# Deriving Improved Plasma Fluid Equations from Collisional Kinetic Theory

Y. S. Dimant ∗

###### Abstract

## 1

Developing a quantitative understanding of wave plasma processes in the lower
ionosphere requires a reasonably accurate theoretical description of the
underlying physical processes. For such highly collisional plasma environment
as the E-region ionosphere, kinetic theory represents the most accurate
theoretical description of wave processes. For the analytical treatment,
however, the collisional kinetic theory is extremely complicated and succeeds
only in a limited number of physical problems. To date, most research applied
oversimplified fluid models that lack a number of critical kinetic aspects, so
that the coefficients in the corresponding fluid equations are often accurate
only to an order of magnitude. This paper presents the derivation for the
highly collisional, partially magnetized case relevant to E-region conditions.
It provides a more accurate reduction of the ion and, especially, electron
kinetic equations to the corresponding 5-moment fluid equations by using a new
set of analytic approximations. This derivation results in more accurate
fluid-model set of equations appropriate for most E-region problems. The
results of this paper could be used for a routine practical analysis when
working with actual data. The improved equations can also serve as a basis for
more accurate plasma fluid computer simulations.

##  2 Keywords:

E-region ionosphere, magnetized plasma, plasma-neutral collisions, kinetic
theory, fluid equations, 5-moment description

##  3 Introduction

At altitudes of the equatorial and high-latitude E-region ionospheres, the
ionosphere is highly collisional in such a way that ions are almost
demagnetized by their frequent collisions with the surrounding neutral
molecules while electrons still remain strongly magnetized. Strong DC electric
fields perpendicular to the geomagnetic field cause there electrojets and give
rise to plasma instabilities whose nonlinear development produces plasma
density irregularities observed by radars and rockets.

Developing a quantitative understanding of wave plasma processes in the lower
ionosphere requires an accurate theoretical description of the underlying
physical processes. For such dissipative environment, collisional plasma
kinetic theory represents the most accurate theoretical description of wave
processes. Particle-in-cell (PIC) simulations present the most advanced way to
apply the kinetic approach, but such massive computer simulations (Oppenheim
and Dimant, 2004; Oppenheim et al., 2008; Oppenheim and Dimant, 2013;
Oppenheim et al., 2020) are usually quite costly. In many cases, simple
estimates and parameter dependencies provided by an analytic approach will
suffice. For the analytical treatment, however, the collisional kinetic theory
is extremely complicated and succeeds only in a limited number of physical
problems. To date, most research applied an oversimplified fluid model that
lacks many critical kinetic aspects. These models mostly apply to weakly
collisional conditions. The coefficients in the simple fluid equations are
often accurate only to an order of magnitude because they were not obtained
using the full kinetic theory. This paper presents the derivation of improved
fluid equations for the highly collisional, partially magnetized case relevant
to E-region conditions, starting from a more consistent kinetic approach. It
provides more accurate values for the fluid-model coefficients.

There are different approaches to analytical description of low-frequency
plasma processes in the E-region ionosphere, including both the kinetic theory
and fluid models. Traditionally, the kinetic theory of the FB instability
applied an oversimplified BGK collision operator (Bhatnagar et al., 1954).
This operator does not follow from an accurate Boltzmann collision operator,
but represents an artificial construct. It simplifies dramatically the
analytical treatment, satisfying the particle number conservation, as well as
the momentum and energy balances (albeit under certain conditions, see below).
This simplified approach is reasonably applicable to the description of the
heavy ions, but it is totally unacceptable to the description of the light
electrons (Dimant and Sudan, 1995a).

More accurate approaches to the kinetic description of electrons under
conditions of the E-region wave processes, such as the Farley-Buneman (FB)
instability, have been developed by a few research groups. Stubbe (1990)
modified the BGK terms to allow for the different rates of the electron energy
and momentum losses. This simple modification, however, does not follow from
the Boltzmann operator and cannot be trusted. Later, two independent research
groups developed more sophisticated and accurate approaches. Kissack and
collaborators (Kissack et al., 1995, 1997, 2008a, 2008b) applied Grad’s method
(Grad, 1949; Rodbard et al., 1995), while Dimant and Sudan (1995a) used an
expansion in Legendre polynomials with respect to the angles in the velocity
space (Gurevich, 1978; Allis, 1982). The latter kinetic approach has allowed
the authors to predict a new electron thermal-driven instability in the lower
E/upper D regions (Dimant and Sudan, 1995b, c), which has been later explained
in terms of a much simpler fluid model (Dimant and Sudan, 1997). This effect
has been verified by others (Robinson, 1998; St. -Maurice and Kissack, 2000).
Later, a similar thermal-instability process has been also suggested for ions
(Kagan and Kelley, 2000; Dimant and Oppenheim, 2004; Dimant et al., 2023).

This paper presents a consistent reduction of the ion and electron kinetic
equations to the 5-moment fluid equations by using a new set of analytic
approximations. This derivation results in a more accurate fluid model
appropriate for most E-region plasma problems.

The paper is organized as follows. Section 4 introduces the collisional
kinetic equation and reviews the generic procedure for obtaining the moment
equations. The collisional parts there are not specified and remain in the
general integral form. Section 5 describes ion momentum equation obtained
using the BGK collision model. The most important part is section 6. It
derives low-frequency electron-fluid equations using a kinetic theory based on
the efficient isotropization of the electron distribution function in the
velocity space (Gurevich, 1978; Dimant and Sudan, 1995a). This requires a more
detailed and sophisticated treatment. Section 6.1 derives the moment equations
where the heat conductivity and frictional heating are given in terms of a
still unspecified small directional part of the velocity distribution
function. To illustrate major ideas of closing the derivation, section 6.2
describes the simplest case of the constant (i.e., velocity-independent)
kinetic collision parameters. Section 6.3 presents the general results
obtained in detail in the Appendix. Compared to the simplest electron-fluid
equations from section 6.2, the general momentum and thermal-balance equations
include a larger number of coefficients, as well as additional heat-
conductivity terms. By their look, the latter may appear collisionless, but
they have arisen exclusively due to the velocity dependence of the kinetic
electron-neutral collision frequency. All these results could be used for a
routine practical analysis when working with actual data. The improved
equations can serve as a basis for more accurate plasma fluid computer
simulations.

##  4 General kinetic treatment

This section discusses a general approach to deriving the fluid model from the
kinetic theory for any plasma particles. To avoid confusion, throughout this
paper we will use the following nomenclature. We denote various kinds of
particles (charged or neutral) by Latin subscripts: $p$, $q$, etc., that stand
for electrons ($e$), ions of various kinds ($i$) and neutrals ($n$), while
denoting vector components by Greek subscripts: $\alpha$, $\beta$, etc.

Non-relativistic kinetics of charged particles of the kind $p$ with the
velocity $\vec{v}_{p}$ at a given location $\vec{r}$ and time $t$ is described
by the Boltzmann kinetic equation,

$\partial_{t}f_{p}+\nabla\cdot\left(\vec{v}_{p}f_{p}\right)+\partial_{\vec{v}_{p}}\cdot\left[\frac{q_{p}}{m_{p}}\left(\vec{E}+\vec{v}_{p}\times\vec{B}\right)f_{p}\right]=\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}},$
(1)

where $f_{p}(\vec{v}_{p},\vec{r},t)$ is the single-particle velocity
distribution function. The left-hand side (LHS) of Eq. (1) describes the
collisionless (Vlasov) dynamics of the $p$-species charged particles in
smoothed over many particles electric ($\vec{E}$) and magnetic ($\vec{B}$)
fields (for simplicity, we ignore here a gravity force); $q_{p}$ and $m_{p}$
are the $p$-particle charge and mass, respectively. The LHS of Eq. (1) is
intentionally written in a conservative (divergence) form which is more
convenient for deriving the moment equations.

The right-hand side (RHS) of Eq. (1), term
$(df_{p}/dt)_{\operatorname{col}}=\sum_{q}S_{pq}$, is the collisional operator
describing binary collisions of the $p$-particles with all available kinds of
charged and neutral particles denoted by $q$ (including the $p$-particles
themselves). In the general case, the partial components $S_{pq}$ represent
integral operators that involve products of $f_{p}(\vec{v}_{p},\vec{r},t)$ by
$f_{q}(\vec{v}_{q},\vec{r},t)$. The partial operator $S_{qq}$ is quadratically
nonlinear, while $S_{pq}$ with $p\neq q$ are linear with respect to $f_{p}$.

The binary collisions can be either elastic or inelastic. Elastic collisions
conserve the total kinetic energy, momentum and angular momentum of the
colliding pair. The corresponding partial collisional operator, $S_{pq}$, can
be described by the well-known Boltzmann collision integral (Shkarofsky et
al., 1966; Gurevich, 1978; Lifshitz and Pitaevskii, 1981; Schunk and Nagy,
2009; Khazanov, 2011). During an inelastic collision of a charged particle
with a neutral particle, a fraction of the total kinetic energy goes to
excitation (de-excitation) of the neutral particle (or ion) or to release of
electrons via ionization. Inelastic processes in the lower ionosphere often
involve molecular dissociation, recombination with ions and electron
attachment, accompanied by photon radiation or absorption. The complete
kinetic description of all these processes is complicated. In many cases,
however, inelastic collisions are close to elastic and one can continue using
Boltzmann’s integral with minor modifications (Gurevich, 1978; Shkarofsky et
al., 1966). Kinetic Eq. (1) with Boltzmann’s collision integral _per ce_
represents a significant simplification over the full multi-particle kinetics,
but it still remains quite difficult for a mathematical treatment and requires
further simplifications.

Being interested in fluid-model equations that follow from kinetic Eq. (1),
right below we review the conventional approach to deriving equations for the
lowest-order moments of the distribution function. The material of this
section will serve as a guide for more specific derivations of the following
sections.

The three lowest-order velocity moments include the $p$-species particle
density,

$n_{p}(\vec{r},t)\equiv\int f_{p}d^{3}v_{p},$ (2)

mean fluid velocity,

$\vec{V}_{p}(\vec{r},t)\equiv\left\langle\vec{v}_{p}\right\rangle=\frac{1}{n_{p}}\int\vec{v}_{p}f_{p}d^{3}v_{p},$
(3)

and effective temperature,

$T_{p}(\vec{r},t)=\frac{m_{p}}{3}\left\langle\left(\vec{v}_{p}-\vec{V}_{p}\right)^{2}\right\rangle=\frac{m_{p}}{3n_{p}}\int\left(\vec{v}_{p}-\vec{V}_{p}\right)^{2}f_{p}d^{3}v_{p}.$
(4)

The derivations below will also involve other velocity-averaged quantities
defined by

$\left\langle\cdots\right\rangle\equiv\frac{1}{n_{p}}\int\left(\cdots\right)f_{p}d^{3}v_{p}.$
(5)

Integrations in Eqs. (2)–(5) are performed over the entire 3-D velocity space.

First, we consider the particle-number balance. Integrating Eq. (1) over the
particle velocities with $f_{p}\rightarrow 0$ as
$v_{p}\equiv|\vec{v}_{p}|\rightarrow\infty$, we easily obtain the continuity
equation for the $p$-particle fluid,

$\partial_{t}n_{p}+\nabla\cdot\left(n_{p}\vec{V}_{p}\right)=\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}.$
(6)

The RHS of Eq. (6) includes various particle sources and losses, like
ionization, recombination and electron attachment. The collisions between the
charged particles of the same species usually conserve the average particle
number and hence do not contribute to the RHS of Eq. (6).

Second, we obtain the momentum-balance equation that involves the mean fluid
drift velocity, $\vec{V}_{p}$. Integrating Eq. (1) with the weighting function
$m_{p}\vec{v}_{p}$, for a given vector-component $\alpha$ of the momentum
density, we obtain

$\displaystyle
m_{p}\partial_{t}\left(n_{p}V_{p_{\alpha}}\right)+\sum_{\beta=1}^{3}\partial_{x_{\beta}}\mathbf{P}_{p_{\alpha\beta}}+m_{p}\sum_{\beta=1}^{3}\partial_{x_{\beta}}\left(n_{p}V_{p_{\alpha}}V_{p_{\beta}}\right)$
$\displaystyle=q_{p}\left[E_{\alpha}+\frac{1}{c}\left(\vec{V}_{p}\times\vec{B}\right)_{\alpha}\right]n_{p}+m_{p}\int
v_{p_{\alpha}}\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p},$
(7)

where $\mathbf{P}_{p}$ is the total pressure tensor with vector components
defined as

$\mathbf{P}_{p_{\alpha\beta}}\equiv
m_{p}\int\left(v_{p_{\alpha}}-V_{p_{\alpha}}\right)\left(v_{p_{\beta}}-V_{p_{\beta}}\right)f_{p}d^{3}v_{p}.$
(8)

It combines the isotropic pressure, $P_{p}\delta_{mn}$ ($\delta_{mn}=1$ if
$m=n$ and $\delta_{mn}=0$ otherwise), $P_{p}=n_{p}T_{p}$, with the viscosity
tensor,
$\mathbf{\Pi}_{p_{\alpha\beta}}\equiv\mathbf{P}_{p_{\alpha\beta}}-P_{p}\delta_{\alpha\beta}$.
Equation (7) includes momentum changes due to various average forces and those
caused by particle density variations. To exclude the latter and separate the
net effect of the total force, we multiply Eq. (6) by $m_{p}\vec{V}_{p}$ and
subtract the resultant equation from Eq. (7). This yields the conventional
momentum-balance equation,

$m_{p}n_{p}\
\frac{D_{p}\vec{V}_{p}}{Dt}=q_{p}n_{p}\left[\vec{E}+\frac{1}{c}\left(\vec{V}_{p}\times\vec{B}\right)\right]-\nabla\cdot\mathbf{P}_{p}+\vec{R}_{p},$
(9)

where $D_{p}/Dt\equiv\partial_{t}+\vec{V}_{p}\cdot\nabla$ is the convective
(also called substantial or material) derivative for the average $p$-particle
flow and

$\vec{R}_{p}\equiv
m_{p}\int\left(\vec{v}_{p}-\vec{V}_{p}\right)\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}.$
(10)

Here and below, the “dot”-products of a vector, $\vec{a}$, with a two-
component tensor, $\mathbf{A}$, depending on the multiplier order, denote
vectors with the components
$\left(\vec{a}\cdot\mathbf{A}\right)_{\alpha}\equiv\sum_{\beta=1}^{3}a_{\beta}\mathbf{A}_{\beta\alpha}$
or
$\left(\mathbf{A}\cdot\vec{a}\right)_{\alpha}\equiv\sum_{\beta=1}^{3}\mathbf{A}_{\alpha\beta}a_{\beta}$.
The tensor divergence, $\nabla\cdot\mathbf{P}_{p}=\nabla
P_{p}+\nabla\cdot\mathbf{\Pi}_{p}$, represents a vector which uses the obvious
symmetry $\mathbf{P}_{p_{\alpha\beta}}=\mathbf{P}_{p_{\beta\alpha}}$ following
from Eq. (8). The RHS of Eq. (9) includes all smooth forces acting on the
average particle flow of the charged particles, such as the total Lorentz
force, pressure gradient, and total friction, $\vec{R}_{p}$. The latter is
associated with collisions of the given $p$-particles with all other charged
or neutral particles. It includes no momentum exchange between the same-
species particles because their mutual collisions automatically conserve the
total momentum, $\int\vec{v}_{p}S_{pp}d^{3}v_{p}=0$.

Third, to obtain the total energy-balance equation, we integrate Eq. (1) with
the weighting function $m_{p}v_{p}^{2}/2$ and obtain

$\partial_{t}\mathcal{E}_{p}+\nabla\cdot\int\frac{m_{p}v_{p}^{2}}{2}\
\vec{v}_{p}f_{p}d^{3}v_{p}=\vec{j}_{p}\cdot\vec{E}+\frac{m_{p}}{2}\int
v_{p}^{2}\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p},$ (11)

where $\mathcal{E}_{p}$ is the $p$-species average kinetic-energy density and
$\vec{j}_{p}$ is their electric-current density,

$\mathcal{E}_{p}\equiv\int\frac{m_{p}v_{p}^{2}}{2}\ \
f_{p}d^{3}v_{p},\qquad\vec{j}_{p}\equiv q_{p}n_{p}\vec{V}_{p}.$ (12)

Before proceeding, we separate from the kinetic particle velocity
$\vec{v}_{p}$ the mean drift velocity $\vec{V}_{p}$, so that (11) becomes

$\displaystyle\partial_{t}\left[n_{p}\left(\frac{m_{p}V_{p}^{2}}{2}+\frac{3T_{p}}{2}\right)\right]$
$\displaystyle+\nabla\cdot\left[n_{p}\left(\frac{m_{p}V_{p}^{2}}{2}+\frac{5T_{p}}{2}\right)\vec{V}_{p}+\mathbf{\Pi}_{p}\cdot\vec{V}_{p}+\frac{n_{p}m_{p}}{2}\left\langle\left(\vec{v}_{p}-\vec{V}_{p}\right)^{3}\right\rangle\right]$
$\displaystyle=\vec{j}_{p}\cdot\vec{E}+\vec{V}_{p}\cdot\vec{R}_{p}+\frac{m_{p}}{2}\int\left(\vec{v}_{p}-\vec{V}_{p}\right)^{2}\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}$
$\displaystyle+\frac{m_{p}V_{p}^{2}}{2}\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}.$
(13)

where
$\left(\vec{v}_{p}-\vec{V}_{p}\right)^{3}=\left|\vec{v}_{p}-\vec{V}_{p}\right|^{2}\left(\vec{v}_{p}-\vec{V}_{p}\right)$.
Equation (13) describes dynamic variations of the total energy density. It
includes a part associated with the average fluid motion,
$n_{p}m_{p}V_{p}^{2}/2$, and the internal thermal energy, $n_{p}T_{p}$. To
extract the equation exclusively for the particle temperature, $T_{p}$, we
multiply Eq. (6) by $\left(m_{p}V_{p}^{2}/2+3T_{p}/2\right)$, take the scalar
product of Eq. (9) with $\vec{V}_{p}$ and subtract the resultant two equations
from Eq. (13). This yields

$\displaystyle\frac{3n_{p}}{2}\
\frac{D_{p}T_{p}}{Dt}+n_{p}T_{p}\nabla\cdot\vec{V}_{p}+\mathbf{\Pi}_{p}\cdot\nabla\cdot\vec{V}_{p}+\nabla\cdot\left[\frac{n_{p}m_{p}}{2}\left\langle\left(\vec{v}_{p}-\vec{V}_{p}\right)^{3}\right\rangle\right]$
$\displaystyle=\frac{m_{p}}{2}\int\left(\vec{v}_{p}-\vec{V}_{p}\right)^{2}\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}+\left(\frac{m_{p}V_{p}^{2}}{2}-\frac{3T_{p}}{2}\right)\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p},$
(14)

where
$\mathbf{\Pi}_{p}\cdot\nabla\cdot\vec{V}_{p}\equiv\sum_{\alpha,\beta=1}^{3}\mathbf{\Pi}_{p_{\alpha\beta}}\nabla_{\alpha}\vec{V}_{p_{\beta}}$.
Note that after this step the electric field has been eliminated from the
energy balance equation. This is a crucial step in deriving the proper form of
the frictional heating, as described below.

Typically, equations like Eq. (14) represent the final form of the thermal-
balance equation. These equations are most convenient for calculations. In
order to clarify the physical meaning of some terms, however, it is helpful to
recast Eq. (14) in a slightly different form. Rewriting the continuity Eq. (6)
as

$\frac{D_{p}n_{p}}{Dt}+n_{p}\nabla\cdot\vec{V}_{p}=\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p},$

we recast the two first terms in the LHS of Eq. (14) as

$\displaystyle\frac{3n_{p}}{2}\
\frac{D_{p}T_{p}}{Dt}+n_{p}T_{p}\nabla\cdot\vec{V}_{p}$
$\displaystyle=\frac{3n_{p}}{2}\ \frac{D_{p}T_{p}}{Dt}-T_{p}\
\frac{D_{p}n_{p}}{Dt}+T_{p}\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p}$
$\displaystyle=n_{p}T_{p}\
\frac{D_{p}s_{p}}{Dt}+T_{p}\int\left(\frac{df_{p}}{dt}\right)_{\operatorname{col}}d^{3}v_{p},$
(15)

where $s_{p}\equiv\ln(T_{p}^{3/2}/n_{p})=\ln(P_{p}^{3/2}/n_{p}^{5/2})$
represents the specific entropy of the $p$-species fluid (Braginskii, 1965)
(for a single-atomic gas, the adiabatic coefficient $\gamma=5/3$). This recast
allows interpreting $n_{p}T_{p}\nabla\cdot\vec{V}_{p}$ as the adiabatic
heating (cooling) term. The two remaining terms in the LHS of Eq. (14)
describe the work performed by viscous forces and the fluid heat conductance.
All these processes are collisionless.

All collisional processes in the thermal balance Eq. (14) are described by its
RHS. After rearranging the last term in Eq. (15) to the RHS of Eq. (14), the
last term there becomes
$\left(m_{p}V_{p}^{2}/2-5T_{p}/2\right)\int\left(df_{p}/dt\right)_{\operatorname{col}}d^{3}v_{p}$.
All integral terms involving $\left(df_{p}/dt\right)_{\operatorname{col}}$
describe the frictional heating and thermal inflows (outflows) associated with
possible emergence (disappearance) of $p$-particles as a result of ionization,
recombination, etc. For the general form of
$\left(df_{p}/dt\right)_{\operatorname{col}}$, calculating the frictional
heating is not an easy task. Below we employ two different kinds of further
approximation: one is more appropriate for heavy single-charged ions (Section
5), while the other is suitable for light electrons (Section 6).

Before proceeding further, we emphasize that in general no truncated chain of
moment equations is closed because starting from the momentum equation every
further moment equation involves higher-order moments. To allow the moment
equation chain to be rigorously truncated, the most appropriate is the near-
equilibrium case when the particle distribution function, along with its small
perturbations, remains reasonably close to Maxwellian (Dimant and Sudan,
1995a; Kissack et al., 1995). This case allows describing the particle
kinetics using a restricted number of spatially and temporarily varying
parameters, such as the particle density, temperature, and average drift
velocity (5-moment equations). In real situations, however, this is not always
the case, that is why inconsistencies in the fluid description often happen
(e.g., García-Colín et al., 2004; Velasco et al., 2002). Higher-order sets of
equations allow more serious deviations from Maxwellian, but still with a
restricted number of additional fluid parameters. Fluid models that include
restricted numbers of equations using approximate closures, such as the 5 or
13-moment models (Schunk and Nagy, 2009), can be successfully employed in
situations when there are no sharp gradients, extreme fields, abundant
superthermal particles, or extremely large temperature differences between
different species of the colliding particles. These conditions are usually met
at the equatorial electrojet. If they cannot be met, then the adequate
description of plasma dynamics may require a direct solution of the
corresponding collisional kinetic equation.

##  5 BGK collision kinetics and the fluid model for ions

In this section, we consider the ion kinetics in low-ionized, highly
collisional plasma. Bearing in mind mainly the lower-E/upper-D regions of the
ionosphere or similar media, we can neglect coulomb collisions between the
charged particles compared to their much more frequent collisions with the
neutrals. For the ion-neutral collision integral, we can use the simple
Bhatnagar-Gross-Krook (BGK) model (Bhatnagar et al., 1954). Disregarding
ionization-recombination processes, we can write the simplest BGK collision
operator in the neutral frame of reference as

$\left(\frac{df_{i}}{dt}\right)_{\operatorname{col}}^{\mathrm{BGK}}=\nu_{i}\left(f_{\mathrm{eff}}-f_{i}\right),$
(16)

where $f_{\mathrm{eff}}$ is a fictitious Maxwellian function, normalized to
the locally varying ion density, $n_{i}(\vec{r},t)$, with the constant neutral
temperature $T_{n}$:

$f_{\mathrm{eff}}(\vec{v}_{i},\vec{r},t)\equiv
n_{i}(\vec{r},t)\left(\frac{m_{i}}{2\pi T_{n}}\right)^{3/2}\exp\left(-\
\frac{m_{i}v_{i}^{2}}{2T_{n}}\right).$ (17)

For the BGK model, it is essential that the ion-neutral collision frequency,
$\nu_{i}$, is assumed constant. The model collision term in the form of Eq.
(16) conserves the local number of particles. Applied to both ions and
neutrals, the BGK model also conserves the total momentum of the two colliding
particles.

The BGK model does not follow from Boltzmann’s collision integral under any
rigorous approximations, but it is a reasonable fit for single-charged ions
that collide, predominantly elastically, with the surrounding neutrals of the
same (or close) mass. Recent 2-D hybrid computer simulations of the Farley-
Buneman instability that used for ions this kinetic equation (Kovalev et al.,
2008) have demonstrated a good agreement with similar results of the more
accurate fully-kinetic PIC or hybrid simulations (Janhunen, 1995; Oppenheim et
al., 2008, 1996, 1995; Oppenheim and Dimant, 2004). There are two major
reasons why this oversimplified model works well for the predominantly ion-
neutral collisions typical for the lower ionosphere. First, within a 1000 K
temperature range, the ion-neutral collision frequency is nearly velocity-
independent (Maxwell molecule collisions) (Schunk and Nagy, 2009). Second,
collisions of ions with neutral particles of the same or close mass have
roughly equal rates of the average momentum and energy transfer, described by
the single parameter $\nu_{i}$. Both these factors distinguish dramatically
the ion-neutral collisions from the electron-neutral ones, as we discuss in
the following section.

For the distribution function of single-charged positive ions,
$f_{i}(\vec{v}_{i},\vec{r},t)$, the BGK kinetic equation in the conservative
(divergence) form is given by

$\partial_{t}f_{i}+\nabla\cdot\left(\vec{v}_{i}f_{i}\right)+\partial_{\vec{v}_{i}}\cdot\left[\left(\frac{e\vec{E}}{m_{i}}+\Omega_{i}\vec{v}_{i}\times\hat{b}\right)f_{i}\right]=\nu_{i}\left(f_{\mathrm{eff}}-f_{i}\right).$
(18)

In this section, we derive the 5-moment ion fluid-model equations for
$n_{i}=\int f_{i}d^{3}v_{i}$,
$\vec{V}_{i}=\left\langle\vec{v}_{i}\right\rangle=\int\vec{v}_{i}f_{i}d^{3}v_{i}$,
and
$T_{i}=m_{i}\left\langle\delta\vec{v}_{i}\right\rangle^{2}/3=(m_{i}/3n_{i})\int\delta
v_{i}^{2}f_{i}d^{3}v_{i}$, where
$\delta\vec{v}_{i}\equiv\vec{v}_{i}-\vec{V}_{i}$. We will largely follow the
major steps of the previous section, but will do this specifically for the
ions in the BGK approximation. This will help us to better understand the
physical meaning of the algebraic manipulations and intermediate results.

Integrating Eq. (18) over velocities, we obtain the ion continuity equation,

$\partial_{t}n_{i}+\nabla\cdot\left(n_{i}\vec{V}_{i}\right)=\frac{D_{i}n_{i}}{Dt}+n_{i}\nabla\cdot\vec{V}_{i}=0.$
(19)

Further, integrating Eq. (18) over ion velocities with the weighting function
$\vec{v}_{i}$, we obtain the total momentum balance equation,

$\partial_{t}\left(n_{i}\vec{V}_{i}\right)+\partial_{x_{m}}\int
v_{i_{k}}v_{i_{m}}fd^{3}v_{i}-n_{i}\left(\frac{e\vec{E}}{m_{i}}+\Omega_{i}\vec{V}_{i}\times\hat{b}\right)=-\nu_{i}n_{i}\vec{V}_{i}.$
(20)

Using Eq. (19) and recalling the definition of the pressure in Eq. (8), we
obtain the final momentum equation,

$m_{i}n_{i}\
\frac{D_{i}\vec{V}_{i}}{Dt}=m_{i}n_{i}\left(\frac{e\vec{E}}{m_{i}}+\Omega_{i}\vec{V}_{i}\times\hat{b}\right)-\nabla\cdot\mathbf{P}_{i}-\nu_{i}n_{i}m_{i}\vec{V}_{i}.$
(21)

Integrating Eq. (18) with the weighting function $m_{i}v_{i}^{2}/2$, we obtain

$\displaystyle\partial_{t}\left(\frac{m_{i}n_{i}V_{i}^{2}}{2}+\frac{3n_{i}T_{i}}{2}\right)+\nabla\cdot\int\vec{v}_{i}f_{i}\
\frac{m_{i}v_{i}^{2}}{2}\ d^{3}v_{i}$
$\displaystyle=n_{i}e\vec{E}\cdot\vec{V}_{i}+\nu_{i}\left[\frac{3}{2}\
n_{i}\left(T_{n}-T_{i}\right)-\frac{m_{i}n_{i}V_{i}^{2}}{2}\right].$ (22)

Using the easily derivable relation

$\int\vec{v}_{i}f_{i}\ \frac{m_{i}v_{i}^{2}}{2}\
d^{3}v_{i}=\vec{V}_{i}\left(\frac{m_{i}n_{i}V_{i}^{2}}{2}+\frac{5n_{i}T_{i}}{2}\right)+\vec{q}_{i},$

where the vector components of $\vec{q}_{i}$ are given by

$q_{i\alpha}\equiv m_{i}\int\delta v_{i_{\alpha}}\ \frac{\delta v_{i}^{2}}{2}\
f_{i}d^{3}v_{i}+\sum_{\beta=1}^{3}V_{i_{\beta}}\mathbf{\Pi}_{i_{\alpha\beta}},$
(23)

we obtain

$\displaystyle\frac{D_{i}}{Dt}\left(\frac{m_{i}n_{i}V_{i}^{2}}{2}+\frac{3n_{i}T_{i}}{2}\right)$
$\displaystyle+\left(\frac{m_{i}n_{i}V_{i}^{2}}{2}+\frac{3n_{i}T_{i}}{2}\right)\nabla\cdot\vec{V}_{i}+\nabla\cdot\left(n_{i}T_{i}\vec{V}_{i}+\vec{q}_{i}\right)$
$\displaystyle=n_{i}e\vec{E}\cdot\vec{V}_{i}+\nu_{i}\left[\frac{3}{2}\
n_{i}\left(T_{n}-T_{i}\right)-\frac{m_{i}n_{i}V_{i}^{2}}{2}\right].$ (24)

Taking a scalar product of Eq. (21) with $\vec{V}_{i}$, we obtain

$n_{i}\
\frac{D_{i}}{Dt}\left(\frac{m_{i}V_{i}^{2}}{2}\right)=n_{i}e\vec{E}\cdot\vec{V}_{i}-\vec{V}_{i}\cdot\nabla\cdot\mathbf{P}_{i}-\nu_{i}m_{i}n_{i}V_{i}^{2}.$
(25)

Using Eq. (19) and $\mathbf{P}_{i}=n_{i}T_{i}\mathbf{I}+\mathbf{\Pi}_{i}$, we
rewrite Eq. (25) as

$\displaystyle\frac{D_{i}}{Dt}\left(\frac{n_{i}m_{i}V_{i}^{2}}{2}\right)$
$\displaystyle=n_{i}e\vec{E}\cdot\vec{V}_{i}-\vec{V}_{i}\cdot\nabla\left(n_{i}T_{i}\right)-\vec{V}_{i_{\alpha}}\nabla_{\beta}\mathbf{\Pi}_{i_{\alpha\beta}}-\nu_{i}m_{i}n_{i}V_{i}^{2}-\frac{n_{i}m_{i}V_{i}^{2}}{2}\
\nabla\cdot\vec{V}_{i}.$ (26)

Rewriting Eq. (19) as

$T_{i}\ \frac{D_{i}}{Dt}\left(\frac{3n_{i}}{2}\right)+\frac{3n_{i}T_{i}}{2}\
\nabla\cdot\vec{V}_{i}=0$ (27)

we subtract both Eqs. (26) and (27) from Eq. (24) and obtain

$\displaystyle\frac{3n_{i}}{2}\
\frac{D_{i}T_{i}}{Dt}+n_{i}T_{i}\cdot\nabla\vec{V}_{i}+\nabla_{\alpha}q_{\alpha}-V_{i\beta}\nabla_{\alpha}\mathbf{\Pi}_{i_{\alpha\beta}}$
$\displaystyle=\frac{\nu_{i}m_{i}n_{i}V_{i}^{2}}{2}+\frac{3}{2}\
\nu_{i}n_{i}\left(T_{n}-T_{i}\right).$

Using

$\nabla_{\alpha}q_{i\alpha}-V_{i\beta}\nabla_{\alpha}\mathbf{\Pi}_{i_{\alpha\beta}}=m_{i}\nabla_{\alpha}\int\delta
v_{i_{\alpha}}\ \frac{\delta v_{i}^{2}}{2}\
f_{i}d^{3}v_{i}+\mathbf{\Pi}_{i_{\alpha\beta}}\nabla_{\alpha}V_{i\beta}$

and $n_{i}\nabla\cdot\vec{V}_{i}=-D_{i}n_{i}/dt$, we obtain the final ion
balance equation

$\displaystyle\frac{3n_{i}}{2}\ \frac{D_{i}T_{i}}{Dt}-T_{i}\
\frac{D_{i}n_{i}}{Dt}+\nabla\cdot\int m_{i}\delta\vec{v}_{i}\ \frac{\delta
v_{i}^{2}}{2}\
f_{i}d^{3}v_{i}+\mathbf{\Pi}_{i_{\alpha\beta}}\nabla_{\alpha}V_{i\beta}$
$\displaystyle=\frac{\nu_{i}m_{i}n_{i}V_{i}^{2}}{2}+\frac{3}{2}\
\nu_{i}n_{i}\left(T_{n}-T_{i}\right).$ (28)

The two last terms in the LHS of the ion thermal balance Eq. (28) describe the
thermal conduction. The thermal flux given by the integral term and
$\mathbf{\Pi}_{i_{\alpha\beta}}$ should be determined from higher-order moment
equations. Alternatively, Eq. (28) could be closed using proper approximations
like Maxwellian $f_{i}$.

The first term in the RHS of Eq. (28) describes the total ion frictional
heating. This term has originated from a subtraction of the frictional energy
loss described by

$\nu_{i}\left(\int\frac{m_{i}v^{2}}{2}\
f_{i}d^{3}v_{i}-\frac{3n_{i}T_{i}}{2}\right)=\frac{\nu_{i}m_{i}n_{i}V_{i}^{2}}{2},$

from the total work performed during the collisional braking of the ion fluid
as a whole, $\nu_{i}m_{i}n_{i}V_{i}^{2}$, given by the last term in Eq. (25).
This means that when the ion fluid is frictionally braked only the half of the
released energy heats the ions, while the other half goes directly to the
colliding neutrals. Incidentally, this is precisely the total amount of
frictional heating for ions and neutrals of the equal mass, $m_{n}=m_{i}$,
following from a more detailed kinetic theory (Schunk and Nagy, 2009; Dimant
and Oppenheim, 2004),

$\frac{\nu_{i}m_{i}m_{n}n_{i}V_{i}^{2}}{m_{i}+m_{n}}=\frac{\nu_{i}m_{i}n_{i}V_{i}^{2}}{2}.$

Thus, the BGK model of ion-neutral collisions describes correctly the ion
frictional heating if the colliding partners have the same or close masses.
This means that one can successfully employ for ions moment equations
(27)–(28) under moderate conditions when the ion distribution function is
reasonably close to Maxwellian. Such conditions mostly apply to the equatorial
E region, rather than to the high-latitude ionosphere.

##  6 Collisional kinetics and the fluid model for electrons

This section is the central piece of this paper. It derives the electron fluid
equations from an approximate, but rigorous, kinetic theory based on
characteristics of the actual physical conditions and wave processes in the
E-region ionosphere. For electrons, the oversimplified BGK collision model
(employed above for ions) can apply only to plasma processes whose
characteristic wave frequencies exceed substantially the electron collision
frequencies. However, for low-frequency processes in the highly collisional
E/D-region ionosphere, where the opposite condition usually holds (see Dimant
and Sudan, 1995a, and references therein), the electron BGK collision model is
totally unsuitable. The main reason is that the rate of electron-neutral
collisional exchange of momentum, $\nu_{en}$, is a few orders of magnitude
larger than the corresponding rate of the energy exchange,
$\delta_{en}\nu_{en}$ (Gurevich, 1978). This means that during collisions with
heavy neutrals the light electrons scatter over angles in the velocity space
much more frequently than they change their kinetic energy. In low-frequency
processes of the lower ionosphere, this leads to an efficient isotropization
of the electron distribution function. The BGK model, however, completely
ignores this feature. In addition, the BGK model does not cover the clearly
pronounced velocity dependence of the kinetic electron-neutral collision
frequency $\nu_{en}(v_{e})$ (Gurevich, 1978; Schunk and Nagy, 2009). This
velocity dependence plays an important role for some E-region instabilities,
(see, e.g., Dimant and Sudan, 1997, and references therein), and it modifies
the instability and wave characteristics.

###  6.1 General kinetic approach and momentum equations

In a weakly ionized plasma of the lower ionosphere, collisions of an electron
with other charged particles, including other electrons, $\nu_{ee},\nu_{ei}$,
are usually negligible compared to electron neutral collisions,
$\nu_{e}\approx\nu_{en}$. At altitudes above 75 km, strongly magnetized
electrons, involved in low-frequency processes with
$\omega\ll\nu_{e}\ll\Omega_{e}$, have an almost isotropic velocity
distribution whose speed dependence can deviate significantly from Maxwellian.
For such processes, an adequate kinetic description is by expanding the
velocity distribution function $f_{e}(\vec{r},t,\vec{v}_{e})$ in Legendre
polynomials with respect to angles in the velocity space (Shkarofsky et al.,
1966; Gurevich, 1978; Khazanov, 2011). To the first-order accuracy with
respect to a small anisotropy of $f_{e}(\vec{r},t,\vec{v}_{e})$, one can
represent the total electron distribution function as a combination of the
major isotropic part, $F_{0}(\vec{r},t,v_{e})$, where
$v_{e}\equiv|\vec{v}_{e}|$, and a relatively small directional part determined
by a single vector-function $\vec{f}_{1}(\vec{r},t,v_{e})$ (Gurevich, 1978;
Dimant and Sudan, 1995a),

$f_{e}(\vec{r},t,\vec{v}_{e})\approx
F_{0}(\vec{r},t,v_{e})+\frac{\vec{f}_{1}(\vec{r},t,v_{e})\cdot\vec{v}_{e}}{v_{e}}=F_{0}+|\vec{f}_{1}|\cos\theta,$
(29)

where $\theta$ is the angle between $\vec{f}_{1}$ and $\vec{v}_{e}$. Here we
assume that $|\vec{f}_{1}|\ll F_{0}$, along with the similar inequalities for
the speed derivatives (see below). The major isotropic part, $F_{0}$,
determines scalar velocity-averaged characteristics of the electron fluid,
such as the electron density and temperature, while the small directional
part, $\vec{f}_{1}$, determines vector characteristics, such as the average
drift velocity and various fluxes. In this approximation, any higher-order
anisotropies are neglected. For electrons in the highly collisional E-region
ionosphere, the higher-order anisotropies usually play no role (see below).

The assumption of $|\vec{f}_{1}|\ll F_{0}$ is well justified for electrons
within the kinetic energy range $\mathcal{E}_{e}<2$ eV ($v_{e}<1000$ km/s).
This range usually includes both the thermal bulk of electrons
($\mathcal{E}_{e}\lesssim 0.03$ eV for the cold E-region ionosphere) and a
significant fraction of superthermal electrons. In this energy rate, the ratio
of the mean, mostly inelastic, collisional energy loss to that of the
predominantly elastic momentum loss,
$\delta_{en}(v_{e})\nu_{en}(v_{e})/\nu_{en}(v_{e})=\delta_{en}(v_{e})$, is
usually quite small: $\delta_{en}(v_{e})\sim(2$–$4)\times 10^{-3}$ (Gurevich,
1978) (although it is two orders of magnitude larger than the corresponding
purely elastic rate, $\delta_{en}^{\mathrm{elas}}\approx 2m_{e}/m_{n}$). The
ratio of $|\vec{f}_{1}|$ to $F_{0}$ is typically $\sim\sqrt{\delta_{en}}$, so
that the directional part of the electron distribution function in Eq. (29)
turns out to be automatically small compared to the major isotropic part,
$|\vec{f}_{1}|\ll F_{0}$. However, this brings up the following question. If
there were an imposed DC electric field, $\vec{E}\perp\vec{B}$, so strong that
the corresponding $\vec{E}\times\vec{B}$-drift velocity,
$\vec{V}_{\mathrm{dr}}=\vec{E}\times\vec{B}/B^{2}$, would be comparable to the
mean electron thermal speed, $v_{Th}^{e}=(T_{e}/m_{e})^{1/2}$, then the
condition of $|\vec{f}_{1}|\ll F_{0}$ would become invalid. As a matter of
fact, however, such strong field would heat electrons so much that the heated
thermal velocity $v_{Th}^{e}$ would automatically exceed
$\vec{V}_{\mathrm{dr}}$. If the new electron temperature is $\lesssim
23,000~{}$K (corresponding to 2 eV) then the approximation (29) still holds.
This is a significant difference of electrons from heavy ions with
$\delta_{in}\simeq 1$.

The kinetic equation with the general electron-neutral collision operator,
$(df_{e}/dt)_{\text{{coll}}}$, leads to the two coupled equations for
$F_{0}(\vec{r},t,v_{e})$ and $\vec{f}_{1}(\vec{r},t,v_{e})$ (Gurevich, 1978;
Dimant and Sudan, 1995a):

$\displaystyle\partial_{t}F_{0}+\frac{v_{e}}{3}\
\nabla\cdot\vec{f}_{1}-\frac{e}{3m_{e}v_{e}^{2}}\ \frac{\partial}{\partial
v_{e}}\left(v_{e}^{2}\vec{E}\cdot\vec{f}_{1}\right)$ $\displaystyle=S_{0},$
(30a)
$\displaystyle\partial_{t}\vec{f}_{1}-\Omega_{e}\hat{b}\times\vec{f}_{1}+v_{e}\nabla
F_{0}-\frac{e\vec{E}}{m_{e}}\ \frac{\partial F_{0}}{\partial v_{e}}$
$\displaystyle=\vec{S}_{1},$ (30b) where
$S_{0}\equiv\frac{1}{2}\int_{-1}^{1}\left(\frac{df_{e}}{dt}\right)_{\operatorname{col}}d\left(\cos\vartheta\right),\qquad\vec{S}_{1}\equiv\frac{3}{2}\int_{-1}^{1}\left(\frac{df_{e}}{dt}\right)_{\operatorname{col}}\frac{\vec{f}_{1}}{\left|\vec{f}_{1}\right|}\left(\cos\vartheta\right)d\left(\cos\vartheta\right)$
(31)

(note that the expressions for $S_{0,1}$ in Dimant and Sudan (1995a) missed
the correct normalization factors). Bearing in mind moderately fast wave
processes, $\tau_{\mathrm{rec}}^{-1}\ll\omega\ll\nu_{e}$, where
$\tau_{\mathrm{rec}}$ is an effective recombination lifetime at a given
altitude, we will ignore ionization-recombination processes, as we did above
for the ions. The kinetic description of electrons based on Eq. (30) differs
dramatically from any kinetic description based on the BGK collision model.

Fluid equations based on Eq. (30), usually implying a nearly-Maxwellian
velocity distribution, have been successfully explored by a number of
researchers (see, e.g., Gurevich, 1978; Dimant and Sudan, 1995a, and
references therein). However, the form of major fluid equations presented in
Gurevich (1978), Chapter 5, does not clearly show the basic structure of
generic Eqs. (9) and (14) or similar ion Eqs. (21) and (28). By this, we mean
that Gurevich’s equations show neither explicit adiabatic heating and cooling,
nor frictional heating $\propto V_{e,i}^{2}$. Adiabatic terms proportional to
$(\gamma_{e,i}-1)$ in Gurevich (1978) Eqs. (5.3) and (5.4) and the
corresponding terms in the following equations appear to have been introduced
“by hand” and are actually extraneous. One can verify that these adiabatic
terms have already been implicitly distributed among other terms of the
temperature balance equations within the corresponding fluxes given by
Gurevich (1978) equations (5.8)–(5.11), so that they are accounted in Eqs.
(5.3)–(5.4) twice.

The explicit adiabatic terms show up naturally in the kinetic approaches based
on small perturbations of the distribution function shifted by the average
particle drift velocity. These approaches clearly differ from that based on
perturbations of the non-shifted velocity distribution, as in Eq. (29)
resulting in Eq. (30). For relatively small drift velocities, however, the two
different approaches should yield the same results. Below we demonstrate that
the kinetic approach based on Eqs. (29)–(30) does reproduce in a rigorous and
natural way the electron fluid equations with the correct adiabatic heating
and cooling, frictional heating, etc. We will also calculate kinetic
corrections associated with the general velocity dependence of the electron-
neutral collision frequency and non-Maxwellian velocity distribution. The
Appendix contains details of these calculations.

In accord with the low-frequency condition of $\omega\ll\nu_{e}\ll\Omega_{e}$,
we neglect in Eq. (30b) the electron inertia term $\partial_{t}\vec{f}_{1}$
and use a standard approximation
$\vec{S}_{1}\approx-\nu_{e}(v_{e})\vec{f}_{1}$ (Gurevich, 1978; Dimant and
Sudan, 1995a). This allows us to close this set of equations in a simple way.
As a result, we obtain

$-\ \frac{e\vec{E}}{m_{e}}\ \frac{\partial F_{0}}{\partial
v_{e}}-\Omega_{e}\hat{b}\times\vec{f}_{1}+v_{e}\nabla
F_{0}=-\nu_{e}(v_{e})\vec{f}_{1}.$ (32)

Resolving this vector equation with respect to $\vec{f}_{1}$, we obtain

$\vec{f}_{1}(v_{e})=-\ \mathbf{N}(v_{e})\cdot\vec{K}F_{0},$ (33)

where the kinetic electron mobility tensor $\mathbf{N}(v)$ and the
differential vector operator $\vec{K}$ are given by

$\mathbf{N}(v_{e})\approx\left[\begin{array}[c]{ccc}\frac{\nu_{e}(v_{e})}{\Omega_{e}^{2}}&\frac{1}{\Omega_{e}}&0\\\
-\ \frac{1}{\Omega_{e}}&\frac{\nu_{e}(v_{e})}{\Omega_{e}^{2}}&0\\\
0&0&\frac{1}{\nu_{e}(v_{e})}\end{array}\right],$ (34) $\vec{K}\equiv
v_{e}\nabla-\frac{e\vec{E}}{m_{e}}\ \frac{\partial}{\partial v_{e}}.$ (35)

Here and elsewhere, we neglect second-order small terms $\sim\nu_{e}^{2}$
compared to $\Omega_{e}^{2}$ and represent all tensors in the matrix form for
the Cartesian system $\hat{x},\hat{y},\hat{z}$ with the $\hat{z}$-axis along
$\vec{B}$. We can write Eqs. (33)–(34) explicitly in terms of the parallel
($\parallel$) and perpendicular ($\perp$) to $\vec{B}$ components as

$f_{1\parallel}=-\ \frac{1}{\nu_{e}(v_{e})}\
K_{\parallel}F_{0},\qquad\vec{f}_{1\perp}=-\left(\frac{\nu_{e}(v)\vec{K}_{\perp}}{\Omega_{e}^{2}}+\frac{\hat{b}}{\Omega_{e}}\times\vec{K}_{\perp}\right)F_{0},$
(36)

where $\hat{b}=\hat{z}$ is the unit vector along $\vec{B}$. The spatial
derivatives in Eq. (33) or (36) express the drift-diffusion approximation in
the collisional kinetic theory, while the velocity derivatives describe
electron energy variations caused by the electric field $\vec{E}$.

Furthermore, expressing the RHS of Eq. (30a) as (Gurevich, 1978)

$S_{0}=\frac{1}{2v_{e}^{2}}\frac{\partial}{\partial
v_{e}}\left[v_{e}^{2}\delta_{en}\nu_{e}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{\partial
F_{0}}{\partial v_{e}}\right)\right],$ (37)

we obtain (Dimant and Sudan, 1995a)

$\partial_{t}F_{0}+\frac{1}{3v_{e}^{2}}\
\vec{K}\cdot\left(v_{e}^{2}\vec{f}_{1}\right)=\frac{1}{2v_{e}^{2}}\frac{\partial}{\partial
v_{e}}\left[v_{e}^{2}\delta_{en}\nu_{e}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{\partial
F_{0}}{\partial v_{e}}\right)\right].$ (38)

Expressing here $\vec{f}_{1}$ in terms of $F_{0}$ via Eq. (33) or (36), we
obtain a closed kinetic equation for the major isotropic distribution
function, $F_{0}(\vec{r},t,v)$. Its solution, with the use of (33) or (36),
provides both parts of the distribution function, so that its scalar and
vector moments can be calculated by a straightforward speed integration.
According to Eq. (29), the lowest-order moments of the electron distribution
function, such as the electron density, mean drift velocity, and temperature,
are given by

$n_{e}\equiv
4\pi\int_{0}^{\infty}\\!\\!F_{0}v_{e}^{2}dv_{e},\qquad\vec{V}_{e}\equiv\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\\!\\!\vec{f}_{1}v_{e}^{3}dv_{e},\qquad
T_{e}\equiv\frac{4\pi
m_{e}}{3n_{e}}\int_{0}^{\infty}\\!\\!F_{0}v_{e}^{4}dv_{e}.$ (39)

A direct solution of the kinetic Eq. (38) would be the most accurate and
general way to describing the electron behavior (Dimant and Sudan, 1995a).
However, the goal of this paper is to obtain a set of the lowest-order fluid
equations in order to properly describe E-region plasma processes, even if
this set of equations is not fully closed due to possible deviations of the
electron velocity distribution function from Maxwellian.

As above, we start from the particle conservation. Using the definitions of
Eq. (39) and integrating Eq. (38) over $v_{e}$ with the weighting function
$4\pi v_{e}^{2}$, we obtain the standard electron continuity equation,

$\partial_{t}n_{e}+\nabla\cdot(n_{e}\vec{V}_{e})=0.$ (40)

Next, we integrate Eq. (32) with the weighting function $4\pi
v_{e}^{3}/(3n_{e})$. Applying the integration by parts, we obtain an equation

$\frac{e\vec{E}}{m_{e}}-\Omega_{e}\hat{b}\times\vec{V}_{e}+\frac{\nabla\left(n_{e}T_{e}\right)}{m_{e}n_{e}}+\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\vec{f}_{1}\nu_{e}v_{e}^{3}dv_{e}=0,$
(41)

that describes the momentum balance of the inertialess electron fluid.
Equation (41) includes the Lorentz force, pressure gradient, and collisional
friction. As we show in the Appendix, in the general case of a velocity-
dependent collision frequency, $\nu_{e}(v)$, the last term in the LHS of Eq.
(41), in addition to the collisional friction, may also include an anisotropic
addition to the total pressure gradient.

Taking a scalar product of Eq. (41) with $m_{e}n_{e}\vec{V}_{e}$, we obtain
the expression

$\vec{V}_{e}\cdot\left[n_{e}e\vec{E}+\nabla\left(n_{e}T_{e}\right)\right]+\frac{4\pi
m_{e}}{3}\
\vec{V}_{e}\cdot\\!\int_{0}^{\infty}\vec{f}_{1}\nu_{e}v_{e}^{3}dv_{e}=0$ (42)

analogous to Eq. (25) for ions. This expression represents the total work done
by the electric field and other forces on the average electron flow. We will
employ this expression right below.

Now we derive an equation describing the total energy balance. Integrating Eq.
(38) with the weighting function $2\pi m_{e}v_{e}^{4}$, we obtain

$\displaystyle\partial_{t}\left(\frac{3n_{e}T_{e}}{2}\right)+\frac{2\pi
m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\
\vec{f}_{1}v_{e}^{5}dv_{e}+n_{e}e\vec{E}\cdot\vec{V}_{e}$ $\displaystyle=-\
2\pi
m_{e}\int_{0}^{\infty}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{dF_{0}}{dv_{e}}\right)\delta_{en}\nu_{e}v_{e}^{3}dv_{e}.$
(43)

Using Eq. (42), we eliminate from Eq. (43) the work done by the electric field
on the average flow, $n_{e}e\vec{E}\cdot\vec{V}_{e}$, and obtain

$\displaystyle\partial_{t}\left(\frac{3n_{e}T_{e}}{2}\right)-\vec{V}_{e}\cdot\nabla\left(n_{e}T_{e}\right)+\frac{2\pi
m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\\!\\!\ \vec{f}_{1}v_{e}^{5}dv_{e}$
$\displaystyle=\frac{4\pi m_{e}}{3}\
\vec{V}_{e}\cdot\int_{0}^{\infty}\\!\\!\vec{f}_{1}\nu_{e}v_{e}^{3}dv_{e}-2\pi
m_{e}\int_{0}^{\infty}\\!\\!\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{dF_{0}}{dv_{e}}\right)\delta_{en}\nu_{e}v_{e}^{3}dv_{e}.$
(44)

Here we have rearranged the terms between the two sides of the equation in
such a way that all terms proportional to the collision frequency remain in
the RHS, while all other terms are put in the LHS. After so doing, it may be
tempting to interpret the first term in the RHS of Eq. (44) as the electron
frictional heating. In the general case of velocity-dependent $\nu_{e}(v)$,
however, this interpretation would not be perfectly accurate, as we show in
Appendix and section 6.3 below.

Equation (44) is not yet the final form of the thermal-balance equation. It
needs to be further transformed into a form similar to Eq. (14) or (28). In
Appendix, we develop this recast for the general case of velocity-dependent
$\nu_{e}(v)$. However, right below we proceed with the simplest model of
constant $\nu_{e}$ and $\delta_{en}$. This model is inaccurate for electron-
neutral collisions of the lower ionosphere (Gurevich, 1978; Schunk and Nagy,
2009), but it will allow us to clarify basic ideas of closing Eq. (44).

###  6.2 Constant collisional parameters

For constant $\nu_{e}$ and $\delta_{en}$, using the definitions of Eq. (39)
and integrating the last term of Eq. (44) by parts, we obtain

$\displaystyle\partial_{t}\left(\frac{3n_{e}T_{e}}{2}\right)-\vec{V}_{e}\cdot\nabla\left(n_{e}T_{e}\right)+\frac{2\pi
m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\ \vec{f}_{1}v_{e}^{5}dv_{e}$
$\displaystyle=m_{e}\nu_{e}n_{e}V_{e}^{2}+\frac{3}{2}\
\delta_{en}\nu_{e}n_{e}\left(T_{n}-T_{e}\right),$ (45)

Using Eq. (33), we rewrite the third term in the LHS as

$\frac{2\pi m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\
\vec{f}_{1}v_{e}^{5}dv_{e}=-\ \frac{5}{2m_{e}}\
\nabla\cdot\mathbf{N\cdot}\left[\nabla\left(\lambda
n_{e}T_{e}^{2}\right)+n_{e}T_{e}e\vec{E}\right].$ (46)

Here the double-dot product involving a tensor means
$\nabla\cdot\mathbf{N}\cdot\nabla\ldots=\sum_{\alpha,\beta=1}^{3}\partial_{x_{\alpha}}(\mathbf{N}_{\alpha\beta}\partial_{x_{\beta}}\ldots)$
(and similar for $\nabla\cdot\mathbf{N}\cdot\vec{E}$) and we have also
introduced a dimensionless parameter of order unity, $\lambda$,

$\lambda\equiv\frac{4\pi
m_{e}^{2}}{15n_{e}T_{e}^{2}}\int_{0}^{\infty}v_{e}^{6}F_{0}dv_{e}=\frac{m_{e}\int_{0}^{\infty}v_{e}^{6}F_{0}dv_{e}}{5T_{e}\int_{0}^{\infty}v_{e}^{4}F_{0}dv_{e}}=\frac{3\left(\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{6}dv_{e}}{5\left(\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}\right)^{2}}.$
(47)

Note that for the Maxwellian isotropic part of the electron distribution
function,

$F_{0}=n_{e}\left(\frac{m_{e}}{2\pi T_{e}}\right)^{3/2}\exp\left(-\
\frac{m_{e}v_{e}^{2}}{2T_{e}}\right),$ (48)

we have $\lambda=1$.

Using Eq. (39) and (35), we obtain

$\vec{V}_{e}=-\ \frac{4\pi}{3n_{e}}\
\mathbf{N}\cdot\int_{0}^{\infty}v_{e}^{3}\vec{K}F_{0}dv_{e}=-\mathbf{N}\cdot\left[\frac{e\vec{E}}{m_{e}}+\frac{\nabla\left(n_{e}T_{e}\right)}{m_{e}n_{e}}\right].$
(49)

Multiplying Eq. (49) by $m_{e}n_{e}T_{e}$, we can rewrite it as

$-\mathbf{N}\cdot\left(n_{e}T_{e}e\vec{E}\right)=m_{e}n_{e}T_{e}\vec{V}_{e}+\mathbf{N}\cdot
T_{e}\nabla\left(n_{e}T_{e}\right).$

This relation allows us to eliminate the electric field from Eq. (46), so that
the latter becomes

$\displaystyle\frac{2\pi m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\
\vec{f}_{1}v_{e}^{5}dv_{e}$ $\displaystyle=\frac{5}{2m_{e}}\
\nabla\cdot\left\\{\mathbf{N\cdot}\left[\left(1-\lambda\right)T_{e}^{2}\nabla
n_{e}-\left(2\lambda-1\right)n_{e}T_{e}\nabla
T_{e}\right]+m_{e}n_{e}T_{e}\vec{V}_{e}\right\\}.$ (50)

Using Eqs. (40) and (50), after a simple algebra,

$\displaystyle\partial_{t}\left(\frac{3n_{e}T_{e}}{2}\right)-\vec{V}_{e}\cdot\nabla\left(n_{e}T_{e}\right)+\frac{5}{2}\
\nabla\cdot\left(n_{e}T_{e}\vec{V}_{e}\right)$
$\displaystyle=\partial_{t}\left(\frac{3n_{e}T_{e}}{2}\right)-\vec{V}_{e}\cdot\nabla\left(n_{e}T_{e}\right)+\frac{5}{2}\
\nabla\cdot\left(n_{e}T_{e}\vec{V}_{e}\right)$
$\displaystyle-\frac{5T_{e}}{2}\left[\partial_{t}n_{e}+\nabla\cdot(n_{e}\vec{V}_{e})\right]=\frac{3n_{e}}{2}\
\frac{D_{e}T_{e}}{Dt}-T_{e}\ \frac{D_{e}n_{e}}{Dt},$ (51)

we obtain the sought-for temperature-balance equation in a more standard form,

$\frac{3n_{e}}{2}\ \frac{D_{e}T_{e}}{Dt}-T_{e}\
\frac{D_{e}n_{e}}{Dt}-\nabla\cdot\vec{q}_{e}=m_{e}\nu_{e}n_{e}V_{e}^{2}+\frac{3}{2}\
\delta_{en}\nu_{e}n_{e}\left(T_{n}-T_{e}\right).$ (52)

Here the electron thermal flux density, $\vec{q}_{e}$, is given by

$\vec{q}_{e}=\frac{5T_{e}}{2m_{e}}\
\mathbf{N}\cdot\left[\left(2\lambda-1\right)n_{e}\nabla
T_{e}+\left(\lambda-1\right)T_{e}\nabla
n_{e}\right]=\vec{q}_{e\parallel}+\vec{q}_{e\mathrm{P}}+\vec{q}_{e\mathrm{H}},$
(53)

where its explicit parallel, Pedersen, and Hall components are given by

$\displaystyle\vec{q}_{e\parallel}$
$\displaystyle=\frac{5T_{e}\left[\left(2\lambda-1\right)n_{e}\nabla_{\parallel}T_{e}+\left(\lambda-1\right)T_{e}\nabla_{\parallel}n_{e}\right]}{2m_{e}\nu_{e}},$
$\displaystyle\vec{q}_{e\mathrm{P}}$
$\displaystyle=\frac{5T_{e}\nu_{e}\left[\left(2\lambda-1\right)n_{e}\nabla_{\perp}T_{e}+\left(\lambda-1\right)T_{e}\nabla_{\perp}n_{e}\right]}{2m_{e}\Omega_{e}^{2}},$
(54) $\displaystyle\vec{q}_{e\mathrm{H}}$
$\displaystyle=\frac{\Omega_{e}}{\nu_{e}}\left(\hat{b}\times\vec{q}_{e\mathrm{P}}\right)=\frac{5T_{e}\hat{b}\times\left[\left(2\lambda-1\right)n_{e}\nabla_{\perp}T_{e}+\left(\lambda-1\right)T_{e}\nabla_{\perp}n_{e}\right]}{2m_{e}\Omega_{e}}.$

The two first terms in the LHS of Eq. (52), as well as the similar ones in Eq.
(14) or (28), describe adiabatic heating or cooling of the electron fluid,
while $\nabla\cdot\vec{q}_{e}$ describes the heat conductivity. Note that the
Hall component of $\vec{q}_{e}$ can contribute into electron heat conductance
only as a quadratically-nonlinear effect and, since
$\nabla\cdot\vec{q}_{e\mathrm{H}}\propto(\nabla_{\perp}n_{e}\times\nabla_{\perp}T_{e})$
only if the gradients of $n_{e}$ and $T_{e}$ are not parallel.

As mentioned above, for Maxwellian $F_{0}(v_{e})$ we have $\lambda=1$, so that
the term in $\vec{q}_{e}$ proportional to $\nabla n_{e}$ disappears. This fact
can be understood as follows. If the major part of the distribution function
remains Maxwellian then it is determined only by two space-dependent
parameters: the density, $n_{e}$, and the temperature, $T_{e}$. If there is a
density gradient but no temperature gradient, then electrons of all energies
will diffuse from denser regions to less dense ones with no redistribution of
the temperature and hence with no heat conductivity.

If the electron velocity distribution deviates from Maxwellian (this happens,
e.g., when a low-ionized plasma heated by strong electric fields is embedded
in an abundant cold neutral atmosphere with a significantly different
temperature (Milikh and Dimant, 2003)) then the situation is more complicated.
The effective electron temperature $T_{e}$, which is proportional to the mean
electron chaotic energy, can be uniformly distributed, but the details of the
electron energy distribution may differ significantly in different regions of
space. The energy transport is stronger for electrons with higher energies
than it is for lower-energy electrons. Hence, if there are spatial gradients
of high-energy distribution tails then more energetic particles provide
stronger energy redistribution. This may make, e.g., some less dense regions
to be on average more energetic than the denser regions, even if they had
initially equal effective temperatures. Moreover, it is even possible to
imagine a situation when electron heat is transferred from cooler regions to
hotter ones, leading to a further electron temperature elevation in the
latter. This counter-intuitive, but theoretically possible, effect should not
surprise because a strongly non-Maxwellian, i.e., strongly non-equilibrium
plasma cannot be adequately described by the conventional equilibrium
thermodynamics.

###  6.3 Velocity-dependent parameters

In the actual lower ionosphere, the electron-neutral kinetic collision
frequency, $\nu_{e}$, and the energy loss fraction, $\delta_{en}$, have
clearly pronounced velocity dependencies (Gurevich, 1978; Schunk and Nagy,
2009). This does not allow $\nu_{e}(v_{e})$ and $\delta_{en}(v_{e})$ to be
factored out from the integrals in Eqs. (41)–(44), making the derivation of
the general momentum and temperature-balance equations more complicated than
that described in section 6.2. Such derivation is developed in detail in the
Appendix, while here we only present the results.

In the general case of velocity-dependent $\nu_{e}$ and $\delta_{en}$,
electron continuity Eq. (40) stays the same. The other two moment equations
have the same basic structure as (49) and (52), but they contain additional
terms and include a large number of dimensionless factors of order unity
listed in Eqs. (63)–(68) below.

The general inertialess expression for the average electron drift velocity
$\vec{V}_{e}$ is given by

$\vec{V}_{e}=-\ \frac{1}{m_{e}}\
\mathbf{M}\cdot\left[e\vec{E}+\frac{\nabla_{\perp}\left(n_{e}T_{e}\right)}{n_{e}}+\frac{\beta_{\parallel}}{\alpha_{\parallel}}\
\frac{\nabla_{\parallel}\left(n_{e}T_{e}\right)}{n_{e}}\right],$ (55)

where

$\mathbf{M}\equiv\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(v_{e}^{3}\mathbf{N}(v_{e})\right)}{dv_{e}}\
F_{0}(v_{e})dv_{e}=\left[\begin{array}[c]{ccc}\frac{\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}^{2}}&\frac{1}{\Omega_{e}}&0\\\
-\
\frac{1}{\Omega_{e}}&\frac{\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}^{2}}&0\\\
0&0&\alpha_{\parallel}\left\langle\frac{1}{\nu_{e}}\right\rangle_{e}\end{array}\right],$
(56)

an the tensor $\mathbf{N}$ is given by Eq. (34) and
$\left\langle\cdots\right\rangle_{e}$ denotes the velocity average over the
major (isotropic) part of the electron distribution function,

$\left\langle\cdots\right\rangle_{e}=\frac{4\pi\int_{0}^{\infty}\left(\cdots\right)F_{0}(v_{e})v_{e}^{2}dv_{e}}{n_{e}}=\frac{\int_{0}^{\infty}\left(\cdots\right)F_{0}(v_{e})v_{e}^{2}dv_{e}}{\int_{0}^{\infty}F_{0}(v_{e})v_{e}^{2}dv_{e}}.$
(57)

The electron current density is given by $\vec{j}_{e}=en_{e}\vec{V}_{e}$, so
that the electric conductivity tensor is given by
$\boldsymbol{\sigma}_{e}=(n_{e}e^{2}/m_{e})\mathbf{M}$. The corresponding
diagonal terms represent the Pedersen ($\propto\alpha_{P}$) and parallel
($\propto\alpha_{\parallel}$), while the antisymmetric off-diagonal terms
($\propto 1/\Omega_{e}$) represent the Hall conductivity.

The general thermal-balance equation is given by

$\displaystyle\frac{3n_{e}}{2}\ \frac{D_{e}T_{e}}{Dt}-T_{e}\
\frac{D_{e}n_{e}}{Dt}$
$\displaystyle+\frac{5}{2}\left(\frac{\rho_{\parallel}-\beta_{\parallel}}{\alpha_{\parallel}}\right)n_{e}T_{e}\nabla_{\parallel}\cdot\vec{V}_{e\parallel}+\left(\frac{5\rho_{\parallel}-3\alpha_{\parallel}-2\beta_{\parallel}}{2\alpha_{\parallel}}\right)\vec{V}_{e\parallel}\cdot\nabla_{\parallel}\left(n_{e}T_{e}\right)-\nabla\cdot\vec{q}_{e}$
$\displaystyle=\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}m_{e}n_{e}\left(V_{e\perp}^{2}+\frac{V_{\parallel}^{2}}{\alpha_{\parallel}\xi}\right)-2\pi
m_{e}\int_{0}^{\infty}\\!\\!v_{e}^{3}\delta_{en}\nu_{e}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{dF_{0}}{dv_{e}}\right)dv_{e},$
(58)

where

$\vec{q}_{e}\equiv\vec{q}_{eP}+\vec{q}_{eH}+\vec{q}_{e\parallel}=\mathbf{X}\cdot\frac{\nabla
T_{e}}{T_{e}}+\left(\mathbf{X}-\mathbf{\Lambda}\right)\cdot\frac{\nabla
n_{e}}{n_{e}},$ (59)

is the thermal-flux density with

$\displaystyle\mathbf{X}$
$\displaystyle=\frac{5n_{e}T_{e}}{2m_{e}}\left[\begin{array}[c]{ccc}\frac{\chi_{P}\left\langle\nu_{e}\right\rangle}{\Omega_{e}^{2}}&\frac{\chi_{H}}{\Omega_{e}}&0\\\
-\
\frac{\chi_{H}}{\Omega_{e}}&\frac{\chi_{P}\left\langle\nu_{e}\right\rangle}{\Omega_{e}^{2}}&0\\\
0&0&\frac{\chi_{\parallel}}{\left\langle\nu_{e}\right\rangle}\end{array}\right],$
(60d) $\displaystyle\mathbf{\Lambda}$
$\displaystyle=\frac{5n_{e}T_{e}}{2m_{e}}\left[\begin{array}[c]{ccc}\frac{\mu_{P}\left\langle\nu_{e}\right\rangle}{\Omega_{e}^{2}}&\frac{\lambda}{\Omega_{e}}&0\\\
-\
\frac{\lambda}{\Omega_{e}}&\frac{\mu_{P}\left\langle\nu_{e}\right\rangle}{\Omega_{e}^{2}}&0\\\
0&0&\frac{\mu_{\parallel}}{\left\langle\nu_{e}\right\rangle}\end{array}\right].$
(60h) The explicit Pedersen, Hall, and parallel components of $\vec{q}_{e}$
are given by  $\displaystyle\vec{q}_{eP}$
$\displaystyle\equiv\frac{5T_{e}\left\langle\nu_{e}\right\rangle\left[\chi_{P}n_{e}\nabla_{\perp}T_{e}+\left(\chi_{P}-\mu_{P}\right)T_{e}\nabla_{\perp}n_{e}\right]}{m_{e}\Omega_{e}^{2}},$
$\displaystyle\vec{q}_{eH}$
$\displaystyle\equiv\frac{5T_{e}\hat{b}\times\left[\chi_{H}n_{e}\nabla_{\perp}T_{e}+\left(\chi_{H}-\lambda\right)T_{e}\nabla_{\perp}n_{e}\right]}{m_{e}\Omega_{e}},$
$\displaystyle\vec{q}_{e\parallel}$
$\displaystyle\equiv\frac{5T_{e}\left[\chi_{\parallel}n_{e}\nabla_{\parallel}T_{e}+\left(\chi_{\parallel}-\mu_{\parallel}\right)T_{e}\nabla_{\parallel}n_{e}\right]}{m_{e}\left\langle\nu_{e}\right\rangle_{e}},$
(61) $\chi_{P}\equiv
2\mu_{P}+\alpha_{P}-\beta_{P}-\rho_{P},\qquad\chi_{H}\equiv
2\lambda-1,\qquad\chi_{\parallel}\equiv 2\mu_{\parallel}-\beta_{\parallel},$
(62)

In addition to $\lambda$ defined by (47), Eqs. (55)–(62) include

$\alpha_{P}\equiv\frac{\int_{0}^{\infty}\frac{d\left(v^{3}\nu_{e}(v_{e})\right)}{dv_{e}}\
F_{0}dv_{e}}{3\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}v_{e}^{2}dv_{e}},\qquad\alpha_{\parallel}\equiv\frac{\int_{0}^{\infty}\frac{d\left(v_{e}^{3}/\nu_{e}(v_{e})\right)}{dv}\
F_{0}dv_{e}}{3\int_{0}^{\infty}\frac{1}{\nu_{e}(v_{e})}F_{0}v^{2}dv_{e}},$
(63) $\displaystyle\rho_{\parallel}$
$\displaystyle\equiv\frac{\left(\int_{0}^{\infty}\frac{d}{dv_{e}}\left(\frac{v_{e}^{5}}{\nu_{e}(v_{e})}\right)F_{0}dv_{e}\right)\left(\int_{0}^{\infty}v_{e}^{2}F_{0}dv_{e}\right)}{5\left(\int_{0}^{\infty}v_{e}^{4}F_{0}dv_{e}\right)\left(\int_{0}^{\infty}\frac{v_{e}^{2}}{\nu_{e}(v_{e})}F_{0}dv_{e}\right)},$
(64a) $\displaystyle\rho_{P}$
$\displaystyle\equiv\frac{\left(\int_{0}^{\infty}\frac{d}{dv_{e}}\left(\nu_{e}(v_{e})v_{e}^{5}\right)F_{0}dv\right)\left(\int_{0}^{\infty}v_{e}^{2}F_{0}dv_{e}\right)}{5\left(\int_{0}^{\infty}v_{e}^{4}F_{0}dv_{e}\right)\left(\int_{0}^{\infty}\nu_{e}(v_{e})v_{e}^{2}F_{0}dv_{e}\right)},$
(64b) $\displaystyle\beta_{P}$
$\displaystyle\equiv\frac{\left(\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}v_{e}^{4}dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}}{\left(\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}v_{e}^{2}dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}},$
(65a) $\displaystyle\beta_{\parallel}$
$\displaystyle\equiv\frac{\left(\int_{0}^{\infty}\frac{F_{0}v_{e}^{4}}{\nu_{e}(v_{e})}\
dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}}{\left(\int_{0}^{\infty}\frac{F_{0}v^{2}}{\nu_{e}(v_{e})}\
dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}},$ (65b)
$\delta_{P}\equiv\frac{\left(\int_{0}^{\infty}\nu_{e}^{2}F_{0}v_{e}^{4}dv_{e}\right)\left(\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}\right)^{2}}{\left(\int_{0}^{\infty}\nu_{e}F_{0}v_{e}^{2}dv_{e}\right)^{2}\left(\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}\right)},$
(66) $\displaystyle\mu_{P}$
$\displaystyle\equiv\frac{3\left(\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}v_{e}^{6}dv_{e}\right)\left(\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}\right)^{2}}{5\left(\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}v_{e}^{2}dv_{e}\right)\left(\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}\right)^{2}},$
(67a) $\displaystyle\mu_{\parallel}$
$\displaystyle\equiv\frac{3\left(\int_{0}^{\infty}\frac{F_{0}v^{6}}{\nu_{e}(v_{e})}\
dv_{e}\right)\left(\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}\right)^{2}}{5\left(\int_{0}^{\infty}\frac{F_{0}v^{2}}{\nu_{e}(v_{e})}\
dv_{e}\right)\left(\int_{0}^{\infty}F_{0}v_{e}^{4}dv_{e}\right)^{2}},$ (67b)
$\xi\equiv\left\langle\frac{1}{\nu_{e}}\right\rangle_{e}\left\langle\nu_{e}\right\rangle_{e}=\frac{\left(\int_{0}^{\infty}\nu_{e}(v_{e})F_{0}(v_{e})v_{e}^{2}dv_{e}\right)\left(\int_{0}^{\infty}\nu_{e}^{-1}(v_{e})F_{0}(v_{e})v_{e}^{2}dv_{e}\right)}{\left(\int_{0}^{\infty}F_{0}(v_{e})v_{e}^{2}dv_{e}\right)^{2}}.$
(68)

For constant $\nu_{e}$ and arbitrary $F_{0}(v)$, we have
$\alpha_{P,\parallel}=\rho_{P,\parallel}=\beta_{P,\parallel}=\delta_{P}=\xi=1$,
$\mu_{P,\parallel}=\lambda$, $\chi_{P,H,\parallel}=2\lambda-1$ and
$\mathbf{M}=\mathbf{N}$, so that Eqs. (58)–(61) reduce to Eqs. (52)–(54). If,
additionally, $F_{0}(v)$ is Maxwellian then we have even simpler parameters:
$\mu_{P,\parallel}=\chi_{P,H,\parallel}=\lambda=1$.

In a broad range of electron energies, $\mathcal{E}_{e}\lesssim 0.3$ eV, the
velocity dependence of $\nu_{e}$ in the lower ionosphere can be approximated
by a simple power-law dependence, $\nu_{e}\propto v_{e}^{2\alpha}$, with
$\alpha\approx 5/6$ [(Gurevich, 1978), Sect. 2.3.1, Fig. 7] or, practically to
the same accuracy, with $\alpha=1$ (Dimant and Sudan, 1995a). For the general
power-law dependent $\nu_{e}\propto v_{e}^{2\alpha}$ with $\alpha$ in the
range between $0$ and $1$ and Maxwellian $F_{0}(v_{e})$, Eqs. (63)–(68)
simplify dramatically,

$\alpha_{P}=\beta_{P}=1+\frac{2\alpha}{3},\qquad\alpha_{\parallel}=\beta_{\parallel}=1-\frac{2\alpha}{3},$
$\rho_{\parallel}=\mu_{\parallel}=\frac{\left(3-2\alpha\right)\left(5-2\alpha\right)}{15},\qquad\rho_{P}=\mu_{P}=\frac{\left(3+2\alpha\right)\left(5+2\alpha\right)}{15},$
(69)
$\delta_{P}=\frac{\sqrt{\pi}\Gamma\left(5/2+2\alpha\right)}{3\Gamma^{2}\left(3/2+\alpha\right)},\qquad\xi=\frac{1-4\alpha^{2}}{\sin\frac{\pi\left(2\alpha+1\right)}{2}}=\frac{4\alpha^{2}-1}{\sin\frac{\pi\left(2\alpha-1\right)}{2}}.$

The case of $\alpha=1/2$ corresponds to hard-sphere collisions. In this case,
the indeterminate expression for $\xi$ yields $4/\pi\approx 1.273$. For
$\alpha=5/6$ (Gurevich, 1978), we have $\alpha_{P}=\beta_{P}\approx 1.556$,
$\alpha_{\parallel}=\beta_{\parallel}\approx 0.444$,
$\rho_{\parallel}=\mu_{\parallel}\approx 0.296$, $\rho_{P}=\mu_{P}\approx
2.074$, $\delta_{P}\approx 3.095,$ and $\xi\approx 2.053$. For $\alpha=1$
(Dimant and Sudan, 1995a), all these factors deviate from unity even further,
e.g., $\alpha_{\parallel}=\beta_{\parallel}\approx 0.333$, $\delta_{P}\approx
3.889$, and $\xi=3$. Thus the quantitative effect of the velocity dependence
of $\nu_{e}(v)$ is significant and should not be ignored. Figure 1 shows the
coefficients given by Eq. (69) for general values of the power-law exponent
$\alpha$ within the physically realistic range of $0\leq\alpha\leq 1$.

Figure 1: Fluid-model coefficients for the power-law dependent _e-n_ collision
frequency, $\nu_{en}(v_{e})\propto v_{e}^{2\alpha}$.

Comparison of general energy-balance Eq. (58) with Eq. (52) shows that the
velocity dependence of the collision parameters results not only in more
complicated heat conductivity, frictional heating, and cooling, but in
additional terms associated with the plasma motion and gradients in the
parallel to $\vec{B}$ direction. It is important that these seemingly
collisionless terms originate fully from electron-neutral collisions due to
the velocity distribution of $\nu_{e}(v)$. Similar effects in the Hall and
Pedersen directions are inconsequential and are neglected here. One should
not, however, neglect Hall and Pedersen components in the heat conductivity
because the plasma temperature and density gradients in those directions can
be much sharper than those in the parallel direction.

Now we discuss the last (cooling) term in the RHS of Eq. (58). For general
velocity-dependent $\delta_{en}(v),$ but a Maxwellian distribution function
$F_{0}\propto\exp[-m_{e}v^{2}/(2T_{e})]$, using (39), we reduce this term to

$-2\pi
m_{e}\int_{0}^{\infty}\\!\\!v_{e}^{3}\delta_{en}\nu_{e}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{dF_{0}}{dv_{e}}\right)dv_{e}=\frac{3}{2}\
\frac{\left\langle\delta_{en}v_{e}^{2}\nu_{e}\right\rangle_{e}n_{e}}{\left\langle
v_{e}^{2}\right\rangle_{e}}\left(T_{n}-T_{e}\right).$ (70)

For general $F_{0}$, but constant $\delta_{en}$, we can rewrite the cooling
term in Eq. (58) as

$-2\pi
m_{e}\int_{0}^{\infty}\\!\\!v_{e}^{3}\delta_{en}\nu_{e}\left(v_{e}F_{0}+\frac{T_{n}}{m_{e}}\frac{dF_{0}}{dv_{e}}\right)dv_{e}=\frac{3}{2}\
\delta_{en}\left\langle\nu_{e}\right\rangle_{e}n_{e}\left(\alpha_{P}T_{n}-\beta_{P}T_{e}\right),$
(71)

where we integrated by parts and used Eqs. (63) and (65a). Equation (71) shows
that for general non-Maxwellian $F_{0}$ the cooling term is not necessarily
proportional to the temperature difference $(T_{n}-T_{e})$. However, for the
power-law dependent $\nu_{e}\propto v_{e}^{2\alpha}$ and Maxwellian $F_{0}$,
according to Eq. (69), we have $\beta_{P}=\alpha_{P}=1+2\alpha/3$. In this
case, the structure of the cooling term proportional to
$\alpha_{P}\left\langle\nu_{e}\right\rangle n_{e}$ matches that of the
frictional heating term for a purely perpendicular field,
$\alpha_{P}\left\langle\nu_{e}\right\rangle m_{e}n_{e}V_{e\perp}^{2}$, as seen
from the first term in the RHS of Eq. (58).

##  7 Discussion

When applying a fluid model, either for analytic calculations or simulations,
it is important to have the corresponding equations with accurate parameters
applicable to the relevant physical conditions. These equations and parameters
are usually derived from the kinetic theory, so that their accuracy is
determined by the accuracy of the underlying kinetic approach.

Based on two different approximate kinetic approaches, this paper derives the
fluid-model equations that describe low-frequency plasma processes in the
highly dissipative E-region ionosphere. The treatment is restricted to
collisions of the plasma particles, ions or electrons, with the neutral
molecules only; no Coulomb collisions are considered. The neglect of Coulomb
collisions at the E-region ionosphere is usually well justified, although
sometimes electron-electron collisions may play a role, resulting in a more
efficient maxwellization of the electron distribution function (Dimant and
Sudan, 1995a). Such maxwellization makes the fluid model (as opposed to the
pure kinetic theory) even more applicable. For the plasma particle collisions
with neutrals (elastic or inelastic), we assume the known cross-sections as
functions of the colliding particle velocities. These velocity dependencies of
the collisional cross-sections can be taken from the literature (e.g., for the
dominant electron-nitrogen collisions, see Itikawa, 2006; Song et al., 2023,
and references therein). Assuming these known cross-sections, we can always
calculate the velocity dependencies of the kinetic collision frequencies,
$\nu_{p}(\vec{v}_{p})$ ($p=i,e,n$) that are used in this paper.

The resultant fluid-model parameters are expressed in general integral forms
through the known velocity dependencies of $\nu_{p}$. For the most important
plasma processes, such as the small- to medium-scale cross-field plasma
instabilities (the thermal Farley-Buneman and gradient drift instabilities),
closed 5-moment multi-fluid models are usually sufficient for the accurate
fluid description. Given the plasma species $p$, the 5-moment set of the
unknowns includes the particle density ($n_{p}$), temperature ($T_{p}$), and
the three components of the mean drift velocity, $\vec{V}_{p}$.

For the ionospheric ions ($p=i$), we have employed the well-known (and fairly
simple) BGK collisional model. For the heavy ions, the applicability of the
BGK collision operator can be justified by the fact that within the thermal
bulk and around, the kinetic ion-neutral collision frequency $\nu_{in}$ is
approximately constant, i.e., velocity-independent; this approximation
corresponds to so-called Maxwell molecule collisions (Schunk and Nagy, 2009).
Additionally, in the E-region the ion masses ionosphere are fairly close to
the neutral-molecule masses, $m_{i}\simeq m_{n}$. As we demonstrate in section
5, in the case of $m_{i}=m_{n}$ the oversimplified BGK model results even in a
quantitatively accurate frictional heating and cooling terms, see the RHS of
equation (28). We should bear in mind, however, that under sufficiently strong
electric field, $E\gtrsim m_{i}\nu_{i}V_{Ti}/e$, the ion distribution function
can be significantly distorted with an appreciable deviation from Maxwellian.
Although the major ion-fluid terms in these cases remain valid, the entire
5-moment model cannot necessarily be easily closed, and hence its validity may
be questionable. The factor of strong electric field is usually of importance
for the high-latitude E region under conditions of severe magnetospheric
perturbations (geomagnetic storms or substorms), while at the equatorial E
region the electric fields are typically much weaker, so that the closed
5-moment ion fluid model is usually more applicable.

The central part of this paper is the derivation of the 5-moment fluid
equations for electrons ($p=e$). For the light electrons, unlike the ions, the
simple BGK model cannot serve even as a crude approximation. As we explained
in section 6, the reasons for the total BGK-model inapplicability are the two
major facts: (1) the mean rate of the collisional loss of the electron energy
is much less than the corresponding loss of the electron momentum, so that the
electron behavior cannot be described by a single collisional parameter; (2)
the kinetic collisional frequency has a pronounced electron velocity
dependence, $\nu_{e}(v_{e})$. The first fact leads to a strong isotropization
of the electron velocity distribution in the velocity space, while the speed
dependence of the electron velocity distribution is effectively decoupled from
its angular dependence. The second fact leads to noticeable modifications of
the electron fluid coefficients and even to occurrence of additional thermal-
diffusion terms. As a result, in the general case the fluid-model coefficients
acquire additional dimensionless multipliers whose values are determined by
some integral relations over the entire electron distribution function, see
Eqs. (59)–(68). For the Maxwellian function, and especially for the power-law
dependencies of the $\nu_{e}$-speed dependence, $\nu_{e}(v_{e})\propto
v_{e}^{2\alpha}$, these general integral relations reduce to simple algebraic
ones, see Eq. (69) and Figure 1. From that figure we see that some numerical
multipliers can deviate from the unity quite significantly, although they
still remain in the same order of magnitude. A better quantitative knowledge
of these fluid coefficients is important for accurate calculations and
predictions of physical characteristics of various plasma wave and other
processes. As our future knowledge of the speed dependence of the kinetic
collision rates becomes more precise, using the more general integral
relationships obtained here one can improved the values of the corresponding
fluid-model coefficients.

The kinetic approach for the electron description employed in this paper is
based on the expansion of the electron velocity distribution in Legendre
polynomials in the velocity space and keeping the two first terms of such
expansion, see Eq.(29): the dominant isotropic part, $F_{0}(\vec{r},t,v_{e})$,
and a small directional part, $|\vec{f}_{1}(\vec{r},t,v_{e})|\cos\theta$,
where $\theta$ is the angle between $\vec{f}_{1}$. This approach is analogous
to that employed by Gurevich (1978), see also Dimant and Sudan (1995a),
although, as we explained in section 6.1, Gurevich’s fluid equations for
electrons, Gurevich (1978, Chapter 5), derived through this kinetic approach,
differ from ours. Gurevich’s equations are written in a form that does not
include explicit adiabatic and frictional heating terms. Also, as wee
mentioned above, the “adiabatic” terms proportional to $(\gamma_{e,i}-1)$ in
Gurevich’s Eqs. (5.3) and (5.4) are extraneous, while their correct equivalent
has already been implicitly distributed within the other terms in Gurevich’s
Eqs. (5.8)–(5.11) (and hence included twice).

An alternative kinetic approach to build an electron fluid description is
based on Grad’s method (Kissack et al., 1995, 1997, 2008a, 2008b). The latter
assumes a finite number of parameters that characterize the velocity
distribution, implying that the electron velocity distribution is reasonably
close to Maxwellian. Our approach is much more general in terms of the
$v_{e}$-dependence, but it restricts the angular distribution of the electron
distribution function to the simplest linear deviation from the isotropy (see
above). This approximation allows calculating vector fluxes like
$n_{e}\vec{V}_{e}$ or energy fluxes (see below), but higher-order tensor
characteristics like an anisotropic pressure, etc., may require an accuracy
beyond its field of applicability. Note, however, that high-order tensor
characteristics for electrons are not expected to be significant due to the
relatively high rate of electron distribution function isotropization
associated with a small value of $\delta_{en}\sim(2$–$4)\times 10^{-3}$ within
the low-energy electron energy range, $\mathcal{E}_{e}\equiv
m_{e}v_{e}^{2}/2<2\mathrm{\ }$eV (Gurevich, 1978). Note also that under
physical conditions when the two methods are applicable both techniques
actually provide reasonable close quantitative results. At the same time, our
kinetic approach provides much simpler, and hence much more practical,
algebraic expressions applicable to various small- and medium-scale E-region
processes.

##  8 Conclusion

Based on relevant physical conditions, we have derived improved fluid
equations for the E-region ionosphere. In this derivation, for the E-region
ions and electrons we have used two different approximate kinetic approaches.

For the ions, we have employed the simple BGK collision operator (section 5).
This resulted in full 5-moment set of the continuity, momentum, and energy
balance equations, see Eqs. (19), (21), and (28). Though these equations look
conventional, our derivation has demonstrated that for the E-region ions with
almost equal masses of the ions and neutrals, the BGK collision operator leads
to the quantitatively accurate frictional and cooling rates.

The central part of this paper is the derivation of the electron fluid
equations. For the electrons, the BGK collisional operator is inapplicable and
we have employed the kinetic approach based on the expansion of the electron
distribution function, $f_{e}(\vec{v}_{e})$, in Legendre polynomials over the
angles in the velocity space. Due to physical conditions resulting in
efficient isotropization of $f_{e}(\vec{v}_{e})$ allowed us to restrict the
treatment to the two highest terms of the Legendre expansion: the dominant
isotropic part, $F_{0}(v_{e})$ and a small directional part
$\vec{f_{1}}\cdot\vec{v}_{e}/v_{e}$. The former is responsible for calculating
the scalar fluid quantities, such as the electron density and temperature,
while the directional part allows one to calculate the electron flux (or
electric current) density. An important factor in our derivations is the fact
that the electron-neutral collisional frequencies are strongly velocity-
dependent. Assuming these velocity dependencies to be known, we have derived
the full set of the 5-moment equations: the continuity equation, the momentum
equation, and the thermal balance equation. Since the E-region electrons in
all relevant low-frequency processes are essentially inertialess, the momentum
equation reduces to an explicit expression for the electron mean drift
velocity given by Eq. (55). The most non-trivial result is the thermal balance
equation given in the general case by Eq. (58), where the parameters are given
by Eqs. (47), (63)–(62). For the Maxwellian distribution function and the
power-law speed dependence of the electron-neutral collision frequency,
$\nu_{e}(v_{e})\propto v_{e}^{2\alpha}$, the integral relationships for the
fluid-model parameters reduce to simple algebraic expressions given by Eq.
(69), see also Fig. 1.

Comparison of general energy-balance Eq. (58) with the corresponding equation
for the velocity-independent electron collision frequency, see Eq. (52), shows
that the velocity dependence of the collision parameters results in a more
complicated heat conductivity, frictional heating, and cooling, as well as in
additional terms associated with the plasma motion and gradients in the
parallel to $\vec{B}$ direction. These terms look collisionless, but they
originate exclusively from the velocity distribution of $\nu_{e}(v)$. Similar
effects in the Hall and Pedersen directions are inconsequential and have been
neglected. However, one should not neglect the Hall and Pedersen components in
the heat conductivity because the plasma temperature and density gradients in
those directions can be much sharper than those in the parallel direction.

The results of this paper could be used for a routine practical analysis when
working with actual data. The improved equations can also serve as a basis for
more accurate plasma fluid computer simulations. In the general case, the
applicability of the closed 5-moment equations is restricted my reasonably
moderate conditions of the equatorial E region. For the high-latitude E-region
ionosphere, the accurate description may require using a fully kinetic
treatment.

## Acknowledgment

The author is deeply grateful to Prof. Oppenheim for his long-term support,
extensive help, and multiple discussions. This work was funded by NSF Grants
1755350, 1007789, 0442075 and NASA Grants 80NSSC21K1322, 80NSSC19K0080.

APPENDIX

## Moment equations for velocity-dependent $\nu_{e}(v_{e})$

In this Appendix, we derive Eqs. (55) and (58). The key step of this
derivation is expressing the electric field, $\vec{E}$, in terms of the
electron average drift velocity, $\vec{V}_{e}$, and the
$n_{e}$,$T_{e}$-gradients.

For velocity-dependent $\nu_{e}(v)$, which cannot be factored out of the
corresponding integrals, we use the following approach. Using Eq. (33) and
integrating by parts, we obtain from Eq. (39)

$\vec{V}_{e}=-\ \frac{4\pi}{3n_{e}}\int_{0}^{\infty}v^{4}\mathbf{N}\cdot\nabla
F_{0}dv-\mathbf{M}\cdot\frac{e\vec{E}}{m_{e}},$ (A.1)

where $\mathbf{M}$ is given by Eq. (56) with $\alpha_{P,\parallel}$ defined by
Eq. (63). By inverting the matrix $\mathbf{M}$, we obtain

$\frac{e\vec{E}}{m_{e}}=-\mathbf{M}^{-1}\cdot\left(\vec{V}_{e}+\frac{4\pi}{3n_{e}}\int_{0}^{\infty}v^{4}\mathbf{N}(v)\cdot\nabla
F_{0}dv\right).$ (A.2)

Using Eq. (33), we obtain

$\vec{f}_{1}(u)=-\mathbf{N}(u)\cdot\left[u\nabla
F_{0}+\mathbf{M}^{-1}\cdot\left(\vec{V}_{e}+\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\mathbf{N}(v)\cdot\nabla
F_{0}(v)v^{4}dv\right)\frac{\partial F_{0}}{\partial u}\right].$ (A.3)

To complete the derivation of the momentum and temperature-balance equations
using Eqs. (41) and (44), we need to calculate there the two integrals
involving $\vec{f}_{1}$. Both integrals have the same structure described by
$\int_{0}^{\infty}\vec{f}_{1}g(v)v^{3}dv$, where the function $g(v)$ is either
$\nu_{e}(v)$ or $v^{2}$. Using integration by parts, changing the variables,
and re-ordering the integration wherever needed, we obtain

$\displaystyle\int_{0}^{\infty}\vec{f}_{1}g(v_{e})v_{e}^{3}dv_{e}=\int_{0}^{\infty}\frac{d\left(g(u)u^{3}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\cdot\vec{V}_{e}\right)}{du}\
F_{0}(u)du$
$\displaystyle+\int_{0}^{\infty}\left[\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(g(u)u^{3}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\right)}{du}\
F_{0}(u)du-g(v)\mathbf{I}\right]$ $\displaystyle\cdot\mathbf{N}(v)\cdot\nabla
F_{0}(v_{e})v_{e}^{4}dv_{e},$

where $\mathbf{I}$ is the unit tensor: $\mathbf{I}_{\alpha\beta}=1$ if
$\alpha=\beta$ and $\mathbf{I}_{\alpha\beta}=0$ otherwise. As a result, for
the integrals in Eqs. (41) and (44) we obtain

$\displaystyle\frac{4\pi
m_{e}}{3}\int_{0}^{\infty}\nu_{e}\vec{f}_{1}v_{e}^{3}dv_{e}$
$\displaystyle=\frac{4\pi
m_{e}}{3}\int_{0}^{\infty}\frac{d\left(\nu_{e}(u)u^{3}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\cdot\vec{V}_{e}\right)}{du}F_{0}(u)du$
$\displaystyle+\frac{4\pi
m_{e}}{3}\int_{0}^{\infty}\left[\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(\nu_{e}(u)u^{3}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\right)}{du}F_{0}(u)du-\nu_{e}(v)\mathbf{I}\right]$
(A.4) $\displaystyle\cdot\mathbf{N}(v)\cdot\nabla
F_{0}(v_{e})v_{e}^{4}dv_{e},$

and

$\displaystyle\frac{2\pi m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\
\vec{f}_{1}v_{e}^{5}dv_{e}=\frac{2\pi m_{e}}{3}\
\nabla\cdot\int_{0}^{\infty}\frac{d\left(u^{5}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\cdot\vec{V}_{e}\right)}{du}F_{0}(u)du$
$\displaystyle+\frac{2\pi m_{e}}{3}\
\nabla\cdot\int_{0}^{\infty}\left[\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(u^{5}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\right)}{du}F_{0}(u)du-v^{2}\mathbf{I}\right]$
(A.5) $\displaystyle\cdot\mathbf{N}(v)\cdot\nabla
F_{0}(v_{e})v_{e}^{4}dv_{e}.$

First, we calculate the integrals in the RHS of Eq. (A.4). After a simple
algebra, we obtain

$\displaystyle\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(\nu_{e}(u)u^{3}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\cdot\vec{V}_{e}\right)}{du}\
F_{0}(u)du$
$\displaystyle\approx\alpha_{P}\left\langle\nu_{e}\right\rangle\vec{V}_{e\perp}+\frac{\left(\alpha_{P}^{2}-\mu\right)\left\langle\nu_{e}\right\rangle^{2}}{\Omega_{e}}\left(\hat{b}\times\vec{V}_{e\perp}\right)+\frac{\left\langle\nu_{e}\right\rangle\vec{V}_{\parallel}}{\alpha_{\parallel}\xi},$
(A.6)

where

$\mu\equiv\frac{\left(\int_{0}^{\infty}\frac{d\left(\nu_{e}^{2}(v_{e})v_{e}^{3}\right)}{dv_{e}}\
F_{0}dv_{e}\right)\int_{0}^{\infty}F_{0}v_{e}^{2}dv_{e}}{3\left(\int_{0}^{\infty}\nu_{e}v_{e}^{2}F_{0}dv_{e}\right)^{2}},$
(A.7)

and similar for the second term in the RHS of (A.5),

$\displaystyle\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\nu_{e}\vec{f}_{1}v_{e}^{3}dv_{e}\approx\alpha_{P}\left\langle\nu_{e}\right\rangle\vec{V}_{e\perp}+\frac{\left(\alpha_{P}^{2}-\mu\right)\left\langle\nu_{e}\right\rangle^{2}}{\Omega_{e}}\left(\hat{b}\times\vec{V}_{e\perp}\right)+\frac{\left\langle\nu_{e}\right\rangle\vec{V}_{\parallel}}{\alpha_{\parallel}\xi}$
$\displaystyle-\
\frac{\left[\alpha_{P}\left(\alpha_{P}-\beta_{P}\right)-\mu+\delta_{P}\right]\left\langle\nu_{e}\right\rangle^{2}\nabla_{\perp}\left(n_{e}T_{e}\right)}{\Omega_{e}^{2}m_{e}n_{e}}$
(A.8)
$\displaystyle+\frac{\left(\alpha_{P}-\beta_{P}\right)\left\langle\nu_{e}\right\rangle\hat{b}\times\nabla_{\perp}\left(n_{e}T_{e}\right)}{\Omega_{e}m_{e}n_{e}}+\left(\frac{\beta_{\parallel}}{\alpha_{\parallel}}-1\right)\frac{\nabla_{\parallel}\left(n_{e}T_{e}\right)}{m_{e}n_{e}}.$

Here and below, all parameters have been defined by Eqs. (63)–(68).
Substituting Eq. (A.8) to Eq. (41) and neglecting the second-order small terms
$\sim\nu_{e}^{2}/\Omega_{e}^{2}$ wherever applicable, we obtain

$\frac{e\vec{E}}{m_{e}}-\Omega_{e}\hat{b}\times\vec{V}_{e\perp}+\left\langle\nu_{e}\right\rangle_{e}\left(\alpha_{P}\vec{V}_{e\perp}+\frac{\vec{V}_{e\parallel}}{\alpha_{\parallel}\xi}\right)+\frac{\nabla_{\perp}\left(n_{e}T_{e}\right)}{m_{e}n_{e}}+\frac{\beta_{\parallel}}{\alpha_{\parallel}}\frac{\nabla_{\parallel}\left(n_{e}T_{e}\right)}{m_{e}n_{e}}=0.$
(A.9)

Expressing the combination of the second and third terms in the LHS of Eq.
(A.9) as a matrix-by-vector product,

$-\Omega_{e}\hat{b}\times\vec{V}_{e\perp}+\left\langle\nu_{e}\right\rangle_{e}\left(\alpha_{P}\vec{V}_{e\perp}+\frac{\vec{V}_{e\parallel}}{\alpha_{\parallel}\xi}\right)=\left[\begin{array}[c]{ccc}\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}&\Omega_{e}&0\\\
-\Omega_{e}&\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}&0\\\
0&0&\frac{\left\langle\nu_{e}\right\rangle_{e}}{\alpha_{\parallel}\xi}\end{array}\right]\vec{V}_{e},$

and resolving the resultant matrix equation for $\vec{V}_{e}$ to the first-
order accuracy with respect to $\nu_{e}/\Omega_{e}$,

$\left[\begin{array}[c]{ccc}\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}&\Omega_{e}&0\\\
-\Omega_{e}&\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}&0\\\
0&0&\frac{\left\langle\nu_{e}\right\rangle_{e}}{\alpha_{\parallel}\xi}\end{array}\right]^{-1}\approx\left[\begin{array}[c]{ccc}\frac{\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}^{2}}&-\
\frac{1}{\Omega_{e}}&0\\\
\frac{1}{\Omega_{e}}&\frac{\alpha_{P}\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}^{2}}&0\\\
0&0&\frac{\alpha_{\parallel}\xi}{\left\langle\nu_{e}\right\rangle_{e}}\end{array}\right]=\mathbf{M}\mathbf{,}$

we arrive at the explicit expression for the electron fluid drift velocity
$\vec{V}_{e}$ given by Eq. (55).

Further, we calculate the integrals in Eq. (A.5). To the first-order accuracy
with respect to $\nu_{e}/\Omega_{e}$, we obtain

$\displaystyle\frac{2\pi m_{e}}{3}\
\nabla\cdot\int_{0}^{\infty}\frac{d}{dv}\left(v_{e}^{5}\mathbf{N}\cdot\mathbf{M}^{-1}\cdot\vec{V}_{e}\right)F_{0}dv_{e}$
$\displaystyle\approx\frac{5}{2}\left\\{\nabla_{\perp}\cdot\left(\vec{V}_{e\perp}n_{e}T_{e}\right)+\nabla_{\parallel}\left(\frac{\rho_{\parallel}V_{e\parallel}n_{e}T_{e}}{\alpha_{\parallel}}\right)\right.$
$\displaystyle\left.+\frac{\left\langle\nu_{e}\right\rangle_{e}\left(\rho_{P}-\alpha_{P}\right)}{\Omega_{e}}\
\nabla_{\perp}\cdot\left[\hat{b}\times\left(\vec{V}_{e\perp}n_{e}T_{e}\right)\right]\right\\}.$
(A.10)

and

$\displaystyle\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(u^{5}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\right)}{du}\
F_{0}(u)du$
$\displaystyle\approx\frac{5T_{e}}{m_{e}}\left[\begin{array}[c]{ccc}1&\frac{\left(\alpha_{P}-\rho_{P}\right)\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}}&0\\\
\frac{\left(\rho_{P}-\alpha_{P}\right)\left\langle\nu_{e}\right\rangle_{e}}{\Omega_{e}}&1&0\\\
0&0&\frac{\rho_{\parallel}}{\alpha_{\parallel}\left\langle
1/\nu_{e}\right\rangle_{e}}\end{array}\right].$ (A.14)

where the dimensionless parameters of order unity $\rho_{\parallel,P}$ are
defined by Eq. (64). Using Eqs. (A.10) and (A.14) and proceeding with
calculations similar to those above, we obtain

$\displaystyle\frac{2\pi m_{e}}{3}\
\nabla\cdot\int_{0}^{\infty}\left[\frac{4\pi}{3n_{e}}\int_{0}^{\infty}\frac{d\left(u^{5}\mathbf{N}(u)\cdot\mathbf{M}^{-1}\right)}{du}F_{0}(u)du-v^{2}\mathbf{I}\right]$
$\displaystyle\cdot\mathbf{N}(v)\cdot\nabla F_{0}(v_{e})v_{e}^{4}dv_{e}$
$\displaystyle=\nabla_{\perp}\cdot\frac{5\left\langle\nu_{e}\right\rangle\left[\left(\beta_{P}+\rho_{P}-\alpha_{P}\right)T_{e}\nabla_{\perp}\left(n_{e}T_{e}\right)-\mu_{P}\nabla_{\perp}\left(n_{e}T_{e}^{2}\right)\right]}{m_{e}\Omega_{e}^{2}}$
$\displaystyle+\nabla_{\perp}\cdot\frac{5\hat{b}\times\left[T_{e}\nabla_{\perp}\left(n_{e}T_{e}\right)-\lambda\nabla_{\perp}\left(n_{e}T_{e}^{2}\right)\right]}{m_{e}\Omega_{e}}$
$\displaystyle+\nabla_{\parallel}\cdot\frac{5\left[\beta_{\parallel}T_{e}\nabla_{\perp}\left(n_{e}T_{e}\right)-\mu_{\parallel}\nabla_{\perp}\left(n_{e}T_{e}^{2}\right)\right]}{m_{e}\left\langle\nu_{e}\right\rangle}$
(A.15)

Separating the temperature and density gradients, after introducing linear
combinations of the numerical coefficients $\chi_{P,H,\parallel}$, as defined
in (62), we obtain from (A.10) and (A.15)

$\displaystyle\frac{2\pi m_{e}}{3}\ \nabla\cdot\int_{0}^{\infty}\
\vec{f}_{1}v_{e}^{5}dv_{e}=\frac{5}{2}\left[\nabla_{\perp}\cdot\left(n_{e}T_{e}\vec{V}_{e\perp}\right)+\frac{\rho_{\parallel}}{\alpha_{\parallel}}\
\nabla_{\parallel}\left(n_{e}T_{e}V_{e\parallel}\right)\right.$
$\displaystyle+\left.\left\langle\nu_{e}\right\rangle_{e}\left(\alpha_{P}-\rho_{P}\right)\frac{\nabla_{\perp}\cdot\left(\hat{b}\times
n_{e}T_{e}\vec{V}_{e\perp}\right)}{\Omega_{e}}\right]-\nabla\cdot\vec{q}_{e},$
(A.16)

where parameters $\beta_{P,\parallel}$, $\mu_{P,\parallel}$ and tensors
$\mathbf{X}$, $\mathbf{\Lambda}$ are defined by (65), (67), and (60). The last
term in (A.16) can be written explicitly as

$\vec{q}_{e}\equiv\vec{q}_{eP}+\vec{q}_{eH}+\vec{q}_{e\parallel}=\mathbf{X}\cdot\frac{\nabla
T_{e}}{T_{e}}+\left(\mathbf{X-}\mathbf{\Lambda}\right)\cdot\frac{\nabla
n_{e}}{n_{e}}$ (A.17)

where

$\displaystyle\vec{q}_{eP}$
$\displaystyle\equiv\frac{5T_{e}\left\langle\nu_{e}\right\rangle\left[\chi_{P}n_{e}\nabla_{\perp}T_{e}+\left(\chi_{P}-\mu_{P}\right)T_{e}\nabla_{\perp}n_{e}\right]}{m_{e}\Omega_{e}^{2}}$
$\displaystyle\vec{q}_{eH}$
$\displaystyle\equiv\frac{5T_{e}\hat{b}\times\left[\chi_{H}n_{e}\nabla_{\perp}T_{e}+\left(\chi_{H}-\lambda\right)T_{e}\nabla_{\perp}n_{e}\right]}{m_{e}\Omega_{e}}$
$\displaystyle\vec{q}_{e\parallel}$
$\displaystyle\equiv\frac{5T_{e}\left[\chi_{\parallel}n_{e}\nabla_{\parallel}T_{e}+\left(\chi_{\parallel}-\mu_{\parallel}\right)T_{e}\nabla_{\parallel}n_{e}\right]}{m_{e}\left\langle\nu_{e}\right\rangle}.$
(A.18)

From Eqs. (44), (51), (A.8), (A.16), neglecting terms that are first- and
second-order small with respect to $\nu_{e}/\Omega_{e}$, we obtain thermal-
balance Eq. (58).

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