SUPPRESSION OF PARALLEL TRANSPORT IN TURBULENT MAGNETIZED PLASMAS AND ITS IMPACT ON THE NON-THERMAL AND THERMAL ASPECTS OF SOLAR FLARES

The pitch-angle diffusion coefficient ${D}_{\mu \mu }$ of charged particles in a magnetized plasma is related to the angular scattering frequency ν (s⁻¹) by

$$\begin{eqnarray}&&{D}_{\mu \mu }=\nu \;\displaystyle \frac{(1-{\mu }^{2})}{2}\;,\end{eqnarray} \tag{ 1 }$$

where μ is the cosine of the pitch-angle relative to the guiding magnetic field. In the case where angular scattering is a superposition of two additive processes, collisional and turbulent, the scattering frequency can be written as

$$\begin{eqnarray}&&\nu (v)={\nu }_{{\rm{C}}}(v)+{\nu }_{{\rm{T}}}(v)\;.\end{eqnarray} \tag{ 2 }$$

Here the collisional contribution is given by

$$\begin{eqnarray}&&{\nu }_{{\rm{C}}}(v)=\displaystyle \frac{4\pi {n}_{e}\;{e}^{4}\;\mathrm{ln}\;{\rm{\Lambda }}}{{m}_{e}^{2}}\;\displaystyle \frac{1}{{v}^{3}}\equiv \displaystyle \frac{v}{{\lambda }_{{\rm{C}}}(v)}\;,\end{eqnarray} \tag{ 3 }$$

where we have introduced the collisional mean free path

$$\begin{eqnarray}&&{\lambda }_{{\rm{C}}}(v)=\displaystyle \frac{{m}_{e}^{2}}{4\pi {n}_{e}\;{e}^{4}\;\mathrm{ln}\;{\rm{\Lambda }}}\;{v}^{4}\equiv {\lambda }_{{\rm{ei}}}{\left(\displaystyle \frac{v}{{v}_{{\rm{te}}}}\right)}^{4}\;.\end{eqnarray} \tag{ 4 }$$

Here e and m_e are the electronic charge (esu) and mass (g), respectively, n_e (cm⁻³) is the ambient electron density, $\mathrm{ln}\;{\rm{\Lambda }}$ is the Coulomb logarithm, and the thermal mean free path

$$\begin{eqnarray}&&{\lambda }_{{\rm{ei}}}=\displaystyle \frac{{(2{k}_{{\rm{B}}})}^{2}}{2\pi {e}^{4}\;\mathrm{ln}\;{\rm{\Lambda }}}\;\displaystyle \frac{{T}_{e}^{2}}{{n}_{e}}\simeq \displaystyle \frac{{10}^{4}\;{T}_{e}^{2}}{{n}_{e}}\;,\end{eqnarray} \tag{ 5 }$$

which, by definition, is the collisional mean free path of electrons with thermal speed $v={v}_{{\rm{te}}}\equiv \sqrt{2{k}_{{\rm{B}}}{T}_{e}/{m}_{e}}$, where k_B is Boltzmann's constant.

Similarly, the turbulent contribution to the angular scattering frequency may be written

$$\begin{eqnarray}&&{\nu }_{{\rm{T}}}(v)=\displaystyle \frac{v}{{\lambda }_{{\rm{T}}}(v)}\;,\end{eqnarray} \tag{ 6 }$$

which involves a (generally velocity-dependent) quantity ${\lambda }_{{\rm{T}}}$, identifiable as the "turbulent mean free path." The physical origin of the turbulence is, for the moment, left unspecified. However, in order to exploit the analogy with collisional scattering we assume a velocity dependence for the turbulent mean free path of the form

$$\begin{eqnarray}&&{\lambda }_{{\rm{T}}}(v)={\lambda }_{0}{\left(\displaystyle \frac{v}{{v}_{{\rm{te}}}}\right)}^{\alpha },\end{eqnarray} \tag{ 7 }$$

corresponding to a turbulent scattering frequency

$$\begin{eqnarray}&&{\nu }_{{\rm{T}}}(v)=\displaystyle \frac{v}{{\lambda }_{0}}{\left(\displaystyle \frac{v}{{v}_{{\rm{te}}}}\right)}^{-\alpha }\;.\end{eqnarray} \tag{ 8 }$$

Using this approach, we can express the total scattering frequency resulting from a combination of collisions and turbulent scattering in the form

$$\begin{eqnarray}&&\nu (v)=\displaystyle \frac{v}{\lambda (v)}\;,\end{eqnarray} \tag{ 9 }$$

where the "effective," or simply the, mean free path $\lambda (v)$ is given by

$$\begin{eqnarray}\displaystyle \frac{1}{\lambda (v)} & = & \displaystyle \frac{1}{{\lambda }_{{\rm{C}}}(v)}+\displaystyle \frac{1}{{\lambda }_{{\rm{T}}}(v)}=\displaystyle \frac{1}{{\lambda }_{{\rm{ei}}}}{\left(\displaystyle \frac{{v}_{{\rm{te}}}}{v}\right)}^{4}\\ & & +\displaystyle \frac{1}{{\lambda }_{{\rm{T}}}(v)}=\displaystyle \frac{1}{{\lambda }_{{\rm{ei}}}}{\left(\displaystyle \frac{{v}_{{\rm{te}}}}{v}\right)}^{4}+\displaystyle \frac{1}{{\lambda }_{0}}{\left(\displaystyle \frac{{v}_{{\rm{te}}}}{v}\right)}^{\alpha }\;.\end{eqnarray} \tag{ 10 }$$

We will find it convenient to introduce the dimensionless ratio

$$\begin{eqnarray}&&R=\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{0}}\;.\end{eqnarray} \tag{ 11 }$$

By Equations (3), (4), (8), and (11), the total scattering frequency can be written in the form

$$\begin{eqnarray}&&\nu (v)=\displaystyle \frac{{v}_{{\rm{te}}}}{{\lambda }_{{\rm{ei}}}}\;\displaystyle \frac{1+R{(v/{v}_{{\rm{te}}})}^{4-\alpha }}{{(v/{v}_{{\rm{te}}})}^{3}}\;,\end{eqnarray} \tag{ 12 }$$

with the corresponding expression

$$\begin{eqnarray}&&\lambda (v)=\displaystyle \frac{{\lambda }_{{\rm{ei}}}\;{(v/{v}_{{\rm{te}}})}^{4}}{1+R{(v/{v}_{{\rm{te}}})}^{4-\alpha }}\end{eqnarray} \tag{ 13 }$$

for the mean free path.

The parameter R is the ratio of two length scales, collisional and turbulent, and here plays the role of a transport reduction factor. In the presence of turbulence, the scattering frequency for thermal electrons with $v\sim {v}_{{\rm{te}}}$ is $\nu ={\nu }_{{\rm{ei}}}(1+R)$, an increase by a factor $(1+R)$ relative to the collisional value ${\nu }_{{\rm{ei}}}={v}_{{\rm{te}}}/{\lambda }_{{\rm{ei}}}$. When R is large, this increased scattering frequency corresponds to a decrease of the the mean free path by a factor ≃R.

Pitch-angle scattering due to turbulence is in general not isotropic; the scattering frequency may also depend on pitch angle, say as

$$\begin{eqnarray}&&{\nu }_{{\rm{T}}}(v,\mu )=\displaystyle \frac{v}{{\lambda }_{0}}{\left(\displaystyle \frac{v}{{v}_{{\rm{te}}}}\right)}^{-\alpha }| \mu {| }^{\beta }\;.\end{eqnarray} \tag{ 14 }$$

In such cases, the mean free path is related to the scattering frequency by the relation

$$\begin{eqnarray}&&{\lambda }_{{\rm{T}}}(v,\mu )=\displaystyle \frac{3v}{4}{\int }_{-1}^{+1}\;\displaystyle \frac{(1-{\mu }^{2})}{\nu (v,\mu )}\;d\mu \;.\end{eqnarray} \tag{ 15 }$$

We caution that in the limit of very large R, one cannot compute the mean free path simply by first taking this limit in Equation (12) and then substituting this into Equation (15). For example, it is easily checked that, for pure turbulent scattering ($\nu ={\nu }_{{\rm{T}}}$) and certain values of β (e.g., $\beta =1$), the expression (15) for the purely turbulent mean free path diverges. Physically, this is because for such values of β turbulent scattering alone is incapable of scattering particles through the 90° ($\mu =0$) "barrier." It is therefore essential, in general, to include collisional effects even in the limit of large $R={\lambda }_{{\rm{ei}}}/{\lambda }_{0}$, so that the mean free path given by Equation (15) remains finite. The corresponding scattering frequency is given (cf. Equation (12)) by

$$\begin{eqnarray}&&\nu (v,\mu )=\displaystyle \frac{{v}_{{\rm{te}}}}{{\lambda }_{{\rm{ei}}}}\;\displaystyle \frac{1+R\;| \mu {| }^{\beta }\;{(v/{v}_{{\rm{te}}})}^{4-\alpha }}{{(v/{v}_{{\rm{te}}})}^{3}}\;.\end{eqnarray} \tag{ 16 }$$

The existence of a turbulent spectrum of magnetic fluctuations transverse to the guiding magnetic field is well-documented (see, e.g., Shalchi 2009, for a review). Such fluctuations can be considered as quasi-static provided the characteristic particle velocity (a few ${v}_{{\rm{te}}}$ for the electrons that carry the bulk of the thermal flux) is larger than ${\lambda }_{{\rm{B}}}/{\tau }_{{\rm{B}}}$, where ${\lambda }_{{\rm{B}}}$ and ${\tau }_{{\rm{B}}}$ are respectively the correlation length and time associated with the fluctuations. Typically, the turbulent mean free path parameter ${\lambda }_{0}$ is proportional to the inverse square of the fractional level of the magnetic fluctuations, i.e.,

$$\begin{eqnarray}&&{\lambda }_{0}\simeq {\lambda }_{{\rm{B}}}{\left(\displaystyle \frac{\delta {B}_{\perp }}{{B}_{0}}\right)}^{-2},\end{eqnarray} \tag{ 17 }$$

so that the value of the ratio R is

$$\begin{eqnarray}&&R=\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{0}}\simeq \left(\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{{\rm{B}}}}\right){\left(\displaystyle \frac{\delta {B}_{\perp }}{{B}_{0}}\right)}^{2}=\displaystyle \frac{{10}^{4}\;{T}_{e}^{2}\;{(\delta {B}_{\perp }/{B}_{0})}^{2}}{{n}_{e}\;{\lambda }_{{\rm{B}}}}\;.\end{eqnarray} \tag{ 18 }$$

We now consider in a semi-quantitative way the impact of adding turbulent scattering on various transport coefficients. The "collisionality" of a plasma is inversely proportional to the ratio of the collisional mean free path ${\lambda }_{{\rm{C}}}$ to the temperature gradient length scale L_T; low values the collisional Knudsen number ${{\rm{Kn}}}_{S}\equiv {\lambda }_{{\rm{C}}}/{L}_{{\rm{T}}}$ imply a high degree of collisionality and vice versa. (Here the subscript S stands for Spitzer (1962).) The collisionality of the solar electron population ranges from quite high (${{\rm{Kn}}}_{S}\sim {10}^{-2}$) close to the Sun to very low (${{\rm{Kn}}}_{S}\sim 1$) in the solar wind at 1 au. For coronal loops, ${L}_{{\rm{T}}}$ is of the order of the loop length L, therefore the collisional Knudsen number ${{\rm{Kn}}}_{S}\simeq {10}^{4}{T}_{e}^{2}/{n}_{e}L$. Taking ${n}_{e}={10}^{10}$ cm⁻³ and $L={10}^{9}$ cm, the collisional Kundsen number ${{\rm{Kn}}}_{S}\;={10}^{-15}{T}_{e}^{2}$, meaning that the range of temperatures ${T}_{e}={10}^{6}\mbox{--}{10}^{7}$ K corresponds to collisional Knudsen numbers ${{\rm{Kn}}}_{S}={10}^{-3}\mbox{--}{10}^{-1}$.

Now let us consider the heat flux density carried by electrons along the field line, which for free-streaming electrons may be straightforwardly written as

$$\begin{eqnarray}&&{q}_{\parallel }={n}_{e}{m}_{e}{v}_{{\rm{te}}}^{3}={n}_{e}{m}_{e}{\left(\displaystyle \frac{2{k}_{{\rm{B}}}{T}_{e}}{{m}_{e}}\right)}^{3/2}\;.\end{eqnarray} \tag{ 19 }$$

However, we know from Fick's law that when the collisional Knudsen number $\lambda /{L}_{{\rm{T}}}$ is small, the thermal flux is driven by the local temperature gradient. Hence we write ${T}_{e}^{3/2}=-{\lambda }_{{\rm{C}}}\;{T}_{e}^{1/2}{{dT}}_{e}/{dz}$, resulting in an approximate expression for the parallel heat flux in a collisional environment:

$$\begin{eqnarray}&&{q}_{\parallel }=-\displaystyle \frac{2{n}_{e}\;{k}_{{\rm{B}}}\;{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}{{m}_{e}^{1/2}}\;{\lambda }_{{\rm{C}}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}\equiv -{\kappa }_{\parallel ,S}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}\;.\end{eqnarray} \tag{ 20 }$$

Adding collisionless scattering effects gives

$$\begin{eqnarray}&&{q}_{\parallel }=-\displaystyle \frac{2{n}_{e}\;{k}_{{\rm{B}}}\;{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}{{m}_{e}^{1/2}}\;\lambda \;\displaystyle \frac{{{dT}}_{e}}{{dz}}\equiv -{\kappa }_{\parallel }\;\displaystyle \frac{{{dT}}_{e}}{{dz}}\;,\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&{\kappa }_{\parallel }=\displaystyle \frac{2{n}_{e}\;{k}_{{\rm{B}}}\;{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}{{m}_{e}^{1/2}}\;\lambda \end{eqnarray} \tag{ 22 }$$

is the thermal conductivity, which, through the form of λ (Equation (13)), includes the effects of both collisional and non-collisional scattering.

When the turbulent transport reduction factor $R={\lambda }_{{\rm{ei}}}/{\lambda }_{0}$ is small, the mean free path takes on its collisional value $\lambda \simeq {\lambda }_{{\rm{ei}}}$ and the parallel heat conductivity correspondingly assumes the standard (e.g., Spitzer 1962) collisional form ${\kappa }_{\parallel S}\;$ $=\;2{n}_{e}{k}_{{\rm{B}}}\;{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}$ $\;{\lambda }_{{\rm{ei}}}/{m}_{e}^{1/2}\;$ $=\;{k}_{{\rm{B}}}{(2{k}_{{\rm{B}}}{T}_{e})}^{5/2}/(\pi {m}_{e}^{1/2}{e}^{4}\mathrm{ln}\;{\rm{\Lambda }})$. (Note that this is density independent because ${\lambda }_{\mathrm{ei}}\propto {T}_{e}^{2}/{n}_{e}$.) On the other hand, when the turbulent reduction factor R is large, the heat conductivity becomes significantly suppressed relative to the collisional value and obeys the (generally density dependent) scaling ${\kappa }_{\parallel }=2{n}_{e}{k}_{{\rm{B}}}{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}{\lambda }_{0}/{m}_{e}^{1/2}={R}^{-1}{\kappa }_{\parallel S}$.

Next we consider the electrical conductivity in a turbulent plasma. As usual, this can be obtained by writing the balance between the electric force and the friction force acting on an electron:

$$\begin{eqnarray}&&-{{eE}}_{\parallel }-\nu \;{m}_{e}{v}_{\parallel }=0\;,\end{eqnarray} \tag{ 23 }$$

where ${E}_{\parallel }$ (statvolt cm⁻¹) is the component of the electric field in the direction of the background guiding magnetic field. From this we find the parallel current density (defined by ${j}_{\parallel }\;=\;-{{en}}_{e}{v}_{\parallel }$) to be ${j}_{\parallel }\;=\;({n}_{e}{e}^{2}/\nu \;{m}_{e})\;{E}_{\parallel }=({n}_{e}{e}^{2}\lambda /{m}_{e}{v}_{{\rm{te}}})\;{E}_{\parallel }$, leading to the Ohm's law

$$\begin{eqnarray}&&{j}_{\parallel }\;=\;\displaystyle \frac{{n}_{e}{e}^{2}\lambda }{{m}_{e}^{1/2}{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}\;{E}_{\parallel }\equiv {\sigma }_{\parallel }\;{E}_{\parallel }\;,\end{eqnarray} \tag{ 24 }$$

where the electrical conductivity

$$\begin{eqnarray}&&{\sigma }_{\parallel }=\displaystyle \frac{{n}_{e}{e}^{2}\lambda }{{m}_{e}^{1/2}{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}\;.\end{eqnarray} \tag{ 25 }$$

Again, when $R\ll 1$, the mean free path takes on its collisional value and the parallel electric conductivity obeys the collisional scaling ${\sigma }_{\parallel S}={n}_{e}{e}^{2}{\lambda }_{{\rm{ei}}}{/m}_{e}{v}_{{\rm{Te}}}={(2{k}_{{\rm{B}}}{T}_{e})}^{3/2}/(2\pi {m}_{e}^{1/2}{e}^{2}\mathrm{ln}\;{\rm{\Lambda }})$. On the other hand, when the turbulent reduction factor $R\gg 1$, the electric conductivity becomes significantly suppressed relative to the collisional value and obeys the scaling ${\sigma }_{\parallel }={n}_{e}{e}^{2}{\lambda }_{0}/{m}_{e}{v}_{{\rm{te}}}={R}^{-1}\;{\sigma }_{\parallel S}$.

We may summarize the above discussion into one simple and rather obvious expression, valid for $v\simeq {v}_{{\rm{te}}}$,

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{{\rm{ei}}}}{\nu }\sim \displaystyle \frac{\lambda }{{\lambda }_{{\rm{ei}}}}\sim \displaystyle \frac{{\rm{Kn}}}{{{\rm{Kn}}}_{S}}\sim \displaystyle \frac{{D}_{\parallel }}{{D}_{\parallel S}}\sim \displaystyle \frac{{\kappa }_{\parallel }}{{\kappa }_{\parallel S}}\sim \displaystyle \frac{{\sigma }_{\parallel }}{{\sigma }_{\parallel S}}\sim \displaystyle \frac{1}{1+R}\;.\end{eqnarray} \tag{ 26 }$$

An enhanced rate of angular scattering yields reduced conductivities compared with the collisional (Spitzer) values.

We first consider the impact of an additional source of isotropic scattering on the transport coefficients. (This will set up the framework for later addressing the problem of angular scattering by a spectrum of transverse magnetostatic fluctuations, which is generally not isotropic.) Let us consider, then, the one-dimensional kinetic equation for a gyrotropic ($\partial /\partial \phi =0$) electron distribution function $f(z,\theta ,v,t)$ under the action of an electric field ${E}_{\parallel }$ parallel to the ambient magnetic field ${\boldsymbol{B}}$,

$$\begin{eqnarray}&&\displaystyle \frac{\partial f}{\partial t}+{v}_{\parallel }\;{\boldsymbol{b}}.{\rm{\nabla }}f-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;{\boldsymbol{b}}.{{\rm{\nabla }}}_{{\boldsymbol{v}}}f={{St}}^{v}(f)+{{St}}^{\theta }(f)\;,\end{eqnarray} \tag{ 27 }$$

where z (cm) is the position of the particle gyrocenter along the magnetic field with direction ${\boldsymbol{b}}={{\boldsymbol{B}}}_{0}/{B}_{0}$, θ is the pitch angle ($\mathrm{cos}\theta ={\boldsymbol{v}}.{{\boldsymbol{B}}}_{0}/{{vB}}_{0}={v}_{\parallel }/v$) and $v=\sqrt{{v}_{\parallel }^{2}+{v}_{\perp }^{2}}$ is the electron speed. Transforming to the variables $(z,\mu ,v,t)$, with $\mu =\mathrm{cos}\theta$ being the pitch-angle cosine, this Fokker–Planck equation may be rewritten as

$$\begin{eqnarray}&&\displaystyle \frac{\partial f}{\partial t}+\mu \;v\;\displaystyle \frac{\partial f}{\partial z}-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\mu \;\displaystyle \frac{\partial f}{\partial v}\\ &&\quad -\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\displaystyle \frac{(1-{\mu }^{2})}{v}\displaystyle \frac{\partial f}{\partial \mu }={{St}}^{v}(f)+{{St}}^{\mu }(f)\;.\end{eqnarray} \tag{ 28 }$$

The velocity scattering operator

$$\begin{eqnarray}&&{{St}}^{v}(f)=\displaystyle \frac{1}{{v}^{2}}\;\displaystyle \frac{\partial }{\partial v}\left[{v}^{2}D(v)\left(\displaystyle \frac{\partial f}{\partial v}+\displaystyle \frac{{m}_{e}}{{k}_{{\rm{B}}}{T}_{e}}\;{fv}\right)\right]\end{eqnarray} \tag{ 29 }$$

describes collisional diffusion in velocity space and collisional drag. The diffusion coefficient in velocity space is given by

$$\begin{eqnarray}&&D(v)=\displaystyle \frac{4\pi {e}^{4}\mathrm{ln}\;{\rm{\Lambda }}\;{n}_{e}{k}_{{\rm{B}}}{T}_{e}}{{m}_{e}^{3}}\;\displaystyle \frac{1}{{v}^{3}}\;,\end{eqnarray} \tag{ 30 }$$

whereas the pitch-angle scattering operator takes the form

$$\begin{eqnarray}&&{{St}}^{\mu }(f)=\displaystyle \frac{\partial }{\partial \mu }\left[{D}_{\mu \mu }\;\displaystyle \frac{\partial f}{\partial \mu }\right]=\displaystyle \frac{\partial }{\partial \mu }\left[\displaystyle \frac{\nu (v)}{2}\;(1-{\mu }^{2})\;\displaystyle \frac{\partial f}{\partial \mu }\right]\;.\end{eqnarray} \tag{ 31 }$$

While the velocity-space scattering operator (29) is responsible for kinetic energy change, the pitch-angle scattering operator (31) is responsible for momentum change at constant kinetic energy. We recall that the scattering frequency $\nu (v)$, and hence (Equation (1)) the pitch-angle diffusion coefficient ${D}_{\mu \mu }$, are sums of two components—collisional and turbulent pitch-angle diffusion; thus ${D}_{\mu \mu }={D}_{\mu \mu }^{C}+{D}_{\mu \mu }^{T}$.

The right hand side of Equation (28) vanishes identically for a Maxwellian distribution, i.e.,

$$\begin{eqnarray}&&{{St}}^{v}({f}_{0})+{{St}}^{\mu }({f}_{0})=0\;,\end{eqnarray} \tag{ 32 }$$

where

$$\begin{eqnarray}&&{f}_{0}(v)={n}_{e}{\left(\displaystyle \frac{{m}_{e}}{2\pi {k}_{{\rm{B}}}{T}_{e}}\right)}^{3/2}\;{e}^{-{m}_{e}{v}^{2}/2{k}_{{\rm{B}}}{T}_{e}}\;.\end{eqnarray} \tag{ 33 }$$

However, if T and/or n are not uniform—${T}_{e}={T}_{e}(z)$ and/or ${n}_{e}={n}_{e}(z)$—then the (local) Maxwellian does not cancel the spatial transport term on the left hand side of the Fokker–Planck Equation (28). This deviation from the homogeneous equilibrium state is responsible for the spatial flux of particles in the plasma and of what they carry (e.g., kinetic energy, electric charge). In this situation, we use a standard Chapman–Enskog expansion of the electron kinetic equation and write the distribution function as the sum of a zeroth order isotropic Maxwellian distribution plus a small flux-carrying anisotropic correction:

$$\begin{eqnarray}&&f={f}_{0}(z,v)+\epsilon \;{f}_{1}(z,\mu ,v)\;,\end{eqnarray} \tag{ 34 }$$

where the expansion parameter is of the order of the (small) Knudsen number in the plasma, i.e., $\epsilon \;\sim$ Kn. All operators being linear, we obtain at order an equation for f₁:

$$\begin{eqnarray}&&{{St}}^{v}({f}_{1})+\displaystyle \frac{\partial }{\partial \mu }\left[{D}_{\mu \mu }(v)\;\displaystyle \frac{\partial {f}_{1}}{\partial \mu }\right]=\mu \;v\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}\\ &&\quad -\mu \left(\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\right)\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial v}\;,\end{eqnarray} \tag{ 35 }$$

where f₀ is the Maxwellian distribution (33). By solving this equation for the lead anisotropic component f₁, we can obtain expressions for both the heat flux and the electric current. This constitutes a standard Spitzer (1962) problem which becomes easily tractable in the Lorentz plasma approximation³ , i.e., if one assumes that the scattering is angular only, i.e., ${{St}}^{v}({f}_{1})=0$. For such a case, Equation (35) can be immediately integrated once over μ to give

$$\begin{eqnarray}\displaystyle \frac{\partial {f}_{1}}{\partial \mu } & = & \displaystyle \frac{{\mu }^{2}}{\nu \;(1-{\mu }^{2})}\left[v\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial v}\right]\\ & & +\displaystyle \frac{C}{\nu \;(1-{\mu }^{2})}\;.\end{eqnarray} \tag{ 36 }$$

The choice of the integration constant

$$\begin{eqnarray*}&&C=-\left[v\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial v}\right]\end{eqnarray*}$$

yields

$$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{1}}{\partial \mu }=-\displaystyle \frac{1}{\nu }\left[v\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial v}\right]\;,\end{eqnarray} \tag{ 37 }$$

which ensures the regularity of f₁ at $\mu =\pm 1$.

Since we are, for now, adopting an isotropic scattering model in which the scattering frequency ν is independent of the pitch angle cosine μ, one can integrate Equation (37) once more to obtain

$$\begin{eqnarray}&&{f}_{1}(z,v,\mu )=-\displaystyle \frac{\mu }{\nu }\left[v\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}-\displaystyle \frac{{{eE}}_{\parallel }}{{m}_{e}}\;\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial v}\right]\;.\end{eqnarray} \tag{ 38 }$$

Evaluating the pertinent space and velocity derivatives of f₀, and assuming a constant pressure along z: ${n}_{e}(z){k}_{{\rm{B}}}{T}_{e}(z)={P}_{e}={\rm{constant}}$, we obtain

$$\begin{eqnarray}&&{f}_{1}(z,v,\mu )=-\displaystyle \frac{\mu v}{\nu }\left[\left(\displaystyle \frac{{m}_{e}{v}^{2}}{2{k}_{{\rm{B}}}{T}_{e}}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{f}_{0}\;,\end{eqnarray} \tag{ 39 }$$

with

$$\begin{eqnarray}&&\nu (v)=\displaystyle \frac{v}{{\lambda }_{{\rm{c}}}(v)}+\displaystyle \frac{v}{{\lambda }_{{\rm{T}}}(v)}=\displaystyle \frac{{v}_{{\rm{te}}}}{{\lambda }_{{\rm{ei}}}}\;\displaystyle \frac{1+R{(v/{v}_{{\rm{te}}})}^{4-\alpha }}{{(v/{v}_{{\rm{te}}})}^{3}}\;.\end{eqnarray} \tag{ 40 }$$

It is convenient to introduce the normalization

$$\begin{eqnarray}&&x=\displaystyle \frac{v}{{v}_{{\rm{te}}}}\;,\end{eqnarray} \tag{ 41 }$$

so that the zero-order Maxwellian distribution (Equation (33)) at a given location (i.e., given values of T_e and ${v}_{{\rm{te}}}$) can be written in the form

$$\begin{eqnarray}&&{f}_{0}(x)={\pi }^{-3/2}{n}_{e}\;{v}_{{\rm{te}}}^{-3}\;{e}^{-{x}^{2}}\;.\end{eqnarray} \tag{ 42 }$$

Further, Equation (40) becomes

$$\begin{eqnarray}&&\nu =\displaystyle \frac{{v}_{{\rm{te}}}}{{\lambda }_{{\rm{ei}}}}\;\displaystyle \frac{{{Rx}}^{4-\alpha }+1}{{x}^{3}}\;,\end{eqnarray} \tag{ 43 }$$

and thus the expression (39) for f₁ becomes

$$\begin{eqnarray}&&{f}_{1}=-\mu {\lambda }_{{\rm{ei}}}\;\displaystyle \frac{{x}^{4}}{{{Rx}}^{4-\alpha }+1}\left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{f}_{0}\;.\end{eqnarray} \tag{ 44 }$$

We can now compute the heat flux from

$$\begin{eqnarray}&&{q}_{\parallel }(z)=2\pi \;{\int }_{0}^{\infty }{dv}\;{v}^{2}{\int }_{-1}^{1}d\mu \;\mu \left(\displaystyle \frac{{m}_{e}{v}^{2}}{2}\right)\;v\;{f}_{1}(z,v,\mu )\;,\end{eqnarray} \tag{ 45 }$$

which yields

$$\begin{eqnarray}{q}_{\parallel }(z) & = & 2\pi {k}_{{\rm{B}}}{T}_{e}\;{v}_{{\rm{te}}}^{4}{\displaystyle \int }_{0}^{\infty }{dx}\;{x}^{5}{\displaystyle \int }_{-1}^{1}d\mu \;\mu \;{f}_{1}(z,x,\mu )\\ & = & -\displaystyle \frac{4}{3\sqrt{\pi }}\;{n}_{e}\;{k}_{{\rm{B}}}{T}_{e}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}{\displaystyle \int }_{0}^{\infty }{dx}\;\displaystyle \frac{{x}^{9}}{{{Rx}}^{4-\alpha }+1}\\ & & \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{e}^{-{x}^{2}}\;.\end{eqnarray} \tag{ 46 }$$

Writing this in the form

$$\begin{eqnarray}&&{q}_{\parallel }=-{\kappa }_{\parallel }\;\displaystyle \frac{{{dT}}_{e}}{{dz}}-{\alpha }_{\parallel }\;{E}_{\parallel }\;,\end{eqnarray} \tag{ 47 }$$

gives the thermoelectric coefficients

$$\begin{eqnarray}&&{\kappa }_{\parallel }=\displaystyle \frac{4}{3\sqrt{\pi }}\;{n}_{e}\;{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{9}}{{{Rx}}^{4-\alpha }+1}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}\end{eqnarray} \tag{ 48 }$$

and

$$\begin{eqnarray}&&{\alpha }_{\parallel }=\displaystyle \frac{4}{3\sqrt{\pi }}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{9}}{{{Rx}}^{4-\alpha }+1}\;{e}^{-{x}^{2}}\;{dx}\;,\end{eqnarray} \tag{ 49 }$$

respectively.

Similarly, substituting for f₁ from Equation (39) in the expression for the parallel current density

$$\begin{eqnarray}&&{j}_{\parallel }(z)=-2\pi \;e{\int }_{0}^{\infty }{dv}\;{v}^{2}{\int }_{-1}^{1}d\mu \;\mu \;v\;{f}_{1}(z,v,\mu )\end{eqnarray} \tag{ 50 }$$

gives

$$\begin{eqnarray}{j}_{\parallel }(z) & = & -2\pi e\;{v}_{{\rm{te}}}^{4}\displaystyle \int {dx}\;{x}^{3}{\displaystyle \int }_{-1}^{1}d\mu \;\mu \;{f}_{1}(z,x,\mu )\\ & = & \;\displaystyle \frac{4}{3\sqrt{\pi }}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;{\displaystyle \int }_{0}^{\infty }{dx}\;\displaystyle \frac{{x}^{7}}{{{Rx}}^{4-\alpha }+1}\\ & & \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{e}^{-{x}^{2}}\;,\end{eqnarray} \tag{ 51 }$$

and writing this in the form

$$\begin{eqnarray}&&{j}_{\parallel }\;=\;{\beta }_{\parallel }\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+{\sigma }_{\parallel }\;{E}_{\parallel }\end{eqnarray} \tag{ 52 }$$

gives the identifications

$$\begin{eqnarray}&&{\beta }_{\parallel }=\displaystyle \frac{4}{3\sqrt{\pi }}\;\displaystyle \frac{{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{T}_{e}}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{7}}{{{Rx}}^{4-\alpha }+1}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}\end{eqnarray} \tag{ 53 }$$

and

$$\begin{eqnarray}&&{\sigma }_{\parallel }=\displaystyle \frac{4}{3\sqrt{\pi }}\;\displaystyle \frac{{n}_{e}\;{e}^{2}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{k}_{{\rm{B}}}{T}_{e}}{\int }_{0}^{\infty }\displaystyle \frac{{x}^{7}}{{{Rx}}^{4-\alpha }+1}\;{e}^{-{x}^{2}}\;{dx}\;.\end{eqnarray} \tag{ 54 }$$

Note that enforcing the current neutrality condition ${j}_{\parallel }\;=\;0$ in Equation (52) shows that ${E}_{\parallel }=(-{\beta }_{\parallel }/{\sigma }_{\parallel })\;{{dT}}_{e}/{dz}$. Substituting this condition into Equation (47) implies that the thermal conductivity coefficient in a current neutralized environment is given by ${\kappa }^{\ast }={\kappa }_{\parallel }-{\alpha }_{\parallel }{\beta }_{\parallel }/{\sigma }_{\parallel }$.

We now provide explicit expressions for these coefficients in the $R\ll 1$ (collision-dominated) and $R\gg 1$ (turbulence-dominated) limiting regimes.

As expected, in the limit $R\ll 1$ we can replace the term $({{Rx}}^{4-\alpha }+1)$ in the denominators of the integrands in expressions (48), (49), (53), and (54) with unity, thus recovering the collisional (Spitzer 1962) values. Noting that the integral

$$\begin{eqnarray}&&{\int }_{0}^{\infty }{dx}\;{x}^{n}\;{e}^{-{x}^{2}}=\displaystyle \frac{1}{2}\;{\rm{\Gamma }}\;\left(\displaystyle \frac{n+1}{2}\right)=\displaystyle \frac{1}{2}\left(\displaystyle \frac{n-1}{2}\right)!\end{eqnarray} \tag{ 55 }$$

gives the purely collisional results

$$\begin{eqnarray}&&{\kappa }_{\parallel }=\displaystyle \frac{2{\rm{\Gamma }}(6)-5{\rm{\Gamma }}(5)}{3\sqrt{\pi }}\;{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}=\displaystyle \frac{40}{\sqrt{\pi }}\;{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;;\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}&&{\alpha }_{\parallel }=\displaystyle \frac{2\;{\rm{\Gamma }}(5)}{3\sqrt{\pi }}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}=\displaystyle \frac{16}{\sqrt{\pi }}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;;\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}&&{\beta }_{\parallel }=\displaystyle \frac{2{\rm{\Gamma }}(5)-5{\rm{\Gamma }}(4)}{3\sqrt{\pi }}\;\displaystyle \frac{{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{T}_{e}}=\displaystyle \frac{6}{\sqrt{\pi }}\;\displaystyle \frac{{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{T}_{e}}\;;\end{eqnarray} \tag{ 58 }$$

and

$$\begin{eqnarray}&&{\sigma }_{\parallel }=\displaystyle \frac{2\;{\rm{\Gamma }}(4)}{3\sqrt{\pi }}\displaystyle \frac{{n}_{e}{e}^{2}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{k}_{{\rm{B}}}{T}_{e}}=\displaystyle \frac{4}{\sqrt{\pi }}\displaystyle \frac{{n}_{e}{e}^{2}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{k}_{{\rm{B}}}{T}_{e}}\;.\end{eqnarray} \tag{ 59 }$$

Finally, in the zero-current regime, the effective thermal conductivity coefficient is

$$\begin{eqnarray}&&{\kappa }^{\ast }\equiv {\kappa }_{\parallel }-\displaystyle \frac{{\alpha }_{\parallel }{\beta }_{\parallel }}{{\sigma }_{\parallel }}=\displaystyle \frac{16}{\sqrt{\pi }}\;{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;.\end{eqnarray} \tag{ 60 }$$

On the other hand, when $R\gg 1$ we replace the term $({{Rx}}^{4-\alpha }+1)$ in the denominators of the integrands in expressions (48), (49), (53), and (54) with ${{Rx}}^{4-\alpha }$. This gives the following expressions:

$$\begin{eqnarray}&&{\kappa }_{\parallel }=\displaystyle \frac{2{\rm{\Gamma }}\left(\tfrac{8+\alpha }{2}\right)-5{\rm{\Gamma }}\left(\tfrac{6+\alpha }{2}\right)}{3\sqrt{\pi }}\;\displaystyle \frac{{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{R}\;,\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{\alpha }_{\parallel }=\displaystyle \frac{2\;{\rm{\Gamma }}\left(\tfrac{6+\alpha }{2}\right)}{3\sqrt{\pi }\;R}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;,\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{\beta }_{\parallel }=\displaystyle \frac{2{\rm{\Gamma }}\left(\tfrac{6+\alpha }{2}\right)-5{\rm{\Gamma }}\left(\tfrac{4+\alpha }{2}\right)}{3\sqrt{\pi }\;R}\;\displaystyle \frac{{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{T}_{e}}\;;\end{eqnarray} \tag{ 63 }$$

and

$$\begin{eqnarray}&&{\sigma }_{\parallel }=\displaystyle \frac{2\;{\rm{\Gamma }}\left(\tfrac{4+\alpha }{2}\right)}{3\sqrt{\pi }\;R}\;\displaystyle \frac{{n}_{e}{e}^{2}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{k}_{{\rm{B}}}{T}_{e}}\;;\end{eqnarray} \tag{ 64 }$$

The case $\alpha =0$ is of special interest as it corresponds (Equation (7)) to a velocity-independent turbulent mean free path ${\lambda }_{{\rm{T}}}$. In this case we have

$$\begin{eqnarray}&&{\kappa }_{\parallel }=\displaystyle \frac{2}{3\sqrt{\pi }\;R}\;{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;,\end{eqnarray} \tag{ 65 }$$

$$\begin{eqnarray}&&{\alpha }_{\parallel }=\displaystyle \frac{4}{3\sqrt{\pi }\;R}\;{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;,\end{eqnarray} \tag{ 66 }$$

$$\begin{eqnarray}&&{\beta }_{\parallel }=-\displaystyle \frac{1}{3\sqrt{\pi }\;R}\;\displaystyle \frac{{n}_{e}\;e\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{T}_{e}}\;;\end{eqnarray} \tag{ 67 }$$

and

$$\begin{eqnarray}&&{\sigma }_{\parallel }=\displaystyle \frac{2}{3\sqrt{\pi }\;R}\;\displaystyle \frac{{n}_{e}{e}^{2}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}}{{k}_{{\rm{B}}}{T}_{e}}\;.\end{eqnarray} \tag{ 68 }$$

In a zero-current scenario, the effective thermal conductivity coefficient is

$$\begin{eqnarray}&&{\kappa }^{\ast }\equiv {\kappa }_{\parallel }-\displaystyle \frac{{\alpha }_{\parallel }{\beta }_{\parallel }}{{\sigma }_{\parallel }}=\displaystyle \frac{4}{3\sqrt{\pi }\;R}\;{n}_{e}{k}_{{\rm{B}}}\;{v}_{{\rm{te}}}\;{\lambda }_{{\rm{ei}}}\;.\end{eqnarray} \tag{ 69 }$$

As expected from the simple dimensional arguments of the previous section, in the turbulence-dominated limit of large R the transport coefficients are suppressed by a factor of R with respect to the Spitzer values.

Figure 1 shows the full dependence on R of all four thermoelectric coefficients for the case $\alpha =0$. We note from this figure that in the turbulence-dominated high-R limit, one cross-transport coefficient (${\beta }_{\parallel }$) has not only changed in magnitude, but also in sign, compared to its collisional value (compare Equations (58) and (67)). Physically, this sign reversal is a consequence of the form of the Maxwellian distribution. In a uniform pressure environment,

$$\begin{eqnarray}{f}_{0}(z,v) & \sim & \displaystyle \frac{{n}_{e}(z)}{{[{T}_{e}(z)]}^{3/2}}{e}^{-{{mv}}^{2}/2{k}_{{\rm{B}}}{T}_{e}(z)}\\ & = & \left(\displaystyle \frac{{P}_{e}}{{k}_{{\rm{B}}}}\right)\;{[{T}_{e}(z)]}^{-5/2}\;{e}^{-{{mv}}^{2}/2{k}_{{\rm{B}}}{T}_{e}(z)}\;,\end{eqnarray} \tag{ 70 }$$

from which we see that there are two contributions to the temperature gradient ${{dT}}_{e}/{dz}$, one involving the derivative of the normalization coefficient ${T}_{e}^{-5/2}$ and the other the derivative of the width of the Maxwellian velocity profile. The relative signs of these terms depend on the value of the speed v: increasing the temperature flattens the Gaussian velocity profile, leading to a general increase in f₀ at a high speeds but a general decrease at low speeds. As a consequence, the spatial derivative

$$\begin{eqnarray}&&\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial z}=\displaystyle \frac{\partial {f}_{0}(z,v)}{\partial {T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}\;,\end{eqnarray} \tag{ 71 }$$

that appears in the expression (38) for the lead anisotropic component ${f}_{1}(z,v,\mu )$ changes sign at some value of v. The overall sign of a particular transport coefficient (e.g., ${\kappa }_{\parallel }$, ${\beta }_{\parallel }$) depends on the relative contributions of the particular moments of ${f}_{1}(v,z,\mu )$ that appear⁴ in the expression for that coefficient. Higher-order moments of the velocity distribution are dominated by larger values of v, so that an increase in temperature generally leads to an increase in that moment, while lower-order moments place a greater weight on lower values of v, leading to a reduced increase (or even a decrease) in that moment. And, since (1) the moments of the velocity distribution involved in the calculation of ${\kappa }_{\parallel }$ and ${\beta }_{\parallel }$ depend on the value of R; and (2) the moments involved in the calculation of ${\kappa }_{\parallel }$ are generally higher than those involved in the calculation of ${\beta }_{\parallel }$, it follows that the signs of the various thermoelectric coefficients are not necessarily invariant with respect to the value of R.

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Consider for definiteness the case of a negative temperature gradient, i.e., a temperature that decreases in the positive z direction (the case of a positive temperature gradient is entirely similar). Also consider first the collisional (low-R) limit, for which the fourth-power dependence of the collision frequency ν causes all the moments involved to be quite high, specifically, the ninth and eleventh moments for ${\kappa }_{\parallel }$ (Equation (48)) and the seventh and ninth moments for ${\beta }_{\parallel }$ (Equation (53)). These high velocity moments are all dominated by large values of v; thus, as we move toward increasing values of z (i.e., toward regions of lower temperature), the predominant effect is a reduction in ${f}_{0}(z,v)$ with z: $\partial {f}_{0}(z,v)/\partial z\lt 0$. Both the heat flux and electron flow are predominantly carried by high-velocity electrons that move in the positive-z direction to fill this relative void in f₀, and hence they are both oriented in the positive-z direction, i.e., along the direction of decreasing temperature. Indeed, we see from Equation (38) that the lead anisotropic term is aligned with the positive-z direction: ${f}_{1}(z,v,\mu )\propto \mu$. The coefficients ${\kappa }_{\parallel }$ and ${\beta }_{\parallel }$ are therefore both positive (the former since ${q}_{\parallel }\sim -{\kappa }_{\parallel }\;{{dT}}_{e}/{dz}$ and the latter since the conventional current ${j}_{\parallel }\;\sim \;{\beta }_{\parallel }\;{{dT}}_{e}/{dz}$ flows in the opposite direction to the electron flow, i.e., in the negative-z direction).

However, in the turbulence-dominated (high-R) limit, the velocity-distribution moments involved are all (four powers) lower because of the weaker velocity dependence of the collision frequency $\nu ={\nu }_{{\rm{T}}}$ (compare Equation (7) [with α = 0] and Equation (4)). Since the expression (48) for ${\kappa }_{\parallel }$ still involves relatively high-order velocity moments (fifth and ninth), this reduction of all the moment orders is not sufficient to reverse the direction of the heat flux; it remains parallel to the direction of the negative temperature gradient. However, the considerably lower-order velocity moments (third and fifth) present in the expression (53) for ${\beta }_{\parallel }$ are such that the value of the pertinent moments are now dominated by lower-velocity electrons. At these velocities a decrease in temperature causes an increase in ${f}_{0}(z,v)$. At the pertinent velocities, the spatial derivative of the isotropic zero-order term is therefore now positive: $\partial {f}_{0}(z,v)/\partial z\gt 0$, and so (Equation (38)) the lead anisotropic term now aligns in the negative-z direction (${f}_{1}(z,v,\mu )\propto -\mu$). The net result is that the predominant (low-velocity) electron flow is now in the negative-z direction, i.e., in the direction of positive temperature gradient, antiparallel to the heat flux. The conventional current and the heat flux are now both in the positive z direction, and so ${\kappa }_{\parallel }\gt 0$ and ${\beta }_{\parallel }\lt 0$.

We remind the reader that all the above results have been derived under the simplifying assumption that the turbulent collision frequency ${\nu }_{{\rm{T}}}$ is independent of pitch angle μ, corresponding to isotropic scattering. While this has provided illustrative results on the impact of turbulent scattering on the various transport coefficients, in actual physical situations the scattering rate ν may depend on the pitch angle μ (and/or on the velocity v). This is indeed the case when angular scattering is produced by the presence of a spectrum of magnetic fluctuations in the plasma, next to be considered.

Except for the additional μ integral which produces a factor that depends on the value of the dimensionless spectral index q, there will be no difference in the scalings for large R compared to those already given above for the case of isotropic scattering. Hence, while the necessary integrals can be evaluated in the large R limit, it is simpler to replace the anisotropic scattering problem by an equivalent isotropic one, i.e., set

$$\begin{eqnarray}&&\nu (v)=\displaystyle \frac{{v}_{{\rm{te}}}}{{\lambda }_{{\rm{ei}}}}\;\displaystyle \frac{1+{{Rx}}^{2q+2}}{{x}^{3}}\;,\end{eqnarray} \tag{ 87 }$$

but with

$$\begin{eqnarray}&&R=\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{{\rm{T}}0}}\end{eqnarray} \tag{ 88 }$$

instead of $R={\lambda }_{{\rm{ei}}}/{\lambda }_{0}$. This will lead to the previously given results of Section 3 with a somewhat different definition of R; these results are valid provided $q\lt 1$, i.e., $\alpha \gt 0$.

In the Lorentzian case q = 1, corresponding to

$$\begin{eqnarray}&&W({k}_{\parallel })=\displaystyle \frac{{(\delta {B}_{\perp })}^{2}}{\pi }\;\displaystyle \frac{(1/{\lambda }_{{\rm{B}}})}{{(1/{\lambda }_{{\rm{B}}})}^{2}+{k}_{\parallel }^{2}}\;,\end{eqnarray} \tag{ 89 }$$

we have

$$\begin{eqnarray}&&{\nu }_{{\rm{T}}}=| \mu | {\left(\displaystyle \frac{\delta {B}_{\perp }}{{B}_{0}}\right)}^{2}\displaystyle \frac{v}{{\lambda }_{{\rm{B}}}}=\displaystyle \frac{| \mu | v}{{\lambda }_{0}}\;,\end{eqnarray} \tag{ 90 }$$

so that

$$\begin{eqnarray}&&R=\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{0}}=\left(\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{{\lambda }_{{\rm{B}}}}\right){\left(\displaystyle \frac{\delta {B}_{\perp }}{{B}_{0}}\right)}^{2}\simeq \displaystyle \frac{{10}^{4}\;{T}_{e}^{2}\;{(\delta {B}_{\perp }/{B}_{0})}^{2}}{{n}_{e}\;{\lambda }_{{\rm{B}}}}\;.\end{eqnarray} \tag{ 91 }$$

Equation (86) can be analytically integrated over μ to give

$$\begin{eqnarray}&&{f}_{1}=\left\{\begin{array}{l}-\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{R}\;\mathrm{ln}(1+{{Rx}}^{4}\mu )\\ \quad \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{f}_{0};\quad \mu \gt 0\\ \quad \displaystyle \frac{{\lambda }_{{\rm{ei}}}}{R}\;\mathrm{ln}(1-{{Rx}}^{4}\mu )\\ \quad \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\;{f}_{0};\quad \mu \lt 0\;.\end{array}\right.\end{eqnarray} \tag{ 92 }$$

This result can now be substituted in the expressions for the normalized heat flux (cf. Equation (46)) and the current density (Equation (51)). Exploiting the hemispherical antisymmetry of Equation (92) in evaluating the integral over μ, we obtain

$$\begin{eqnarray}{q}_{\parallel } & = & -\displaystyle \frac{4{n}_{e}{k}_{{\rm{B}}}{T}_{e}{v}_{{\rm{Te}}}{\lambda }_{{\rm{ei}}}}{\sqrt{\pi }\;R}{\displaystyle \int }_{0}^{\infty }{dx}\;{x}^{5}\\ & & \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\\ & & \times \;{e}^{-{x}^{2}}\;{\displaystyle \int }_{0}^{1}d\mu \;\mu \mathrm{ln}(1+{{Rx}}^{4}\mu )\;\;\end{eqnarray} \tag{ 93 }$$

and

$$\begin{eqnarray}{j}_{\parallel } & = & \displaystyle \frac{4{n}_{e}\;e\;{v}_{{\rm{Te}}}{\lambda }_{{\rm{ei}}}}{\sqrt{\pi }\;R}{\displaystyle \int }_{0}^{\infty }{dx}\;{x}^{3}\\ & & \times \left[\left({x}^{2}-\displaystyle \frac{5}{2}\right)\displaystyle \frac{1}{{T}_{e}}\;\displaystyle \frac{{{dT}}_{e}}{{dz}}+\displaystyle \frac{{{eE}}_{\parallel }}{{k}_{{\rm{B}}}{T}_{e}}\right]\\ & & \times \;{e}^{-{x}^{2}}\;{\displaystyle \int }_{0}^{1}d\mu \;\mu \;\mathrm{ln}(1+{{Rx}}^{4}\mu )\;.\end{eqnarray} \tag{ 94 }$$

Making the substitution $y=1+{{Rx}}^{4}\mu$ and then integrating by parts, the integral over μ evaluates to

$$\begin{eqnarray}&&{\displaystyle \int }_{0}^{1}d\mu \;\mu \mathrm{ln}(1+{{Rx}}^{4}\mu )=\displaystyle \frac{1}{2{({{Rx}}^{4})}^{2}}\\ &&\quad \times \left\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})+{{Rx}}^{4}\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;.\end{eqnarray} \tag{ 95 }$$

The values of the thermoelectric transport coefficients ${\kappa }_{\parallel },{\alpha }_{\parallel }$, ${\beta }_{\parallel }$ and ${\sigma }_{\parallel }$ may now be obtained by substituting the expression (95) for the μ integral in the expressions (93) and (94). These transport coefficients are best represented in relation to their Spitzer (R = 0) values. We find

$$\begin{eqnarray}&&\displaystyle \frac{{\kappa }_{\parallel }}{{\kappa }_{\parallel ,S}}=\\ &&\displaystyle \frac{3}{2R}\;\displaystyle \frac{\begin{array}{l}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{1}{{({{Rx}}^{4})}^{2}}\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})\\ +{{Rx}}^{4}\left.\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;{x}^{5}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}\end{array}}{{\displaystyle \int }_{0}^{\infty }{x}^{9}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}\;;\end{eqnarray} \tag{ 96 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\alpha }_{\parallel }}{{\alpha }_{\parallel ,S}}=\\ &&\displaystyle \frac{3}{2R}\;\displaystyle \frac{\begin{array}{l}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{1}{{({{Rx}}^{4})}^{2}}\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})\\ +{{Rx}}^{4}\left.\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;{x}^{5}\;{e}^{-{x}^{2}}\;{dx}\end{array}}{{\displaystyle \int }_{0}^{\infty }{x}^{9}\;{e}^{-{x}^{2}}\;{dx}}\;;\end{eqnarray} \tag{ 97 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\beta }_{\parallel }}{{\beta }_{\parallel ,S}}=\\ &&\displaystyle \frac{3}{2R}\;\displaystyle \frac{\begin{array}{l}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{1}{{({{Rx}}^{4})}^{2}}\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})\\ +{{Rx}}^{4}\left.\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;{x}^{3}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}\end{array}}{{\displaystyle \int }_{0}^{\infty }{x}^{7}\left({x}^{2}-\displaystyle \frac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}\;;\end{eqnarray} \tag{ 98 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{\parallel }}{{\sigma }_{\parallel ,S}}=\\ &&\displaystyle \frac{3}{2R}\;\displaystyle \frac{\begin{array}{l}{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{1}{{({{Rx}}^{4})}^{2}}\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})\\ +{{Rx}}^{4}\left.\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;{x}^{3}\;{e}^{-{x}^{2}}\;{dx}\end{array}}{{\displaystyle \int }_{0}^{\infty }{x}^{7}\;{e}^{-{x}^{2}}\;{dx}}\;.\end{eqnarray} \tag{ 99 }$$

For $R\ll 1$, we have a collision-dominated regime, and the results (96) through (99) should approach unity. Indeed, for $R\ll 1$, the logarithm may be expanded in a Taylor series $\mathrm{ln}(1+q)=q-{q}^{2}/2+{q}^{3}/3-\ldots$, giving

$$\begin{eqnarray}&&\displaystyle \frac{1}{2{({{Rx}}^{4})}^{2}}\phantom{\rule{0ex}{3.15ex}}\{[{({{Rx}}^{4})}^{2}-1]\mathrm{ln}(1+{{Rx}}^{4})\\ &&\quad +\left.{{Rx}}^{4}\left(1-\displaystyle \frac{1}{2}{{Rx}}^{4}\right)\right\}\;\overset{R\ll 1}{\longrightarrow }\;\displaystyle \frac{1}{3}\;{{Rx}}^{4}\end{eqnarray} \tag{ 100 }$$

in place of Equation (95). (This result may also be obtained quickly by expanding the logarithm to first order in the integral ${\int }_{0}^{1}d\mu \;\mu \mathrm{ln}(1+{{Rx}}^{4}\mu )$.) Using the limiting form in the numerators of (96) through (99) yields unity for all four ratios, as it should.

On the other hand, for the turbulence-dominated transport regime characterized by $R\gg 1$, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{\kappa }_{\parallel }}{{\kappa }_{\parallel ,S}}\;\overset{R\gg 1}{\longrightarrow }\;\displaystyle \frac{3{\int }_{0}^{\infty }\;{x}^{5}\left({x}^{2}-\tfrac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}{2{\int }_{0}^{\infty }{x}^{9}\left({x}^{2}-\tfrac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)=\displaystyle \frac{1}{40}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)\;;\end{eqnarray} \tag{ 101 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\alpha }_{\parallel }}{{\alpha }_{\parallel ,S}}\;\overset{R\gg 1}{\longrightarrow }\;\displaystyle \frac{3{\int }_{0}^{\infty }\;{x}^{5}\;{e}^{-{x}^{2}}\;{dx}}{2{\int }_{0}^{\infty }{x}^{9}\;{e}^{-{x}^{2}}\;{dx}}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)=\displaystyle \frac{1}{8}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)\;;\end{eqnarray} \tag{ 102 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\beta }_{\parallel }}{{\beta }_{\parallel ,S}}\;\overset{R\gg 1}{\longrightarrow }\;\displaystyle \frac{3{\displaystyle \int }_{0}^{\infty }\;{x}^{3}\left({x}^{2}-\tfrac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}{2{\displaystyle \int }_{0}^{\infty }{x}^{7}\left({x}^{2}-\tfrac{5}{2}\right)\;{e}^{-{x}^{2}}\;{dx}}\\ &&\quad \times \left(\displaystyle \frac{\mathrm{ln}R}{R}\right)=-\displaystyle \frac{1}{12}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)\;;\end{eqnarray} \tag{ 103 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{\parallel }}{{\sigma }_{\parallel ,S}}\;\overset{R\gg 1}{\longrightarrow }\;\displaystyle \frac{3{\int }_{0}^{\infty }\;{x}^{3}\;{e}^{-{x}^{2}}\;{dx}}{2{\int }_{0}^{\infty }{x}^{7}\;{e}^{-{x}^{2}}\;{dx}}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)=\displaystyle \frac{1}{4}\left(\displaystyle \frac{\mathrm{ln}R}{R}\right)\;.\end{eqnarray} \tag{ 104 }$$

Note both the $\mathrm{ln}R/R$ dependence of these expressions and also the change of sign of ${\beta }_{\parallel }$ at large R (which occurs for the same reasons given in the discussion at the end of Section 3).

A solar flare is ubiquitously characterized by enhanced emission in soft X-rays (e.g., Culhane 1969; Culhane & Acton 1970; Antonucci et al. 1982); indeed, the commonly used GOES classification of flares is based on the soft X-ray flux in the $(1\mbox{--}8)$ Å soft X-ray waveband. The plasma responsible for emitting these soft X-rays is generally located in the corona and has a temperature $\gtrsim {10}^{7}$ K (e.g., Reeves & Golub 2011). The in situ heating (Cargill & Priest 1983; Feldman et al. 2004; Raymond et al. 2007) to these temperatures is generally believed to be due to a combination of ohmic heating associated with current dissipation in the primary site of magnetic reconnection, plus collisional heating by accelerated nonthermal particles, notably electrons. Ohmic heating by passage of the beam-neutralizing return current (see Section 5.2) through the flaring corona may also play a role.

The heat energy transported by both nonthermal electrons and thermal conduction (Heinzel & Karlický 2003) to the chromosphere causes local heating and a corresponding increase in the emission measure of 10⁷ K gas, enhancing the overall soft X-ray emission. Further, the pressure gradients established by this rapid heating of chromospheric material through the region of radiative instability from $T\simeq {10}^{5}$ K to $T\simeq {10}^{7}$ K (Cox & Tucker 1969), drive a significant hydrodynamic response of the solar atmosphere (e.g., Nagai & Emslie 1984; Mariska et al. 1989; Allred et al. 2005), in particular an upward motion of soft-X-ray-emitting plasma into the corona, a process somewhat incorrectly, but nevertheless ubiquitously, termed "chromospheric evaporation."

The hot 10⁷ K plasma thus created subsequently cools through radiation, conduction to the chromosphere, and flow of enthalpy. Estimating the pertinent cooling times shows that, under the assumption of classical, collisional (Spitzer 1962) conductivity, thermal conduction generally dominates the cooling time at the highest temperatures. However, in many events the thermal plasma is sustained well beyond the duration of the impulsive hard X-ray burst (and hence heating by nonthermal electrons), for times much longer than the conductive cooling time. This has led to the suggestion (e.g., Moore et al. 1980; Emslie et al. 2012) that energy is somehow injected, by unknown processes, into the corona after the impulsive phase has ceased. However, it is important to realize that this conclusion is based on thermal conductive losses governed by a purely collisional model of heat transport, which may not be valid if, as suggested above, turbulent processes play a significant role in the region of electron transport.

To illustrate, let us consider a flare volume of $V\simeq {10}^{27}$ cm³, with a plasma of density ${n}_{e}={10}^{10}$ cm⁻³ embedded in a magnetic field $B\simeq {10}^{3}$ Gauss. The magnetic energy density is ${B}^{2}/8\pi \simeq {10}^{5}$ erg cm⁻³ and the total available magnetic energy is $({B}^{2}/8\pi )V\simeq {10}^{32}$ erg, which is a good estimate of the amount of energy released by magnetic reconnection in a large flare (e.g., Emslie et al. 2012). Taking the flare duration as $\tau \simeq {10}^{2}\;{\rm{s}}$, this corresponds to a released power of 10³⁰ erg s⁻¹. This energy is transformed into kinetic energy of the particles, thermal and non thermal. A fraction, say 10%, of this energy goes into the kinetic energy of non-thermal electrons (Emslie et al. 2012), giving a power 10²⁹ erg s⁻¹ in accelerated electrons, a number close to that inferred from observations of hard X-ray emission in flares (e.g., Holman et al. 2011). This corresponds to a volumetric heating rate due to fast electrons $Q\simeq 100$ erg cm⁻³ s⁻¹.

With these parameters established, we can now consider the heating (by fast electrons) and cooling (by conduction) of the soft X-ray emitting plasma, governed by the energy equation

$$\begin{eqnarray}&&\displaystyle \frac{\partial ({n}_{e}{k}_{{\rm{B}}}{T}_{e})}{\partial t}=-\displaystyle \frac{\partial {q}_{\parallel }}{\partial z}+Q\;.\end{eqnarray} \tag{ 105 }$$

As we have seen above, the conductive heat flux takes the form

$$\begin{eqnarray}&&{q}_{\parallel }=-\displaystyle \frac{2{n}_{e}{k}_{{\rm{B}}}\;{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}{{m}_{e}^{1/2}}\;\lambda \;\displaystyle \frac{{{dT}}_{e}}{{dz}}\;,\end{eqnarray} \tag{ 106 }$$

where λ is the mean free path, representing the combined effect of Coulomb collisions and turbulent scattering,

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{\lambda }_{{\rm{ei}}}}{1+R}\;,\end{eqnarray} \tag{ 107 }$$

and where the turbulent reduction factor

$$\begin{eqnarray}&&R=\displaystyle \frac{{\lambda }_{\mathrm{ei}}}{{\lambda }_{0}}=\displaystyle \frac{{10}^{4}\;{T}_{e}^{2}\;{(\delta {B}_{\perp }/{B}_{0})}^{2}}{{n}_{e}\;{\lambda }_{{\rm{B}}}}\;.\end{eqnarray} \tag{ 108 }$$

When $R\gg 1$, i.e., when the turbulent parameter ${\lambda }_{0}$ is much smaller than the collisional mean free path ${\lambda }_{{\rm{ei}}}$ (corresponding to ${\lambda }_{0}\ll {10}^{4}{T}_{e}^{2}/{n}_{e}$), heat transport is regulated predominantly by turbulence. We call this a "turbulence-dominated regime," although we again stress (cf. Section 2.1) that the role of collisions in maintaining a near Maxwellian distribution of background electrons remains important. On the other hand, turbulence plays a negligible role when $R\ll 1$.

Let us first balance the terms on the right side of Equation (105) to obtain an estimate of the steady-state temperature in the presence of a heat source Q balanced by thermal conduction:

$$\begin{eqnarray}&&2{n}_{e}{k}_{{\rm{B}}}{\left(\displaystyle \frac{2{k}_{{\rm{B}}}{T}_{e}}{{m}_{e}}\right)}^{1/2}\lambda \;\left(\displaystyle \frac{{T}_{e}}{{L}^{2}}\right)\simeq Q\;,\end{eqnarray} \tag{ 109 }$$

from which follows the scaling law

$$\begin{eqnarray}&&{T}_{e}\simeq \displaystyle \frac{{m}_{e}^{1/3}}{2{k}_{{\rm{B}}}}{\left(\displaystyle \frac{{{QL}}^{2}}{{n}_{e}\lambda }\right)}^{2/3}\;.\end{eqnarray} \tag{ 110 }$$

The vast majority of theoretical studies concerning the response of coronal loops to heating use the collisional (Spitzer 1962) conductivity and corresponding mean free path ${\lambda }_{{\rm{ei}}};$ this includes the determination of equilibrium scaling laws and temperature distribution for active region loops (e.g., Craig et al. 1978; Rosner et al. 1978; Martens 2010) and the modeling of evaporative cooling and enthalpy-based response to coronal heating (e.g., Antiochos & Sturrock 1978; Cargill et al. 1995; Reale 2010; Cargill et al. 2012). When the mean free path is indeed close to its collisional value ${\lambda }_{\mathrm{ei}}$ (Equation (5)), Equation (110) gives the scaling law appropriate to a collisional regime of transport (Rosner et al. 1978) :

$$\begin{eqnarray}&&{T}_{e}\simeq \displaystyle \frac{{m}_{e}^{1/7}}{2{k}_{{\rm{B}}}}\;{(2\pi {e}^{4}\mathrm{ln}{\rm{\Lambda }})}^{2/7}{Q}^{2/7}\;{L}^{4/7}\simeq 50\;{Q}^{2/7}\;{L}^{4/7}\;.\end{eqnarray} \tag{ 111 }$$

Note that this is independent on the density n_e, and that this scaling follows straightforwardly from the familiar balancing relation $Q\propto {T}_{e}^{7/2}/{L}^{2}$ appropriate to a collisional environment. Substituting the values $Q\simeq 100$ erg cm⁻³ s⁻¹ and $L\simeq {10}^{9}\;{\rm{cm}}$, we obtain

$$\begin{eqnarray}&&{T}_{e}\simeq 3\times {10}^{7}\;{\rm{K}}\;,\end{eqnarray} \tag{ 112 }$$

a value nicely consistent with soft X-ray emission.

On the other hand, returning to Equation (110) with $R\gg 1$, we now obtain the fundamentally different scaling law for the equilibrium temperature in a turbulence dominated regime:

$$\begin{eqnarray}&&{T}_{e}\simeq \displaystyle \frac{{m}_{e}^{1/3}}{2{k}_{{\rm{B}}}}\;{\left(\displaystyle \frac{Q}{{n}_{e}}\right)}^{2/3}{L}^{4/3}{\lambda }_{{\rm{B}}}^{-2/3}\;{\left(\displaystyle \frac{\delta {B}_{\perp }}{{B}_{0}}\right)}^{4/3}\;,\end{eqnarray} \tag{ 113 }$$

which now depends on the density n_e. With a magnetic field perturbation ratio $\delta {B}_{\perp }/{B}_{0}\simeq 0.1$ and a magnetic correlation length ${\lambda }_{{\rm{B}}}\simeq {10}^{6}\;{\rm{cm}}$ (${\lambda }_{0}\simeq {10}^{8}$ cm) gives the significantly larger temperature

$$\begin{eqnarray}&&{T}_{e}\simeq 1\times {10}^{8}\;{\rm{K}}\;.\end{eqnarray} \tag{ 114 }$$

Hence, for a given heating rate Q and loop properties n_e and L, the turbulent suppression of heat transport associated with large values of R leads to a higher steady-state temperature than that obtained by using Spitzer conductivity. Plasma at temperatures ∼10⁸ K ($\simeq 10$ keV) can make a meaningful contribution to the hard X-ray emission from the flare. Since thermal conduction is inhibited, possibly also by other collective plasma processes (Brown et al. 1979; Smith & Brown 1980), such 10⁸ K temperatures will likely be confined rather than extending along the entire magnetic loop. For a given heating rate Q, Equations (111) and (113) give quite different dependencies of the flaring coronal temperature T_e on the loop length L (${L}^{4/7}$ and ${L}^{4/3}$, respectively), a result that should be observationally testable.

We note that the steady state temperature ${T}_{e}=3\times {10}^{7}$ K in Equation (112) above was obtained by assuming a collisional transport regime. Such an assumption is valid only under the dual conditions that both the Knudsen number ${\rm{Kn}}={\lambda }_{{\rm{ei}}}/{L}_{{\rm{T}}}$ and the turbulent reduction factor R are $\ll 1$. Consistency therefore demands that we evaluate the validity of these two assumptions. We find that at these temperatures (and densities ${n}_{e}\sim {10}^{10}$ cm⁻³), the collisional mean free path ${\lambda }_{{\rm{ei}}}$ is actually of the order of the loop length $L\simeq {10}^{9}\;{\rm{cm}};$ thus the Knudsen number is of order unity and so the use of a collision-related expression for the heat flux is somewhat questionable. In such conditions the heat flux is instead determined (e.g.. Brown et al. 1979) by a flux-limited value equal to a fraction of the free-streaming limit (Equation (19)), leading to a generally higher equilibrium temperature and conductive cooling time than is appropriate for the collisional case. With regard to the second assumption, the main thrust of the present paper is that, whether or not the Knudsen number is small, the heat flux may be further limited (by a factor $\simeq R$) by the presence of collisionless pitch-angle scattering. Thus, if the turbulent mean free path is substantially less than the collisional mean free path (or, for Knudsen numbers of order unity or more, the length of the flaring loop), then the much larger temperatures (${T}_{e}\sim {10}^{8}$ K; Equation (114)) appropriate to a turbulence-dominated regime apply.

We now turn to a consideration of the role of conduction in cooling the coronal plasma after the energy input Q has ceased. Balancing the heating and conductive cooling in Equation (105)), and using the expression (106), we obtain an expression for the cooling time-scale ${\tau }_{{\rm{c}}}$:

$$\begin{eqnarray}&&{\tau }_{{\rm{c}}}\simeq \displaystyle \frac{{m}_{e}^{1/2}}{{(2{k}_{{\rm{B}}}{T}_{e})}^{1/2}}\;\displaystyle \frac{{L}^{2}}{\lambda }=\left(\displaystyle \frac{L}{{v}_{{\rm{te}}}}\right)\left(\displaystyle \frac{L}{\lambda }\right)=\left(\displaystyle \frac{L}{{v}_{{\rm{te}}}}\right)\left(\displaystyle \frac{L}{{\lambda }_{{\rm{ei}}}}\right)\;(1+R)\;.\end{eqnarray} \tag{ 115 }$$

The cooling time ${\tau }_{{\rm{c}}}$ is thus the free-streaming transport time-scale $L/{v}_{{\rm{te}}}$ of thermal electrons multiplied by the inverse of the Knudsen number ${\rm{Kn}}=\lambda /L$. For a temperature ${T}_{e}\simeq 3\times {10}^{7}$ K (Equation (112)), the electron thermal speed ${v}_{{\rm{Te}}}\simeq 3\times {10}^{9}$ cm s⁻¹, so that for a loop length $L\simeq {10}^{9}\;{\rm{cm}}$, the free-streaming escape time is $L/{v}_{{\rm{te}}}\;\simeq$ 0.3 s. The collisional mean free path ${\lambda }_{{\rm{ei}}}\simeq {10}^{9}\;{\rm{cm}}$ and hence $L/{\lambda }_{\mathrm{ei}}\sim 1$, which when $R\ll 1$, yields a cooling time of the order of the free-streaming time scale. A smaller initial temperature ${T}_{e}\simeq {10}^{7}$ K has the collisional mean free path decreased by an order of magnitude giving a cooling time ${\tau }_{{\rm{c}}}\sim 3\;{\rm{s}}$. These timescales are much shorter than the duration of the soft X-ray emission, which has led to the realization that the coronal plasma will cool very rapidly after the cessation of the energy input term Q, to the point where some form of post-impulsive-phase energy input to the corona is needed to sustain the soft X-ray emission for the observed times $\gtrsim 100\;{\rm{s}}$ (see, e.g., Moore et al. 1980; Emslie et al. 2012). However, in the presence of turbulence the cooling time ${\tau }_{{\rm{c}}}$ is further enhanced by a factor $(1+R)$, and so cooling by thermal conduction parallel to the guiding magnetic field can thus be significantly inhibited. This increase in the cooling time significantly reduces the previously assumed requirement (Moore et al. 1980; Emslie et al. 2012) for post-impulsive phase heating of the coronal plasma.

Before moving on, we parenthetically note that similar issues regarding thermal conductivity (and its suppression) in stochastic magnetic fields arise in the context of galaxy cluster formation and in the theory of cooling flows (Fabian 1994; Chandran & Cowley 1998; Malyshkin & Kulsrud 2001).

The very significant electrical currents associated with the injection of non-thermal electrons necessitate a current-neutralizing return current, set up by a combination of electrostatic and inductive processes, the relative role of which has been a matter of some debate (Knight & Sturrock 1977; Emslie 1980; Spicer & Sudan 1984; Holman 1985; Larosa & Emslie 1989; van den Oord 1990; Zharkova et al. 1995; Zharkova & Gordovskyy 2005, 2006; Battaglia & Benz 2008; Codispoti et al. 2013). However, irrespective of the detailed physics responsible for establishing this return current, driving it through the finite resistivity of the ambient medium requires a local electric field ${{ \mathcal E }}_{\parallel }={j}_{\parallel }/{\sigma }_{\parallel }$, which in turn causes both an Ohmic heating rate ${Q}_{{\rm{rc}}}={j}_{\parallel }\;{{ \mathcal E }}_{\parallel }={j}_{\parallel }^{2}/{\sigma }_{\parallel }$ and an additional energy loss rate $| {dE}/{dt}| =e\;{{ \mathcal E }}_{\parallel }\;v={j}_{\parallel }\;{{ \mathcal E }}_{\parallel }/{n}_{e}={j}_{\parallel }^{2}/{n}_{e}\;{\sigma }_{\parallel }$ for each of the accelerated electrons.

Now, the transport of non-thermal electrons is dominated by non-diffusive cold-target energy losses (see, e.g., Brown 1971, 1973; Emslie 1978), and hence we expect that the transport of such electrons, and hence the current density ${j}_{\parallel }$ that they carry, is largely unaffected by collisionless pitch-angle scattering. (Although Kontar et al. (2014) have shown that the direct beam current ${j}_{\parallel }$ is also reduced somewhat due to the presence of pitch-angle scattering, the reduction factor is not as large as the transport coefficient reduction factors R considered here, so that we may assume that the current density ${j}_{\parallel }$ associated with the injected electrons is essentially the same as in the purely collisional case.) Any change in ohmic energy losses is therefore driven primarily by changes in the parallel electrical conductivity ${\sigma }_{\parallel }$. Reducing the value of ${\sigma }_{\parallel }$ through turbulence results in a greater rate of Ohmic heating ${Q}_{{\rm{rc}}}$ (and hence higher coronal heating rates) and also a greater energy loss rate $| {dE}/{dt}|$ for the accelerated electrons.

This enhancement of the return-current electron energy loss rates affects the heating rate as a function of position (Emslie 1980) and hence the hydrodynamic response of the atmosphere (e.g., Nagai & Emslie 1984; Mariska et al. 1989). It also reduces the amount of energy precipitating into the chromospheric footpoints, thus possibly accounting for the "gentle" evaporation observed by, e.g., Zarro & Lemen (1988). Enhanced return current energy losses also result in a more effective confinement of hard-X-ray-producing electrons in the corona, which may offer an alternative explanation for loop-top coronal sources (Kane & Hurford 2003; Xu et al. 2008; Guo et al. 2012a, 2012b; Jeffrey et al. 2014).

Estimates of the ratio of return current heating to collisional energy loss in the flaring corona show that, for moderately large flares they are comparable (see Figure 3 of Emslie 1980). The same figure shows that the ratio of return current heating to collisional energy loss in the chromosphere, where most of the electron heating occurs, can be up to several percent. Thus, enhancing the return current heating/energy loss rate by even an order of magnitude through turbulent modification of the electrical conductivity ${\sigma }_{\parallel }$ and could possibly transform the flaring corona into a return-current-dominated regime (Knight & Sturrock 1977; Larosa & Emslie 1989; van den Oord 1990; Zharkova et al. 1995). This has very significant implications, ranging from the spatial distribution of hard X-ray emission and electron heating, to the total number of accelerated electrons required to produce a given hard X-ray intensity (Zharkova & Gordovskyy 2005, 2006).

A more dominant role for return current losses in the energy loss rate for accelerated electrons has a possibly even more interesting effect. Since the energy loss rate for an individual electron $| {dE}/{dt}| =e\;{{ \mathcal E }}_{\parallel }\;v=e\;v\;{j}_{\parallel }/{\sigma }_{\parallel }$, which is proportional to the injected current ${j}_{\parallel }$ and hence the electron injection rate, and since the total hard X-ray yield is proportional to the injection rate divided by the energy loss rate (Brown & Emslie 1988), it follows that the hard X-ray yield in a return-current-loss dominated regime is independent of the injected number of electrons (Emslie 1980). Such a possible saturation of hard X-ray flux with increasing flare intensity has been reported by Alexander & Daou (2007).