2. Radiative cooling in electron–positron plasmas

It was shown in (I) that a plasma that is optically thin to cyclotron emission will radiate its perpendicular kinetic energy on a timescale given by the radiation time:

(2.1a,b)\begin{gather} \frac{\textrm{d} w_{\perp}}{\textrm{d} t} = - \frac{{e}^{4}B^{2}}{3{\rm \pi}\epsilon_{0}(mc)^{3}} w_{\perp}, \quad \tau_{r} = \frac{3{\rm \pi}\epsilon_{0}(mc)^{3}}{{e}^{4}B^{2}} = \frac{3{\rm \pi} \rho_{e}}{\varOmega_{e}^{2}c}, \end{gather}

with $\varOmega _{e} = eB/mc$ electron cyclotron frequency and $\rho _{e} = v_{\text {th}e}/\varOmega _{e}$ the electron gyroradius.

2.1. Timescales in laboratory pair plasmas

In the aforementioned laboratory efforts to create and confine the first terrestrial electron–positron plasma, high-field devices with magnetic fields of the order of 1 T are being investigated as possible candidates for confinement. The radiative cooling time is inversely proportional to the square of the magnetic field strength and, hence, devices with larger magnetic fields will be able to dissipate perpendicular energy more quickly. As such, in systems of interest, the radiative cooling time can be relatively short compared with the other timescales in the system.

It is clear that collisions can mediate this cooling process owing to the scattering of the velocity vector. A simple estimate of the collision time is given by

(2.2)\begin{gather} \tau_{c} = \frac{3}{4\sqrt{\rm \pi}nv_{\text{th}}b_{\text{min}}^{2}\ln \varLambda}, \end{gather}

where $b_{\text {min}} = {e^{2}}/{2{\rm \pi} \epsilon _{0}mv_{\text {th}}^{2}}$ is the classical distance of closest approach, $v_{\text {th}} = \sqrt {2T/m}$ is the thermal velocity and $\ln \varLambda$ is the Coulomb logarithm. The factor $3/4\sqrt {{\rm \pi} }$ has been inserted to make $\tau _{c}$ equal to the conventional definition of the electron collision time by Braginskii (Reference Braginskii1965). From this estimate, and the target parameters given in the text, one can see that

(2.3)\begin{gather} \epsilon = \frac{\tau_{r}}{\tau_{c}} = \frac{nm \ln \varLambda}{\sqrt{\rm \pi} \epsilon_{0}B^{2}} \left(\frac{c}{v_{\text{th}}}\right)^{3} \ll 1. \end{gather}

In the following analysis, this parameter surfaces repeatedly, and usually turns up inside a logarithm. Our results hold to logarithmic accuracy in epsilon, whose exact definition is, thus, only important within a multiplicative factor of order unity. Such factors will therefore usually be ignored. An equivalent definition (again up to an order unity factor) of our expansion parameter is given by

(2.4)\begin{gather} \epsilon = \frac{\ln \varLambda}{\sigma_{M}}\left(\frac{c}{v_{\text{th}}}\right)^{3}, \end{gather}

where $\sigma _{M} = B^{2}/nmc^{2}\mu _{0}$. This quantity is the ratio of the magnetic energy to the rest mass energy and is sometimes referred to as the magnetisation in the astrophysical literature (e.g. Sironi & Spitkovsky Reference Sironi and Spitkovsky2010). From this form, one sees immediately the constraints which must be imposed on the magnetisation and temperature in order to satisfy the ordering requirements.

A caveat here is that, of course, the collision time itself will decrease as the plasma cools and, hence, this assumption will be violated after sufficient time has elapsed.

3. Radiative cooling in collisional plasmas

In § 2, we used simple estimates to show that, to leading order, the plasma is effectively collisionless. However, this is an assumption which, of course, will not be true indefinitely. In particular, the collision frequency, $\nu _{c}$ scales with $T^{-3/2}$ and this means that as the plasma loses thermal energy through radiative cooling, the collision frequency will increase.

After sufficient time has elapsed, the collision time will become comparable with the radiation time and collisional effects will become important. That is, eventually the assumption $\tau _{r}\nu _{c} \ll 1$ will be violated. On even longer timescales, the collision time will be shorter than the radiation time.

We wish to gain an understanding of how the plasma energy evolves as a function of time in these regimes. To this end, we introduce the total, perpendicular and parallel thermal energy of the plasma through the appropriate moments of the total (electron + positron) distribution function

(3.1a–c)\begin{gather} W = \int \frac{mv^{2}}{2} f \, \textrm{d}^{3}{\boldsymbol{v}}, \quad W_{\perp} = \int \frac{mv_{\perp}^{2}}{2} f \, \textrm{d}^{3}{\boldsymbol{v}}, \quad W_{\parallel} = \int \frac{mv_{\parallel}^{2}}{2} f \, \textrm{d}^{3}{\boldsymbol{v}}. \end{gather}

It is clear that this evolution of these quantities can be partitioned into three separate regimes based on the relative sizes of the collision time $\tau _{c}$ and the radiation time $\tau _{r}.$

1. (I) Initially, $0 < \tau _{r} < t < \tau _{c}(0),$ the initial collision time. In this regime the behaviour is described as in (I), § 2 therein, and we recover the result that the perpendicular energy will decay exponentially on the radiation timescale, whilst the parallel kinetic energy will remain constant:

   (3.2a,b)\begin{gather} W_{\perp} = W_{\perp0} \exp (-t/\tau_{r}), \quad W_{\parallel} = \text{constant.} \end{gather}

2. (II) After some time, $t \sim \tau _{c}(t)$ and collisional scattering becomes important. The remainder of this paper is devoted to studying this process.

3. (III) Eventually, $t \gg \tau _{r}, \tau _{c}(t)$. In this regime, the distribution function is isotropic (Maxwellian), and any remaining parallel kinetic energy will be converted into perpendicular kinetic energy via collisional scattering and then radiated:

   (3.3)\begin{gather} W \propto \exp (-t/\tau_{r}). \end{gather}

4. Collisions in strongly magnetised pair plasmas

Guided by our experience in (I), we write down the collisional kinetic equation in the varying magnetic field case

(4.1)\begin{gather} \frac{\partial f}{\partial t} + v_{\parallel}\frac{\partial f}{\partial l} - \frac{\partial}{\partial\mu}\left(\frac{\mu}{\tau_{r}}f\right) - \frac{\mu}{m}\boldsymbol{\nabla}_{\parallel}B \frac{\partial f}{\partial v_{\parallel}} = C[\,f,f], \end{gather}

where $l$ parameterises the length of a magnetic field line, and we retain the assumption that the parallel electric field is negligible. We have also introduced $C[\,f,f]$ as the appropriate collision operator. It is, of course, pertinent to discuss the collisions that occur in such exotic plasmas.

Great care must be taken as we recall that our plasmas are in an unusually strongly magnetised regime, $\rho \ll \lambda _{D}$. As such, the standard collision operators cannot be used immediately as these invoke the opposite ordering and, therefore, do not ‘see’ the gyromotion of the colliding particles. When the collisions are described by a Debye shielded interaction, particles can be separated by a distance even as large as $\lambda _{D}$ and still exchange momentum and energy.

In the limit $\rho \ll \lambda _{D},$ the relevant collision operator is non-standard and must include the effects of helical trajectories. Such collisions are prevalent in highly magnetised non-neutral plasma experiments and have attracted some attention in the literature, being invoked to explain enhanced heat and particle transport in Penning–Malmberg trap experiments (e.g. O'Neil Reference O'Neil1985; Dubin & O'Neil Reference Dubin and O'Neil1988; Dubin Reference Dubin1998). However, a rigorous collision operator has never been written down. The reason for this is the presence of a novel effect, namely that there can be multiple binary collisions between the same pair of particles, rendering the usual techniques fruitless (Dubin Reference Dubin1997a). Earlier work, starting from that of Ichimaru & Rosenbluth (Reference Ichimaru and Rosenbluth1970), ignored this circumstance.

However, all hope is not lost. During such long-range collisions, particles execute many gyrations around the magnetic field lines. As a result of this, it follows that the magnetic moment is an adiabatic invariant (Dubin Reference Dubin1997b):

(4.2)\begin{gather} \rho \ll \lambda _{D} \implies \mu = \frac{mv_{\perp}^{2}}{2B} = \text{constant}. \end{gather}

As such, the perpendicular kinetic energy will be conserved and these relatively long-range collisions result in negligible velocity scattering. The physical reasoning for this is as follows. When guiding centres on different field lines interact, the long-range Coulomb force causes an exchange in momentum. The force transverse to the magnetic field does almost no work, because, at leading order, the guiding centres are constrained to follow the magnetic field lines. However, the force parallel to the magnetic field can do work, causing a transfer of energy between the guiding centres.

It is in light of this remarkable feature of the long-range collisions that we are able to make progress without knowing the full collision operator.

Of course, standard short-range collisions, with impact parameter $b \ll \rho ,$ i.e. those that do not ‘see’ the gyromotion, are still present and, in fact, it is these collisions that will be responsible for velocity scattering. We may therefore decompose the collision operator as

(4.3)\begin{gather} C[\,f,f] = C_{1}(f) + C_{2}(f), \end{gather}

where $C_{1}(f)$ is our unknown collision operator describing long-range collisions that preserve $v_{\perp }$ and $\mu ,$ and $C_{2}(f)$ is the standard Landau operator describing short-range collisions (Landau Reference Landau1936). Each of these collision operators has an associated collision time, $\tau _{1}$ and $\tau _{2},$ respectively.

We make the assumption that long-range collisions are much more frequent than short-range collisions. Our aim is to prove three properties about the unknown collision operator $C_{1}$ and how it acts in tandem with the Landau collision operator $C_{2}.$

1. (P ₁) The collision operator $C_{1}$ Maxwellianises $v_{\parallel }$ for each $v_{\perp }.$

2. (P ₂) Although $\tau _{1} \ll \tau _{2}$ initially, eventually the short-range collisions become important.

3. (P ₃) The previous properties also hold in general magnetic geometry.

These properties will allow us to make progress without explicitly writing down the collision operator $C_{1}.$

4.1. $P_{1}$: $C_{1}$ Maxwellianises $v_{\parallel }$ for each $v_{\perp }$

The second law of thermodynamics demands that the action of any collision operator must not decrease the entropy of a system, $C_{1}$ is of course no exception.

Consider the entropy functional

(4.4)\begin{gather} S[\,f] = - \int f \ln \,f \, \textrm{d}^{3} {\boldsymbol{v}}. \end{gather}

Then any distribution function $f_{0}(\boldsymbol {r},\boldsymbol {v})$ say, which has been allowed to evolve under the influence of long-range Coulomb collisions, must maximise this functional subject to certain constraints.

As with any collision operator, the total particle number and the total energy must be conserved. That is,

(4.5)\begin{gather} \int f_{0} \, \textrm{d}^{3}\boldsymbol{v} = n(\boldsymbol{r}), \quad \text{fixed}, \end{gather}

(4.6)\begin{gather}\int \frac{1}{2}mv^{2}f_{0} \, \textrm{d}^{3}\boldsymbol{v} = \frac{3n(\boldsymbol{r})T(\boldsymbol{r})}{2}, \quad \text{fixed}. \end{gather}

We also know that the magnetic moment is conserved during long-range collisions and, hence, an additional constraint is

(4.7)\begin{gather} \int f_{0} \, \textrm{d} v_{\parallel} = f_{\perp}(v_{\perp}), \quad\text{fixed}. \end{gather}

In order to find the distribution function $f_{0}$ which maximises (4.4) subject to the constraints (4.5)–(4.7), we introduce sets of Lagrange multipliers $\kappa (\boldsymbol {r})$, $\lambda (\boldsymbol {r})$ and $\xi (v_{\perp })$ and seek to maximise the functional

(4.8)\begin{align} W[\,f_{0},\kappa,\lambda,\xi] &= - \int f_{0} \ln \,f_{0} \, \textrm{d}^{3}\boldsymbol{v} + \kappa(\boldsymbol{r}) \left[ \int f_{0} \, \textrm{d}^{3}\boldsymbol{v} - n(\boldsymbol{r}) \right] \nonumber\\ &\quad + \lambda(\boldsymbol{r}) \left[ \int \frac{1}{2}mv^{2}f_{0} \, \textrm{d}^{3}\boldsymbol{v} - \frac{3n(\boldsymbol{r})T(\boldsymbol{r})}{2}\right] \nonumber\\ &\quad + \int \xi(v_{\perp}) \left[ \int f_{0} \, \textrm{d} v_{\parallel} - f_{\perp}(v_{\perp}) \right] \textrm{d} v_{\perp}. \end{align}

The first-order variation is given by

(4.9)\begin{gather} \delta W = \int \delta f (-(1+\ln \,f_{0}) + \kappa(\boldsymbol{r}) + \tfrac{1}{2}mv^{2}\lambda(\boldsymbol{r}) + \xi(v_{\perp})) \textrm{d}^{3} \boldsymbol{v}. \end{gather}

In order to ensure that this variation vanishes, the integrand must vanish point-wise and hence we obtain

(4.10)\begin{align} f_{0}(\boldsymbol{r},\boldsymbol{v}) &= \exp({-\kappa(\boldsymbol{r}) + 1}) \exp({-({1}/{2})mv_{\parallel}^{2}\lambda(\boldsymbol{r})}) \exp({-({1}/{2})mv_{\perp}^{2}\lambda(\boldsymbol{r})}) \exp({-\xi(v_{\perp})}) \nonumber\\ &= f_{\parallel}(v_{\parallel})f_{\perp}(v_{\perp}). \end{align}

Indeed, this implies that the distribution function can be decomposed as

(4.11a,b)\begin{gather} f(v_{\parallel},v_{\perp}) = n f_{\parallel}(v_{\parallel})f_{{\perp}}(v_{\perp}), \quad f_{\parallel}(v_{\parallel}) = \left(\frac{m}{2{\rm \pi} T_{\parallel}}\right)^{1/2}\exp\left( -\frac{mv_{\parallel}^{2}}{2T_{\parallel}}\right). \end{gather}

4.2. $P_{2}$: short-range collisions are still important

In a straight field, the kinetic equation is

(4.12)\begin{gather} \frac{\partial f}{\partial t} - \frac{1}{\tau_{r}}\frac{\partial}{\partial\mu} (\mu f) = C_{1}(f) + C_{2}(f) \end{gather}

where $C_{1}$ describes collisions that Maxwellianise $v_{\parallel }$ for each $v_{\perp }$. We write

(4.13a,b)\begin{gather} f = g \,\mathrm{e}^{t/\tau_{r}}, \quad \frac{v_{\perp}^{2}}{2} = y\,\mathrm{e}^{-t/\tau_{r}} \end{gather}

and find that

(4.14)\begin{gather} \left(\frac{\partial f}{\partial t}\right)_{\mu} = \left(\frac{\partial g}{\partial t}\right)_{y}\,\mathrm{e}^{t/\tau_{r}}, \end{gather}

from which it then follows that

(4.15)\begin{gather} \frac{\partial g}{\partial t} = C_{1}(g) + C_{2}(g). \end{gather}

If

(4.16a–c)\begin{gather} C_{1}(f) \sim \frac{f}{\tau_{1}}, \quad C_{2}(f) \sim \frac{f}{\tau_{2}}, \quad \tau_{1}\ll \tau_{2}, \end{gather}

then $g$ will approach a parallel Maxwellian on the timescale $\tau _{1},$ that is

(4.17)\begin{gather} f(v_{\parallel},y,t)\rightarrow f_{_{\perp}}(y,t) \sqrt{\frac{m}{2{\rm \pi} T_{\parallel}(t)}} \exp \left(-\frac{mv_{\parallel}^{2}}{2T_{\parallel}(t)}\right), \quad t \gg \tau_{1}. \end{gather}

The ‘thermal’ (i.e. typical) velocity in the perpendicular direction is thus given by

(4.18a,b)\begin{gather} v_{\perp} \sim v_{\parallel} \,\mathrm{e}^{-t/\tau_{r}}, \quad v_{\parallel} = \sqrt{\frac{2T_{\parallel}}{m}}. \end{gather}

It can be shown (see (I)) that the operator $C_{2}$ then approaches

(4.19a,b)\begin{gather} C_{2}(f) \rightarrow \frac{\sigma}{n} \sqrt{\frac{m}{{\rm \pi} T_{\parallel}}} |\ln \epsilon| \frac{1}{v_{\perp}}\frac{\partial}{\partial v_{\perp}}v_{\perp} \frac{\partial f}{\partial v_{\perp}}, \quad \sigma = \frac{ne^{4}\ln \varLambda}{8{\rm \pi}\epsilon_{0}^{2}m^{2}}, \end{gather}

where $\ln \varLambda$ is the Coulomb logarithm. It follows that in the new coordinate system

(4.20)\begin{gather} C_{2}(f) \rightarrow \frac{2\sigma}{n} |\ln \epsilon| \sqrt{\frac{m}{{\rm \pi} T_{\parallel}}} \frac{\partial}{\partial y} \left(y\frac{\partial f}{\partial y}\right) \mathrm{e}^{t/\tau_{r}} \sim \frac{\mathrm{e}^{t/\tau_{r}}}{\tau_{2}} |\ln \epsilon |\,f, \end{gather}

where $\tau _{2}$ is equal to $\tau _{c}$ evaluated at the temperature $T_{_{\parallel }}.$

Hence, it is clear that $C_{2}(f)$ is only smaller than the radiation term as long as

(4.21)\begin{gather} \mathrm{e}^{t/\tau_{r}} |\ln \epsilon| \ll \frac{\tau_{2}}{\tau_{r}} \implies t \ll \tau_{r} \ln \left( \frac{\tau_{2}}{\tau_{r}|\ln\epsilon|} \right). \end{gather}

At later times, $C_{2}(f)$ cannot be neglected. The distribution will then stop contracting in the perpendicular direction.

4.3. $P_{3}$: general magnetic geometry

In (I), it was shown that the theory of collisional scattering in strongly anisotropic plasmas could be developed for general magnetic geometry. This result also holds in strongly magnetised plasmas.

We were able to show in (I), albeit with different collision operators, that bounce-averaging the collisional kinetic equation with a varying magnetic field results in the equation

(4.22)\begin{gather} \frac{\partial\bar{f}}{\partial t} - \frac{1}{\tau_{r}}\overline{\frac{\partial}{\partial\mu}(\mu f)} - \frac{\mu B_{0}}{\tau_{r}}\overline{\frac{\partial f}{\partial w}} = \overline{C_{1}(f)} + \overline{C_{2}(f)}, \end{gather}

where the bounce average is defined as

(4.23a,b)\begin{gather} \bar{Q} = \frac{1}{\tau_{b}} \int Q \, \frac{\textrm{d} l}{v_{\parallel}}, \quad \tau_{b} = \int \frac{\textrm{d} l}{v_{\parallel}} \end{gather}

and we are to understand that the $B^{2}$ term in Larmor's formula is replaced with its average along a field line.

We again take $C_{1},$ denoting $\mu$-preserving collisions, to be dominant, and expand $\bar {f} = \overline {f_{0}} + \overline {f_{1}} + \cdots$ to obtain

(4.24)\begin{gather} \frac{\partial\overline{f_{0}}}{\partial t} - \frac{1}{\tau_{r}}\overline{\frac{\partial}{\partial\mu}(\mu f_{0})} - \frac{\mu B_{0}}{\tau_{r}}\overline{\frac{\partial f_{0}}{\partial w}} = \overline{C_{1}(f_{0})}. \end{gather}

We again switch to a stretched coordinate system and write

(4.25a,b)\begin{gather} \overline{f_{0}} = g\,\mathrm{e}^{t/\tau_{r}}, \quad \mu = B_{0}^{-1}y\,\mathrm{e}^{-t/\tau_{r}}, \end{gather}

such that

(4.26)\begin{gather} \overline{\left(\frac{\partial f_{0}}{\partial t}\right)_{\mu} - \frac{1}{\tau_{r}}\frac{\partial}{\partial\mu}(\mu f_{0})} = \mathrm{e}^{t/\tau_{r}} \left(\frac{\partial\bar{g}}{\partial t}\right)_{y} \end{gather}

and we thus obtain

(4.27)\begin{gather} \frac{\partial\bar{g}}{\partial t} - \frac{y\,\mathrm{e}^{-t/\tau_{r}}}{\tau_{r}} \frac{\partial\bar{g}}{\partial w} = \overline{C_{1}(g)}. \end{gather}

Equation (4.27) suggests that $g$ will approach a distribution function in the null space of the bounce-averaged collision operator $\overline {C_{1}}.$ However, any such function must satisfy

(4.28)\begin{align} 0 &= - \int \tau_{b} \frac{4{\rm \pi}}{m^{2}} \overline{C_{1}(g)} \ln g \, \textrm{d} w\, \textrm{d} \mu = - \frac{4{\rm \pi}}{m^{2}} \int \ln g \, \textrm{d} w \,\textrm{d} \mu \oint C_{1}(g) \, \frac{\textrm{d} l}{v_{\parallel}} \nonumber\\ &= - \oint \frac{\textrm{d} l}{B} \int \ln g C(g) \, \textrm{d}^{3}\boldsymbol{v} \geq 0 \end{align}

with equality if, and only if, $g$ is of the form

(4.29)\begin{gather} g = h (\mu,t) \exp({-w/T(\mu,t)}). \end{gather}

Once this relaxation has occurred, on the time scale $\tau _{1},$ $f$ will be Maxwellian in the parallel direction for each $\mu ,$ and the distribution will have shrunk so much in the $\mu$ direction that $C_{2}$ becomes important.

We have now proven that properties ${P}_{1}$ and ${P}_{2}$ also hold in general magnetic geometry. We have also dealt with the general magnetic geometry case for the Landau collision operator in (I).

4.4. Collision operator in a strongly magnetised plasma

The short-range collisions, which have not yet been taken into account, can be described by the usual Landau collision operator acting on the decomposed distribution functions. This leads us to a result that will be the key in unlocking this problem.

Namely, the appropriate treatment of the collision operator in a strongly magnetised plasma is simply

(4.30a,b)\begin{gather} C(f) = C_{1}(f) + C_{2}(f) \simeq C_{2}(n f_{\parallel}(v_{\parallel})f_{\perp}(v_{\perp})), \quad t \gg \tau_{1} \gg \tau_{r} \end{gather}

with

(4.31a,b)\begin{gather} f_{\parallel} = \left(\frac{m}{2{\rm \pi} T_{\parallel}}\right)^{1/2} \exp \left({-\frac{mv_{\parallel}^2}{2T_{\parallel}}} \right), \quad \int f_{\perp} \, \textrm{d}^{2}v_{\perp} = 1. \end{gather}

We can now turn our attention to solving the collisional kinetic equation.

5. Collisional scattering in pair plasmas

The kinetic equation is

(5.1)\begin{gather} \frac{\partial f}{\partial t} + v_{\parallel}\frac{\partial f}{\partial l} + \frac{\partial}{\partial\mu}\left(\frac{\mu}{\tau_{r}}f\right) - \frac{\mu}{m}\boldsymbol{\nabla}_{\parallel}B \frac{\partial f}{\partial v_{\parallel}} \simeq C_{2}(f), \end{gather}

where we now understand that the long-range collisions have been taken into account by invoking (4.30a,b).

In the interest of adopting a pedagogical approach, let us specialise to straight field-line geometry, noting once more that the results should hold more generally after associating $B^{2}$ with its field line average.

One then arrives at the equation

(5.2)\begin{gather} \frac{\partial f}{\partial t} - \frac{1}{\tau_{r}}\frac{\partial}{\partial\mu}(\mu f) = \frac{{\sigma}}{n} \int \boldsymbol{\nabla} \boldsymbol{\cdot} [\boldsymbol{\mathsf{U}} \boldsymbol{\cdot} (f^{\prime} \boldsymbol{\nabla} f - f \nabla^{\prime} f^{\prime})] \,\textrm{d}^{3}\boldsymbol{v}^{\prime}. \end{gather}

The gradient operators are defined by

(5.3a,b)\begin{gather} \boldsymbol{\nabla} = \frac{\partial}{\partial\boldsymbol{v}}, \quad \nabla^{\prime} = \frac{\partial}{\partial\boldsymbol{v}^{\prime}}. \end{gather}

We have also introduced $\boldsymbol{\mathsf{U}},$ the second-rank tensor

(5.4)\begin{gather} \boldsymbol{\mathsf{U}}(\boldsymbol{u}) = \boldsymbol{\nabla}\boldsymbol{\nabla} u= \frac{u^{2} \boldsymbol{\mathsf{I}} - \boldsymbol{u}\boldsymbol{u}}{u^{3}}, \end{gather}

where $\boldsymbol {u} = \boldsymbol {v} - \boldsymbol {v}^{\prime }$ is the difference in velocity vectors between colliding particles, and $\boldsymbol{\mathsf{I}}$ is the identity matrix.

The energy moments introduced in (3.1a–c) satisfy the evolution equations

(5.5a,b)\begin{gather} \frac{\textrm{d} W}{\textrm{d} t} = - \frac{W_{\perp}}{\tau_{r}}, \quad \frac{\textrm{d} W_{\perp}}{\textrm{d} t} = - \frac{W_{\perp}}{\tau_{r}}+ S_{c}, \end{gather}

where the scattering term $S_{c}$ is given by

(5.6)\begin{gather} S_{c} = \int \frac{mv_{\perp}^{2}}{2} C[\,f,f] \, \textrm{d}^{3}\boldsymbol{v} = -\frac{\sigma m}{n} \iint \boldsymbol{v}_{\perp} \boldsymbol{\cdot} [\boldsymbol{\mathsf{U}} \boldsymbol{\cdot} (f^{\prime} \boldsymbol{\nabla} f - f \nabla^{\prime} f^{\prime})] \textrm{d}^{3}\boldsymbol{v} \, \textrm{d}^{3}\boldsymbol{v}^{\prime}. \end{gather}

We can now take advantage of the large anisotropy in the distribution function, which is a result of the perpendicular kinetic energy being radiated before collisions become important, by expanding the scattering operator in powers of the small parameter

(5.7)\begin{gather} \epsilon = \frac{u_{\perp}}{u_{\parallel}} \ll 1. \end{gather}

To lowest order, this yields

(5.8)\begin{align} S_{c} &= - \frac{{\sigma}m}{n} \iint \boldsymbol{v}_{\perp} \boldsymbol{\cdot} \frac{u^{2} \boldsymbol{\mathsf{I}} - \boldsymbol{u}\boldsymbol{u}}{u^{3}} (\,f^{\prime} \boldsymbol{\nabla} f - f \nabla^{\prime} f^{\prime})\, \textrm{d}^{3}\boldsymbol{v} \, \textrm{d}^{3}\boldsymbol{v}^{\prime}, \nonumber\\ &\simeq -\frac{\sigma m}{n} \iint \frac{\boldsymbol{v}_{\perp}}{u_{_{\parallel}}} \boldsymbol{\cdot} f^{\prime}\boldsymbol{\nabla} f \, \textrm{d}^{3} \boldsymbol{v}\, \textrm{d}^{3} \boldsymbol{v}^{\prime},\nonumber\\ &\simeq \frac{2\sigma m}{n} \iint \frac{ff^{\prime}}{u_{\parallel}} \, \textrm{d}^{3}\boldsymbol{v} \, \textrm{d}^{3}\boldsymbol{v}^{\prime} \end{align}

and one can evaluate

(5.9)\begin{align} \iint \frac{ff^{\prime}}{u_{\parallel}} \, \textrm{d}^{3}\boldsymbol{v} \, \textrm{d}^{3}\boldsymbol{v}^{\prime} &= n^{2} \left(\frac{m}{2{\rm \pi} T_{\parallel}}\right) \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp \left(-\frac{m(v_{\parallel}^{2} + v_{\parallel}^{\prime 2})}{2T_{\parallel}}\right) \frac{\textrm{d} v_{\parallel} \, \textrm{d} v_{\parallel}^{\prime}}{| v_{\parallel} - v_{\parallel^{\prime}}|} \end{align}

(5.10)\begin{align} &= \frac{n^{2}}{\rm \pi} \sqrt{\frac{m}{2T_{\parallel}}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp (-x^{2} - y^{2}) \frac{\textrm{d}\kern0.7pt x \, \textrm{d} y}{|x-y|}, \end{align}

where we have made the substitution

(5.11a,b)\begin{gather} x = v_{\parallel}\sqrt{\frac{m}{2T_{\parallel}}}, \quad y = v_{\parallel}^{\prime}\sqrt{\frac{m}{2T_{\parallel}}}, \end{gather}

and note that a small region around $x=y$ needs to be excluded from the integration range because the ordering (5.7) does not hold.

Upon making a further change of variables

(5.12)\begin{gather} x - y = u\sqrt{2}, \end{gather}

(5.13)\begin{gather}x + y = v\sqrt{2}, \end{gather}

we obtain

(5.14)\begin{gather} \iint \frac{ff^{\prime}}{u} \, \textrm{d}^{3}\boldsymbol{v} \, \textrm{d}^{3}\boldsymbol{v}^{\prime} = \frac{n^{2}}{2{\rm \pi}}\sqrt{\frac{m}{T_{\parallel}}} \int_{-\infty}^{\infty} \frac{\exp(-u^2)}{u} \, \textrm{d} u \int_{-\infty}^{\infty}\exp(-v^{2}) \, \textrm{d} v = \frac{n^{2}}{2}\sqrt{\frac{m}{{\rm \pi} T_{\parallel}}} | \ln \epsilon |. \end{gather}

Here it is pertinent to remark that we had a choice in whether to perform the integration over $v_{\parallel }$ or $v_{\perp }$ first in (5.8). As we know $f_{\perp },$ we could perform the $v_{\parallel }$ integrals first, giving a Bessel function instead of the divergent integral (5.10), thus avoiding the ‘ad hoc’ cut off of the latter. This procedure is displayed explicitly in the next section.

It follows from (5.14), that provided $w_{\perp } \ll w,$ we have

(5.15)\begin{gather} S_{c} = \sigma m n \sqrt{\frac{m}{{\rm \pi} T_{\parallel}}} |\ln \epsilon| = \frac{3}{\sqrt{2}} \frac{W}{\tau_{2}} |\ln \epsilon| ,\end{gather}

where $W \simeq nT_{_{\parallel }}/2$ and $\tau _{2} \propto T_{_{\parallel }}^{3/2}$.

It then follows from (5.5a,b) that

(5.16)\begin{gather} \frac{\textrm{d} W}{\textrm{d} t} = - \frac{W_{\perp}}{\tau_{r}} = - S_{c} \simeq - \frac{3}{\sqrt{2}}\frac{W}{\tau_{0}}\left(\frac{W_{0}}{W}\right)^{3/2} | \ln \epsilon |, \end{gather}

where $\tau _{0}$ denotes the collision time $\tau _{c}$ when $W = W_{0}.$ In deriving this equation, we have appealed to the properties of the two collision operators discussed in § 4.

Equation (5.16) suggests that

(5.17)\begin{gather} \frac{W}{W_{0}} = \left(1 - \frac{9|\ln \epsilon| t}{2^{3/2}\tau_{0}}\right)^{2/3} \end{gather}

and, thus, $W \rightarrow 0$ after a few collision times, after which $W_{\perp } \ll W$ will no longer hold. Instead

(5.18)\begin{gather} \frac{\textrm{d} W}{\textrm{d} t} \simeq - \frac{2W}{3\tau_{r}} \end{gather}

and $W$ will fall exponentially.

Equation (5.16) describes a self-accelerating process and the plasma will only remain in this regime for a few collision times $\tau _{c}.$

5.1. Radiative cooling in electron–positron plasmas and implications in the laboratory

One can now elucidate the different cooling regimes which were introduced at the beginning of this section.

1. (I) Initially, $0 < \tau _{r} < t < \tau _{c}(0),$ the initial collision time:

   (5.19a,b)\begin{gather} W_{\perp} = W_{\perp,0} \exp (-t/\tau_{r}), \quad W_{\parallel} = \text{constant.} \end{gather}

2. (II) Then, $t \sim \tau _{c}(t)$ and scattering will occur. Equation (5.16) will hold and, hence,

   (5.20)\begin{gather} W = W_{0}\left(1 - \frac{9 | \ln \epsilon| t}{2^{3/2}\tau_{0}}\right)^{2/3}. \end{gather}
   This is a self-accelerating process. The power radiated by the plasma in this regime can be calculated through
   (5.21)\begin{gather} -\frac{\textrm{d} W}{\textrm{d} t} \simeq \frac{3}{\sqrt{2}}\frac{W}{\tau_{0}}\left(\frac{W_{0}}{W}\right)^{3/2} | \ln \epsilon |. \end{gather}

3. (III) Eventually, $t \gg \tau _{r}, \tau _{c}(t).$ In this regime any remaining parallel kinetic energy will be converted into perpendicular kinetic energy via collisional scattering and then radiated:

   (5.22)\begin{gather} W \propto \exp (-t/\tau_{r}). \end{gather}

Thus, it seems as though the exploitation of cyclotron cooling at high field will provide a very efficient mechanism to dissipate heat in the plasma as the radiation will simply be absorbed by the vessel walls.

This rapid cooling will have another benefit for laboratory experiments. As stated previously, there are fairly stringent conditions on the number of positrons that one can accumulate and store in the laboratory. As such, this means that the laboratory plasmas will have extremely low densities. In order for the system to be classified as a plasma, as opposed to simply a collection of charged particles, there must be collective behaviour that places a requirement on the Debye length compared with the system size $L.$ Namely, one must ensure that

(5.23)\begin{gather} \lambda_{D} = \sqrt{\frac{\epsilon_{0}T}{2ne^{2}}} \ll L. \end{gather}

The conditions placed on both $L$ and $n$ ensure that meeting this requirement could prove difficult. However, cyclotron cooling opens up a new avenue through which this condition might be satisfied. Simply put, the cyclotron cooling process will lower the Debye length on the time scale $\max (\tau _{r},\tau _{c}(0))$ and convert what might (and indeed likely will) be initially a collection of electron–positron pairs into a plasma.

For a collisionless pair plasma, we require a condition on the plasma parameter:

(5.24)\begin{gather} \varLambda = n \lambda_{D}^{3} \gg 1, \end{gather}

i.e. that there are many particles in a Debye sphere. For a fixed density, this parameter scales with $T^{3/2}$ and so this condition will become less and less well satisfied. Inevitably, there will be some trade-off between the density and temperature, which dictates the kind of plasma we are able to explore. In the upcoming series of laboratory experiments, this will not be a serious limitation; the Debye length will not be sufficiently small as to violate (5.24).

5.2. A note on entropy

It is interesting to look at the rate of change of entropy in optically thin electron–positron plasmas. Defining the entropy $S$ by

(5.25)\begin{gather} S= -\int f \ln \,f \, \textrm{d}^{3} \boldsymbol{r} \, \textrm{d}^{3} \boldsymbol{v}, \end{gather}

it follows immediately from the kinetic equation that

(5.26)\begin{gather} \frac{\textrm{d} S}{\textrm{d} t} + \frac{n}{\tau_{r}} = -\int C[\,f,f] \ln \,f \, \textrm{d}^{3}\boldsymbol{r} \, \textrm{d}^{3}\boldsymbol{v} \geq 0 \end{gather}

where the final inequality follows from Boltzmann's H-theorem.

This tells us that, on the timescale $t \sim \tau _{r} \ll \tau _{c}/|\ln \epsilon |$, the entropy of the plasma decreases at a constant rate that is proportional to the density. There is a simple physical explanation of this result, namely, that the entropy of the plasma will decrease through the loss of energy and that the loss rate will be proportional to the number of the particles in the plasma. Note also that the entropy loss rate thus remains constant as the plasma loses energy, which can be understood from the fact that the energy loss $\textrm {d} Q,$ in the thermodynamic relation

(5.27)\begin{gather} \textrm{d} S = \frac{\textrm{d} Q}{T}, \end{gather}

is proportional to temperature for cyclotron radiation.

6. The perpendicular component of the distribution function

We have already argued that, for times exceeding $\tau _{1},$

(6.1a,b)\begin{gather} f(v_{\parallel},v_{\perp}) = n f_{\parallel}(v_{\parallel})f_{{\perp}}(v_{\perp}), \quad f_{\parallel}(v_{\parallel}) = \left(\frac{m}{2{\rm \pi} T_{\parallel}}\right)^{1/2}\exp\left( -\frac{mv_{\parallel}^{2}}{2T_{\parallel}}\right), \end{gather}

where the long-range guiding centre collisions have driven $f_{\parallel }$ to a Maxwellian. It is instructive to ask what can be deduced about $f_{\perp }.$

We begin from the kinetic equation

(6.2)\begin{gather} \frac{\partial f}{\partial t} - \frac{1}{\tau_{r}}\frac{\partial}{\partial\mu}(\mu f) = \frac{\sigma}{n} \int \boldsymbol{\nabla} \boldsymbol{\cdot} \left[\boldsymbol{\mathsf{U}} \boldsymbol{\cdot} \left(f^{\prime} \boldsymbol{\nabla} f - f \nabla^{\prime} f^{\prime}\right)\right] \,\textrm{d}^{3}\boldsymbol{v}^{\prime}. \end{gather}

Integrating over the parallel velocity to remove the $f_{\parallel }$ terms on the left-hand side, one obtains

(6.3)\begin{gather} \frac{\partial f_{\perp}}{\partial t} - \frac{1}{2\tau_{r}v_{\perp}}\frac{\partial}{\partial v_{\perp}}\left(v_{\perp}^{2}f_{\perp}\right) = C_{\perp}[\,f,f], \end{gather}

where we have now introduced the operator

(6.4)\begin{gather} C_{\perp}[\,f,f] = \frac{\sigma}{n^{2}} \frac{1}{v_{\perp}} \frac{\partial}{\partial v_{\perp}} \boldsymbol{v}_{\perp} \boldsymbol{\cdot}\int_{-\infty}^{\infty} \textrm{d} v_{\parallel} \int f f^{\prime} \boldsymbol{\mathsf{U}} \boldsymbol{\cdot} \left(\boldsymbol{\nabla}\ln \,f - \nabla^{\prime} \ln \,f^{\prime}\right) \textrm{d}^{3}\boldsymbol{v}^{\prime}. \end{gather}

Now, using the factorisation of the distribution function, we have

(6.5)\begin{align} C_{\perp} &= \sigma \frac{1}{v_{\perp}} \frac{\partial}{\partial v_{\perp}} \boldsymbol{v}_{\perp} \boldsymbol{\cdot} \int f_{\perp}\,f_{\perp}^{\prime} \textrm{d}^{2}\boldsymbol{v}_{\perp}^{\prime} \iint f_{\parallel}\, f_{\parallel}^{\prime} \boldsymbol{\mathsf{U}}\nonumber\\ &\quad \boldsymbol{\cdot} \left(\boldsymbol{\nabla}\ln \,f_{\parallel} - \nabla^{\prime} \ln \,f_{\parallel}^{\prime} + \boldsymbol{\nabla}_{\perp} \ln \,f_{\perp} - \nabla_{\perp}^{\prime} \ln \,f_{\perp}^{\prime}\right) \textrm{d} v_{\parallel} \textrm{d} v_{\parallel}^{\prime} \end{align}

and we can also exploit that $f_{\parallel }$ is a Maxwellian to obtain

(6.6)\begin{gather} \boldsymbol{\nabla}_{\parallel} \ln \,f_{\parallel} - \nabla^{\prime}_{\parallel} \ln \,f^{\prime}_{\parallel} = \frac{m}{T_{\parallel}}(\boldsymbol{v}_{\parallel}^{\prime} - \boldsymbol{v}_{\parallel}) = - \frac{m\boldsymbol{u}_{\parallel}}{T_{\parallel}}. \end{gather}

We thus arrive at

(6.7)\begin{align} C_{\perp} &= \sigma \frac{1}{v_{\perp}} \frac{\partial}{\partial v_{\perp}} \boldsymbol{v}_{\perp} \boldsymbol{\cdot} \int f_{\perp}\,f_{\perp}^{\prime} \textrm{d}^{2}\boldsymbol{v}_{\perp}^{\prime} \iint f_{\parallel} \, f_{\parallel}^{\prime} \left[ \frac{m}{T_{\parallel}} \left(-\frac{u^{2} \boldsymbol{u}_{\parallel} - u_{\parallel}^{2}\boldsymbol{u}}{u^{3}}\right) \right. \nonumber\\ &\left.\vphantom{\left(-\frac{u^{2} \boldsymbol{u}_{\parallel} - u_{\parallel}^{2}\boldsymbol{u}}{u^{3}}\right)}\quad+ \boldsymbol{\mathsf{U}} \boldsymbol{\cdot} (\boldsymbol{\nabla}_{\perp} \ln\, f_{\perp} - \nabla_{\perp}^{\prime} \ln \,f_{\perp}^{\prime}) \right] \textrm{d} v_{\parallel} \,\textrm{d} v_{\parallel}^{\prime}, \end{align}

and the leading-order (in $\epsilon \ll 1$) term is given by

(6.8)\begin{gather} C_{\perp} \simeq \sigma \frac{1}{v_{\perp}} \frac{\partial}{\partial v_{\perp}} v_{\perp}\frac{\partial f_{\perp}}{\partial v_{\perp}} \int f_{\perp}^{\prime} \, \textrm{d}^{2} \boldsymbol{v}_{\perp}^{\prime} \iint \frac{f_{\parallel} \,f_{\parallel}^{\prime}}{u} \, \textrm{d} v_{\parallel} \, \textrm{d} v_{\parallel}^{\prime}. \end{gather}

We first turn our attention to

(6.9)\begin{gather} I_{1} := \iint \frac{f_{\parallel} \,f_{\parallel}^{\prime}}{u} \, \textrm{d} v_{\parallel} \, \textrm{d} v_{\parallel}^{\prime}. \end{gather}

Upon making the change of variables

(6.10)\begin{gather} v_{\parallel} - v_{\parallel}^{\prime} = u_{\parallel}, \end{gather}

(6.11)\begin{gather}v_{\parallel} + v_{\parallel}^{\prime} = w_{\parallel}, \end{gather}

we see that we may write

(6.12)\begin{gather} f_{\parallel}\,f_{\parallel}^{\prime} = \frac{m}{2{\rm \pi} T_{\parallel}} \exp \left(-\frac{m}{4T_{\parallel}} \left(u_{\parallel}^{2} + w_{\parallel}^{2}\right)\right), \end{gather}

and, thus, our integral becomes

(6.13)\begin{gather} I_{1} = \frac{m}{2 {\rm \pi} T_{\parallel}} \int_{-\infty}^{\infty} \frac{\exp \left(-\dfrac{mu_{\parallel}^{2}}{4T_{\parallel}}\right)}{\sqrt{u_{\parallel}^{2} + u_{\perp}^{2}}} \, \textrm{d} u_{\parallel} \int_{-\infty}^{\infty} \exp \left(-\dfrac{mw_{\parallel}^{2}}{4T_{\parallel}}\right) \, \textrm{d} w_{\parallel}. \end{gather}

Upon evaluating the second integral term and making a further change of variables $x^{2} = mu_{\parallel }^{2}/4T_{\parallel },$ we see that

(6.14)\begin{gather} I_{1} = \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}} }}\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-x^2}}{\sqrt{x^{2} + \dfrac{mu_{\perp}^{2}}{4T_{\parallel}}}}\, \textrm{d}\kern0.7pt x = \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}}}} \,\mathrm{e}^{{mu_{\perp}^{2}}/{8T_{\parallel}}} \textrm{K}_{0} \left(\frac{mu_{\perp}^{2}}{8T_{\parallel}}\right), \end{gather}

where $\textrm{K}_{0}$ is the modified Bessel function of the second kind.

In order to calculate $C_{\perp }$ it remains to evaluate $I_{2}$ given by

(6.15)\begin{gather} I_{2} := \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}}}}\int f_{\perp}^{\prime} \mathrm{e}^{{mu_{\perp}^{2}}/{8T_{\parallel}}} \textrm{K}_{0} \left(\frac{mu_{\perp}^{2}}{8T_{\parallel}}\right) \textrm{d}^{2} \boldsymbol{v}_{\perp}^{\prime}. \end{gather}

Introducing the change of variables

(6.16a,b)\begin{gather} x_{\perp}^{2} = \frac{mv_{\perp}^{2}}{2T_{\perp}}, \quad x_{\perp}^{\prime 2} = \frac{mv_{\perp}^{\prime 2}}{2T_{\perp}}, \end{gather}

we obtain

(6.17)\begin{gather} \frac{mu_{\perp}^{2}}{8T_{\parallel}} = \delta (x_{\perp}^{2} + x_{\perp}^{\prime 2} - 2x_{\perp}x_{\perp}^{\prime}\cos\theta) =: \delta g(x_{\perp},x_{\perp}^{\prime},\theta), \end{gather}

where $\theta$ is the angle between $\boldsymbol {v}_{\perp }$ and $\boldsymbol {v}_{\perp }^{\prime }$ and we have introduced another small parameter

(6.18)\begin{gather} \delta = \frac{T_{\perp}}{4T_{\parallel}} \ll 1. \end{gather}

We may now approximate

(6.19)\begin{gather} I_{2} = \frac{2T_{\perp}}{m} \int_{0}^{\infty} f_{\perp}^{\prime} x_{\perp}^{\prime} \, \textrm{d}\kern0.7pt x_{\perp}^{\prime} \int_{0}^{2{\rm \pi}} \exp({\delta g(x_{\perp},x_{\perp}^{\prime},\theta) }) \textrm{K}_{0} (\delta g(x_{\perp},x_{\perp}^{\prime},\theta)) \, \textrm{d} \theta. \end{gather}

To leading order in $\delta ,$ we thus obtain, to logarithmic accuracy,

(6.20)\begin{gather} I_{2} \simeq \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}}}} |\ln \delta|, \end{gather}

and, hence,

(6.21)\begin{gather} C_{\perp} = \sigma \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}}}} | \ln \delta| \frac{1}{v_{\perp}}\frac{\partial}{\partial v_{\perp}}\left(v_{\perp} \frac{\partial f_{\perp}}{\partial v_{\perp}} \right). \end{gather}

Now, (6.3) becomes

(6.22)\begin{gather} -\frac{1}{2\tau_{r}v_{\perp}}\frac{\partial}{\partial v_{\perp}}(v_{\perp}^{2}f_{\perp}) = \sigma \sqrt{\frac{m}{{\rm \pi} T_{_{\parallel}}}} | \ln \delta| \frac{1}{v_{\perp}}\frac{\partial}{\partial v_{\perp}}\left(v_{\perp} \frac{\partial f_{\perp}}{\partial v_{\perp}} \right), \end{gather}

which can be solved for the perpendicular distribution function via direct integration to give

(6.23)\begin{gather} f_{\perp}(v_{\perp}) = \frac{m}{2{\rm \pi} T_{\perp}} \exp \left( -\frac{mv_{\perp}^{2}}{2T_{_{\perp}}} \right). \end{gather}

That is, in the scattering regime (II), $f_{\perp }$ is also a Maxwellian, but with a perpendicular temperature given through the equation

(6.24)\begin{gather} \frac{T_{\perp}}{T_{\parallel}} = \frac{8}{\sqrt{2{\rm \pi}}} \frac{\tau_{r}}{\tau_{2}} |\ln \delta |. \end{gather}

That is, we see that the cooling during the scattering regime occurs on the timescale of parallel (to the magnetic field) collisions, and is thus a self-accelerating process as the temperature $T_{\parallel }$ falls.

We can also see from (6.24) that, in fact, $\delta$ and $\epsilon$ coincide within a factor of $A|\ln \epsilon |,$ where $A$ is some multiplicative $O(1)$ factor, and can therefore be treated as identical if we are willing to accept a relative error of order $1/ | \log \epsilon |$.

Note, here (6.23) is simply a specialisation of the result obtained in (I) when the distribution function can be factored into perpendicular and parallel components.

7. Stability of anisotropic electron–positron plasmas in straight field-line geometry

Having derived a kinetic theory for the radiative cooling of strongly anisotropic pair plasmas, we finish by modifying one of the important calculations of Helander (Reference Helander2014). We show that stability of electron–positron plasmas to low-frequency waves in slab geometry (i.e. a plasma threaded by a straight and constant magnetic field) still holds if the distribution function is non-Maxwellian.

We write the gyrokinetic distribution function for our plasma as

(7.1)\begin{gather} f_{a}(\boldsymbol{r}) = F_{a} + e_{a} \phi (\boldsymbol{r}) \frac{\partial F_{a}}{\partial E} + \frac{e_{a}}{B_{0}}\left[\phi(\boldsymbol{r}) - \langle \phi(\boldsymbol{r})\rangle_{\boldsymbol{R}} \right]\frac{\partial F_{a}}{\partial\mu} + g_{a}(\boldsymbol{R}), \end{gather}

where $F_{a}$ is the a priori arbitrary equilibrium distribution function of species $a,$ $\phi$ is the electrostatic potential and $E$ is the energy of the plasma. We take care to distinguish between quantities which are evaluated at the particle position $\boldsymbol {r}$ and those which are evaluated at the guiding centre position $\boldsymbol {R}$. The function $g_{a}$ satisfies the linearised, electrostatic, collisionless gyrokinetic equation (Catto, Tang & Baldwin Reference Catto, Tang and Baldwin1981):

(7.2)\begin{gather} iv_{\parallel}\boldsymbol{\nabla}_{\parallel}g_{a} + (\omega - \omega_{da})g_{a} = - e_{a} J_{0}(k_{\perp}\rho) \phi(\boldsymbol{R}) \left[ \omega \frac{\partial F_{a}}{\partial E} + \frac{1}{e_{a}B_{0}} \boldsymbol{k}\times \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} F_{a} \right], \end{gather}

where we have introduced $\omega _{da} = \boldsymbol {k}_{\perp } \boldsymbol {\cdot } \boldsymbol {v}_{da}$ the magnetic drift frequency. We have also introduced the gyroaverage $\langle \alpha \rangle _{\boldsymbol {R}}$ of any function of the gyroangle $\alpha (\theta ,\ldots )$ at fixed guiding centre $\boldsymbol {R},$ defined by

(7.3)\begin{gather} \langle \alpha \rangle_{\boldsymbol{R}} = \left.\frac{1}{2{\rm \pi}} \int_{0}^{2{\rm \pi}} \alpha(\theta,\ldots) \right\vert_{\boldsymbol{R}=\text{constant}} \, \textrm{d}\theta. \end{gather}

The quasineutrality condition demands

(7.4)\begin{gather} \sum_{a} e_{a} \int_{\boldsymbol{r} = \text{constant}} \textrm{d}^{3}\boldsymbol{v} \, f_{a} = 0, \end{gather}

which for a pure electron–positron plasma requires

(7.5)\begin{gather} \frac{1}{2e} \int_{\boldsymbol{R}=\text{constant}} \textrm{d}^{3} \boldsymbol{v} (g_{p} - g_{e}) J_{0} + \phi \int_{\boldsymbol{R}=\text{constant}} \textrm{d}^{3}\boldsymbol{v} \left( \mathrm{e}^{\textrm{i}{\boldsymbol{k}}_{\perp}\boldsymbol{\cdot}\boldsymbol{\rho}} \frac{\partial F}{\partial E} + \frac{1}{B_{0}}[\mathrm{e}^{\textrm{i}{\boldsymbol{k}}_{\perp}\boldsymbol{\cdot}\boldsymbol{\rho}} - J_{0}] \frac{\partial F}{\partial\mu} \right) J_{0} = 0. \end{gather}

In the limit of straight field lines we can set $\omega _{da} = 0$ and obtain the solution of the gyrokinetic equation as

(7.6)\begin{gather} g_{a}(\boldsymbol{R}) = - \frac{e_{a} J_{0}\phi(\boldsymbol{R})}{\omega - k_{\parallel}v_{\parallel}} \left[ \omega \frac{\partial F_{a}}{\partial E} + \frac{1}{e_{a}B_{0}} \boldsymbol{k}\times \boldsymbol{b} \boldsymbol{\cdot} \boldsymbol{\nabla} F_{a} \right], \end{gather}

hence one obtains

(7.7)\begin{gather} g_{p} - g_{e} = -\frac{2\omega e J_{0} \phi}{\omega - k_{\parallel}v_{\parallel}} \frac{\partial F}{\partial E}. \end{gather}

Substituting into Poisson's equation gives the dispersion relation

(7.8)\begin{gather} \int \textrm{d}^{3} \boldsymbol{v} \, \frac{\omega }{\omega - k_{\parallel}v_{\parallel}} \frac{\partial F}{\partial E} J_{0}^{2} = \int \textrm{d}^{3}\boldsymbol{v} \left( \mathrm{e}^{\textrm{i}{\boldsymbol{k}}_{\perp}\boldsymbol{\cdot}\boldsymbol{\rho}} \frac{\partial F}{\partial E} + \frac{1}{B_{0}}[\mathrm{e}^{\textrm{i}{\boldsymbol{k}}_{\perp}\boldsymbol{\cdot}\boldsymbol{\rho}} - J_{0}] \frac{\partial F}{\partial\mu} \right) J_{0}. \end{gather}

One can see that, just as in Helander (Reference Helander2014), the gradients, which can drive microinstabilities, simply cancel owing to the mass symmetry and the result that any instability must involve magnetic curvature still holds for anisotropic plasmas.