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 remain strongly magnetized. Strong DC electric fields perpendicular to the geomagnetic field cause electrojets and give rise to plasma instabilities whose nonlinear development produces plasma density irregularities that can be 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 a dissipative environment, collisional plasma kinetic theory represents the most accurate theoretical description of wave processes. Particle-in-cell (PIC) simulations apply the kinetic approach as a comprehensive numeric experiment, 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 has 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 of electron-neutral collisions. 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 analytically describing low-frequency plasma processes in the E-region ionosphere, including both the kinetic theory and fluid models. Traditionally, the kinetic theory of the Farley–Buneman (FB) instability applied an oversimplified Bhatnagar–Gross–Krook (BGK) collision operator (Bhatnagar et al., 1954). This operator does not follow from an accurate Boltzmann collision operator [except assuming special conditions (St-Maurice and Schunk, 1977)] but represents an artificial construct. It dramatically simplifies the analytical treatment and satisfies the particle number conservation and 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 FB instability, have been developed by a few research groups. Stubbe (1990) modified the BGK terms to allow for different rates of electron energy and momentum losses. This simple modification, however, does not follow from the Boltzmann operator, and its applicability for given physical conditions should be verified. Later, two independent research groups developed more sophisticated and accurate approaches. Kissack and collaborators (Kissack et al., 1995; 1997; 2008a; b) 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 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 main contribution of this work comes from relaxing the assumption of constant electron-neutral collision frequency and allowing significant deviations of the electron velocity distribution from the Maxwellian distribution (although the pitch-angle anisotropy of the electron distribution function always remains weak, as described in the text).

The results of this work 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. Concerning the latter, we note the following. These improved fluid equations include no Landau damping, so they cannot properly model the FB instability in the short-wavelength range of turbulence (of the order of the ion-neutral collisional mean free path and shorter), where this kinetic effect plays an important role. However, these improved fluid equations can successfully model plasma waves generated by the larger-scale gradient drift and thermal instabilities; see Dimant et al. (2023) and references therein. Even more so, as PIC simulations demonstrate (Oppenheim and Dimant, 2013), after the brief evolution of the FB instability to its dynamic nonlinear saturation, the energetically dominant part of the developed turbulence spectrum usually moves to longer wavelengths. In this later stage, the kinetic effects of Landau damping become less important, so the improved set of fluid equations could also be successfully employed for reasonably accurate modeling of the FB instability. A recent work has demonstrated the satisfactory applicability of fluid modeling to FB instability, both in the E-region ionosphere (Rojas et al., 2023) and in the solar chromosphere (Evans et al., 2023). Furthermore, the improved fluid equations can also model the dynamics of such plasma objects as quickly ionized chemically released gas clouds, sporadic E-layers, long-lived meteor plasma, etc.

The paper is organized as follows. Section 2 introduces the collisional kinetic equation and reviews the generic procedure for obtaining the moment equations. The collisional parts are not specified and remain in the general integral form. Section 3 describes the ion momentum equation obtained using the BGK collision model. The most important part is Section 4. 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 4.1 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 4.2 describes the simplest case of the constant (i.e., velocity-independent) kinetic collision parameters. Section 4.3 presents the general results obtained in detail in the Supplementary Appendix. Compared to the simplest electron fluid equations from Section 4.2, the general momentum and thermal-balance equations include more coefficients, as well as additional heat conductivity terms. The latter may appear collisionless, but they have arisen exclusively due to the velocity dependence of the kinetic electron-neutral collision frequency.

This section discusses a general approach to deriving the fluid model from the kinetic theory for any plasma particles. To avoid confusion, we will use the following nomenclature throughout this article. 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: α , β , etc.

The non-relativistic kinetics of charged particles of the kind p with the velocity v ⃗ p at a given location r ⃗ and time t are described by the Boltzmann kinetic equation, ∂ t f p + ∇ ⋅ v ⃗ p f p + ∂ v ⃗ p ⋅ q p m p E ⃗ + v ⃗ p × B ⃗ f p = d f p d t col , (1)

where f p ( v ⃗ p , r ⃗ , t ) is the single-particle velocity distribution function. The left-hand side (LHS) of Equation (1) describes the collisionless (Vlasov) dynamics of the p -species charged particles in smoothed-over-many-particles electric ( E ⃗ ) and magnetic ( 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 Equation (1) is intentionally written in a conservative (divergence) form that is more convenient for deriving the moment equations.

The right-hand side (RHS) of Equation (1), term ( d f p / d t ) col = ∑ q S p q , 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 p q represent integral operators that involve products of f p ( v ⃗ p , r ⃗ , t ) by f q ( v ⃗ q , r ⃗ , t ) . The partial operator S q q is quadratically nonlinear, while S p q with p ≠ q are linear with respect to f p . The linear integral operators describe electron-neutral (e-n) and ion-neutral (i-n) collisions, while the quadratically nonlinear operators describe electron–electron (e-e) and ion–ion (i-i) collisions. The latter redistribute the energy and momentum within the same-species population. In the E-region ionosphere, where the Coulomb collisions are usually relatively weak, the e-e and i-i collisions can often be neglected. In a sufficiently dense day-time ionosphere, the e-e collisions can sometimes play a role, resulting mostly in the evolution of f e ( v ⃗ q , r ⃗ , t ) to a “more Maxwellian” distribution. This only helps improve the applicability of the fluid model compared to the more complicated kinetic theory (Dimant and Sudan, 1995a).

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 p q , 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 the excitation (de-excitation) of the neutral particle (or ion) or to the 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 Equation (1) with Boltzmann’s collision integral per se 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 the fluid-model equations that follow from kinetic Equation (1), we review the conventional approach to deriving equations for the lowest-order moments of the distribution function below. The material in 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 r ⃗ , t ≡ ∫ f p d 3 v p , (2)

the mean fluid velocity, V ⃗ p r ⃗ , t ≡ v ⃗ p = 1 n p ∫ v ⃗ p f p d 3 v p , (3)

and the effective temperature, T p r ⃗ , t = m p 3 v ⃗ p − V ⃗ p 2 = m p 3 n p ∫ v ⃗ p − V ⃗ p 2 f p d 3 v p . (4)

Note that in all equations, here and below, the temperatures are given in the energy units; that is, we imply that the temperatures in Kelvin (K) units are multiplied by the Boltzmann constant, although the K units will also be used in the text. The derivations below will also involve other velocity-averaged quantities defined by ⋯   ≡ 1 n p ∫ ⋯   f p d 3 v p . (5)

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

First, we consider the particle number balance. Integrating Equation 1 over the particle velocities with f p → 0 as v p ≡ | v ⃗ p | → ∞ , we easily obtain the continuity equation for the p -particle fluid, ∂ t n p + ∇ ⋅ n p V ⃗ p = ∫ d f p d t col d 3 v p . (6)

The RHS of Equation 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 Equation 6.

Second, we obtain the momentum balance equation that involves the mean fluid drift velocity, V ⃗ p . Integrating Equation 1 with the weighting function m p v ⃗ p , for a given vector-component α of the momentum density, we obtain m p ∂ t n p V p α + ∑ β = 1 3 ∂ x β P p α β + m p ∑ β = 1 3 ∂ x β n p V p α V p β = q p E α + 1 c V ⃗ p × B ⃗ α n p + m p ∫ v p α d f p d t col d 3 v p , (7)

where P p is the total pressure tensor with vector components defined as P p α β ≡ m p ∫ v p α − V p α v p β − V p β f p d 3 v p . (8)

It combines the isotropic pressure, P p δ m n ( δ m n = 1 if m = n ; otherwise δ m n = 0 ), P p = n p T p , with the viscosity tensor, Π p α β ≡ P p α β − P p δ α β . 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 Equation 6 by m p V ⃗ p and subtract the resultant equation from Equation 7. This yields the conventional momentum balance equation, m p n p D p V ⃗ p D t = q p n p E ⃗ + 1 c V ⃗ p × B ⃗ − ∇ ⋅ P p + R ⃗ p , (9)

where D p / D t ≡ ∂ t + V ⃗ p ⋅ ∇ is the convective (also called substantial or material) derivative for the average p -particle flow and R ⃗ p ≡ m p ∫ v ⃗ p − V ⃗ p d f p d t col d 3 v p . (10)

Here and below, the “dot”-products of a vector, a ⃗ , with a two-component tensor, A , depending on the multiplier order, denote vectors with the components a ⃗ ⋅ A α ≡ ∑ β = 1 3 a β A β α or A ⋅ a ⃗ α ≡ ∑ β = 1 3 A α β a β . The tensor divergence, ∇ ⋅ P p = ∇ P p + ∇ ⋅ Π p , represents a vector which uses the obvious symmetry P p α β = P p β α following from Equation 8. The RHS of Equation 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, 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, ∫ v ⃗ p S p p d 3 v p = 0 .

Third, to obtain the total energy balance equation, we integrate Equation 1 with the weighting function m p v p 2 / 2 and obtain ∂ t E p + ∇ ⋅ ∫ m p v p 2 2 v ⃗ p f p d 3 v p = j ⃗ p ⋅ E ⃗ + m p 2 ∫ v p 2 d f p d t col d 3 v p , (11)

where E p is the p -species average kinetic energy density and j ⃗ p is their electric current density, E p ≡ ∫ m p v p 2 2 f p d 3 v p , j ⃗ p ≡ q p n p V ⃗ p . (12)

Note that the particle gyromotion does not contribute to the kinetic energy balance. Before proceeding, we separate the mean drift velocity V ⃗ p from the kinetic particle velocity v ⃗ p so that (11) becomes ∂ t n p m p V p 2 2 + 3 T p 2 + ∇ ⋅ n p m p V p 2 2 + 5 T p 2 V ⃗ p + Π p ⋅ V ⃗ p + n p m p 2 v ⃗ p − V ⃗ p 3 = j ⃗ p ⋅ E ⃗ + V ⃗ p ⋅ R ⃗ p + m p 2 ∫ v ⃗ p − V ⃗ p 2 d f p d t col d 3 v p + m p V p 2 2 ∫ d f p d t col d 3 v p , (13)

where v ⃗ p − V ⃗ p 3 = v ⃗ p − V ⃗ p 2 v ⃗ p − V ⃗ p . 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 Equation 6 by m p V p 2 / 2 + 3 T p / 2 , take the scalar product of Equation 9 with V ⃗ p , and subtract the resultant two equations from Equation 13. This yields 3 n p 2 D p T p D t + n p T p ∇ ⋅ V ⃗ p + Π p ⋅ ∇ ⋅ V ⃗ p + ∇ ⋅ n p m p 2 v ⃗ p − V ⃗ p 3 = m p 2 ∫ v ⃗ p − V ⃗ p 2 d f p d t col d 3 v p + m p V p 2 2 − 3 T p 2 ∫ d f p d t col d 3 v p , (14)

where Π p ⋅ ∇ ⋅ V ⃗ p ≡ ∑ α , β = 1 3 Π p α β ∇ α V ⃗ p β . 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 Equation 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 Equation 14 in a slightly different form. Rewriting the continuity Equation 6 as D p n p D t + n p ∇ ⋅ V ⃗ p = ∫ d f p d t col d 3 v p ,

we recast the two first terms in the LHS of Equation 14 as 3 n p 2 D p T p D t + n p T p ∇ ⋅ V ⃗ p = 3 n p 2 D p T p D t − T p D p n p D t + T p ∫ d f p d t col d 3 v p = n p T p D p s p D t + T p ∫ d f p d t col d 3 v p , (15)

where s p ≡ 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, this is the adiabatic coefficient γ = 5 / 3 ). This recast allows interpreting n p T p ∇ ⋅ V ⃗ p as the adiabatic heating (cooling) term. The two remaining terms in the LHS of Equation 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 Equation 14 are described by its RHS. After rearranging the last term in Equation 15 to the RHS of Equation 14, the last term there becomes m p V p 2 / 2 − 5 T p / 2 ∫ d f p / d t col d 3 v p . All integral terms involving d f p / d t 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 d f p / d t col , calculating the frictional heating is not an easy task. Below, we use two different kinds of further approximation: one is more appropriate for heavy single-charged ions (Section 3), while the other is suitable for light electrons (Section 4).

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 have a restricted number of additional fluid parameters. Fluid models that include restricted numbers of equations using approximate closures, such as the 5-, 8-, 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.

For ionospheric ions, an accurate fluid theory has been developed by several authors who derived the fluid-model equations using a rigorous collisional kinetic approach; see, for example, Schunk and Walker (1971), Schunk and Walker (1972), and St-Maurice and Schunk (1977); for references, see Shkarofsky et al. (1966), Gurevich (1978), and Schunk and Nagy (2009). This theory results, for example, in a comprehensive set of 13-moment fluid equations that contain many transport terms (Schunk and Nagy, 2009). For typical plasma processes in the E-region ionosphere, however, such comprehensive equations are often excessive, and a much simpler set of 5-moment ion equations would usually suffice.

The goal of this section is to demonstrate that the derivation of the 5-moment ion fluid equations that have been successfully used, for example, for the treatment of the E-region instabilities (Dimant and Oppenheim, 2004; Kovalev et al., 2008; Makarevich, 2020), does not require a full and rigorous kinetic theory. This set of equations can be derived from the ion kinetic equation, where the complicated Boltzmann collision integral is replaced by a much simpler and more practical model discussed below. Under certain conditions, usually fulfilled automatically in the E-region ionosphere, the resultant 5-moment ion equations provide quantitatively accurate frictional heating and cooling terms.

In the lower-E/upper-D regions of the ionosphere (or similar media), one can usually neglect Coulomb collisions between the charged particles, compared to their much more frequent collisions with the dominant neutrals. For the ion-neutral collision integral, one can use the simple BGK model (Bhatnagar et al., 1954). Disregarding ionization-recombination processes and assuming in the general case a neutral wind with the local velocity V ⃗ n , for the laboratory frame of reference, we write the simplest BGK collision operator as d f i d t col BGK = ν i f eff − f i , (16)

where f i is the real ion distribution function (IDF), while f eff is a fictitious Maxwellian function, normalized to the locally varying ion density, n i ( r ⃗ , t ) , with the constant neutral temperature T n : f eff v ⃗ i , r ⃗ , t ≡ n i r ⃗ , t m i 2 π T n 3 / 2 ⁡ exp − m i v ⃗ i − V ⃗ n 2 2 T n . (17)

This simple linear algebraic form of the model collision operator has also been called the “relaxation collision model” (St-Maurice and Schunk, 1973; St-Maurice and Schunk, 1974; St-Maurice and Schunk, 1977), the “Krook collision model” (Schunk and Nagy, 2009), the “model integral of elastic collisions” (Aleksandrov et al., 1984), and by some other terms. Note that the BGK collision model noticeably exaggerates the IDF distortion effect (Schunk and Nagy, 2009; Koontaweepunya et al., 2024). This happens for several reasons (Schunk and Nagy, 2009); in particular, because the BGK operator does not include any collisional angular scattering and hence does not include particle redistribution in the velocity space between the preferred direction of the imposed electric field and the two perpendicular directions.

For the BGK model, it is essential that the ion-neutral collision frequency, ν i , is assumed constant. The standard justification for this is that at sufficiently low energies, the ion-neutral collisions are dominated by the long-range polarization interaction (Dalgarno et al., 1958; Schunk and Walker, 1971; Schunk and Walker, 1972), which results in the approximate constancy of ν i (“Maxwell molecule collisions”) (Schunk and Nagy, 2009). The model collision term in the form of Equation 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 and, after some adjustment to the temperature in f eff for unequal masses of the colliding species, conserves the total energy of the colliding particles as well (Aleksandrov et al., 1984).

Generally, the BGK model does not follow from Boltzmann’s collision integral under any rigorous approximations, although this becomes possible under certain conditions (St-Maurice and Schunk, 1977). This model is a reasonable and simple 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 this kinetic equation for ions (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; Oppenheim et al., 1996; Oppenheim et al., 1995; Oppenheim and Dimant, 2004; Koontaweepunya et al., 2024). There are two major reasons why this oversimplified model works well for the ion-neutral collisions typical for the lower ionosphere. First, within a 1000 K temperature range, the ion-neutral collision frequency is almost velocity-independent (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 ν 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 ( v ⃗ i , r ⃗ , t ) , the BGK kinetic equation in the conservative (divergence) form is given by ∂ t f i + ∇ ⋅ v ⃗ i f i + ∂ v ⃗ i ⋅ e E ⃗ m i + Ω i v ⃗ i × b ̂ f i = ν i f eff − f i . (18)

In this section, we derive the 5-moment ion fluid model equations for n i = ∫ f i d 3 v i , V ⃗ i = v ⃗ i = ∫ v ⃗ i f i d 3 v i , and T i = m i δ v ⃗ i 2 / 3 = ( m i / 3 n i ) ∫ δ v ⃗ i 2 f i d 3 v i , where δ v ⃗ i ≡ v ⃗ i − V ⃗ i .

Following the steps described in the preceding section and assuming the laboratory frame of reference, we obtain from Equation 18 the ion continuity, momentum, and energy-balance equation, ∂ t n i + ∇ ⋅ n i V ⃗ i = D i n i D t + n i ∇ ⋅ V ⃗ i = 0 , (19) m i n i D i V ⃗ i D t = m i n i e E ⃗ m i + Ω i V ⃗ i × b ̂ − ∇ ⋅ P i − ν i n i m i V ⃗ i − V ⃗ n , (20) 3 n i 2 D i T i D t − T i D i n i D t + ∇ ⋅ ∫ m i δ v ⃗ i δ v i 2 2 f i d 3 v i + Π i α β ∇ α V i β = ν i m i n i V i − V ⃗ n 2 2 + 3 2 ν i n i T n − T i . (21)

The two last terms in the LHS of Equation 21 describe heat conduction. Generally, the thermal flux is given by the integral term, and Π i α β should be determined from higher-order moment equations. In the strongly collisional lower ionosphere, assuming sufficiently long-wavelength processes (so that the fluid theory is applicable), these two terms can usually be neglected. This makes Equations 19–21 a closed set of the 5-moment equations for the ion density, n i , temperature, T i , and the three components of the ion drift velocity, V ⃗ i . We should bear in mind, however, that the IDF may deviate from an isotropic Maxwellian function so that T i is an effective temperature determined in the general case by Equation 4 (substituting p = i ). For example, if the ion velocity distribution is approximated by a bi-Maxwellian function ∝ exp [ −   ( m i / 2 ) ( V ⊥ 2 / T ⊥ + V ‖ 2 / T ‖ ) ] , then T i = ( 2 T ⊥ + T ‖ ) / 3 .

The first term in the RHS of Equation 21 describes the total rate of ion frictional heating. This term equals the rate that follows from a more detailed kinetic theory (Schunk and Nagy, 2009), ν i m i m n n i ( V i − V n ) 2 / ( m i + m n ) , provided m i = m n . Coincidentally, in the E region, the masses of the major ions ( N O + and O 2 + ) and neutrals ( N 2 and O 2 ) are indeed close to each other, m i ≃ m n ≃ 30 amu. Thus, in the E-region ionosphere, the BGK model of ion-neutral collisions should correctly describe the ion frictional heating so that one can successfully use it for ions Equations 19–21. The applicability of the fluid equations is better under moderate conditions when the IDF is reasonably close to Maxwellian so that their closing is better justified. Such moderate conditions mostly occur at the equatorial E region rather than at the high-latitude ionosphere, especially during the events of the strongly disturbed magnetosphere–ionosphere–thermosphere system.

To conclude this section, note that closed Equations 19–21 are mostly applicable to moderately disturbed conditions when the IDF is reasonably close to Maxwellian; otherwise, more equations for the higher-order moments are required. For strongly perturbed conditions, however, even higher-order closed sets of fluid equations are not fully applicable because, unlike the original kinetic Equation 18, any closed fluid equations do not include the important kinetic effect of Landau damping and hence they have limited applicability, for example, to describe the Farley–Buneman instability in the short-wavelength range of the turbulence spectrum where the wavelengths become comparable to, or shorter than, the ion mean free path.

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 substantially exceed 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, ν e , is a few orders of magnitude larger than the corresponding rate of the energy exchange, δ e n ν e (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 (EDF). 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 ν e ( v e ) (Gurevich, 1978; Schunk and Nagy, 2009). This velocity dependence plays an important role in some E-region instabilities (see, e.g., Dimant and Sudan, 1997, and references therein), and it modifies the instability and wave characteristics.