\begin{gather} \frac{\xi}{M} \partial_t (n_\textrm{e} u_\textrm{e}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e}u_\textrm{e} \otimes u_\textrm{e}) + \frac{1}{(M\varepsilon)^2} \left( \boldsymbol{\nabla}_x p_\textrm{e} + \eta n_\textrm{e} \left(E +\frac{\beta M}{\xi}u_\textrm{e} \times B\right)\right) =0, \end{gather}

(3.25c)\begin{gather} \frac{\xi}{M} \partial_t W_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{e} + p_\textrm{e})u_\textrm{e} ) + \eta n_\textrm{e} E\cdot u_\textrm{e} = 0. \end{gather}

A similar system is derived for the ions, however with $\varepsilon =1$ and $\eta$ replaced by $-\eta$. These two sets of conservation laws are coupled to the Maxwell system (2.21). Performing the quasi-neutral limit in this system $(\lambda =\alpha )\to 0$ and focusing on the electronic dynamics with $\xi \varepsilon =1$ yields the quasi-neutral bi-fluid Euler–Maxwell system. This model is similar to that of § 3.2.1 but with a fluid description for the plasma. Following the same terminology, this model is referred to as Electron-MHD system (also used in Kingsep et al. (Reference Kingsep, Chukbar and Yank'kov1990) though different terminology may be used by other authors). It is implemented and investigated numerically in the framework of Asymptotic-Preserving methods in Degond et al. (Reference Degond, Deluzet and Savelief2012a).

3.3.2 The Hall MHD regime

The Hall MHD regime (see Lighthill Reference Lighthill1960; Witalis Reference Witalis1986; Schnack Reference Schnack2009) is recovered from the assumptions of the previous section but with a typical velocity equal to the plasma mean flow yielding $\xi =M$. The fast electronic dynamics is filtered out from the equations to provide a low-frequency modelling for the plasma driven by the evolution of the massive ions. The plasma velocity, denoted $u$, is defined as that of the heavy species $u=u_\textrm {i}$. The other parameters obey the classical scaling relations of MHD models: $\eta =1$ and $\beta =1$.

In the drift regime ($M\varepsilon \to 0)$, the electronic energy reduces to the internal energy

(3.26a)\begin{equation} \mathcal{E}_\textrm{e}= p_\textrm{e}/(\gamma-1), \end{equation}

with the electronic momentum and energy verifying

(3.26b)\begin{gather} \boldsymbol{\nabla}_x p_\textrm{e} = - n \left(E + u_\textrm{e} \times B \right), \end{gather}

(3.26c)\begin{gather}\partial_t \mathcal{E}_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( (\mathcal{E}_\textrm{e}+p_\textrm{e}) u_\textrm{e} ) = - n E\cdot u_\textrm{e}. \end{gather}

The generalized Ohm's law (3.26b) is harnessed to compute the electric field. The electronic velocity $u_\textrm {e}$ is substituted by $u_\textrm {e}=u - J/n$.

The electric field is computed via the generalized Ohm's law, giving rise to

(3.27)\begin{equation} E = - u \times B+ \frac{\boldsymbol{\nabla}_x \times B}{n}\times B - \frac{\boldsymbol{\nabla}_x p_\textrm{e}}{n}. \end{equation}

The first term on the right-hand side of equation (3.27) is the classical frozen field term, explaining the convection of the magnetic field together with the plasma. The second and third contributions are the so-called Hall and diamagnetic terms. Inserting this definition in the Faraday law (3.6b), the magnetic induction equation can be constructed, with

(3.28)\begin{equation} \partial_t B + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (u \otimes B - B \otimes u) = - \boldsymbol{\nabla}_x \times \left(\frac{\boldsymbol{\nabla}_x \times B}{n}\times B \right) + \boldsymbol{\nabla}_x \times\left( \frac{\boldsymbol{\nabla}_xp_\textrm{e}}{n}\right). \end{equation}

Finally the plasma mass density, momentum and total pressure $p$ and energy $W$ defined by

(3.29a)\begin{gather} p = p_\textrm{i} +p_\textrm{e} = n (T_\textrm{i}+T_\textrm{e}), \end{gather}

(3.29b)\begin{gather}W = W_\textrm{i} + \mathcal{E}_\textrm{e} = (M)^2 \frac{1}{2} n {|u|^2} + \frac{p}{\gamma-1} \end{gather}

verify

(3.29c)\begin{gather} \partial_t n + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n u) = 0, \end{gather}

(3.29d)\begin{gather}\partial_t (n u) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (nu \otimes u) + \frac{1}{M^2} \boldsymbol{\nabla}_x p = \frac{1}{M^2} J \times B, \end{gather}

(3.29e)\begin{gather}\partial_t W + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( (W + p) u ) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( (\mathcal{E}_\textrm{e}+p_\textrm{e}) \text{v}_\text{H} ) = E\cdot J. \end{gather}

The Hall velocity $\text {v}_\text {H}$, which can be interpreted as the electron velocity in the ion frame, is defined by

(3.30)\begin{equation} \text{v}_\text{H} = - \frac{J}{n}. \end{equation}

The ideal MHD equations are classically written under a conservative form using the system total pressure and energy:

(3.31a)\begin{equation} p_\text{TOT} = p + \frac{B^2}{2},\quad W_\text{TOT} = W + \frac{B^2}{2}, \end{equation}

writing the system as

(3.31b)\begin{gather} \partial_t n + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n u) = 0, \end{gather}

(3.31c)\begin{gather}\partial_t (n u) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} \left( nu \otimes u - \frac{1}{M^2} B \otimes B\right) + \frac{1}{M^2} \boldsymbol{\nabla}_x p_\text{TOT} = 0, \end{gather}

(3.31d)\begin{gather}\partial_t W_\text{TOT} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( (W_\text{TOT} + p_\text{TOT}) u - (B\cdot u) B ) = - \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( (\mathcal{E}_\textrm{e}+p_\textrm{e}) \text{v}_\text{H} ), \end{gather}

(3.31e)\begin{gather}\partial_t B - \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( B\otimes (u+\text{v}_\text{H}) - (u+\text{v}_\text{H}) \otimes B ) = \boldsymbol{\nabla}_x \times \left( \frac{\boldsymbol{\nabla}_xp_\textrm{e}}{n}\right). \end{gather}

This set of equations is supplemented with the electronic energy conservation (3.26c).

The ideal MHD equations are recovered from this system assuming an ideal Ohm's law where the current density is assumed small compared to the ion mean velocity and therefore neglected. However, in this simplified framework (omitting the unlike particle collisions), there are no mechanisms preventing the electron mean velocity from departing from that of the ions. Consequently, the generalized Ohm's law incorporates the Hall velocity in complement to the so-called ideal Ohm's law. The effect of the resistivity should be considered to derive the ideal MHD regime.

The drift approximation operated for the electrons amounts to vanishing of the electronic Mach numbers $(M\varepsilon )$. The scale separation introduced by the small electron to ion mass ratio $\varepsilon$ is not always sufficient to consider this limit independently of vanishing ionic Mach numbers $M \to 0$. For low ionic Mach number, a low-frequency filtering may be operated performing the limit of vanishing electronic Mach numbers jointly with ionic Mach numbers. This asymptotic defines the massless MHD regime (Besse et al. Reference Besse, Degond, Deluzet, Claudel, Gallice and Tessieras2004).

4.1 Electrostatic limit of the Maxwell system

The electrostatic regime is recovered from the dimensionless Maxwell system (2.21) by letting $\alpha$ go to zero, independently of $\lambda$. This assumption shall be interpreted as a typical velocity negligible compared to the speed of light. From Ampère's law

(4.1)\begin{equation} \lambda^2 \eta \frac{\partial {E}}{\partial t} + \frac{M}{\xi} J =\frac{\beta \lambda^2 \eta}{\alpha^2} \boldsymbol{\nabla}_x \times B , \end{equation}

the limit $\alpha \to 0$ provides $\boldsymbol {\nabla } \times B=0$. Nonetheless, the right-hand side of (4.1) remains an undetermined form. Therefore, Ampère's law is not well suited for the computation of the electric field in the electrostatic limit. However, subjected to convenient boundary conditions, the property $\boldsymbol {\nabla }_x \times B = 0$ together with $\boldsymbol {\nabla }_x \boldsymbol {\cdot } B = 0$ yields $\partial _t B = 0$. Inserting this property into the Faraday equation (2.21b) provides an electrostatic electric field: $\boldsymbol {\nabla } \times E = 0$. Furthermore, the undetermined form in (4.1) is divergence-free. Therefore, computing the divergence of Ampère's law provides

(4.2)\begin{equation} \lambda^2 \eta \frac{\partial \boldsymbol{\nabla}_x \boldsymbol{\cdot} {E}}{\partial t} = - \frac{M}{\xi} \boldsymbol{\nabla}_x \boldsymbol{\cdot} J , \end{equation}

which is a well-posed problem for the electric field under the condition $\boldsymbol {\nabla }_x \times E = 0$. Note that, owing to the continuity equation

(4.3)\begin{equation} \frac{\partial \rho}{\partial t} + \frac{M}{\xi} \boldsymbol{\nabla}_x \boldsymbol{\cdot} J = 0, \end{equation}

originating from the conservation of the particle densities (3.3a) and (3.4a), the divergence of Ampère's law is equivalent to the time derivative of Gauss's law, with

(4.4)\begin{equation} \frac{\partial}{\partial t} (\lambda^2 \eta \boldsymbol{\nabla}_x \boldsymbol{\cdot} E -{\rho})=0. \end{equation}

Therefore, in the electrostatic regime, the Gauss equation is used to compute the electric field.

4.2 Quasi-neutral models at the electronic scale

This analysis is carried out under the assumption of a vanishing magnetic field ($B=0$). The plasma description considered in the following is therefore

(4.5a)\begin{gather} \xi \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{i} + \eta E \boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} = \frac{ \xi}{\kappa} ( \nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}) ), \end{gather}

(4.5b)\begin{gather}\xi \varepsilon \partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x\, f_\textrm{e} - \eta E\boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} = \frac{ \xi}{\kappa} ( \nu_{\textrm{ee}} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}} - f_\textrm{e})) \end{gather}

coupled to the Gauss equation

(4.5c)\begin{equation} -\lambda^2 \eta {\rm \Delta} \phi= n_\textrm{i} - n_\textrm{e}, \end{equation}

$\phi$ being the electrostatic potential, with $E = - \boldsymbol {\nabla } \phi$ and $n_\alpha = \int f_\alpha \,\textrm {d} v$.

The quasi-neutrality of the plasma is recovered for vanishing scaled Debye length $\lambda \to 0$. In this regime, similarly to the electromagnetic case, an equation needs to be manufactured from the motion of the particles, to compute the electric field. This is classically obtained using the equation of the electronic momentum conservation. The electric field is computed in order for this conservation to be satisfied. In Degond et al. (Reference Degond, Deluzet, Navoret, Sun and Vignal2010) an equivalent approach is proposed. It consists of using the time derivatives of Gauss's law to produce

(4.6)\begin{equation} -\lambda^2 \eta \frac{\partial^2}{\partial t^2} {\rm \Delta} \phi = \frac{\partial^2 }{\partial t^2} \left( n_\textrm{i} -n_\textrm{e}\right). \end{equation}

In the quasi-neutral limit ($\lambda \to 0$), the electrostatic field is the Lagrangian multiplier of the quasi-neutrality constraint

(4.7)\begin{equation} \frac{\partial^2 }{\partial t^2} \left( n_\textrm{i} -n_\textrm{e}\right) = 0. \end{equation}

From the system (3.4) (with $B=0$), formally computing the time derivative of the density conservation together with the divergence of the momentum equation, the following identity is recovered:

(4.8)\begin{gather} \left(\frac{\xi}{M}\right)^2\frac{\partial^2 n_\textrm{e}}{\partial t^2} = -\boldsymbol{\nabla}_x \boldsymbol{\cdot} (\boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e} \otimes u_\textrm{e}) )- \frac{1}{(M\varepsilon)^2} \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\boldsymbol{\nabla}_x P_\textrm{e} + \eta n_\textrm{e} E); \end{gather}

(4.9)\begin{gather}P_\textrm{e} = p_\textrm{e} \mathbb{I}{\textrm{d}} + \kappa \left\langle v \otimes v g_\textrm{e} \right\rangle, \quad {\text{with } \left\langle v \otimes v g_\textrm{e} \right\rangle = \int (v\otimes v) g_\textrm{e} \,\textrm{d} v.} \end{gather}

Resuming the scaling relations of § 3.2.1, $\xi =1/\varepsilon$ and $\kappa \varepsilon > 1$, assuming that the ions are at rest, the evolution of the charge density $n_\textrm {i} -n_\textrm {e}$ is governed by (4.8) with

(4.10)\begin{equation} \frac{\partial^2 n_\textrm{e}}{\partial t^2} = -(M\varepsilon)^2 \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e} \otimes u_\textrm{e}) )- \boldsymbol{\nabla}_x \boldsymbol{\cdot} (\boldsymbol{\nabla}_x P_\textrm{e} + \eta n_\textrm{e} E). \end{equation}

The evolution of the density is barely independent of the mean flow velocity but relies on the balance between the pressure and the electric forces. The equation providing the electric potential $\phi$ is obtained by inserting this relation into (4.6) and passing to the limit $\lambda \to 0$.

This yields the following quasi-neutral kinetic plasma description:

(4.11a)\begin{gather} \partial_t\, f_\textrm{e} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{e} + \eta \boldsymbol{\nabla}_x \phi \boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{e} = \frac{1}{\varepsilon\kappa} ( \nu_{ee} (\mathcal{M}_{n_\textrm{e},u_\textrm{e},T_\textrm{e}} - f_\textrm{e}), \end{gather}

(4.11b)\begin{gather}-\eta \boldsymbol{\nabla}_x \boldsymbol{\cdot} \left( n_\textrm{e} \boldsymbol{\nabla}_x \phi \right) = - \boldsymbol{\nabla}_x \boldsymbol{\cdot} ( \boldsymbol{\nabla}_x \boldsymbol{\cdot} P_\textrm{e} - (M\varepsilon)^2 \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e} \otimes u_\textrm{e}) ). \end{gather}

According to the values of $\kappa$, the collision term in (4.11a) may be disregarded, defining therefore a non-collisional kinetic description. Contrariwise, letting $\kappa \to 0$, a fluid description for the electrons may be derived, with

(4.12a)\begin{gather} \frac{1}{(M\varepsilon)} \partial_t n_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e} u_\textrm{e}) = 0, \end{gather}

(4.12b)\begin{gather}\frac{1}{(M\varepsilon)} \partial_t (n_\textrm{e} u_\textrm{e}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{e}u_\textrm{e} \otimes u_\textrm{e}) + \frac{1}{(M\varepsilon)^2} \left( \boldsymbol{\nabla}_x p_\textrm{e} -\eta n_\textrm{e} \boldsymbol{\nabla}_x \phi \right) =0, \end{gather}

(4.12c)\begin{gather}\frac{1}{(M\varepsilon)} \partial_t W_\textrm{e} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{e} + p_\textrm{e})u_\textrm{e} ) - \eta n_\textrm{e} \boldsymbol{\nabla}_x\phi\boldsymbol{\cdot} u_\textrm{e} = 0. \end{gather}

Note that a similar equation to (4.10), but with $P_\textrm {e} = p_\textrm {e} \mathbb {I}{\textrm {d}}$, may be worked out of the conservation of the electronic density (4.12a) and momentum (4.12b). This outlines that the electronic dynamics, in particular the electronic speed of sound, is resolved in this model. Comparable models are implemented and numerically explored in the context of asymptotic-preserving methods in Crispel, Degond & Vignal (Reference Crispel, Degond and Vignal2007) for fluid plasma description and Manfredi, Hirstoaga & Devaux (Reference Manfredi, Hirstoaga and Devaux2011) for kinetic equations.

4.3 Quasi-neutral models at the ionic scale

The typical velocity is chosen to be that of the ions with $\xi =1$ for the kinetic descriptions of the ions and $\xi =M$ for macroscopic models.

The hybrid modelling investigated in § 3.2.2 is defined by the scaling relations $\xi =1$, $(\varepsilon \kappa )\ll 1$ and $\eta =1$. The equilibria stated by (3.22) yield

(4.13)\begin{equation} T_\textrm{e} \boldsymbol{\nabla}_x n_\textrm{e} = n_\textrm{e} \boldsymbol{\nabla}_x \phi, \end{equation}

with a homogeneous electronic temperature. This equation is integrated to provide the so-called Boltzmann relation

(4.14)\begin{equation} n_\textrm{e} = n_0 \exp\left( - \frac{\phi}{T_\textrm{e}}\right), \end{equation}

$n_0$ being a constant (independent of the space variable $x$) that should be determined from adequate conditions (Hagelaar Reference Hagelaar2007). Due to the Boltzmann relation, the quasi-neutral limit is no longer singular. Indeed plugging the Boltzmann relation (4.14) into the Gauss equation yields

(4.15)\begin{equation} -\lambda^2 {\rm \Delta} \phi= n_\textrm{i} - n_0 \exp\left(-\frac{\phi}{T_\textrm{e}}\right). \end{equation}

This equation is not degenerate for the computation of the electric potential for vanishing $\lambda$. Indeed, the nonlinear part of the equation provides a means of computing $\phi$ in the quasi-neutral limit. This property is thoroughly investigated in Degond et al. (Reference Degond, Liu, Savelief and Vignal2012b).

The hybrid electrostatic model may be recast into

(4.16a)\begin{gather} \partial_t\, f_\textrm{i} + v \boldsymbol{\cdot} \boldsymbol{\nabla}_x f_\textrm{i} - \boldsymbol{\nabla}_x \phi \boldsymbol{\cdot} \boldsymbol{\nabla}_v\, f_\textrm{i} = \frac{1}{\kappa} ( \nu_{\textrm{ii}} (\mathcal{M}_{n_\textrm{i},u_\textrm{i},T_\textrm{i}} - f_\textrm{i}) ), \end{gather}

(4.16b)\begin{gather}\phi = - T_\textrm{e} \ln\left(\frac{n_\textrm{i}}{n_0}\right). \end{gather}

Letting $\kappa \to 0$ together with $\kappa /(M\varepsilon )\ll 1$ and $\xi =M$ yields the quasi-neutral fluid model:

(4.17a)\begin{gather} \partial_t n_\textrm{i} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{i} u_\textrm{i}) = 0, \end{gather}

(4.17b)\begin{gather}\partial_t (n_\textrm{i} u_\textrm{i}) + \boldsymbol{\nabla}_x \boldsymbol{\cdot} (n_\textrm{i}u_\textrm{i} \otimes u_\textrm{i}) + \frac{1}{M^2} \left(\boldsymbol{\nabla}_x p_\textrm{i} + n_\textrm{i} \boldsymbol{\nabla}_x\phi\right) =0 , \end{gather}

(4.17c)\begin{gather}\partial_t W_\textrm{i} + \boldsymbol{\nabla}_x \boldsymbol{\cdot} ((W_\textrm{i} + p_\textrm{i})u_\textrm{i} ) + n_\textrm{i} \boldsymbol{\nabla}_x\phi\boldsymbol{\cdot} u_\textrm{i} = 0, \end{gather}

(4.17d)\begin{gather}\phi = - T_\textrm{e} \ln\left(\frac{n_\textrm{i}}{n_0}\right). \end{gather}

In the models (4.16) and (4.17) following the evolution of the plasma at the ionic scale, the fast electronic dynamics introduced by the electron inertia is filtered out of the equations by performing the low-frequency limit $(M\varepsilon ) \to 0$.