2.1. Field structure

We use the 3-D magnetic field and electric potential solutions to the steady-state, kinematic, resistive MHD equations given by Wyper & Jain (Reference Wyper and Jain2010) and Wyper & Jain (Reference Wyper and Jain2011). For the torsional fan case, the twist in the magnetic field is localized to two flat, disk-like regions above and below the fan plane and is given in cylindrical coordinates ($r$, $\theta$, $z$ ). This magnetic field is of the form,

(2.1)\begin{equation} \boldsymbol{B} = B_{0} [ r, jr(zr^{2})^{\beta}z^{\gamma}\, \textrm{e}^{-(z^{2}/l^{2})-(c^{2}/l^{2})(zr^{2})^{2}}, -2z ], \end{equation}

where distances are scaled to a characteristic length described in §§ 2.2 and 3.2, $j$, $l$ and $c$ govern the degree and spatial extent of the twist, and $\beta$ and $\gamma$ are positive integers. Throughout this paper, we set $j = 5$, $\gamma = 3$, $l = 1$ and $\beta = c = 0$, corresponding to a generic radially linear perturbation, and consistent with the torsional fan reconnection simulations of Hosseinpour (Reference Hosseinpour2015), Hosseinpour (Reference Hosseinpour2014a) and Hosseinpour et al. (Reference Hosseinpour, Mehdizade and Mohammadi2014). Using these values, (2.1) reduces to

(2.2)\begin{equation} \boldsymbol{B} = B_{0} [ r, 5rz^{3}\, \textrm{e}^{{-}z^{2}}, -2z ].\end{equation}

Figure 2 displays $XZ$ and $XY$ streamlines of (2.2), with the distance scaled to a characteristic length (see below). The location of greatest twist is in the $XY$ plane at height $Z = \sqrt {1.5}$ defined by the $\theta$ term in (2.2). To approximate a controlled injection of plasma, we apply a resistivity profile $\eta$ depending only on height $z$ as

(2.3)\begin{equation} \eta = \eta_{0} |z|^{\delta} \, \mathrm{e}^{{-}z^{2}/l^{2}},\end{equation}

where $\eta _{0} = m_{e} \nu _{e,i} / n_{e} \, \textrm {e}^{2}$ is the plasma resistivity calculated from the characteristic parameters of electron mass $m_{e}$, electron–ion collision frequency $\nu _{e,i}$, electron number density $n_{e}$ and the elementary charge $e$ (see § 3.2). We set $\delta = 2$ to localize the resistivity profile to the height $Z = \sqrt {1.5}$ of maximum magnetic field twist predicted by (2.2). This choice, as will be shown in § 3.3, inherently confines and localizes the injected plasma sheath to the region of stressed, non-potential magnetic field. This choice also satisfies the constraints for realistic solutions when the sum of parameters $\gamma + \delta \ge 3$ and is odd (Wyper & Jain Reference Wyper and Jain2011). The electric potential in torsional fan reconnection with the above resistivity profile is of the form

(2.4)\begin{equation} \varPhi = \varPhi_{0} \left[ \left( \gamma G(z, \gamma + \delta -2) - \frac{2}{l^{2}} G(z, \gamma + \delta) \right) zr^{2} + 4 G(z, \gamma + \delta + 1) \right], \end{equation}

with the recurrence relation for $G(z, \lambda )$ being

(2.5)\begin{equation} G(z, \lambda+2) = \frac{l^{2}}{2} \left( \lambda G(z, \lambda) + \frac{z^{\lambda}}{2} \, \textrm{e}^{{-}z^{2}/l^{2}} \right), \end{equation}

and solutions for all values of $\lambda$ found using

(2.6)\begin{equation} G(z, 1) ={-}\frac{l \sqrt{\rm \pi}}{4} \mathrm{erf}\left( \frac{z}{l} \right) \end{equation}

and

(2.7)\begin{equation} G(z, 2) = \frac{l^{2}}{4} \, \textrm{e}^{{-}z^{2}/l^{2}}.\end{equation}

As in Wyper & Jain (Reference Wyper and Jain2011), the initial electric potential $\varPhi _{0}$ is in units of volts and depends on the current density $j_{0}$ and normalized distances as $\varPhi _{0} = j_{0} \eta _{0} L_{0}$. The electric field can then be derived using $\boldsymbol {E} = - \boldsymbol {\nabla } \varPhi$. Using our numerical parameters, these equations reduce to

(2.8)\begin{equation} \varPhi = \frac{\varPhi_{0}}{2} [ (2 - r^{2}) z^{4} + 4 z^{2} + 4 ] \, \textrm{e}^{{-}z^{2}} \end{equation}

and

(2.9)\begin{equation} \boldsymbol{E} = E_{0} [ r z^{4}, 0, (2 - r^{2}) z^{5} + 2 r^{2} z^{3} ] \, \textrm{e}^{{-}z^{2}}. \end{equation}

Figure 2. Magnetic field streamlines of torsional fan reconnection. (a) In the $XZ$ plane, the greatest azimuthal twist is localized to $Z = \sqrt {1.5}$. (b) Spine field lines can be seen to twist in the $XY$ plane.

Figure 3 displays cross-sections of the vector and scalar fields of the torsional fan magnetic field (figure 3a,b, (2.2)), electric potential (figure 3c,d, (2.8)) and the electric field (figure 3e,f, (2.9)). Blue contours denote the extents of the plasma sheath injected into these fields, and will be described in more detail in § 3.3. Figure 4 shows the vector drift velocity (top row) calculated from

(2.10)\begin{equation} v_{d} = \frac{\boldsymbol{E} \times \boldsymbol{B}}{B^{2}},\end{equation}

with the bottom row displaying the scalar resistivity profile of (2.3) localized to the region of maximum magnetic twist at $Z = \sqrt {1.5}$.

Figure 3. Vector and scalar fields of torsional fan reconnection using $B_{0}=1$ T, $L_{0}=10$ cm, $\varPhi _{0}=6$ kV and $E_{0}=60$ kV m$^{-1}$. Vertical extents of panels’ (a,c,e) images are expanded to [$-2 \sqrt {1.5}$, $2 \sqrt {1.5}$]. In all images, blue lines show extent of particle input sheath (see § 3.3). Here (a) $XZ$ magnetic field at $Y = 0$ with units of Tesla; (b) $XY$ magnetic field twist at $Z = \sqrt {1.5}$; (c) $XZ$ electric potential at $Y = 0$; (d) $XY$ electric potential at $Z = \sqrt {1.5}$; (e) $XZ$ electric field at $Y = 0$; (f) $XY$ electric field at $Z = \sqrt {1.5}$.

Figure 4. Vector and scalar quantities of torsional fan reconnection fields in figure 3 using $B_{0}=1$ T, $L_{0}=10$ cm, $\varPhi _{0}=6$ kV and $E_{0}=60$ kV m$^{-1}$. The plasma resistivity $\eta _{0}$ is calculated from parameters in § 3.2. Vertical extents of panels’ (a,c) images are expanded to [$-2 \sqrt {1.5}$, $2 \sqrt {1.5}$]. In all images, blue lines show extent of particle input sheath (see § 3.3). Here (a) $XZ$ drift velocity at $Y = 0$; (b) $XY$ drift velocity at $Z = \sqrt {1.5}$; (c) $XZ$ resistivity profile at $Y = 0$; (d) uniform $XY$ resistivity profile at $Z = \sqrt {1.5}$.

2.2. Particle trajectories

Particle trajectories are computed using the relativistic equations of motion for momentum $\boldsymbol {p}$, namely

(2.11)\begin{equation} \frac{\textrm{d} \boldsymbol{r}}{\textrm{d} t} = \frac{\boldsymbol{p}}{\gamma A m_{p}},\end{equation}

and the Lorentz force in a collisional plasma

(2.12)\begin{equation} \frac{\textrm{d} \boldsymbol{p}}{\textrm{d} t} = e \left( \boldsymbol{E} + \frac{\boldsymbol{p}}{\gamma A m_{p}} \times \boldsymbol{B} \right) - \nu_{in} \boldsymbol{p},\end{equation}

where $\boldsymbol {r}$ is the 3-D Cartesian position, $e$ is the charge of a singly ionized particle, $A$ is the atomic mass, $m_{p}$ is the proton rest mass in MeV c$^{-2}$ and $\nu _{in}$ is the ion–neutral collision frequency (see below). Within the code, (2.11) and (2.12) are solved in dimensionless form, with distance scaled to a characteristic length $\boldsymbol {r} = L_{0} \boldsymbol {x}$, momentum scaled to the initial momentum $\boldsymbol {p} = p_{0} \boldsymbol {n}$, magnetic field to a characteristic strength $\boldsymbol {B} = B_{0} \boldsymbol {b}$, time to the characteristic non-relativistic gyroperiod $t = \tau _{0} t^{\prime }$, with $\tau _{0} = 2 {\rm \pi}m_{0} / e B_{0}$ (Dalla & Browning Reference Dalla and Browning2005, Reference Dalla and Browning2006, Reference Dalla and Browning2008). The initial momentum in MeV c$^{-1}$ is calculated from the initial particle kinetic energy $K_{0}$ in eV as $p_{0} = (K_{0}^{2} + 2 A m_{p} K_{0} )^{1/2}$. Our choices of $L_{0}$ and $B_{0}$ will be described in § 3.2.

Since full orbit particle tracing in plasmas is not limited by inelastic collisions (e.g. Marchand Reference Marchand2010; Homma et al. Reference Homma, Hoshino, Tokunaga, Yamoto, Hatayama, Asakura, Sakamoto and Tobita2018), the effects of collisions are accounted for in the final term on the right-hand side of (2.12). The addition of this collisional term has been shown to be effective in particle tracking studies of magnetic reconnection in high density laser plasmas up to $\sim 10^{25}$ m$^{-3}$ (Zhong et al. Reference Zhong2016). The collision frequency is determined by our choice of plasma parameters in § 3.2. Most laboratory plasmas are weakly ionized, where the degree of ionization is dependent on the ion ($n_{i}$) and neutral gas ($n_{g}$) number densities $\chi = n_{i} / ( n_{g} + n_{i} )$ and $\chi \ll 1$. Weakly ionized plasmas are further defined by the electron–ion collision frequency $\nu _{ei}$ being less than the electron–neutral collision frequency $\nu _{en}$ ($\nu _{ei} < \nu _{en}$; Park et al. Reference Park, Choe, Moon and Yoo2019). In the case of low-energy electrons (a few eV), the main collisional process in a weakly ionized plasma is dominated by polarization scattering of ions against neutral particles $\nu _{in}$ (Lieberman & Lichtenberg Reference Lieberman and Lichtenberg2005). Electron and ion polarization scattering rate constants in units of cm$^{3}$ s$^{-1}$ are calculated from

(2.13)\begin{equation} K_{Le} = 3.85 \times 10^{{-}8} \alpha_{R}^{1/2} \end{equation}

and

(2.14)\begin{equation} K_{Li} = 8.99 \times 10^{{-}10} \left( \frac{\alpha_{R}}{A} \right)^{1/2},\end{equation}

respectively, where $\alpha _{R}$ is the species-dependent relative polarizability ($\alpha _{R, He} = 1.384$ for helium;

Schwerdtfeger & Nagle (Reference Schwerdtfeger and Nagle2019)). The charged particle collision frequencies against neutrals are then calculated using

(2.15)\begin{equation} \nu_{e,i-n} = n_{g} K_{L-e,i}.\end{equation}

Parameters using typical laboratory values given in table 1 demonstrate numerically that the chosen helium plasma is weakly ionized ($\chi = 1\,\%$ and $\nu _{ei} < \nu _{en}$) and polarization scattering is the dominant collisional process.

Table 1. Helium plasma parameters used for the CPG plasma regime. Assumes a capacitor bank discharge of 6 kV over 10 cm ($E=60$ kV m$^{-1}$).

The dimensionless forms of (2.11) and (2.12) are solved numerically using the scipy.integrate.odeint package which uses LSODA from the FORTRAN library to dynamically monitor data and reduce errors by automatically switching between stiff and non-stiff methods. Numerical validation of this routine first verified the relativistic ion gyroradius $r_{g,i} = \gamma A m_{p} v_{\bot } / e B$ within uniform magnetic fields ($\boldsymbol {E} = 0$ V m$^{-1}$). Then, summing the kinetic and electric potential energies throughout test runs with collisionless single particle trajectories when $\boldsymbol {E} \neq 0$ V m$^{-1}$ demonstrated total energy conservation to five significant figures, of the order of $0.1$ eV, using (Dalla & Browning Reference Dalla and Browning2005, Reference Dalla and Browning2006, Reference Dalla and Browning2008; Gascoyne Reference Gascoyne2015)

(2.16)\begin{equation} W = K + e \varPhi.\end{equation}

Monitoring the total energy in the collisional plasma regime will elucidate the energy loss and energy loss rate due to collisions and scattering.

3.1. Electrodynamic simulation of hardware

Here, we motivate our choice of particle injection geometry by simulating the electrodynamics of a hardware infrastructure that mimics the analytical requirements for torsional fan reconnection. In torsional fan reconnection, it is an azimuthal perturbation to the spine axis of a 3-D null point topology that provides the driver. In previous investigations of the torsional fan mode, the $r$ and $z$ components of the fan-spine topology are subjected to a non-zero $\theta$ component (Galsgaard, Priest & Titov Reference Galsgaard, Priest and Titov2003; Pontin & Galsgaard Reference Pontin and Galsgaard2007; Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011; Wyper & Jain Reference Wyper and Jain2011, Reference Wyper and Jain2010) as in (2.2). Consider a region of a spine axis consisting of vertical, potential magnetic field lines where the perturbation is to be imposed (such as in figure 1). A localized plasma injection imposing this azimuthal perturbation upon these spine axis field lines has the effect of pushing the vertical spine axis field lines toward their nearest neighbour (such as in figure 2), thus becoming a localized non-potential field with current density $\boldsymbol {J} \neq 0$. The stronger the azimuthal component, the more horizontal these spine axis field lines become in the localized region, and are thus in very close proximity to one another. If the magnetic field is initially frozen into the plasma, as characterized by a Lundquist number $S > 1$ (§ 3.2), then a sufficiently narrow spatial extent exists within which current sheets can form (as defined by solving for the transition length from frozen-in condition effects while $S > 1$ to resistive diffusion effects when $S < 1$; $L_{cs} = m_{e} \nu _{ei} / \mu _{0} v_{A} n_{e,i} \, \textrm {e}^{2}$ when $S = 1$). It is within these narrow current sheets that $S \lesssim 1$ and where diffusion of magnetic field lines through the plasma and non-ideal effects take over. In these current sheet regions, the magnetic field lines can diffuse through the plasma and re-form the initial potential spine axis configuration. This reconnection to the original potential configuration has the effect of directly converting the stored magnetic field energy into kinetic energy of the surrounding plasma due to the strong electric fields that are generated by the current sheet in the twisted region.

Expanding on the general design of Chesny et al. (Reference Chesny, Orange, Oluseyi and Valletta2017), a first-order, 3-D fan-spine magnetic null point topology is created by a four-coil solenoid as shown in figure 5(a,b). The coils are distributed axially every 10 cm, each with radius 15 cm and carrying constant current $I = 200$ kA. The current is reversed for each symmetric half to create a central, positive magnetic null ( Parnell et al. (Reference Parnell, Smith, Neukirch and Priest1996); field vectors point in along the spine, out along the fan) with magnitude chosen to generate a field approaching 1 T at 10 cm from the null point. This will correspond directly to our choice of characteristic length in the system of MHD equations discussed below. We stress that the considered solenoid is only one of many possible coil configurations for producing a qualitative fan-spine topology (Chesny & Orange Reference Chesny and Orange2020). As such, the authors leave quantitative analyses of more complex coil configurations and modulated currents as a topic for a future study. The fan-spine topology of the magnetic field of this considered configuration is solved via the analytical expressions for circular, current carrying coils given by Simpson et al. (Reference Simpson, Lane, Immer and Youngquist2001).

Figure 5. (a) Radial view of streamlines for an arbitrary, solenoidal fan-spine topology, spaced over 40 cm, each with radius 15 cm and $I=100$ kA. (b) Axial view of fan plane. (c) Computer-aided design of axially aligned CPG at an arbitrary spine axis location with plasma sheath illustrated in green. The CPG discharge produces an azimuthal magnetic field. (d) Axial view of azimuthal CPG magnetic field. (e) Simulated coil fan-spine topology with localized CPG field superimposed. (f) Simulated superposition of localized CPG magnetic field upon coil fan plane magnetic field at $Z=12.25$ cm with units of Tesla.

To provide an azimuthal perturbation to one side of this coil-generated spine axis, we consider the magnetic field of a generalized coaxial plasma gun (CPG) (e.g. Larson, Liebing & Dethlefsen Reference Larson, Liebing and Dethlefsen1966; Scheuer et al. Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994; Thio et al. Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002) with plasma sheath parameters and run-down phase dynamics identical to a dense plasma focus apparatus (DPF) (see below). A CPG is constructed from a thin, cylindrical anode a few centimetres in length insulated from an outer, concentric cathode. The cathode is often a solid cylindrical shell surrounding the anode, but for visual purposes we illustrate the outer electrodes as long, thin rods, as common in DPF geometries (Krishnan Reference Krishnan2012). The CPG is aligned axially with one side of the spine axis within the coils as shown in the computer-aided design in figure 5(c,d). Here, we describe the general process of CPG plasma sheath generation for argument purposes only, as it is well documented in literature. A high voltage capacitor bank discharge ionizes the gas in-between the CPG electrodes, thus forming a plasma sheath at the insulating base (Esaulov et al. Reference Esaulov, Makhin, Bauer, Siemon, Sotnikov, Paraschiv, Presura, Freeman, Hagen, Ziegler, Lindemuth and Sheehey2003). This plasma current density and the azimuthal magnetic field of the CPG generate a $\boldsymbol {J} \times \boldsymbol {B}$ force so the plasma sheath lifts off of the insulator and propagates down the electrode axis (e.g. Krishnan & Thompson Reference Krishnan and Thompson2010). This propagating plasma sheath sustains and amplifies an azimuthal magnetic field internal to the plasma (Lindberg & Jacobsen Reference Lindberg and Jacobsen1961), which is centred around the spine axis of the conducting coil field. We consider this description to be the source of both the plasma and in situ azimuthal perturbation magnetic field (figure 2) required for the breakdown of ideal MHD in torsional fan reconnection as described earlier. In the hardware configuration of figure 5(c,d), the thin plasma sheath (green) is injected axially along the spine, subjecting it to a magnetic twist.

Chesny et al. (Reference Chesny, Orange, Oluseyi and Valletta2017) modelled the dynamic magnetic field of a capacitor bank discharge across CPG electrodes as

(3.1)\begin{equation} B_{\theta} (r, t) \mathrm{[T]} = \frac{ \mu_{0} V_{0}}{2 {\rm \pi}R r} \, \mathrm{e}^{{-}t/RC}, \end{equation}

where $R$ is the resistance, $C$ is the capacitance, $r$ is the radial distance and $t$ is the time elapsed since the discharge. Typical values of CPG discharges, $V_{0} = 6$ kV, $R = 1$ m$\varOmega$ and $C = 330 \mathrm {\mu }$F, were adopted to approximate this magnetic field (Lindberg & Jacobsen Reference Lindberg and Jacobsen1961; Larson et al. Reference Larson, Liebing and Dethlefsen1966; Schaer Reference Schaer1994; Scheuer et al. Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994; Heo & Park Reference Heo and Park2002; Thio et al. Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002; Witherspoon et al. Reference Witherspoon, Case, Messer, Bomgardner, Phillips, Brockington and Elton2009). Figures 5(e) and 5(f) show the simulated electrodynamics of the hardware architecture and demonstrate the feasibility of this configuration to provide the required driving of torsional fan reconnection under proper MHD plasma conditions. Time $t \approx 8.2 \times 10^{-7}$ s was solved from (3.1) under the constraints of a magnetic field of $B_{\theta }=1$ T located $r=10$ cm from the null. This magnetic field is solved in a plane at $Z=10$ cm and superimposed upon the initial potential spine axis of the coil-generated fan-spine topology (figure 5a). Figure 5(e) shows streamlines of this localized magnetic twist upon the spine axis of the coil field in figure 5(a) and mimics the azimuthal twist of figure 2, i.e. the analytical perturbation imposed in (2.2). Vector field superposition in this $Z=10$ cm plane is plotted in figure 5(f), showing the magnitude of the twist applied to the fan-spine topology from an approximated CPG discharge, which is consistent with ${\approx }1$ T magnetic fields internal to the plasma sheath in robust MHD models (Esaulov et al. Reference Esaulov, Makhin, Bauer, Siemon, Sotnikov, Paraschiv, Presura, Freeman, Hagen, Ziegler, Lindemuth and Sheehey2003, Reference Esaulov, Bauer, Lindemuth, Makhin, Presura, Ryutov, Sheehey, Siemon and Sotnikov2004). A full description of the superposition of these fields would require a full 2-D or 3-D MHD or kinetic simulation where the background fan-spine topology is threaded within the initial plasma formation and evolution. The following section will present the quantitative plasma parameters that satisfy the kinetic and MHD requirements for reconnection to predict particle response to the torsional fan mode.

3.2. Plasma parameters

Kinetic and MHD plasma parameters are calculated in table 1. The parameters listed in the top portion of the table give the characteristic values used to numerically solve (2.1) to (2.16). Coaxial plasma gun specifications from previous experiments (Larson et al. Reference Larson, Liebing and Dethlefsen1966; Scheuer et al. Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994; Thio et al. Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002) are used to show in table 1 that such a configuration can satisfy the requirements for reconnection to occur from an induced breakdown of ideal MHD plasma conditions (Büchner Reference Büchner1999; Ryutov, Drake & Remington Reference Ryutov, Drake and Remington2000; Kallenrode Reference Kallenrode2004). Helium CPG plasma sheaths ($A=4$) are formed at discharge voltages of $\lesssim 10$ kV and densities of $\sim 10^{20}$ m$^{-3}$ (Lindberg & Jacobsen Reference Lindberg and Jacobsen1961; Larson et al. Reference Larson, Liebing and Dethlefsen1966; Schaer Reference Schaer1994; Scheuer et al. Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994; Heo & Park Reference Heo and Park2002; Thio et al. Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002). The electric potential value $\varPhi _{0}$ is derived from a capacitor bank discharge voltage $V_{0} = 6$ kV typical of CPG helium gas breakdown across ${\approx }10$ cm (Larson et al. Reference Larson, Liebing and Dethlefsen1966; Scheuer et al. Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994; Thio et al. Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002), corresponding to electric fields of ${\approx }60$ kV m$^{-1}$. A characteristic magnetic field $B_{0} = 1$ T, as the one simulated in § 3.1, is achievable with high-temperature superconductors (Goodzeit, Meinke & Ball Reference Goodzeit, Meinke and Ball2005; Levin et al. Reference Levin, Barnes, Murphy, Brunke, Long, Horwath and Turgut2008; Squire et al. Reference Squire, Carter, Chang Diaz, Giambusso, Ilin, Ogilve-Araya, Olsen, Bering and Longmier2013). In figure 3, the simulated particle sheath is immersed at the location of maximum twist $Z=\sqrt {1.5}$ (§ 3.3). The initial MHD condition of this helium plasma is demonstrated by a Lundquist number $S \approx 3000$ with motions dominated by the magnetic field pressure (plasma $\beta < 1$). The plasma resistivity $\eta _{0} = 4.5 \times 10^{-5} \varOmega$-m is used to solve for the localized resistivity profile and the solutions of Wyper & Jain (Reference Wyper and Jain2011) and Wyper & Jain (Reference Wyper and Jain2010), within which ideal MHD can break down and reconnection can be initiated. The calculation of each relevant collision frequency demonstrates that the condition for a weakly ionized laboratory plasma is satisfied ($\nu _{ei} < \nu _{en}$) and that the ion–neutral scattering frequency becomes the dominant term in (2.12). The collision time scale compared with the gyroperiod is $1/\nu _{in} \tau _{0} \approx 0.7$, demonstrating the need for the collisional term in (2.12), which is further highlighted by the ion mean free path of 0.2 mm being significantly less than the chosen length scale. External fields can penetrate the plasma, as the electron skin depth $d_{e} = 0.5$ mm is of the same order of the ion gyroradius $r_{g,i} = 1$ mm (Fridman & Kennedy Reference Fridman and Kennedy2004). These are typical order-of-magnitude values and by no means limit the possible ranges of CPG/DPF parameters.

3.3. Particle injection

While previous authors considered the response of a randomized injection of a collection of particles to torsional fan reconnection (e.g. within the boundaries of a sphere and centred around the null;

Dalla & Browning (Reference Dalla and Browning2006), Dalla & Browning (Reference Dalla and Browning2008) and Hosseinpour (Reference Hosseinpour2015)), we consider the geometrically controlled injection of particles typical of a CPG device. The initial kinetic energy and momentum ($K_{0} \approx 10$ eV; Larson et al. (Reference Larson, Liebing and Dethlefsen1966), Schaer (Reference Schaer1994), Scheuer et al. (Reference Scheuer, Schoenberg, Henins, Gerwin, Moses, Garcia, Gribble, Hoyt, Black and Mayo1994) ,Thio et al. (Reference Thio, Eskridge, Lee, Smith, Martin, Markusic and Cassibry2002) and Witherspoon et al. (Reference Witherspoon, Case, Messer, Bomgardner, Phillips, Brockington and Elton2009)) of each helium ion in the sheath will follow the generalized snowplough model of a CPG plasma sheath rundown phase (Lee Reference Lee1983; Lee & Serban Reference Lee and Serban1996; Ziemer & Choueiri Reference Ziemer and Choueiri2001; Tang, Adams & Rusnak Reference Tang, Adams and Rusnak2010), and be taken to be in the $+Z$ direction. The sheath injection height is set to $Z = \sqrt {1.5}$, corresponding to the $+Z$ location of greatest magnetic field twist (see § 5 for a discussion of the exclusion of the $-Z$ location twist). It will be confined in the radial $XY$ plane to an inner ($r_{in} = 0.1$) and outer ($r_{out} = 1.0$) radius, corresponding to the inner (1 cm) and outer electrode (10 cm) radii, respectively, of an approximated CPG device. Since the initial plasma temperature $T$ will be much smaller than the discharge electric potential $\varPhi _{0}$, the finite thickness of the sheath $s$ can be modelled to follow the Child sheath law (Fridman & Kennedy Reference Fridman and Kennedy2004; Lieberman & Lichtenberg Reference Lieberman and Lichtenberg2005)

(3.2)\begin{equation} s = \frac{\sqrt{2}}{3} \lambda_{D} \left( \frac{2 V_{0}}{T} \right)^{3/4},\end{equation}

where $\lambda _{D}=(\epsilon _{0} T_{eV} / e n_{e,i})^{1/2}$ is the Debye length. In table 1, the sheath thickness $s \sim 10^{2} \lambda _{D}$, is consistent with expectations for high voltage discharges (Fridman & Kennedy Reference Fridman and Kennedy2004; Lieberman & Lichtenberg Reference Lieberman and Lichtenberg2005). The positions and momenta of $10\,000$ input particles are shown in figure 6 with streamlines of the magnetic field present for comparison.

Figure 6. Particle injection geometry. (a) The $10\,000$ particle sheath disk is confined to a radius of $0.1 \leq r \leq 1.0$ in the $XY$ plane at $Z = \sqrt {1.5}$, with Child sheath thickness $s$. (b) Each helium ion has initial kinetic energy of 10 eV with its momentum in the $+Z$ direction.

An important note on the injection position of this sheath refers back to the localized resistivity profile shown in figure 4(c,d) calculated from (2.3). The vertical position ($Z = \sqrt {1.5}$) of the input sheath is centred within the peak resistivity, i.e. the diffusion region, where the non-ideal effects of magnetic field connectivity changes must occur in our investigated geometry.