2.1 Relativistic drift-kinetic equation

The relativistic equations of motion for the electron guiding centre, ${\boldsymbol X}_{\rm gc}$, can be written as follows (Littlejohn Reference Littlejohn1984):

(2.1a)\begin{gather} \dot{\boldsymbol X}_{\rm gc} = v_\parallel\hat{\boldsymbol B} + {\boldsymbol v}_{\rm dr}, \end{gather}

(2.1b)\begin{gather}{\boldsymbol v}_{\rm dr} = \frac{c}{B^2} [{\boldsymbol E}\times{\boldsymbol B}] - \frac{m_e c u^2(1+\xi^2)}{2 e B^3 \gamma}[{\boldsymbol B}\times\boldsymbol{\nabla} B] \equiv {\boldsymbol v}_E + {\boldsymbol v}_{\boldsymbol{\nabla} B}. \end{gather}

Here, $u = p_e/m_e=v\gamma$ is the electron momentum per unit mass and $\gamma =\sqrt {1+u^2/c^2}$ is the relativistic factor; E and B are the electric and magnetic field; $v_\parallel =u_\parallel /\gamma$ and $u_\parallel =({\boldsymbol u}\boldsymbol{\cdot }\hat {\boldsymbol B})$ with $\hat {\boldsymbol B}={\boldsymbol B}/B$ as the unit vector and $\xi = u_\parallel /u$ the pitch; the drift velocities ${\boldsymbol v}_E$ and ${\boldsymbol v}_{\boldsymbol {\nabla } B}$ correspond in the toroidal plasma to the poloidal precession due to ${\boldsymbol E}\times {\boldsymbol B}$ and the vertical drift due to $\boldsymbol {\nabla } B$, respectively. The adiabatic invariance of the magnetic moment, $\lambda =m_e u_\perp ^2/2B$, is also taken into account. Here and below, the sign of electron charge is taken into account explicitly, i.e. $e=|e|$. All other notations are standard.

Using the pair of variables $(u,\xi )$, one can write also

(2.2a)\begin{gather} \dot{u} =-\frac{e\gamma}{m_e u}(\dot{\boldsymbol X}_{\rm gc}\cdot{\boldsymbol E}) =-\frac{e\xi}{m_e}E_\parallel+ \frac{cu(1+\xi^2)}{2B^3} ({\boldsymbol E}\boldsymbol{\cdot }[{\boldsymbol B}\times\boldsymbol{\nabla} B]), \end{gather}

(2.2b)\begin{gather}\dot{\xi} =(1-\xi^2)\left[-\frac{e}{m_eu}E_\parallel+ \frac{c\xi}{2B^3}({\boldsymbol E}\cdot[{\boldsymbol B}\times\boldsymbol{\nabla} B]) - \frac{u}{2\gamma B^2}({\boldsymbol B}\cdot\boldsymbol{\nabla} B)\right], \end{gather}

where $E_\parallel$ is the inductive electric field. With these variables the relativistic drift-kinetic equation (rDKE) can be written as

(2.3)\begin{equation} \displaystyle{\frac{\partial f_e}{\partial t}} + \boldsymbol{\nabla}\boldsymbol{\cdot }(\dot{\boldsymbol X}_{\rm gc} \,f_e) + \frac{1}{u^2} \displaystyle{\frac{\partial }{\partial u}} (u^2\dot{u}f_e) + \displaystyle{\frac{\partial }{\partial \xi}} (\dot{\xi}f_e) = C_e(\,f_e), \end{equation}

which has practically the same form as in the classical approach. The collision operator in the right-hand side, $C_e=C_{\rm ee}+C_{\rm ei}$, describes Coulomb collisions of electrons with themselves and with ions, where ions are considered as non-relativistic particles. The fully relativistic formulation for $C_e$ is taken from Beliaev & Budker (Reference Beliaev and Budker1956) and Braams & Karney (Reference Braams and Karney1989). The source and loss terms are omitted since they do not play any role in the present consideration. Below, the case of low characteristic velocities of the electron flows, $V_{\rm flow}\ll v _{\mathrm {Te}}$, is considered, while the temperature is assumed to be arbitrarily high.

The distribution function can be assumed to be a perturbed thermodynamic equilibrium, $f_e=f_{e0}+f_{e1}$. The relativistic thermal equilibrium, $\,f_{e0}$, is given by the Jüttner distribution function, also called the relativistic Maxwellian (de Groot et al. Reference de Groot, van Leewen and van Weert1980; Braams & Karney Reference Braams and Karney1989)

(2.4)\begin{equation} f_{e0}=\frac{n_e m_e}{4{\rm \pi} cT_e K_2(\mu_e)} \exp(-\mu_e\gamma), \end{equation}

where $K_n(x)$ is a modified Bessel function of the second kind, and $\mu _e=m_ec^2/T_e$ (typically, $\mu _e \simeq 10\unicode{x2013}20$ for D+T fusion plasmas, while for a $p+^{11}$B (Putvinski et al. Reference Putvinski, Ryutov and Yushmanov2019) aneutronic scheme it can be approximately $\mu _e\simeq 1.7$).

It seems more convenient to represent the Maxwell–Jüttner distribution function in classic-like form

(2.5)\begin{equation} f_{e0}=C_{\rm MJ}\frac{n_e}{{\rm \pi}^{3/2}u_{\mathrm{Te}}^3} \exp[-\mu_e(\gamma-1)], \end{equation}

with $u_{\mathrm{Te}}=\sqrt {2T_e/m_e}$. Note that $u_{\mathrm{Te}}$ only formally coincides with the classical thermal velocity $v_{\mathrm{Te}}$, while its physical meaning is different: it is the thermal momentum per unit mass, $u_{\mathrm{Te}}\equiv p _{\mathrm{Te}}/m_e$ with $p_{\mathrm{Te}}=\sqrt {2m_eT_e}$, which, unlike $v_{\mathrm{Te}}$, is not limited by the speed of light. It is also possible to find a simple relation

(2.6)\begin{equation} \mu_e(\gamma-1) = \frac{u^2}{u_{\mathrm{Te}}^2} \frac{2}{\gamma+1}, \end{equation}

that is particularly useful for numerical calculations.

Since the Maxwellian is normalized by density, $n_e=\int f_{e0}\, {\rm d}^3u$, the normalizing factor can accordingly be written as

(2.7)\begin{equation} C_{\rm MJ}(\mu_e)=\sqrt{\frac{\rm \pi}{2\mu_e}} \frac{\mathrm{e}^{-\mu_e}}{K_2(\mu_e)} = 1-\frac{15}{8\mu_e}+\frac{345}{128\mu_e^2}- \frac{3285}{1024\mu_e^3} + O(\mu_e^{-4}), \end{equation}

where the asymptotic series for $K_2(x)$ is applied (Abramowitz & Stegun Reference Abramowitz and Stegun1972).

2.2 Moments of distribution function and fluxes

Here, the necessary set of moments for $f_e$ is given in the relativistic formulation. As usual, we assume that the first two even moments, density and thermal energy, are given by the zeroth order of the distribution function, $f_{e0}$, while the particle and heat fluxes come from the linear perturbation, $f_{e1}$, which must be found by solving the linearized rDKE.

In classical transport theory, the energy balance is usually written for thermal energy as $W=\frac {3}{2}nT$, while in the relativistic approach the use of this definition makes the energy balance equation non-divergent.

In the literature devoted to relativistic kinetics (Dzhavakhishvili & Tsintsadze Reference Dzhavakhishvili and Tsintsadze1973; de Groot et al. Reference de Groot, van Leewen and van Weert1980; Mettens & Balescu Reference Mettens and Balescu1990), the total energy is typically used. For the relativistic Maxwellian, it can be written as

(2.8)\begin{equation} \mathcal{E}_{\rm total} = \int m_ec^2\gamma f_{e0} \, {\rm d}^3u = n_em_ec^2 \frac{K_3(\mu_e)}{K_2(\mu_e)}-n_eT_e. \end{equation}

Alternatively, it is convenient to represent the thermal energy, $W_e = \mathcal {E}_{\rm total} - n_e m_ec^2$, in the classic-like form (Marushchenko et al. Reference Marushchenko, Azarenkov and Marushchenko2013)

(2.9)\begin{equation} W_e =(\tfrac{3}{2}+\mathcal{R})n_e T_e, \end{equation}

where $\mathcal {R}$ is the relativistic correction

(2.10)\begin{equation} \mathcal{R}(\mu_e) = \mu_e\left(\frac{K_3(\mu_e)}{K_2(\mu_e)}-1\right)-\frac{5}{2} = \frac{15}{8\mu_e} - \frac{15}{8\mu_e^2} + \frac{135}{128\mu_e^3} + O(\mu_e^{-4}), \end{equation}

This is a monotonic function increasing with $T_e$. One can find that, for the range $T_e$ from 25 to 75 keV, the correction term $\mathcal {R}$ varies from 0.09 to 0.24.

As usual in neoclassical transport theory, we assume that even moments defined only by the zeroth order of the distribution function, $f_{e0}$, give quantities such as $n_e$ and $W_e$, while odd moments defined by $f_{e1}$ give fluxes

(2.11a)\begin{gather} {\pmb{\varGamma}_e} = \displaystyle\int\dot{\boldsymbol X}_{\rm gc} f_{e1} \, {\rm d}^3u, \end{gather}

(2.11b)\begin{gather}{\boldsymbol Q}_e = \displaystyle\int\dot{\boldsymbol X}_{\rm gc} m_ec^2 (\gamma-1) f_{e1} \, {\rm d}^3u, \end{gather}

where ${\pmb \varGamma }_e$ and ${\boldsymbol Q}_e$ are the fluxes of particles and energy, respectively. It follows that the heat flux can also be written in the classic-like form (Marushchenko et al. Reference Marushchenko, Azarenkov and Marushchenko2013)

(2.12)\begin{equation} {\boldsymbol q}_e = {\boldsymbol Q}_e - (\tfrac{5}{2}+\mathcal{R})T_e{\pmb\varGamma}_e. \end{equation}

This expression can also be obtained from the 4-vector formalism (see Appendix A). Again, the difference from the standard non-relativistic definition (Hinton & Hazeltine Reference Hinton and Hazeltine1976; Helander & Sigmar Reference Helander and Sigmar2002) consists of the presence of the relativistic correction term $\mathcal {R}$.

2.3 Relativistic transport equations for toroidal plasmas

To obtain the equations of radial transport in toroidal plasmas, the standard procedure of flux-surface averaging is applied (Hinton & Hazeltine Reference Hinton and Hazeltine1976; Helander & Sigmar Reference Helander and Sigmar2002)

(2.13)\begin{equation} \langle F\rangle = \frac{1}{V^\prime}\iint F\sqrt{g}\,{\rm d} \theta \,{\rm d} \phi, \end{equation}

with $\theta$ and $\phi$ as the poloidal and toroidal angles, respectively, $V^\prime \equiv {\rm d} V/ {\rm d} \rho$ and $V=\iint {\rm d} \theta \,{\rm d} \phi \int _0^\rho \sqrt {g}\,{\rm d} \rho$ as the volume enclosed within the magnetic surface labelled $\rho$ and where $\sqrt {g}=(\boldsymbol {\nabla }\rho \boldsymbol{\cdot }\boldsymbol {\nabla }\theta \times \boldsymbol {\nabla }\phi )^{-1}$ is the Jacobian.

It seems convenient to interpret the flux-surface label $\rho$ as the effective radius $r_{\mathrm{eff}}$, making $\boldsymbol {\nabla }\rho$ a dimensionless quantity. The geometric factor $\langle |\boldsymbol {\nabla }\rho |\rangle$, which formally should be present in the equations for the averaged moments, plays no role here and is omitted for simplicity.

In order to obtain the continuity equation, one needs to integrate (2.3) with averaging over the magnetic surface

(2.14)\begin{equation} \displaystyle{\frac{\partial n_e}{\partial t}} + \frac{1}{V^\prime}\displaystyle{\frac{\partial }{\partial \rho}} (V^\prime \varGamma_e^\rho) = 0, \end{equation}

where the radial flux is defined as

(2.15)\begin{equation} \varGamma_e^\rho = \left\langle \int ({\boldsymbol v}_{\boldsymbol{\nabla} B}\boldsymbol{\cdot }\boldsymbol{\nabla}\rho) \,f_{e1} \, {\rm d}^3u \right\rangle. \end{equation}

Although the continuity equation has the same form as in the non-relativistic limit, the radial flux contains the relativistic corrections (Marushchenko et al. Reference Marushchenko, Azarenkov and Marushchenko2013; Kapper et al. Reference Kapper, Kasilov, Kernbichler and Aradi2018).

The next equation is the power balance, which can be obtained by integrating (2.3) with weight $m_ec^2(\gamma -1)$ and using (2.9)

(2.16)\begin{equation} \displaystyle{\frac{\partial }{\partial t}} \left[\left(\frac{3}{2}+\mathcal{R}\right) n_e T_e\right] + \frac{1}{V^\prime}\displaystyle{\frac{\partial }{\partial \rho}} (V^\prime Q_e^\rho) = P_{\rm ei} + P_E, \end{equation}

where the radial energy flux is

(2.17)\begin{equation} Q_e^\rho = \left\langle \int ({\boldsymbol v}_{\boldsymbol{\nabla} B}\boldsymbol{\cdot }\boldsymbol{\nabla}\rho) m_ec^2(\gamma-1) f_{e1}\, {\rm d}^3 u \right\rangle = q_e^\rho +\left(\frac{5}{2}+\mathcal{R}\right)T_e\varGamma_e^\rho. \end{equation}

On the right-hand side of (2.16), $P_{\rm ei}$ is the rate of heat exchange between electrons and ions (Marushchenko et al. Reference Marushchenko, Azarenkov and Marushchenko2012)

(2.18)\begin{equation} P_{\rm ei} = \left\langle \int m_ec^2(\gamma-1) C_{\rm ei}(\,f_{e0},f_{i0}) \, {\rm d}^3u \right\rangle = C_{\rm MJ}(\mu_e)\left(1+\frac{2}{\mu_e}+\frac{2}{\mu_e^2}\right) P_{\rm ei}^{\rm (nr)}, \end{equation}

where both electrons and ions are assumed to be Maxwellian (relativistic and classical, respectively) and $P_{\rm ei}^{\rm (nr)}$ is the non-relativistic expression (Braginskii Reference Braginskii1965)

(2.19)\begin{equation} P_{\rm ei}^{\rm (nr)} =-3 \frac{m_e}{M_i} \frac{n_e}{\tau_{\rm ei}} (T_e-T_i), \end{equation}

with $\tau _{\rm ei} = ({3\sqrt {{\rm \pi} }}/{4})({n_e}/{n_i Z_i^2}) ({\ln \varLambda ^{e/e}}/{\ln \varLambda ^{e/i}})\nu _{e0}^{-1}$ and $\nu _{e0}={4{\rm \pi} n_e e^4\ln \varLambda ^{e/e}}/{m_e^2u_{\mathrm{Te}}^3}$ as the characteristic $e/i$ collisional time and the simplest electron collision frequency, respectively. The formulation remains valid as long as the Landau approximation for small scattering angles is valid.

The next term, $P_E$, represents the work of the electric field

(2.20)\begin{equation} P_E =-\left\langle \displaystyle\int m_ec^2(\gamma-1)\frac{1}{u^2}\displaystyle{\frac{\partial }{\partial u}} (u^2\dot{u}f_{e}) \, {\rm d}^3u \right\rangle = P_{E_\parallel} + P_{E_\rho}, \end{equation}

where $\dot {u}$ is given by (2.2a). The terms $P_{E_\parallel }$ and $P_{E_\rho }$ correspond to the work of the parallel (inductive) and radial (ambipolar) components of the electric field, $E_\parallel \hat {\boldsymbol B}$ and $E_\rho \boldsymbol {\nabla }\rho$, respectively. Given that $-e\varGamma _{e\parallel }=j_\parallel$ (ion current is not considered here) with $j_\parallel /B=\text {const}$ on the magnetic surface, these quantities can be written as

(2.21a,b)\begin{equation} P_{E_\parallel} = \langle \,j_\parallel B\rangle \frac{\langle E_\parallel B\rangle}{\langle B^2\rangle}\quad \text{and}\quad P_{E_\rho} =-e\varGamma_e^\rho E_\rho, \end{equation}

with $E_\rho =-\varPhi ^\prime$ and $\varPhi (\rho )$ as plasma potential.

To close the (2.14) and (2.16), it is necessary to define the fluxes, $\varGamma _e^\rho$, $q_e^\rho$ and $j_\parallel$, as functions of the plasma parameters, their gradients and the electric field.