Approximate Formulae Representing Effective-Charge-Number Dependence of Collisional Relaxation Coefficients for Electron Fluid Moment Equations in Open Magnetic Field System

Abstract

We study the collisional relaxation processes of electron fluid moments in open magnetic field systems, such as scrape-off layer and divertor plasmas, by using a simplified Fokker-Planck equation. Anisotropies in temperature and heat flux in directions parallel and perpendicular to the magnetic field are taken into account. The thermal force is not given by the temperature gradient but by the heat flux damping. The relaxation of heat-flux anisotropy is found to play a role in the heat flux transport. For the convenience of plasma fluid simulations, approximate formulae are devised to represent the dependence of relaxation coefficients on the effective charge number.

1.  Introduction

Heat transports in peripheral plasmas of magnetic confinement systems have been investigated by using plasma fluid models [1–3]. Magnetic field lines in the scrape-off layer (SOL) plasma hit directly plasma facing components, such as divertor plates. The outflowing power from a core plasma is transported mainly along the open magnetic field $\mathit{B}$ in the SOL-divertor plasma. Major channel of the heat transport along $\mathit{B}$ is the electron heat conduction, which is governed by Coulomb collisions. In the general electron fluid equations, the parallel conductive heat flux is given by $q_{\mathrm{e}}=-{\kappa }_{\parallel \mathrm{e}}{\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}$. The thermal conductivity is defined as ${\kappa }_{\parallel \mathrm{e}}={\mathrm{\gamma }}_0{n}_{\mathrm{e}}{T}_{\mathrm{e}}{\tau }_{\mathrm{e}0}/m_{\mathrm{e}}$, where the subscript $\parallel$ denotes the parallel to $\mathit{B}$, $T_{\mathrm{e}}$ is the electron temperature, $n_{\mathrm{e}}$ the electron density and $m_{\mathrm{e}}$ the electron mass. The electron collision time ${\tau }_{\mathrm{e}0}$ is proportional to ${T_{\mathrm{e}}}^{3/2}$ and inversely proportional to the effective charge number of background ions $Z_{\text{eff}}$ as well as to n_e; ${\tau }_{\mathrm{e}0}$ = 3.5 × 10¹¹ ${T_{\mathrm{E}}}^{3/2}/n_{\mathrm{e}}{Z}_{\text{eff}}\Lambda$ (${\tau }_{\mathrm{e}0}$: s, $T_{\mathrm{E}}$: eV, $n_{\mathrm{e}}$: m⁻³, $\Lambda$: Coulomb logarithm) [4]. A constant called “thermal-conductivity coefficient” is ${\mathrm{\gamma }}_0$ = 3.16 for $Z_{\text{eff}}$ = 1, and is increased gradually with $Z_{\text{eff}}$.

The friction force, $F_{\text{fr}}=-{\mathrm{\alpha }}_0{n}_{\mathrm{e}}{m}_{\mathrm{e}}U/{\tau }_{\mathrm{e}0}$, and the thermal force, $F_{\text{th}}=-{\mathrm{\beta }}_0{n}_{\mathrm{e}}{\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}$, are also dependent on $Z_{\text{eff}}$ nonlinearly ($U$: electron mean flow velocity relative to ion flow). A constant called “friction-force coefficient” ${\mathrm{\alpha }}_0$ = 0.51 for $Z_{\text{eff}}$ = 1 is decreased gradually with $Z_{\text{eff}}$, while another constant called “thermal-force coefficient” ${\mathrm{\beta }}_0$ = 0.71 for $Z_{\text{eff}}$ = 1 is increased gradually with $Z_{\text{eff}}$ [4]. For the convenience of simulation studies, we devise approximate formulae representing the $Z_{\text{eff}}$ dependence of these relaxation coefficients.

In Sec. 2, a simplified Fokker-Planck equation is applied to study the collisional relaxation processes of an electron fluid in an open magnetic field system, such as SOL-divertor plasma. The anisotropy in $T_{\mathrm{e}}$ and $q_{\mathrm{e}}$ in directions parallel and perpendicular to $\mathit{B}$ is taken into account. Relaxation terms for fluid moment equations are derived in Sec. 3. Approximate formulae representing the $Z_{\text{eff}}$ dependence of relaxation coefficients are devised. Summary and discussion are given in Sec. 4.

2.  Basics of Fluid Moment Equations

We derive electron fluid moment equations in an open magnetic field system, where the transport along $\mathit{B}$ is dominant. Five fluid moments are taken into account in this paper, a mean flow (or electric current), two anisotropic temperatures (or effective isotropic temperature and temperature anisotropy), and two anisotropic heat fluxes (or total heat flux and heat flux anisotropy). Relaxation terms are derived by taking account of a simplified Fokker-Planck equation.

2.1  Distortion of velocity distribution function

Consider a plasma in a uniform magnetic field $\mathit{B}$. Electrons are perturbed from a thermal equilibrium state. Mean flow, anisotropic temperature and heat flux are the results of the perturbation. The velocity distribution function of electrons is distorted a little from a Maxwellian, $f_{\mathrm{M}}=n_{\mathrm{e}}{(m_{\mathrm{e}}/2\mathrm{\pi }{T}_{\mathrm{e}})}^{3/2}exp(-m_{\mathrm{e}}{v}^2/2T_{\mathrm{e}})$, and is expressed as

  

$$f_{\mathrm{e}}=f_{\mathrm{M}}+\mathrm{\delta }{f}_{\mathrm{e}}=f_{\mathrm{M}}+(\mathrm{\delta }{f}_U+\mathrm{\delta }{f}_{\Delta T}+\mathrm{\delta }{f}_q).$$(1)

The “mean-flow (MF) distortion” $\mathrm{\delta }{f}_U$ and the “anisotropic-temperature (AT) distortion” $\mathrm{\delta }{f}_{\Delta T}$ are derived by expanding a shifted bi-Maxwellian distribution, $f_{\text{SBM}}\sim exp\{-m_{\mathrm{e}}{(v_{\parallel }-U)}^2/2T_{\mathrm{e}\parallel }\}\times exp\{-m_{\mathrm{e}}{v_{\perp }}^2/2T_{\mathrm{e}\perp }\}$,

  

$$\mathrm{\delta }{f}_U=(zvU/{v_{\mathrm{te}}}^2){\mathrm{G}}_U{f}_{\mathrm{M}},$$(2)

  

$$\mathrm{\delta }{f}_{\Delta T}={\mathrm{\Delta }}_T(3z^2-1)(v^2/4{v_{\mathrm{te}}}^2)f_{\mathrm{M}},$$(3)

where $v$ is the absolute velocity, $z=\cos \theta$ is a pitch angle variable ($\theta$ = 0 is at the direction of $\mathit{B}$) and $v_{\mathrm{te}}\equiv {(T_{\mathrm{e}}/m_{\mathrm{e}})}^{1/2}$ is the electron thermal speed. The parallel velocity is $v_{\parallel }=vz$, and the perpendicular velocity $v_{\perp }=v{(1-z^2)}^{1/2}$ (subscripts $\parallel$ and $\perp$ denote the components parallel and perpendicular to $\mathit{B}$, respectively).

Fluid moments are obtained by integrating $f_{\mathrm{e}}$ in the velocity space, ${\mathit{M}}^{(\mathrm{n})}=⟨{\mathit{v}}^{\mathrm{n}}⟩=(1/n_{\mathrm{e}})\int d\mathit{v}({\mathit{v}}^{\mathrm{n}}{f}_{\mathrm{e}})=(2\mathrm{\pi }/n_{\mathrm{e}})\int \int dz{v}^{2}dv({\mathit{v}}^{\mathrm{n}}{f}_{\mathrm{e}})$. The 1^st order moment is the electron mean flow velocity $U$ relative to the ion flow velocity, which comes from a MF distortion $\mathrm{\delta }{f}_U$; $U=⟨v_{\parallel }⟩=(1/n_{\mathrm{e}})\int d\mathit{v}(v_{\parallel }\mathrm{\delta }{f}_U)$. In the parallel direction to $\mathit{B}$, a shifted Maxwellian distribution is further deformed by the competitive relaxation due to electron-ion (e/i) and electron-electron (e/e) collisions. We conveniently put an adjustment function in Eq. (2), ${\mathrm{G}}_U={\mathrm{G}}_U(\xi ,Z_{\text{eff}})$ with $\xi \equiv v/v_{\text{te}}$, so that the friction force agrees with that given in [4] (see next Sec. 3 and Appendix). The 2^nd order moment is the electron temperature, which is anisotropic in the parallel and perpendicular to $\mathit{B}$ directions; $T_{\mathrm{e}\parallel }=⟨m_{\mathrm{e}}{v_{\parallel }}^2⟩$, and $T_{\mathrm{e}\perp }=⟨m_{\mathrm{e}}{v_{\perp }}^2/2⟩$. The effective isotropic temperature is given by $T_{\mathrm{e}}=(T_{\mathrm{e}\parallel }+2T_{\mathrm{e}\perp })/3$, and the anisotropy in the temperature, ${\mathrm{\Delta }}_T=2(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/3T_{\mathrm{e}}$, is generated by an AT distortion $\mathrm{\delta }{f}_{\Delta T}$. Kinetic simulations for SOL-divertor plasmas by a particle modeling (PARASOL) showed a remarkable discrepancy between $T_{\mathrm{e}\parallel }$ and $T_{\mathrm{e}\perp }$ (${\mathrm{\Delta }}_T$ ~ −0.5) for low collisionality [5, 6]. The 4^th order moments can be expressed with the square of the 2^nd order moment including ${\mathrm{\Delta }}_T$ effect as,

  

$$⟨m_{\mathrm{e}}{v_{\parallel }}^4/2⟩=(3/2+3{\mathrm{\Delta }}_T){T_{\mathrm{e}}}^2/m_{\mathrm{e}}\approx (3/2){T_{\mathrm{e}\parallel }}^2/m_{\mathrm{e}},$$(4)

  

$$⟨m_{\mathrm{e}}{v_{\parallel }}^2{v_{\perp }}^2/2⟩=(1+{\mathrm{\Delta }}_T/2){T_{\mathrm{e}}}^2/m_{\mathrm{e}}\approx T_{\mathrm{e}\parallel }{T}_{\mathrm{e}\perp }/m_{\mathrm{e}},$$(5)

which are necessary for the transport equation of the heat flux $q_{\mathrm{e}}$ [7].

The conductive heat flux $q_{\mathrm{e}}$ is generated by the asymmetry in the velocity spread, i.e., skewness. One of major sources of the velocity-spread asymmetry (VSA) is the temperature gradient, ${\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}$. At first, we assume a skewed Maxwellian distribution for an isotropic VSA, $f_{\text{SkM}}\sim exp\{-m_{\mathrm{e}}{v}^2/2(1+\mathrm{\epsilon }zv/v_t)T_{\mathrm{e}}\}$. The isotropic VSA is in proportion to $zv$ ($\mathrm{\epsilon }$ is a small parameter). The “heat-flux (HF) distortion”, $\mathrm{\delta }{f}_q$, is derived by expanding $f_{\text{SkM}}$; $\mathrm{\delta }{f}_q\sim z(v^3/5{v_{\mathrm{te}}}^3-v/v_{\mathrm{te}})f_{\mathrm{M}}$. It is noted that the HF distortion is set not to generate the mean flow; $⟨v_{\parallel }⟩=(1/n_{\mathrm{e}})\int d\mathit{v}(v_{\parallel }\mathrm{\delta }{f}_q)=0$. A function, $(v^3/5{v_{\mathrm{te}}}^3-v/v_{\mathrm{te}})$, makes $⟨v_{\parallel }⟩=0$ by $\int dv$ integration. The total heat flux can be divided into parallel component and perpendicular component, $q_{\mathrm{e}}=q_{\mathrm{e}\parallel }+q_{\mathrm{e}\perp }$; $q_{\mathrm{e}}=\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel }{v}^2/2)\mathrm{\delta }{f}_q$ and $q_{\mathrm{e}\parallel }=\int d\mathit{v}(m_{\mathrm{e}}{v_{\parallel }}^3/2)\mathrm{\delta }{f}_q$. For the case of isotropic VSA, their percentages are fixed as $q_{\mathrm{e}\parallel }/q_{\mathrm{e}}$ = 3/5 and $q_{\mathrm{e}\perp }/q_{\mathrm{e}}$ = 2/5.

Next, we consider the anisotropic VSA; $f_{\text{SkB}}\sim exp\{-m_{\mathrm{e}}{v_{\parallel }}^2/2(1+{\mathrm{\epsilon }}_{\parallel }zv/v_{\mathrm{te}})T_{\mathrm{e}}\}\times exp\{-m_{\mathrm{e}}{v_{\perp }}^2/2(1+{\mathrm{\epsilon }}_{\perp }zv/v_{\mathrm{te}})T_{\mathrm{e}}\}$. The above percentages are varied by the anisotropic VSA as $q_{\mathrm{e}\parallel }/q_{\mathrm{e}}=3/5+r_q$ and $q_{\mathrm{e}\perp }/q_{\mathrm{e}}=2/5-r_q$, where $r_q=(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/5q_{\mathrm{e}}$ is the “HF anisotropy” in the range of −3/5 < $r_q$ < 2/5. An HF distortion function consists of two parts, $\mathrm{\delta }{f}_q=\mathrm{\delta }{f_q}_1+\mathrm{\delta }{f_q}_2$, total flux part $\mathrm{\delta }{f_q}_1$ and anisotropic flux part $\mathrm{\delta }{f_q}_2$;

  

$$\mathrm{\delta }{f}_{q1}=(q_{\mathrm{e}}/{q_{\mathrm{e}}}^{\mathrm{FS}})z(v^3/5{v_{\mathrm{te}}}^3-v/v_{\mathrm{te}})G_{q1}{f}_{\mathrm{M}},$$(6)

  

$$\mathrm{\delta }{f}_{q2}=r_q(q_{\mathrm{e}}/{q_{\mathrm{e}}}^{\mathrm{FS}})(5z^3-3z)(v^3/6{v_{\mathrm{te}}}^3)G_{q2}{f}_{\mathrm{M}}.$$(7)

The magnitude of $q_{\mathrm{e}}$ is based on the free-streaming-energy flux, ${q_{\mathrm{e}}}^{\text{FS}}=v_{\mathrm{te}}{n}_{\mathrm{e}}{T}_{\mathrm{e}}$. A skewed Maxwellian distribution is further deformed by collisions. Similarly to the MF dissipation for reasonable convenience, we put functions in Eqs. (6) and (7), ${\mathrm{G}}_{q1,2}={\mathrm{G}}_{q1,2}(\xi ,Z_{\text{eff}})$, so that the heat conductivity and the thermal force agree with those given in [4]. As noted above, the HF distortion does not create the mean flow. A function, $(5z^3-3z)$, for $\mathrm{\delta }{f_q}_2$ promises $⟨v_{\parallel }⟩$ = 0 by $\int dz$ integration.

On the other hand, the current-carrying heat flux $q^J$ comes from $\mathrm{\delta }{f}_U$, and its components are approximately given by ${q^J}_{\parallel }=3U{n}_{\mathrm{e}}{T}_{\mathrm{e}}/2$ and ${q^J}_{\perp }=U{n}_{\mathrm{e}}{T}_{\mathrm{e}}$, respectively. The percentages are ${q^J}_{\parallel }/q^J$ = 3/5 and ${q^J}_{\perp }/q^J$ = 2/5, which coincide with those of conductive heat flux for the isotropic condition, $q_{\mathrm{e}\parallel }/q_{\mathrm{e}}$ = 3/5 and $q_{\mathrm{e}\perp }/q_{\mathrm{e}}$ = 2/5.

2.2  Simplified Fokker-Planck equation

The above distortions are dissipated by Coulomb collisions. Non-thermal-equilibrium quantities, $U$, ${\mathrm{\Delta }}_T$, $q_{\mathrm{e}}$ and $r_q$, are relaxed in the fluid moment equations. In order to evaluate quantitatively the relaxation processes, we apply a Fokker-Planck equation for electron distribution function $f_{\mathrm{e}}$ in a one-dimensional system along $\mathit{B}$.

  

$${\mathit{\partial }}_t{f}_{\mathrm{e}}+v_{\parallel }{\mathit{\partial }}_s{f}_{\mathrm{e}}-(e{E}_{\parallel }/m_{\mathrm{e}}){\mathit{\partial }}_{v\parallel }{f}_{\mathrm{e}}=C(f_{\mathrm{e}})+S,$$(8)

where $t$ is the time and $s$ is a coordinate parallel to $\mathit{B}$. Partial differential operators are ${\partial }_t\equiv \partial /\partial t$, ${\partial }_s\equiv \partial /\partial s$ and ${\partial }_{v\parallel }=\partial /\partial v_{\parallel }$, respectively. Variables are the elementary charge $e$, the electric field $E_{\parallel }$ parallel to $\mathit{B}$ and the source/sink term $S$ for particle, momentum, energy and heat flux.

The simplified collision term consists of two terms, $C(f_{\mathrm{e}})=C^{\mathrm{e}/\mathrm{i}}(f_{\mathrm{e}})+C^{\mathrm{e}/\mathrm{e}}(f_{\mathrm{e}})$,

  

$$C^{\mathrm{e}/\mathrm{i}}(f_{\mathrm{e}})=(Z_{\text{eff}}/2{\tau }_{\mathrm{e}1}){\mathit{\partial }}_z[(1-z^2){\mathit{\partial }}_z\mathrm{\delta }{f}_{\mathrm{e}}],$$(9)

  

$$C^{\mathrm{e}/\mathrm{e}}(f_{\mathrm{e}})=(1/c_z⟨{\tau }_{\mathrm{e}1}⟩){\mathit{\partial }}_z[(1-z^2){\mathit{\partial }}_z\mathrm{\delta }{f}_{\mathrm{e}}],$$(10)

where Eq. (9) corresponds to the pitch-angle scattering (PAS) with ions, and Eq. (10) corresponds to the PAS within electrons. The energy diffusion term is ignored. A basic collision time depending on $v^3$, ${\tau }_{\mathrm{e}1}=4\mathrm{\pi }{{\epsilon }_0}^2{m_{\mathrm{e}}}^2{v}^3/e^4{n}_{\mathrm{e}}\Lambda$ (${\epsilon }_0$: permittivity of free space) [8], is adopted for e/i collision, while an averaged collision time, $⟨{\tau }_{\mathrm{e}1}⟩\equiv {(2{v_{\mathrm{t}}}^2/v^2)}^{3/2}{\tau }_{\mathrm{e}1}=(4/3{\mathrm{\pi }}^{1/2})Z_{\text{eff}}{\tau }_{\mathrm{e}0}$ defined at $m_{\mathrm{e}}{v}^2/2=T_{\mathrm{e}}$, is used for e/e collision, for simplicity.

The AT distortion $\mathrm{\delta }{f}_{\Delta T}$ is dissipated by the PAS. The self-relaxation time for the anisotropic temperature is defined as $d{T}_{\mathrm{e}\parallel }/dt=-2(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/{{\tau }_{\text{rel}}}^{\mathrm{e}/\mathrm{e}}$ with ${{\tau }_{\text{rel}}}^{\mathrm{e}/\mathrm{e}}=15{(2\mathrm{\pi })}^{1/2}⟨{\tau }_{\mathrm{e}1}⟩/8=5Z_{\text{eff}}{\tau }_{\mathrm{e}0}/2^{1/2}$ [8]. The RHS of the above equation is given by the 2^nd moment of the e/e collision term; $\int d\mathit{v}(m_{\mathrm{e}}{v}^2{z}^2)C^{\mathrm{e}/\mathrm{e}}(\mathrm{\delta }{f}_{\Delta T})=-6n_{\mathrm{e}}{\mathrm{\Delta }}_T{T}_{\mathrm{e}}/{\mathrm{c}}_z⟨{\tau }_{\mathrm{e}1}⟩=-4n_{\mathrm{e}}(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/{\mathrm{c}}_z⟨{\tau }_{\mathrm{e}1}⟩$. The coefficient is then determined as ${\mathrm{c}}_z=15{(2\mathrm{\pi })}^{1/2}/4$.

3.  Relaxation Terms for Fluid Moment Equations

3.1  Momentum balance equation

The electron fluid momentum equation is simplified to a generalized Ohm’s law by neglecting the inertia and source terms;

  

$${\mathit{en}}_{\mathrm{e}}{E}_{\parallel }=-{\mathit{\partial }}_s(n_{\mathrm{e}}{T}_{\mathrm{e}\parallel })+F_{\text{fr}}+F_{\text{th}},$$(11)

where $-{\partial }_s(n_{\mathrm{e}}{T}_{\mathrm{e}\parallel })$ is the pressure-gradient force. The friction force $F_{\text{fr}}$ parallel to $\mathit{B}$ comes from the e/i collision term as

  

$$F_{\text{fr}}\equiv \int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel })C^{\mathrm{e}/\mathrm{i}}(\mathrm{\delta }{f}_U)=-{\mathrm{\alpha }}_0{n}_{\mathrm{e}}{m}_{\mathrm{e}}U/{\tau }_{\mathrm{e}0}.$$(12)

The “friction-force coefficient” is given by ${\mathrm{\alpha }}_0$. Its value is ${\mathrm{\alpha }}_0$ = 0.51 for $Z_{\text{eff}}$ = 1, and is decreased gradually with $Z_{\text{eff}}$. The nonlinear dependence on $Z_{\text{eff}}$ comes from the fact that the relaxation process is affected by both the e/e collision, $C^{\mathrm{e}/\mathrm{e}}\propto {Z_{\text{eff}}}^0/v^0$, and the e/i collision, $C^{\mathrm{e}/\mathrm{i}}\propto Z_{\text{eff}}/v^3$ (see Sec. 2.2). The value of α₀ was shown in TABLE 2 of [4] by “Braginskii”. Hereafter we denote this listed value as ${{\mathrm{\alpha }}_0}^{\text{BR}}$. For the availability on the continuous $Z_{\text{eff}}$ dependence of ${\mathrm{\alpha }}_0$, we devise a simple and convenient expression,

  

$${{\alpha }_0}^{\mathrm{\text{fit}}}=(0.77+0.35Z_{\mathrm{\text{eff}}})/(1+1.2Z_{\mathrm{\text{eff}}}).$$(13)

Table 1 compares ${{\mathrm{\alpha }}_0}^{\text{fit}}$ and ${{\mathrm{\alpha }}_0}^{\text{BR}}$ for several $Z_{\text{eff}}$ values ($Z_{\text{eff}}$ = 1, 2, 3, 4 and ∞). There are also listed “exact” values at $Z_{\text{eff}}$ = 1 and ∞, which were written nearby TABLE 2 of [4]. One sees satisfactory agreement between them. How to construct this fitting function is presented in Appendix.

Table 1. $Z_{\text{eff}}$ dependences of ${\mathrm{\alpha }}_0$, ${\mathrm{\beta }}_0$, ${\mathrm{\gamma }}_0$, and ${\mathrm{A}}_{q*}=5(1-{\mathrm{\beta }}_0)/2{\mathrm{\gamma }}_0$.

$Z_{\text{eff}}$	1	2	3	4	∞
${{\mathrm{\alpha }}_0}^{\text{BR}}$	0.5129	0.441	0.397	0.375	0.2949
exact	0.5063				3π/32
${{\mathrm{\alpha }}_0}^{\text{fit}}$	0.509	0.432	0.396	0.374	0.292
${{\mathrm{\beta }}_0}^{\text{BR}}$	0.7110	0.905	1.016	1.090	1.521
exact	0.7033				3/2
${{\mathrm{\beta }}_0}^{\text{fit}}$	0.7083	0.894	1.009	1.087	1.500
${{\mathrm{\gamma }}_0}^{\text{BR}}$	3.162	4.890	6.064	6.920	12.47
exact	3.203				128/3π
${{\mathrm{\gamma }}_0}^{\text{fit}}$	3.200	4.900	6.114	7.025	13.40
${{\mathrm{A}}_{q*}}^{\text{BR}}$	0.2285	0.0485	−0.007	−0.033	−0.104
exact	0.2316				−0.092
${{\mathrm{A}}_{q*}}^{\text{fit}}$	0.233	0.067	0.011	−0.017	−0.100

The thermal force $F_{\text{th}}$ is generated by the relaxation of HF distortion $\mathrm{\delta }{f}_q$ due to e/i PAS depending on $1/v^3$. This force is linearly proportional to $q_{\mathrm{e}}$ and inversely proportional to ${\tau }_{\mathrm{e}0}$,

  

$$F_{\text{th}}\equiv \int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel })C^{\mathrm{e}/\mathrm{i}}(\mathrm{\delta }{f}_q)={\mathrm{A}}_{\text{th}}{q}_{\mathrm{e}}/{v_{\mathrm{te}}}^2{\tau }_{\mathrm{e}0},$$(14)

where we call ${\mathrm{A}}_{\text{th}}$ the “basic thermal-force coefficient”. It is noted that the HF anisotropy $r_q$ does not appear in Eq. (14) (see Appendix). This formula is explicitly different from the conventional one [4],

  

$$F_{\text{th,}{\nabla }_{\parallel }T}=-{\mathrm{\beta }}_0{n}_{\mathrm{e}}{\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}},$$(15)

which is linearly proportional to ${\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}$ and independent of ${\tau }_{\mathrm{e}0}$. Though Eqs. (14) and (15) seem fully different each other, they are equivalent when the conductive heat flux is given by the conventional formula so called “Spitzer-Hahm (SH)” heat flux,

  

$$q_{\mathrm{e}}={q_{\mathrm{e}}}^{\text{SH}}\equiv -{\mathrm{\gamma }}_0{n}_{\mathrm{e}}{v_{\mathrm{te}}}^2{\tau }_{\mathrm{e}0}{\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}.$$(16)

Equation (14) is rewritten as $F_{\text{th}}=-{\mathrm{A}}_{\text{th}}{\mathrm{\gamma }}_0{n}_{\mathrm{e}}{\nabla }_{\parallel }{T}_{\mathrm{e}}$, and an essential relation is found;

  

$${\mathrm{A}}_{\text{th}}={\mathrm{\beta }}_0/{\mathrm{\gamma }}_0.$$(17)

In a rare-collision plasma, however, ${q_{\mathrm{e}}}^{\text{SH}}$ can become much larger than ${q_{\mathrm{e}}}^{\text{FS}}$ unphysically, and some kinetic effects (or flux limiting effects) on $q_{\mathrm{e}}$ have been added to the fluid simulation modeling, such as a harmonic average model, $1/q_{\mathrm{e}}=1/{q_{\mathrm{e}}}^{\text{SH}}+1/{\alpha }_{\mathrm{e}}{q_{\mathrm{e}}}^{\text{FS}}$ (${\alpha }_{\mathrm{e}}$: electron heat-flux-limiting factor) [6, 7, 9]. Therefore, it is necessary to use Eq. (14) directly instead of the conventional formula, Eq. (15).

Values of ${\mathrm{\beta }}_0$ and ${\mathrm{\gamma }}_0$ were listed in TABLE 2 of [4]. Dependences of ${\mathrm{\beta }}_0$ and ${\mathrm{\gamma }}_0$ on $Z_{\text{eff}}$ are approximated by the expressions similar to Eq. (13) for ${\mathrm{\alpha }}_0$,

  

$${{\mathrm{\beta }}_0}^{\text{fit}}=(0.36+0.66Z_{\text{eff}})/(1+0.44Z_{\text{eff}}),$$(18)

  

$${{\mathrm{\gamma }}_0}^{\text{fit}}=(0.65+3.35Z_{\text{eff}})/(1+0.25Z_{\text{eff}}).$$(19)

Figure 1, as well as Table 1, shows that approximate formulae of ${{\mathrm{\alpha }}_0}^{\text{fit}}$, ${{\mathrm{\beta }}_0}^{\text{fit}}$ and ${{\mathrm{\gamma }}_0}^{\text{fit}}$ fit very well those values given in [4]. The approximation of ${\mathrm{A}}_{\text{th}}$ is simply given by ${{\mathrm{A}}_{\text{th}}}^{\text{fit}}={{\mathrm{\beta }}_0}^{\text{fit}}/{{\mathrm{\gamma }}_0}^{\text{fit}}$.

Fig. 1.  Fitting curves of relaxation coefficients, ${{\mathrm{\alpha }}_0}^{\text{fit}}$, ${{\mathrm{\beta }}_0}^{\text{fit}}$ and ${{\mathrm{\gamma }}_0}^{\text{fit}}$. Open diamonds are those listed in TABLE 2 of [4], and closed circles are those written as “exact” values in [4].

3.2  Transport equations for temperature

Basing on the above simplified Boltzmann transport equation, the 2^nd order moment equation of an electron fluid for the effective isotropic temperature $T_{\mathrm{e}}$ is obtained as

  

$$(3/2)n_{\mathrm{e}}{\mathit{\partial }}_t{T}_{\mathrm{e}}+{\mathit{\partial }}_s{q}_{\mathrm{e}}=Q,$$(20)

where $Q$ is the total heating/cooling power density. The current-carrying heat transport is neglected, for simplicity. When taking into account the anisotropic temperatures, two transport equations for $T_{\mathrm{e}\parallel }$ and $T_{\mathrm{e}\perp }$ are treated separately;

  

$$(1/2)n_{\mathrm{e}}{\mathit{\partial }}_t{T}_{\mathrm{e}\parallel }+{\mathit{\partial }}_s{q}_{\mathrm{e}\parallel }=Q_{\parallel }-{\mathrm{A}}_{\Delta T}{n}_{\mathrm{e}}(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/{\tau }_{\mathrm{e}0},$$(21)

  

$$n_{\mathrm{e}}{\mathit{\partial }}_t{T}_{\mathrm{e}\perp }+{\mathit{\partial }}_s{q}_{\mathrm{e}\perp }=Q_{\perp }+{\mathrm{A}}_{\Delta T}{n}_{\mathrm{e}}(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/{\tau }_{\mathrm{e}0},$$(22)

where $Q_{\parallel }$ heats up $T_{\mathrm{e}\parallel }$ and $Q_{\perp }$ heats up $T_{\mathrm{e}\perp }$, separately. For the case of isotropic heating/cooling, $Q$ is divided into the parallel component $Q_{\parallel }=Q/3$ and the perpendicular component $Q_{\perp }=2Q/3$, respectively. The “AT relaxation coefficient” ${\mathrm{A}}_{\Delta T}$ is given by

  

$${\mathrm{A}}_{\Delta T}=2/5+2^{1/2}/5Z_{\text{eff}},$$(23)

where the 2^nd part is equal to ${\tau }_{\mathrm{e}0}/{{\tau }_{\text{rel}}}^{\mathrm{e}/\mathrm{e}}$. These set of equations for $T_{\mathrm{e}\parallel }$ and $T_{\mathrm{e}\perp }$ are rewritten to a transport equation of ${\mathrm{\Delta }}_T{T}_{\mathrm{e}}$;

  

$$(3/2)n_{\mathrm{e}}{\mathit{\partial }}_t({\mathrm{\Delta }}_T{T}_{\mathrm{e}})+{\mathit{\partial }}_s(2q_{\mathrm{e}\parallel }-q_{\mathrm{e}\perp })=(2Q_{\parallel }-Q_{\perp })-9{\mathrm{A}}_{\Delta T}{n}_{\mathrm{e}}{\mathrm{\Delta }}_T{T}_{\mathrm{e}}/2{\tau }_{\mathrm{e}0}.$$(24)

Even for the isotropic heating/cooling case ($2Q_{\parallel }=Q_{\perp }$), or no heating/cooling case ($Q_{\parallel }$ = $Q_{\perp }$ = 0), the temperature anisotropy ${\mathrm{\Delta }}_T{T}_{\mathrm{e}}$ can be driven by the 2^nd term of LHS; unbalance in the heat flux, $2q_{\mathrm{e}\parallel }-q_{\mathrm{e}\perp }=(4/5+3r_q)q_{\mathrm{e}}$. This unbalance exists naturally for the isotropic heat flux, $r_q$ = 0. Accordingly, we have to solve the transport of heat fluxes, $q_{\mathrm{e}\parallel }$ and $q_{\mathrm{e}\perp }$, simultaneously with Eq. (24).

3.3  Transport equations for heat flux

The 3^rd order moment equation for the total conductive heat flux $q_{\mathrm{e}}$ is obtained as

  

$${\mathit{\partial }}_t{q}_{\mathrm{e}}+{\mathit{\partial }}_s(n_{\mathrm{e}}{M}_4)+(5/2+{\mathrm{\Delta }}_T)e{E}_{\parallel }{n}_{\mathrm{e}}{v_{\mathrm{te}}}^2=-{\mathrm{A}}_{q^{*}}{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0},$$(25)

where $M_4\equiv ⟨m_{\mathrm{e}}{v_{\parallel }}^2{v}^2/2⟩$ is a 4^th order moment explained below in detail. We neglect the current-carrying heat transport and the source/sink term for $q_{\mathrm{e}}$, for simplicity.

The collision term $\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel }{v}^2/2)C(\mathrm{\delta }{f}_q)=\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel }{v}^2/2)C(\mathrm{\delta }{f_q}_1)$ is expressed by $-{\mathrm{A}}_{q*}{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}$ in the RHS. It is interestingly noted that the “HF rate coefficient” ${\mathrm{A}}_{q*}$ in Eq. (25) is not always the damping one (${\mathrm{A}}_{q*}$ > 0) but sometimes the growing one (${\mathrm{A}}_{q*}$ < 0) due to the $v^3$ dependence of the e/i collision time ${\tau }_{\mathrm{e}1}$. Considering $q_{\mathrm{e}}$ > 0, $\mathrm{\delta }{f}_q$ consists of “forward” higher-energy part with $v_{\parallel +}$ > 0 and “backward” lower-energy part with $v_{\parallel -}$ < 0. The larger scattering of low-energy electrons by ions reduces faster the backward heat flux, while smaller scattering of high-energy electrons reduces slower the forward heat flux. As a result, an increase in the net forward heat flux can arise. When $Z_{\text{eff}}$ ≫ 1, the PAS by e/i collision with ${\tau }^{\mathrm{e}/\mathrm{i}}\propto v^3$ is dominant. We evaluate $q_{\mathrm{e}}\sim (n_{+}{v}_{\parallel +}{T}_{+}+n_{-}{v}_{\parallel -}{T}_{-})$, and obtain $q_{\mathrm{e}}\sim n_{\mathrm{e}}{v}_{\mathrm{te}}(T_{+}-T_{-})$ under a condition of $⟨v_{\parallel }⟩$ = 0. Then we can approximate the temporal change of $q_{\mathrm{e}}$ for $Z_{\text{eff}}$ ≫ 1 as, $(1/n_{\mathrm{e}}{v}_{\mathrm{te}})d{q}_{\mathrm{e}}/dt\sim d/dt(T_{+}-T_{-})\sim -(T_{+}/{\tau }_{+}-T_{-}/{\tau }_{-})\sim -({T_{\mathrm{e}}}^{3/2}/{\tau }^{\mathrm{e}/\mathrm{i}})(1/{T_{+}}^{1/2}-1/{T_{-}}^{1/2})\sim (T_{+}-T_{-})/{\tau }^{\mathrm{e}/\mathrm{i}}\sim +(1/n_{\mathrm{e}}{v}_{\mathrm{te}})(q_{\mathrm{e}}/{\tau }^{\mathrm{e}/\mathrm{i}})$. When $Z_{\text{eff}}$ approaches to unity and the PAS by e/e collision is effective, ${\mathrm{A}}_{q*}$ becomes positive. The above collisional growth of $q_{\mathrm{e}}$ was not clearly mentioned in the previous works, e.g., [7]. It is noted again a well-known fact that the thermal force is generated due to $v^3$ dependence of ${\tau }^{\mathrm{e}/\mathrm{i}}$; $F_{\text{th}}=d/dt(m_{\mathrm{e}}{n}_{\mathrm{e}}⟨v_{\parallel }⟩)={\mathrm{A}}_{\text{th}}{q}_{\mathrm{e}}/{v_{\mathrm{te}}}^2{\tau }^{\mathrm{e}/\mathrm{i}}$ (see Eq. (14)).

The 4^th order moment, $n_{\mathrm{e}}{M}_4\equiv n_{\mathrm{e}}⟨m_{\mathrm{e}}{v_{\parallel }}^2{v}^2/2⟩$, in the LHS of Eq. (25) is given from Eqs. (4) and (5) as; $n_{\mathrm{e}}{M}_4=(5/2+7{\mathrm{\Delta }}_T/2)n_{\mathrm{e}}{T}_{\mathrm{e}}{v_{\mathrm{te}}}^2=(5/2)n_{\mathrm{e}}{T}_{\mathrm{e}\parallel }{v_{\mathrm{te}}}^2+{\mathrm{\Delta }}_T{n}_{\mathrm{e}}{T}_{\mathrm{e}}{v_{\mathrm{te}}}^2$. Its gradient is expanded as; ${\mathit{\partial }}_s(n_{\mathrm{e}}{M}_4)\approx (5/2+{\mathrm{\Delta }}_T){v_{\mathrm{te}}}^2{\mathit{\partial }}_s(n_{\mathrm{e}}{T}_{\mathrm{e}\parallel })+(5/2)(1+{\mathrm{\Delta }}_T)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s{T}_{\mathrm{e}}+(n_{\mathrm{e}}{v_{\mathrm{te}}}^2){\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})$. Considering the force balance of the generalized Ohm’s law, Eq. (11), the above ${\mathit{\partial }}_s(n_{\mathrm{e}}{M}_4)$ is rewritten as follows; ${\mathit{\partial }}_s(n_{\mathrm{e}}{M}_4)+(5/2+{\mathrm{\Delta }}_T)e{E}_{\parallel }{n}_{\mathrm{e}}{v_{\mathrm{te}}}^2\approx (5/2+{\mathrm{\Delta }}_T){\mathrm{A}}_{\text{th}}{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}+(5/2)(1+{\mathrm{\Delta }}_T)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s{T}_{\mathrm{e}}+(n_{\mathrm{e}}{v_{\mathrm{te}}}^2){\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})$.

Taking into account this thermal-force effect on ${\mathit{\partial }}_s(n_{\mathrm{e}}{M}_4)$, Eq. (25) is reduced to

  

$${\mathit{\partial }}_t{q}_{\mathrm{e}}+(5/2)(1+{\mathrm{\Delta }}_T)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s{T}_{\mathrm{e}}+n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})=-{\mathrm{A}}_q{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}.$$(26)

A new relaxation coefficient, ${\mathrm{A}}_q\equiv {\mathrm{A}}_{q*}+(5/2+{\mathrm{\Delta }}_T){\mathrm{A}}_{\text{th}}$, is called “HF relaxation coefficient”. This collision term is always the damping type (${\mathrm{A}}_q$ > 0). When we force ${\mathrm{\Delta }}_T$ = 0, we obtain a steady-state relation similar to the SH heat flux; $q_{\mathrm{e}}=-(5/2{\mathrm{A}}_q)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\tau }_{\mathrm{e}0}{\mathit{\partial }}_s{T}_{\mathrm{e}}$. By comparing with Eq. (16), we immediately find ${\mathrm{A}}_q{|}_{{\Delta }_T=0}=5/2{\mathrm{\gamma }}_0$. The HF rate coefficient is then determined as ${\mathrm{A}}_{q*}=5/2{\mathrm{\gamma }}_0-5{\mathrm{A}}_{\text{th}}/2=5(1-{\mathrm{\beta }}_0)/2{\mathrm{\gamma }}_0$. Although ${\mathrm{A}}_{q*}$ is approximated by ${\mathrm{A}}_{q*}\approx 5(1-{{\mathrm{\beta }}_0}^{\text{fit}})/2{{\mathrm{\gamma }}_0}^{\text{fit}}$, we apply a much simpler expression, a linear function of $1/Z_{\text{eff}}$,

  

$${{\mathrm{A}}_{q^{*}}}^{\text{fit}}=(1-0.3Z_{\text{eff}})/3Z_{\text{eff}}.$$(27)

This $Z_{\text{eff}}$ dependence agrees fairly well with that of ${\mathrm{A}}_{q*}\equiv 5(1-{\mathrm{\beta }}_0)/2{\mathrm{\gamma }}_0$ (see Table 1). The HF relaxation coefficient ${\mathrm{A}}_q$ is approximated by applying the fitting expressions, ${{\mathrm{\gamma }}_0}^{\text{fit}}$ and ${{\mathrm{A}}_{\mathrm{th}}}^{\text{fit}}={{\mathrm{\beta }}_0}^{\text{fit}}/{{\mathrm{\gamma }}_0}^{\text{fit}}$,

  

$${{\mathrm{A}}_q}^{\text{fit}}=5/2{{\mathrm{\gamma }}_0}^{\text{fit}}+{\mathrm{\Delta }}_T{{\mathrm{A}}_{\text{th}}}^{\text{fit}}.$$(28)

In order to calculate ${\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})$ in Eq. (26), the transport of anisotropic heat fluxes, $q_{\mathrm{e}\parallel }$ and $q_{\mathrm{e}\perp }$, have to be solved simultaneously. The transport equations for $q_{\mathrm{e}\parallel }$ and $q_{\mathrm{e}\perp }$ are given separately as follows,

  

$$\begin{array}{c}{\mathit{\partial }}_t{q}_{\mathrm{e}\parallel }+(3/2)(1+{\mathrm{\Delta }}_T)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s{T}_{\mathrm{e}}+(3/2)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})\hfill \\ =-3{\mathrm{A}}_q{q}_{\mathrm{e}}/5{\tau }_{\mathrm{e}0}-{\mathrm{A}}_r(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/5{\tau }_{\mathrm{e}0},\hfill \end{array}$$(29)

  

$$\begin{array}{c}{\mathit{\partial }}_t{q}_{\mathrm{e}\perp }+(1+{\mathrm{\Delta }}_T)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s{T}_{\mathrm{e}}-(1/2)n_{\mathrm{e}}{v_{\mathrm{te}}}^2{\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})\hfill \\ =-2{\mathrm{A}}_q{q}_{\mathrm{e}}/5{\tau }_{\mathrm{e}0}+{\mathrm{A}}_r(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/5{\tau }_{\mathrm{e}0}.\hfill \end{array}$$(30)

The 1^st collision terms in RHS of Eqs. (29) and (30) concern the HF distortion $\mathrm{\delta }{f_q}_1$; $\int d\mathit{v}(m_{\mathrm{e}}{v_{\parallel }}^3/2)C(\mathrm{\delta }{f_q}_1)=-3{\mathrm{A}}_{q*}{q}_{\mathrm{e}}/5{\tau }_{\mathrm{e}0}$ with ${\mathrm{A}}_q\equiv {\mathrm{A}}_{q*}+(5/2+{\mathrm{\Delta }}_T){\mathrm{A}}_{\text{th}}$. On the other hand, the 2^nd collision terms concern the anisotropic part $\mathrm{\delta }{f_q}_2$; $\int d\mathit{v}(m_{\mathrm{e}}{v_{\parallel }}^3/2)C(\mathrm{\delta }{f_q}_2)=-{\mathrm{A}}_r{r}_q{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}$ with $r_q\equiv (2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/5q_{\mathrm{e}}$. It is interesting that the relaxation processes are fully coupled between $q_{\mathrm{e}\parallel }$ and $q_{\mathrm{e}\perp }$ due to the PAS, $(1/\tau ){\mathit{\partial }}_z[(1-z^2){\mathit{\partial }}_z\mathrm{\delta }{f}_q]$. In the previous works, e.g. [7], this coupling was not taken into account. Because we cannot find suitable reference values for the curve fitting on the “HF anisotropy relaxation coefficient” ${\mathrm{A}}_r$, we create an approximate formula similar to Eq. (27) for ${{\mathrm{A}}_{q*}}^{\text{fit}}$.

  

$${{\mathrm{A}}_r}^{\text{cre}}=(2+0.8Z_{\text{eff}})/Z_{\text{eff}}.$$(31)

To create this linear function of $1/Z_{\text{eff}}$, we apply two values of ${\mathrm{A}}_r=-({\tau }_{\mathrm{e}0}/r_q{q}_{\mathrm{e}})\int d\mathit{v}(m_{\mathrm{e}}{v_{\parallel }}^3/2)C(\mathrm{\delta }{f_q}_2)$ with ${\mathrm{G}}_{q2}\propto {\mathit{\xi }}^3$ for $Z_{\text{eff}}$ ≫ 1 and ${\mathrm{G}}_{q2}\propto {\mathit{\xi }}^2$ for $Z_{\text{eff}}$ ≈ 2.7 (see Appendix).

Figure 2 shows fitting curves (dashed lines) of the relaxation coefficients, ${{\mathrm{\alpha }}_0}^{\text{fit}}$, ${{\mathrm{A}}_{\text{th}}}^{\text{fit}}$ and ${{\mathrm{A}}_{q*}}^{\text{fit}}$ dependent on $1/Z_{\text{eff}}$. Open circles are obtained by setting $\mathrm{G}\propto {\xi }^3$ for $1/Z_{\text{eff}}$ = 0 and $\mathrm{G}\propto {\xi }^2$ for $1/Z_{\text{eff}}$ = 1/2.7, which agree well with own fitting curve. A red solid line for ${{\mathrm{A}}_r}^{\text{cre}}$ passes through two points at $1/Z_{\text{eff}}$ = 0 and 1/2.7 (red closed circles). The slope 2 of ${{\mathrm{A}}_r}^{\text{cre}}$ is set just six times larger than 1/3 of ${{\mathrm{A}}_{q*}}^{\text{fit}}$, because $C^{\mathrm{e}/\mathrm{e}}(\mathrm{\delta }{f}_{q1})=-(2^{1/2}/5{\mathrm{Z}}_{\text{eff}}{\tau }_{\mathrm{e}0})\mathrm{\delta }{f_q}_1$ and $C^{\mathrm{e}/\mathrm{e}}(\mathrm{\delta }{f}_{q2})=-6\times (2^{1/2}/5{\mathrm{Z}}_{\text{eff}}{\tau }_{\mathrm{e}0})\mathrm{\delta }{f_q}_2$.

Fig. 2.  Fitting curves of relaxation coefficients, ${{\mathrm{\alpha }}_0}^{\text{fit}}$, ${{\mathrm{A}}_{\text{th}}}^{\text{fit}}={{\mathrm{\beta }}_0}^{\text{fit}}/{{\mathrm{\gamma }}_0}^{\text{fit}}$ and ${{\mathrm{A}}_{q*}}^{\text{fit}}$. Diamonds are those given in [4]. Circles are analytical ones. HF anisotropy relaxation coefficient ${{\mathrm{A}}_r}^{\text{cre}}$ is created so that the line passes through two points (red circles at $1/Z_{\text{eff}}$ = 0 and 1/2.7).

It is noted that a sum of Eqs. (29) and (30) becomes the total heat flux transport equation of Eq. (26). Further, a subtraction, (2/5) × Eq. (29) − (3/5) × Eq. (30), becomes the transport equation for the anisotropic heat flux;

  

$${\mathit{\partial }}_t(r_q{q}_{\mathrm{e}})+(9/10)(n_{\mathrm{e}}{v_{\mathrm{te}}}^2){\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})=-{\mathrm{A}}_r{r}_q{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}.$$(32)

This equation for $r_q{q}_{\mathrm{e}}$ includes ${\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})$ term but does not include ${\mathit{\partial }}_s{T}_{\mathrm{e}}$ term, while Eq. (26) for $q_{\mathrm{e}}$ includes both ${\mathit{\partial }}_s{T}_{\mathrm{e}}$ and ${\mathit{\partial }}_s({\mathrm{\Delta }}_T{T}_{\mathrm{e}})$ terms.

We complete the five transport equations of electron fluid moments, Eq. (11) of the force balance, Eq. (20) for $T_{\mathrm{e}}$, Eq. (24) for ${\mathrm{\Delta }}_T{T}_{\mathrm{e}}$, Eq. (26) for $q_{\mathrm{e}}$ and Eq. (32) for $r_q{q}_{\mathrm{e}}$. Anisotropies in $T_{\mathrm{e}}$ and $q_{\mathrm{e}}$ are taken into account. Another set of five equations is equivalently available; Eqs. (11) and (21) for $T_{\mathrm{e}\parallel }$, Eq. (22) for $T_{\mathrm{e}\perp }$, Eq. (29) for $q_{\mathrm{e}\parallel }$ and Eq. (30) for $q_{\mathrm{e}\perp }$. The electron continuity equation is not always necessary to be solved, but the ion continuity equation is solved under the condition of charge neutrality.

4.  Summary and Discussion

In this paper, we study the collisional relaxation processes of electron fluid moments in open magnetic field systems, such as SOL-divertor plasmas, by using a simplified Fokker-Planck equation. The thermal force is given not by the temperature gradient, $F_{\text{th}}=-{\mathrm{\beta }}_0{n}_{\mathrm{e}}{\mathrm{\nabla }}_{\parallel }{T}_{\mathrm{e}}$, but by the heat flux damping, $F_{\text{th}}={\mathrm{A}}_{\text{th}}{q}_{\mathrm{e}}/{v_{\mathrm{te}}}^2{\tau }_{\mathrm{e}0}$. We devise approximate formulae representing the $Z_{\text{eff}}$ dependence of relaxation coefficients, “friction-force coefficient” ${\mathrm{\alpha }}_0$, “thermal-force coefficient” ${\mathrm{\beta }}_0$ and “thermal-conductivity coefficient” ${\mathrm{\gamma }}_0$;

  

$${{\mathrm{\alpha }}_0}^{\mathrm{\text{fit}}}=(0.77+0.35Z_{\mathrm{\text{eff}}})/(1+1.2Z_{\mathrm{\text{eff}}}),$$

  

$${{\mathrm{\beta }}_0}^{\mathrm{\text{fit}}}=(0.36+0.66Z_{\mathrm{\text{eff}}})/(1+0.44Z_{\mathrm{\text{eff}}}),$$

  

$${{\mathrm{\gamma }}_0}^{\mathrm{\text{fit}}}=(0.65+3.35Z_{\mathrm{\text{eff}}})/(1+0.25Z_{\mathrm{\text{eff}}}).$$

These expressions can be easily utilized in general divertor simulation codes using conventional plasma fluid equations.

Anisotropies in the temperature, ${\mathrm{\Delta }}_T\equiv 2(T_{\mathrm{e}\parallel }-T_{\mathrm{e}\perp })/3T_{\mathrm{e}}$, and in the conductive heat flux, $r_q=(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/5q_{\mathrm{e}}$, are taken into account. A relaxation between $q_{\mathrm{e}\parallel }$ and $q_{\mathrm{e}\perp }$ is definitely described in the transport equation, ${\mathit{\partial }}_t(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })\sim -{\mathrm{A}}_r(2q_{\mathrm{e}\parallel }-3q_{\mathrm{e}\perp })/{\tau }_{\mathrm{e}0}$. The “heat-flux anisotropy relaxation coefficient” ${\mathrm{A}}_r$ is derived and its approximation formula is created, ${{\mathrm{A}}_r}^{\text{cre}}=(2+0.8Z_{\text{eff}})/Z_{\text{eff}}$. We complete a set of five electron fluid moment equations; a force balance equation, and four transport equations of $T_{\mathrm{e}}$, ${\mathrm{\Delta }}_T{T}_{\mathrm{e}}$, $q_{\mathrm{e}}$ and $r_q{q}_{\mathrm{e}}$. We will study, in the near future, the electron heat transport along $\mathit{B}$ in open magnetic field systems by utilizing these electron fluid moment equations. The conductive heat flux is not simply be expressed as $q_{\mathrm{e}}={q_{\mathrm{e}}}^{\text{SH}}$ nor $1/q_{\mathrm{e}}=1/{q_{\mathrm{e}}}^{\text{SH}}+1/{\alpha }_{\mathrm{e}}{q_{\mathrm{e}}}^{\text{FS}}$. The flux-limiting effect in a rare-collision plasma, which has been sometimes called kinetic effect, can be described to some extent by solving directly the transport equation of $q_{\mathrm{e}}$.

We have developed the anisotropic-ion-pressure (AIP) fluid model for the SOL-divertor plasma simulation [10, 11]. To simulate more accurately the plasma transport in a rare-collision plasma, we are going to extend the AIP model by adding the heat flux transport equations. The present methodology will be useful. For the ion transport in SOL plasmas, however, the Mach number and temperature anisotropy become remarkably large [5, 6, 10]. Resultantly the present functional forms of $\mathrm{\delta }f$ generate the unacceptably-wide negative $f$ region. Alternative proper forms of $\mathrm{\delta }{f}_{\mathrm{i}}$ should be found out to complete the ion fluid moment equations.

 Acknowledgements

We acknowledge Dr. K. Ibano of Starlight Engine Ltd., Dr. Y. Homma of QST and Prof. M. Sakamoto of University of Tsukuba for their collaborations and encouragements on our research activities. This work is partly supported by the NIFS Collaboration Research Program (NIFS25KFFT001).

Appendix. Relaxation Coefficients

The collision term given by Eqs. (9) and (10) for an MF distortion, $\mathrm{\delta }{f}_U=(zvU/{v_{\mathrm{te}}}^2){\mathrm{G}}_U{f}_{\mathrm{M}}$, is written as $C(\mathrm{\delta }{f}_U)=-2(Z_{\text{eff}}/2{\tau }_{\mathrm{e}1}+1/{\mathrm{c}}_z⟨{\tau }_{\mathrm{e}1}⟩)\mathrm{\delta }{f}_U$, because ${\mathit{\partial }}_z[(1-z^2){\mathit{\partial }}_z(z)]=-2z$. A stationary Fokker-Planck equation, $-(e{E}_{\parallel }/m_{\mathrm{e}}){\mathit{\partial }}_{v\parallel }{f}_{\mathrm{M}}=C(\mathrm{\delta }{f}_U)$, is then reduced to a relation, $e{E}_{\parallel }/m_{\mathrm{e}}=(2U/⟨{\tau }_{\mathrm{e}1}⟩)\{Z_{\text{eff}}{(2{v_{\mathrm{te}}}^2/v^2)}^{3/2}+1/{\mathrm{c}}_z\}{\mathrm{G}}_U$. We find that the adjustment function has a form of ${\mathrm{G}}_U(\xi ,Z_{\text{eff}})\sim 1/(2^{3/2}{Z}_{\text{eff}}/{\xi }^3+1/{\mathrm{c}}_z)$, where $\xi \equiv v/v_{\mathrm{te}}$. The friction force is given by $F_{\text{fr}}=\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel })C^{\mathrm{e}/\mathrm{i}}(\mathrm{\delta }{f}_U)=-{\mathrm{\alpha }}_0{n}_{\mathrm{e}}{m}_{\mathrm{e}}U/{\tau }_{\mathrm{e}0}$. In case of $Z_{\text{eff}}$ ≫ 1, ${\mathrm{G}}_U$ becomes simply proportional to ${\xi }^3$ and $\mathrm{\delta }{f}_U[{\mathrm{G}}^{(3)}]=\{{(2\mathrm{\pi })}^{1/2}U/32v_{\mathrm{te}}\}{z\xi }^4{f}_{\mathrm{M}}$. Brackets $[{\mathrm{G}}^{(3)}]$ denote a quantity calculated by setting $\mathrm{G}\propto {\xi }^3$, and later $[{\mathrm{G}}^{(2)}]$ denote a quantity calculated by setting $\mathrm{G}\propto {\xi }^2$. One obtains analytically $F_{\text{fr}}[{\mathrm{G}}^{(3)}]=-(3\mathrm{\pi }/32)n_{\mathrm{e}}{m}_{\mathrm{e}}U/{\tau }_{\mathrm{e}0}$, i.e., ${\mathrm{\alpha }}_0[{\mathrm{G}}^{(3)}]$ = 3π/32. This was written nearby TABLE 2 of [4] as an exact value for $Z_{\text{eff}}$ = ∞.

The above ${\mathrm{G}}_U$ is not enough accurate for $Z_{\text{eff}}$ ~ O(1), because the present e/e collision term is a simplified model without taking account of the relative-velocity effect and the energy-relaxation effect. Therefore, we apply an approximate expression of the “friction-force coefficient” ${\mathrm{\alpha }}_0$ representing the $Z_{\text{eff}}$ dependence. Considering the form, ${\mathrm{G}}_U\sim 1/\{1+\mathrm{g}(\xi )Z_{\text{eff}}\}$, we introduce a variable, $\mathrm{X}(\mathrm{h},Z_{\text{eff}})=1/(1+\mathrm{h}{Z}_{\text{eff}})$, and carry out a linear fitting, ${{\mathrm{\alpha }}_0}^{\text{fit}}=\mathrm{a}+\mathrm{b}\mathrm{X}(\mathrm{h},Z_{\text{eff}})$, against 5 points of ${{\mathrm{\alpha }}_0}^{\text{BR}}$ values in TABLE 2 of [4]. We obtain 3 coefficients, $\mathrm{h}$ = 1.2, $\mathrm{a}$ ≈ 0.29 and $\mathrm{b}$ ≈ 0.48, i.e., ${{\mathrm{\alpha }}_0}^{\text{fit}}=(0.77+0.35Z_{\text{eff}})/(1+1.2Z_{\text{eff}})$. Approximate formulae for ${\mathrm{\beta }}_0$ and ${\mathrm{\gamma }}_0$ are also obtained by this linear fitting technique; ${{\mathrm{\beta }}_0}^{\text{fit}}=(0.36+0.66Z_{\text{eff}})/(1+0.44Z_{\text{eff}})$ and ${{\mathrm{\gamma }}_0}^{\text{fit}}=(0.65+3.35Z_{\text{eff}})/(1+0.25Z_{\text{eff}})$, respectively. To obtain values of a and b for ${{\mathrm{\gamma }}_0}^{\text{fit}}$, an exact value of ${\mathrm{\gamma }}_0$ = 128/3π at $Z_{\text{eff}}$ = ∞ is regarded.

In accordance with the above analysis, we calculate the “basic thermal-force coefficient” ${\mathrm{A}}_{\text{th}}$. The collision terms for $\mathrm{\delta }{f}_{q1}$ and $\mathrm{\delta }{f}_{q2}$ are written as $C(\mathrm{\delta }{f}_{q1})=-2(Z_{\text{eff}}/2{\tau }_{\mathrm{e}1}+1/{\mathrm{c}}_z⟨{\tau }_{\mathrm{e}1}⟩)\mathrm{\delta }{f_q}_1\propto z$ and $C(\mathrm{\delta }{f}_{q2})=-12(Z_{\text{eff}}/2{\tau }_{\mathrm{e}1}+1/c_z⟨{\tau }_{\mathrm{e}1}⟩)\mathrm{\delta }{f}_{q2}\propto (5z^3-3z)$, respectively. Because an integration, $\int dz(m_{\mathrm{e}}{v}_{\parallel })C(\mathrm{\delta }{f}_{q2})$, is always null, the HF anisotropy $r_q$ does not appear in Eq. (14) for $F_{\text{th}}$. In case of $Z_{\text{eff}}$ ≫ 1 and ${\mathrm{G}}_{q1}\propto {\xi }^3$, the HF distortion is given by $\mathrm{\delta }{f}_{q1}[{\mathrm{G}}^{(3)}]=\{{(2\mathrm{\pi })}^{1/2}/32\}(q_{\mathrm{e}}/{q_{\mathrm{e}}}^{\text{FS}})z({\xi }^6/8-{\xi }^4)f_{\mathrm{M}}$. The thermal force is analytically obtained as $F_{\text{th}}[{\mathrm{G}}^{(3)}]=\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel })C(\mathrm{\delta }{f}_{q1})=(9\mathrm{\pi }/256)q_{\mathrm{e}}/{v_{\mathrm{te}}}^2{\tau }_{\mathrm{e}0}$, i.e., ${\mathrm{A}}_{\text{th}}[{\mathrm{G}}^{(3)}]$ = 9π/256 ≈ 0.110. This is smaller a little than ${{\mathrm{\beta }}_0}^{\text{BR}}/{{\mathrm{\gamma }}_0}^{\text{BR}}$ ≈ 0.122 based on TABLE 2 of [4], but just the same as one based on exact values, ${\mathrm{\beta }}_0$ = 3/2 and ${\mathrm{\gamma }}_0$ = 128/3π, wrote in [4]. The approximation of ${\mathrm{A}}_{\text{th}}$ is simply given by ${{\mathrm{A}}_{\text{th}}}^{\text{fit}}={{\mathrm{\beta }}_0}^{\text{fit}}/{{\mathrm{\gamma }}_0}^{\text{fit}}$.

The “HF-rate coefficient” ${\mathrm{A}}_{q*}$ can be approximated as ${\mathrm{A}}_{q*}\approx 5(1-{{\mathrm{\beta }}_0}^{\text{fit}})/2{{\mathrm{\gamma }}_0}^{\text{fit}}$, but we apply a much simpler expression, a linear function of $1/Z_{\text{eff}}$, ${{\mathrm{A}}_{q*}}^{\text{fit}}=(1-0.3Z_{\text{eff}})/3Z_{\text{eff}}$, by fair agreements. The analytical evaluation, $-{\mathrm{A}}_{q*}{q}_{\mathrm{e}}/{\tau }_{\mathrm{e}0}=\int d\mathit{v}(m_{\mathrm{e}}{v}_{\parallel }{v}^2/2)C(\mathrm{\delta }{f_q}_1)$ for $Z_{\text{eff}}$ ≫ 1 and ${\mathrm{G}}_{q1}\propto {\xi }^3$, gives ${\mathrm{A}}_{q*}[{\mathrm{G}}^{(3)}]$ = −(15π/512) ≈ −0.092, which agrees with one based on exact values wrote in [4].

Since there was no suitable reference value for the curve fitting on the “HF anisotropy relaxation coefficient” ${\mathrm{A}}_r$, we create an approximate formula similar to ${{\mathrm{A}}_{q*}}^{\text{fit}}$. To create a linear function of $1/Z_{\text{eff}}$, two points of ${\mathrm{A}}_r=-({\tau }_{\mathrm{e}0}/r_q{q}_{\mathrm{e}})\int d\mathit{v}(m_{\mathrm{e}}{v_{\parallel }}^3/2)C(\mathrm{\delta }{f}_{q2})$ are necessary. One point is at $1/Z_{\text{eff}}$ = 0, which is calculated by setting ${\mathrm{G}}_{q2}\propto {\xi }^3$ and $\mathrm{\delta }{f}_{q2}[{\mathrm{G}}^{(3)}]=\{7{(2\mathrm{\pi })}^{1/2}/3072\}{r}_q(q_{\mathrm{e}}/{q_{\mathrm{e}}}^{\text{FS}})(5z^3-3z){\xi }^6{f}_{\mathrm{M}}$. The result is ${\mathrm{A}}_r[{\mathrm{G}}^{(3)}]$ = 63π/256 ≈ 0.77. Another point is calculated by setting artificially ${\mathrm{G}}_{q2}\propto {\xi }^2$ and $\mathrm{\delta }{f}_{q2}[{\mathrm{G}}^{(2)}]=(1/54)r_q(q_{\mathrm{e}}/{q_{\mathrm{e}}}^{\text{FS}})(5z^3-3z){\xi }^5{f}_{\mathrm{M}}$. The result is ${\mathrm{A}}_r[{\mathrm{G}}^{(2)}]=32/35+6(2^{1/2}/5Z_{\text{eff}})$. In order to select a proper $Z_{\text{eff}}$ value corresponding to ${\mathrm{G}}_{q2}\propto {\xi }^2$, we calculate former relaxation coefficients for $\mathrm{G}\propto {\xi }^2$; ${\mathrm{\alpha }}_0[{\mathrm{G}}^{(2)}]$ = 2/5, ${\mathrm{A}}_{\text{th}}[{\mathrm{G}}^{(2)}]$ = 6/35, and ${\mathrm{A}}_{q*}[{\mathrm{G}}^{(2)}]=4/35-2^{1/2}/5Z_{\text{eff}}$. The MF distortion is $\mathrm{\delta }{f}_U[{\mathrm{G}}^{(2)}]=(U/5v_{\mathrm{te}}){z\xi }^3{f}_{\mathrm{M}}$. Each value coincides with each fitting expression at a certain $Z_{\text{eff}}$ value; $Z_{\text{eff}}$ ≈ 2.9 for ${\mathrm{\alpha }}_0$, $Z_{\text{eff}}$ ≈ 2.7 for ${\mathrm{A}}_{\text{th}}$, and $Z_{\text{eff}}$ ≈ 2.7 for ${\mathrm{A}}_{q*}$. By putting ${\mathrm{A}}_r[{\mathrm{G}}^{(2)}]$ ≈ 1.54 at $Z_{\text{eff}}$ ≈ 2.7, we obtain the slope for $1/Z_{\text{eff}}$ is about 2.7(1.54 − 0.77) ≈ 2, which is six times larger than that of ${{\mathrm{A}}_{q*}}^{\text{fit}}$. This magnification is quite reasonable, because e/e collision terms, Eq. (10), are related as $C^{\mathrm{e}/\mathrm{e}}(\mathrm{\delta }{f}_{q1})=-(2^{1/2}/5{\mathrm{Z}}_{\text{eff}}{\tau }_{e0})\mathrm{\delta }{f}_{q1}$ and $C^{\mathrm{e}/\mathrm{e}}(\mathrm{\delta }{f}_{q2})=-6\times (2^{1/2}/5{\mathrm{Z}}_{\text{eff}}{\tau }_{e0})\mathrm{\delta }{f}_{q2}$. Finally, we chose a very simple form of ${{\mathrm{A}}_r}^{\text{cre}}$ with an intercept 0.8; ${{\mathrm{A}}_r}^{\text{cre}}=(2+0.8Z_{\text{eff}})/Z_{\text{eff}}$ (see Fig. 2 in Sec. 3.3).

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