2.1. Electron kinetic equation

Given the electron distribution function $f$, the kinetic equation in the absence of transverse motion of the guiding centre is given by

(2.1)\begin{equation} \frac{\partial f}{\partial t} + v_\parallel \frac{\partial f}{\partial z} + \frac{e}{m_e}\frac{\partial \phi}{\partial z}\frac{\partial f}{\partial v_\parallel} = C, \end{equation}

for $\phi$ the electric potential, $z$ the coordinate parallel to the magnetic field line, $v_\parallel$ the parallel velocity, $e$ the absolute electron charge, $m_e$ the electron mass and $C$ the collision operator. We split up the collision operator into self-collisions (electron–electron, ‘$e,e$’) and collisions with ions (electron–ion (species $k$), ‘$e,ik$’):

(2.2)\begin{equation} C = C_{e,e}(f) + \sum_k C_{e,ik}(f). \end{equation}

Then, we express the left-hand side of the kinetic equation in terms of energy $\mathcal {E} = m_ev^2/2 -e\phi$, magnetic moment $\mu = m_ev_\perp ^2/(2B)$ and $z$:

(2.3)\begin{equation} \frac{\partial f}{\partial t} + v_\parallel \frac{\partial f}{\partial z} - e \frac{\partial \phi}{\partial t}\frac{\partial f}{\partial \mathcal{E}} = C_{e,e}(f) + \sum_k C_{e,ik}(f). \end{equation}

This form of the kinetic equation highlights the fact that any collisionless change in electron energy is associated with explicit time variation of the electric potential. As the problem is symmetric in $z$, $f$ is an even function of $v_\parallel$, so we do not need to account for the part of $f$ that is odd in $v_\parallel$ when changing variables from $(v_\parallel, v_\perp, z, t)$ to $(\mathcal {E},\mu,z,t)$.

The self-collision operator is given by

(2.4)\begin{equation} C_{e,e}(f) = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e^2}\boldsymbol{\nabla}_{\boldsymbol{v}}\boldsymbol{\cdot}\left(\boldsymbol{\nabla}_{\boldsymbol{v}}\varphi f - \left(\boldsymbol{\nabla}_{\boldsymbol{v}}\boldsymbol{\nabla}_{\boldsymbol{v}} \psi\right)\boldsymbol{\nabla}_{\boldsymbol{v}}f\right) \end{equation}

(Helander & Sigmar Reference Helander and Sigmar2002, pp. 29–30, (3.15), (3.16), (3.21)), for $\varphi$ and $\psi$ the Rosenbluth potentials

(2.5a–c)\begin{equation} \varphi ={-}\frac{1}{4 {\rm \pi}} \int \frac{f\left(\boldsymbol{v}^\prime\right)}{u}\,\mathrm{d}^3v^\prime,\quad \psi ={-}\frac{1}{8{\rm \pi}}\int u f\left(\boldsymbol{v}^\prime\right)\mathrm{d}^3v^\prime,\quad \boldsymbol{u} = \boldsymbol{v} - \boldsymbol{v}^\prime, \end{equation}

the double nabla notation being

(2.6)\begin{equation} \boldsymbol{\nabla}_{\boldsymbol{v}}\boldsymbol{\nabla}_{\boldsymbol{v}}h := \hat{\boldsymbol{v}}_i\hat{\boldsymbol{v}}_j\frac{\partial^2 h}{\partial v_i \partial v_j}. \end{equation}

We stress that henceforth the integral notation will be such that, for integration variable $\lambda$, the integrand is to the right of $\int$ and to the left of $\mathrm {d}\lambda$.

Collisions with ions are well-approximated by pitch-angle scattering against stationary charges:

(2.7)\begin{equation} C_{e,ik}(f) = \frac{e^4 \ln \varLambda}{4 {\rm \pi}\varepsilon_0^2 m_e^2} Z_k^2 n_{ik} \frac{1}{v^3}\mathcal{L}(f) \end{equation}

for Lorentz scattering operator

(2.8)\begin{equation} \mathcal{L} = \frac{1}{2\sin\theta}\frac{\partial }{\partial \theta}\left(\sin\theta \frac{\partial}{\partial \theta}\right), \end{equation}

which serves to alter only the pitch angle $\theta = \arctan {(v_\perp /v_\parallel )}$, leaving the energy of the electron unaffected. We assume no variation of $f$ in the gyroangle. In the variables $(\mathcal {E}, \mu, z)$, the Lorentz scattering operator is

(2.9)\begin{equation} \mathcal{L} = m_e v_\parallel \frac{\partial }{\partial \mu} \left(\frac{\mu v_\parallel}{B} \frac{\partial }{\partial \mu}\right), \end{equation}

indicating that anisotropy in $f$ is associated with its dependence on $\mu$.

We note that the terms proportional to $\partial f/\partial t$ and $\partial f/\partial \mathcal {E}$ on the left-hand side of (2.3) are associated with the time variation of the potential, whereas $v_\parallel (\partial f/\partial z)$ is associated with bounce motion. We seek to further simplify this kinetic equation in the context of rapid pitch-angle scattering. To this end, turning our attention first to the electron self-collision operator, we observe that in spherical coordinates (with no dependence on the gyroangle),

(2.10)\begin{gather} \boldsymbol{\nabla}_{\boldsymbol{v}} \varphi = \hat{\boldsymbol{v}} \frac{\partial \varphi}{\partial v} + \frac{\hat{\boldsymbol{\theta}}}{v} \frac{\partial \varphi}{\partial \theta}, \end{gather}

(2.11)\begin{gather}\boldsymbol{\nabla}_{\boldsymbol{v}}\boldsymbol{\nabla}_{\boldsymbol{v}} \psi = \left(I - \hat{\boldsymbol{v}}\hat{\boldsymbol{v}}\right)\frac{1}{v}\frac{\partial \psi}{\partial v} + \hat{\boldsymbol{v}}\hat{\boldsymbol{v}} \frac{\partial^2 \psi}{\partial v^2} - \frac{2\hat{\boldsymbol{\theta}}\hat{\boldsymbol{v}}}{v^2}\frac{\partial \psi}{\partial \theta} + \left(\hat{\boldsymbol{v}}\hat{\boldsymbol{\theta}} + \hat{\boldsymbol{\theta}}\hat{\boldsymbol{v}}\right)\frac{1}{v}\frac{\partial^2 \psi}{\partial \theta \partial v} + \frac{\hat{\boldsymbol{\theta}}\hat{\boldsymbol{\theta}}}{v^2}\frac{\partial^2 \psi}{\partial \theta^2}, \end{gather}

(2.12)\begin{gather}\boldsymbol{\nabla}_{\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{F} = \frac{1}{v^2}\frac{\partial }{\partial v}\left(\boldsymbol{F}\boldsymbol{\cdot}\hat{\boldsymbol{v}}\,v^2\right) + \frac{1}{v\sin\theta}\frac{\partial }{\partial \theta}\left(\boldsymbol{F}\boldsymbol{\cdot}\hat{\boldsymbol{\theta}}\sin\theta\right), \end{gather}

which gives

(2.13)\begin{equation} C_{e,e}(f) = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e^2}\left\{\frac{1}{v^2}\frac{\partial }{\partial v}\left[v^2\left(\frac{\partial \varphi}{\partial v}f - \frac{\partial^2 \psi}{\partial v^2}\frac{\partial f}{\partial v}\right)\right] - \frac{2}{v^3}\frac{\partial \psi}{\partial v}\mathcal{L}(f) + R(f,\varphi,\psi)\right\} \end{equation}

for

(2.14)\begin{align} R(f,\varphi,\psi) & ={-}\frac{1}{v^3}\frac{\partial^2 \psi}{\partial \theta \partial v}\frac{\partial f}{\partial \theta} - \frac{1}{v^2} \frac{\partial }{\partial v}\left(\frac{\partial^2 \psi}{\partial \theta \partial v}\frac{\partial f}{\partial \theta}\right)\nonumber\\ & \quad +\frac{1}{v\sin\theta}\frac{\partial }{\partial \theta}\left[\sin\theta\left(\frac{1}{v}\frac{\partial \varphi}{\partial \theta}f - \left(\frac{1}{v}\frac{\partial^2 \psi}{\partial \theta \partial v} - \frac{2}{v^2}\frac{\partial \psi}{\partial \theta}\right)\frac{\partial f}{\partial v} - \frac{1}{v^3}\frac{\partial^2 \psi}{\partial \theta^2}\frac{\partial f}{\partial \theta}\right)\right], \end{align}

noting that all terms in $R$ are proportional to a $\theta$ derivate of one of the Rosenbluth potentials. Thus,

(2.15)\begin{align} C_{e,e}(f) + \sum_k C_{e,ik}(f) & = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e^2} \left\{\frac{1}{v^2}\frac{\partial }{\partial v}\left[v^2\left(\frac{\partial \varphi}{\partial v}f - \frac{\partial^2 \psi}{\partial v^2}\frac{\partial f}{\partial v}\right)\right]\right.\nonumber\\ & \quad \left. + \left(\frac{1}{4{\rm \pi}}\sum_k Z_k^2 n_{ik} - 2\frac{\partial \psi}{\partial v}\right)\frac{1}{v^3}\mathcal{L}(f) + R(f,\varphi,\psi)\right\}. \end{align}

In this form, collisions are split up into the following effects: energy-altering terms arising from self-collisions, pitch-angle scattering arising from self-collisions and collisions with ions, and the remainder term $R$ which is composed of terms due to angular variation of the Rosenbluth potentials.

We suppose that in the tail of the distribution function pitch-angle scattering and electron bounce motion are faster than collisions that alter energy. That is, the tail distribution follows the ordering

(2.16)\begin{equation} \left|\frac{\partial }{\partial t}\right| \sim \left|C_\mathrm{energy-altering}\right| \sim \left|C_R\right| \ll \left|v_\parallel \frac{\partial }{\partial z}\right| \sim \left|C_\mathrm{p.a.\,scattering}\right|, \end{equation}

where $C_\mathrm {p.a.\,scattering}$ is the term proportional to $\mathcal {L}$ in (2.15), $C_\mathrm {energy-altering}$ is the term proportional to $(1/v^2)(\partial /\partial v)$ and $C_R$ is the term proportional to $R$.

We correspondingly split up the distribution:

(2.17)\begin{equation} f = f_0 + f_1,\quad f_0 \gg f_1, \end{equation}

giving the lowest-order kinetic equation

(2.18)\begin{equation} v_\parallel \frac{\partial f_0}{\partial z} = v_\parallel G(\mathcal{E},\mu,z) \frac{\partial }{\partial \mu}\left(\frac{\mu v_\parallel}{v^3 B}\frac{\partial f_0}{\partial \mu}\right), \end{equation}

where

(2.19)\begin{equation} G = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e}\left(\frac{1}{4{\rm \pi}}\sum_k Z_k^2 n_{ik} - 2\frac{\partial \psi}{\partial v}\right). \end{equation}

Dividing through by $v_\parallel$ and orbit integrating, assuming that $f_0$ is symmetric in $z$, yields

(2.20)\begin{equation} \oint G \frac{\partial }{\partial \mu}\left(\frac{\mu v_\parallel}{v^3 B}\frac{\partial f_0}{\partial \mu}\right)\mathrm{d}z = 0. \end{equation}

As we are in the context of fast pitch-angle scattering, the above and (2.18) are solved by

(2.21)\begin{equation} \frac{\partial f_0}{\partial \mu} = \frac{\partial f_0}{\partial z} = 0. \end{equation}

However, we note that for the lowest-energy ‘core’ electrons, which are deeply trapped in the potential well, self-collisions dominate. The distribution of these electrons will be near-Maxwellian, with the distribution of Maxwellian core electrons being independent of both $z$ and $\mu$.

Therefore, (2.17) and (2.21) hold for the total distribution function labelled $f$. The kinetic equation to next order is given by

(2.22)\begin{align} \frac{\partial f_0}{\partial t} + v_\parallel \frac{\partial f_1}{\partial z} - e \frac{\partial \phi}{\partial t}\frac{\partial f_0}{\partial \mathcal{E}} & = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e^2} \left\{\frac{1}{v^2}\frac{\partial }{\partial v}\left[v^2\left(\frac{\partial \varphi_0}{\partial v}f_0 - \frac{\partial^2 \psi_0}{\partial v^2}\frac{\partial f_0}{\partial v}\right)\right] \right.\nonumber\\ & \quad + \left. m_e v_\parallel \left(\frac{1}{4{\rm \pi}}\sum_k Z_k^2 n_{ik} - 2\frac{\partial \psi_0}{\partial v}\right)\frac{\partial }{\partial \mu}\left(\frac{\mu v_\parallel}{v^3 B}\frac{\partial f_1}{\partial \mu}\right)\right\}, \end{align}

where we henceforth treat $f$ as the total electron distribution function. In the above, we note that $R$ vanishes in this order, and we denote $\varphi _0$ and $\psi _0$ the Rosenbluth potentials calculated from $f_0$.

Dependence upon $f_1$ can be eliminated from the above by applying the following integration to both sides:

(2.23)\begin{equation} \int^{({\mathcal{E} + e\phi_m})/{B}}_0\oint\,\frac{\cdot}{v_\parallel}\,\mathrm{d}z\,\mathrm{d}\mu, \end{equation}

which corresponds to first integrating over bounce motion, then over all possible values of $\mu$, the variable representing anisotropy, for a given energy $\mathcal {E}$.

We observe that the volume element is given by $\mathrm {d}^3v\,\mathrm {d}z = (4{\rm \pi} B)/(m_e^2v_\parallel )\,\mathrm {d}\mathcal {E}\,\mathrm {d}\mu \,\mathrm {d}z$, so the above may be interpreted as multiplying by the volume element and integrating over $z$ and $\mu$, leaving only dependence on $\mathcal {E}$. That is, the resulting kinetic equation describes how electrons may move between shells of constant energy due to adiabatic change in the potential and collisions. In essence, the zeroth moment of the kinetic equation is taken with one integration variable in momentum space and another in physical space, which is in contrast to the usual moment over all of momentum space.

Dropping the subscript from $f_0$, the resulting kinetic equation is

(2.24)\begin{equation} \frac{\partial f}{\partial t} = \left\langle C\right\rangle \end{equation}

for the integrated collision operator

(2.25)\begin{equation} \left\langle C\right\rangle = \frac{e^4 \ln \varLambda}{\varepsilon_0^2 m_e^2}\frac{\partial }{\partial K}\left[f \int^K_0 f^\prime \,\mathrm{d}K^\prime + H\frac{\partial f}{\partial K}\left(\int^K_0 f^\prime \frac{K^\prime}{H^\prime}\,\mathrm{d}K^\prime + K\int^\infty_K \frac{f^\prime}{H^\prime}\,\mathrm{d}K^\prime\right) \right], \end{equation}

where

(2.26)\begin{gather} K = \oint_s \frac{m_e^2 v^3}{3}\, \mathrm{d}z, \end{gather}

(2.27)\begin{gather}H = \frac{\partial K}{\partial \mathcal{E}} = \oint_s m_e v\,\mathrm{d}z. \end{gather}

We note that $f = f(K,t)$ in (2.24). Here $K$ is the invariant associated with electrons undergoing rapid pitch-angle scattering and bounce motion inside a slowly varying electric potential well while experiencing no energy-altering collisions. The definition of $\oint _s$ and the details of the integration procedure with its associated simplification of the kinetic equation are contained in Appendix A.

2.2. Trapped and passing electrons

Equation (2.24) expresses the fact that in the absence of collisions that alter energy, the quantity $K$ is constant for an individual electron. The fact that $K$ involves the orbit integral of $v^3$ rather than $v$ corresponds to the energy of an individual electron being equipartitioned into all three degrees of freedom by pitch-angle scattering.

At this point it is useful to make the distinction between passing and trapped electrons, and those with energy $\mathcal {E}$ above or below zero. By convention, trapped electrons are those that, when experiencing collisionless motion, are unable to escape the well; their position does not tend to infinity in infinite time. The conditions for trapped/passing electrons are

(2.28)\begin{equation} \begin{cases} \mathcal{E} < \mu B, & \mathrm{trapped,} \\ \mathcal{E} \geqslant \mu B, & \mathrm{passing.} \end{cases} \end{equation}

Consequently, bounce-averaged kinetic problems may have a separatrix at $\mathcal {E} = \mu B$.

However, the kinetic equation (2.24) has also been integrated over $\mu$, the variable responsible for anisotropy. Physically speaking, any electron with $\mathcal {E} > 0$ will experience enough pitch-angle scattering to become untrapped within any timescale present in (2.24). Conversely, electrons with $\mathcal {E} < 0$ are never able to be pitch-angle scattered such that they become passing: they simply do not have enough kinetic energy; if all their kinetic energy were in the parallel degree of freedom, their parallel speed would not equal or exceed $\sqrt {2e\phi /m_e}$.

Therefore, we introduce the terminology energy-trapped and energy-passing

(2.29)\begin{equation} \begin{cases} \mathcal{E} < 0, & \mathrm{energy-trapped,} \\ \mathcal{E} \geqslant 0, & \mathrm{energy-passing,} \end{cases} \end{equation}

in order to avoid confusion with the conventional definitions of trapped and passing.

When dealing with bounce-averaged problems on infinitely long magnetic field lines, is it very important to distinguish between trapped and passing distributions, because the bounce average of any quantity on such a field line takes on the value it has in the limit $|z|\rightarrow \infty$. This is a consequence of the orbit of a passing electron being infinitely long. Such problems have a passing–trapped separatrix, the passing distribution typically being static, with the trapped distribution equal to the passing at the separatrix to ensure continuity.

However, for closed field lines, the orbit of a passing electron is simply the whole field line. In order to account for a closed field line, it is enough to define the turning points of an orbit integral to be $\pm z_c$, where

(2.30)\begin{equation} z_c(\mathcal{E},\mu,t): \begin{cases} \mathcal{E} - \mu B(z_c,t) + e\phi(z_c,t) = 0, & \mathcal{E} < \mu B,\\ L_F/2, & \mathcal{E} \geqslant \mu B, \end{cases} \end{equation}

for connection length $L_F$. Then, electrons of all energies and magnetic moments can be considered in the bounce-integrated kinetic equation.

2.3. Alternative forms of the integrated collision operator

The integrated collision operator can be expressed directly in terms of phase-space moments of $f$:

(2.31)\begin{equation} \left\langle C\right\rangle = \frac{e^4 \ln \varLambda}{2{\rm \pi}\varepsilon_0^2} \frac{\partial }{\partial K}\left(N_Kf + H \frac{\partial f}{\partial K} \left(\frac{2}{3}E_K + K M_K\right)\right), \end{equation}

for $N_K$, $E_K$ and $M_K$ defined in Appendix B (see (B5), (B7) and (B8)). Here $N_K$ corresponds to the line-integrated density of electrons with invariant less than $K$, $E_K$ is the line-integrated kinetic energy of electrons with invariant less than $K$ and $K M_K$ is not so easily interpreted, but has the same dimensions as $E_K$ and is associated with electrons with invariant larger than $K$.

The above is a friction-diffusion form of the collision operator (namely, Rosenbluth potential form), and the contributions to each effect are somewhat intuitive: an electron with invariant $K$ experiences friction proportional to the number of electrons with invariant less than $K$, and experiences diffusion proportional to two quantities, one being the kinetic energy of electrons with invariants smaller than $K$, the other being associated with electrons with invariants larger than $K$.

The collision operator in the form (2.31) is convenient for numerical implementation and is clearly represented as a friction-diffusion operator. However, it may also be written in the form

(2.32)\begin{equation} \left\langle C\right\rangle = \frac{e^4 \ln\varLambda}{\varepsilon_0^2 m_e^2} \frac{\partial }{\partial K}\int^\infty_{{-}e\phi_m} Q\left(f^\prime \frac{\partial f}{\partial \mathcal{E}} - f \frac{\partial f^\prime}{\partial \mathcal{E}^\prime}\right)\mathrm{d}\mathcal{E}^\prime, \end{equation}

where

(2.33)\begin{equation} Q = K^\prime \varTheta\left(\mathcal{E} - \mathcal{E}^\prime\right) + K \varTheta\left(\mathcal{E}^\prime - \mathcal{E}\right) \end{equation}

for $\varTheta$ the Heaviside step function. Equation (2.32) closely resembles the Landau collision operator as presented in the original paper (Landau Reference Landau1936). In this form it is clear that the integrated collision operator vanishes if the distribution is Maxwellian.

2.4. Particle conservation and the $H$ theorem

As the integrated kinetic equation was obtained from the drift kinetic equation, it must conserve particles and have an $H$ theorem. Both may be proven within the framework of the invariant $K$. Equation (2.32) provides the most convenient form for proving these properties. We write $\left \langle C\right \rangle = A (\partial I/\partial K)$ for brevity, where $A = (e^4\ln \varLambda /\varepsilon _0^2 m_e^2)$ and

(2.34)\begin{equation} I = \int^\infty_{{-}e\phi_m} Q\left(f^\prime \frac{\partial f}{\partial \mathcal{E}} - f \frac{\partial f^\prime}{\partial \mathcal{E}^\prime}\right)\mathrm{d}\mathcal{E}^\prime. \end{equation}

The line-integrated electron density (B5) is

(2.35)\begin{equation} N = \frac{2{\rm \pi}}{m_e^2}\int^\infty_0 f\,\mathrm{d}K, \end{equation}

thus,

(2.36)\begin{equation} \frac{\mathrm{d} N}{\mathrm{d} t} = \frac{2{\rm \pi} A}{m_e^2}\int^\infty_0 \frac{\partial I}{\partial K}\,\mathrm{d}K = 0, \end{equation}

proving particle conservation. The line-integrated electron entropy density is

(2.37)\begin{equation} S ={-} \frac{2{\rm \pi}}{m_e^2}\int^\infty_0 f \ln f\,\mathrm{d}K, \end{equation}

thus,

(2.38)\begin{equation} \frac{\mathrm{d} S}{\mathrm{d} t} ={-}\frac{2{\rm \pi}}{m_e^2}\int^\infty_0 \frac{\partial f}{\partial t}\left(1 + \ln f\right)\mathrm{d}K. \end{equation}

Substituting (2.24), integrating by parts and changing the integration variable to $\mathcal {E}$ yields

(2.39)\begin{equation} \frac{\mathrm{d} S}{\mathrm{d} t} = \frac{2{\rm \pi} A}{m_e^2} \int^\infty_{{-}e\phi_m}\int^\infty_{{-}e\phi_m} Q ff^\prime\left(\frac{\partial \ln f}{\partial \mathcal{E}} - \frac{\partial \ln f^\prime}{\partial \mathcal{E}^\prime}\right)\frac{\partial \ln f}{\partial \mathcal{E}}\,\mathrm{d}\mathcal{E}\,\mathrm{d}\mathcal{E}^\prime. \end{equation}

Using the fact that above expression holds when exchanging primed and unprimed terms (noting that $Q^\prime = Q$), we see that

(2.40)\begin{equation} \frac{\mathrm{d} S}{\mathrm{d} t} = \frac{{\rm \pi} A}{m_e^2} \int^\infty_{{-}e\phi_m}\int^\infty_{{-}e\phi_m} Q ff^\prime\left(\frac{\partial \ln f}{\partial \mathcal{E}} - \frac{\partial \ln f^\prime}{\partial \mathcal{E}^\prime}\right)^2\,\mathrm{d}\mathcal{E}\,\mathrm{d}\mathcal{E}^\prime \geqslant 0, \end{equation}

proving the $H$ theorem. It is also clear that the line-integrated entropy density is constant when $f$ is Maxwellian.

2.5. Generalisation to an electric–magnetic potential well

The preceding section may be generalised to include a symmetric, time-dependent magnetic field. If we include the magnetic mirror force $-\mu (\partial B/\partial z)$ in the kinetic equation (2.3), the calculation may be carried out in the same fashion as in the theory section and Appendix A (making sure not to neglect the $(z,t)$ dependence of $B$, and replacing $(\mathcal {E}+e\phi _m)/B$ with $(\mathcal {E}+e\phi _m)/(B(z=0))$ in (2.23)). We find that now the invariant $K$ has a contribution from the magnetic well:

(2.41)\begin{equation} K = \oint_s \frac{m_e^2 v^3}{3B}\,\mathrm{d}z. \end{equation}

The form of the integrated kinetic equation and integrated collision operator in terms of $K$ is identical. With regards to the conservation properties and $H$ theorem, one subtlety must be observed: the integration of particle density and entropy density must be carried out over an infinitesimal flux tube rather than over a field line. Indeed, a time-varying magnetic field changes the cross-section of the flux tube, moving the guiding centres of electrons along with the field lines, resulting in compression or expansion of the plasma due to collisionless transverse motion.

As this investigation concerns itself with parallel dynamics, the effect of magnetic field variation has been neglected, allowing us to be agnostic about the transverse profile of the plasmoid, ambient plasma and magnetic field.

5.1. Dimensionless scaling

For the purpose of numerics, we use a dimensionless scaling

(5.1a–e)\begin{equation} f^* = \frac{2{\rm \pi} K_m}{m_e^2}f,\quad K^* = \frac{K}{K_m},\quad H^* = \frac{H}{H_m},\quad \phi^* = \frac{\phi}{\phi_m},\quad \mathcal{E}^* = \frac{\mathcal{E}}{e\phi_m}, \end{equation}

where $K_m$ and $H_m$ are the values of $K$ and $H$ at $\mathcal {E} = 0$, respectively (their maximum values for an energy-trapped electron). The resulting kinetic equation is

(5.2)\begin{align} \frac{\partial f^*}{\partial t} & = \frac{\partial }{\partial K^*}\left[\left(K^* \frac{\mathrm{d} \ln K_m}{\mathrm{d} t} + \frac{e^4 \ln \varLambda}{2 {\rm \pi}\varepsilon_0^2 K_m}N^*_{K^*}\right)f^*\right]\nonumber\\ & \quad + \frac{\partial }{\partial K^*}\left[\frac{e^4 \ln \varLambda}{2{\rm \pi} \varepsilon_0^2 K_m}H^*\left(\frac{2}{3}E^*_{K^*} + K^*M^*_{K^*}\right)\frac{\partial f^*}{\partial K^*}\right], \end{align}

where

(5.3)\begin{equation} \left.\begin{gathered} K^* = \left. \oint_s \left(\mathcal{E}^* + \phi^*\right)^{3/2} \,\mathrm{d}z \right/ \oint_s \left(\phi^*\right)^{3/2} \mathrm{d}z, \\ K_m = \frac{2}{3} \sqrt{2m_e} \left(e\phi_m\right)^{3/2} \oint_s \left(\phi^*\right)^{3/2}\, \mathrm{d}z, \\ H^* = \left.\oint_s \left(\mathcal{E}^* + \phi^*\right)^{1/2} \,\mathrm{d}z \right/ \oint_s \left(\phi^*\right)^{1/2}\, \mathrm{d}z, \\ H_m = \sqrt{2m_e} \left(e\phi_m\right)^{1/2} \oint_s \left(\phi^*\right)^{1/2} \,\mathrm{d}z, \\ N^*_{K^*} = \int^{K^*}_0 {f^*}^\prime\,\mathrm{d}{K^*}^\prime = N_K, \\ E^*_{K^*} = \frac{3}{2}\int^{K^*}_0 {f^*}^\prime \frac{{K^*}^\prime}{{H^*}^\prime}\,\mathrm{d}{K^*}^\prime = \frac{H_m}{K_m}E_K, \\ M^*_{K^*} = \int^\infty_{K^*} {f^*}^\prime \frac{\mathrm{d}{K^*}^\prime}{{H^*}^\prime} = H_m M_K, \end{gathered}\right\} \end{equation}

and $\oint _s$ in the absence of $\mathcal {E}^*$ in the integrand corresponds to $2\int ^{\infty }_{-\infty }$.

5.2. Numerical scheme and the self-consistent electric potential

The numerical package FiPy (Guyer, Wheeler & Warren Reference Guyer, Wheeler and Warren2009) was used to solve (5.2). The solver uses explicit time-stepping, and is based on the finite-volume method, so manifestly conserves particles. Energy is not manifestly conserved, and is a significant challenge with such a nonlinear collision operator. Obtaining sufficient energy conservation typically requires a very fine $K^*$ grid, small time-stepping and multiple linear sweeps within a timestep.

Supposing that the electric potential is self-consistent at a given timestep, it is no longer self-consistent after $f^*$ has been advanced by (5.2). The new self-consistent electric potential must be found before the next timestep. Therefore, we introduce a ‘dummy’ time variable $s$ between timesteps, which is used in the procedure that finds the self-consistent potential.

As $n_i$ is specified, and $n_e(z) = g[f](\phi (z))$ for the functional $g[f]$ (namely, (B9)), we may simply invert $g[f]$ to find the $\phi$ that gives $n_e = n_i$ at each point. However, the distribution function must change if the potential changes. Therefore, any change in the potential requires solving the kinetic equation excluding the effect of collisions that alter energy:

(5.4)\begin{equation} \frac{\partial f^*}{\partial s} = \frac{\partial }{\partial K^*}\left(K^* \frac{\mathrm{d} \ln K_m}{\mathrm{d} t}f^*\right). \end{equation}

Thus, we iteratively solve for $\phi$ via inversion of $g[f]$, then update $f$ via the above equation. After many iterations, $n_e$ approaches $n_i$ and, consequently, $\phi$ approaches the self-consistent value given by quasineutrality.

5.3. Ion profile

Any symmetric single-peak ion profile may be used within the model. In this paper, we take the Gaussian profile from the self-similar plasmoid expansion in Aleynikov et al. (Reference Aleynikov, Breizman, Helander and Turkin2019), which provides a qualitative picture for the shape of a pellet plasmoid. We choose the ion density to be this Gaussian profile plus a constant density of ambient ions:

(5.5)\begin{equation} n_i = n_a + N_{ip} \frac{1}{\sqrt{\rm \pi}}\frac{1}{L_p} \mathrm{e}^{-\left({z}/{L_p}\right)^2}, \end{equation}

where $n_i$ is the effective density of singly charged ions, $N_{ip}$ is the line-integrated density of ‘plasmoid ions’ (associated with the excess density) and $L_p$ is the length of the plasmoid. The ion profile qualitatively corresponds to that shown in figure 1.