2 A candidate TSR system

At the most basic level, reconnection experiments can be described by a two-step process. First, the system is set-up with plasma frozen into a static (potential) magnetic field with cusp or null regions where $\boldsymbol{B}=0$ . Second, an externally driven instability is introduced that forces the breakdown of the frozen-in condition, allowing current sheets to form and reconnection to occur (Yamada et al. Reference Yamada, Ji, Hsu, Carter, Kulsrud, Bretz, Jobes, Ono and Perkins1997a ; Egedal et al. Reference Egedal, Fasoli, Porkolab and Tarkowski2000). Robust simulations of the TSR mode (Pontin & Galsgaard Reference Pontin and Galsgaard2007; Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011) provide an infrastructure in which to identify existing technologies that can mimic such behaviour.

The driven reconnection mode of TSR first requires an isolated potential 3-D null point field following the $\boldsymbol{B}=\boldsymbol{M}\boldsymbol{\cdot }\boldsymbol{r}$ matrix formulation of Parnell et al. (Reference Parnell, Smith, Neukirch and Priest1996), where $\boldsymbol{B}$ is the magnetic field, $\boldsymbol{M}$ is a $3\times 3$ magnetization tensor, and $\boldsymbol{r}$ is the 3-D position vector. The potential field parametrization of Pontin et al. (Reference Pontin, Al-Hachami and Galsgaard2011) using $p=2$ and $q=j_{\Vert }=j_{\bot }=0$ produced an improper radial null point field, identical to that shown in figure 2, with fan field lines concentrated along the $y$ -axis of the null. It is important to note that the parameters $p$ and $q$ describe the potential part of the null point magnetic field, and $j_{\Vert }$ and $j_{\bot }$ affect the twist and inclination of the fan plane, respectively. Both spheromak and tokamak-style technologies for X-line experiments (e.g. Yamada et al. Reference Yamada, Ji, Hsu, Carter, Kulsrud, Bretz, Jobes, Ono and Perkins1997a , Egedal et al. Reference Egedal, Fasoli, Tarkowski and Scarabosio2001) and 3-D EMHD (Stenzel et al. Reference Stenzel, Urrutia, Griskey and Strohmaier2001) used multiple relatively large ( ${\sim}70~\text{cm}$ diameter) flux cores for the production of their potential fields. Such conditions were required for their large system sizes $L$ to satisfy the conditions of MHD plasma (see § 3). Fortunately, state-of-the-art multi-layer helical conducting coil technology (e.g. Meinke, Senti & Stelzer Reference Meinke, Senti and Stelzer1997, Goodzeit et al. Reference Goodzeit, Meinke and Ball2005) can be optimized to produce unique null point fields of high purity (such as figure 2), and machined to fit within a volume of ${\lesssim}1~\text{m}^{3}$ . The use of such concentric multi-layer coils have wide ranging applications for high-order multipole fields in particle accelerator beam steering applications and MHD thrust devices. While smaller volumes reduce the global scale length $L$ parameter space within which ideal MHD conditions may prevail, the described TSR experiment is designed to overcome this shortcoming with the use of a dense plasma and still satisfy the frozen-in condition (see below and § 3). This is possible since magnetic reconnection is fundamentally a multi-scaled self similar process (Chang Reference Chang1999; Raouafi et al. Reference Raouafi, Georgoulis, Rust and Bernasconi2010; Chesny et al. Reference Chesny, Oluseyi, Orange and Champey2015; Chesny, Oluseyi & Orange Reference Chesny, Oluseyi and Orange2016; Wyper, Antiochos & DeVore Reference Wyper, Antiochos and DeVore2017).

In the TSR simulations of Pontin & Galsgaard Reference Pontin and Galsgaard2007 and Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011, a rotational magnetic field perturbation to the field lines of the fan plane, centred around the spine axis (figure 3), caused the propagation of Alfvén waves toward and away from the null. The TSR mode of interest to this work occurs due to the ingoing wave as it propagates along the background magnetic field lines to form a concentric cylindrical current tube centred around the spine. The bulk plasma flow is rotational, with the peak spine-directed current increasing as the wave localizes around the vicinity of the spine. This collapsed wavefront induces the rotational slippage of the perturbed field lines and is observed as a current intensification close to the spine. This helical Alfvén wave (Galsgaard, Priest & Titov Reference Galsgaard, Priest and Titov2003) is dominated by the current component oriented parallel to the spine. The bulk rotational plasma flows associated with TSR peak as field line diffusion maximizes, and then reverse direction once the rotational slippages relax back to their low-energy potential state.

Figure 3.

Fan spine magnetic null point topology (black vectors) with applied circular magnetic field perturbation (red vectors). This configuration was shown to lead to TSR (Pontin & Galsgaard Reference Pontin and Galsgaard2007; Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011). Hardware that can mimic these magnetic field dynamics may open up the possibility for a laboratory set-up that can induce TSR.

To design an experiment for inducing TSR featuring these observational signatures, an apparatus placed in the conducting coil-produced potential null point field must provide a circular magnetic field perturbation to the fan plane, centred around the spine. A dense plasma focus (DPF) is a cylindrical apparatus that produces a dense (Guillory, Rose & Lerner Reference Guillory, Rose, Lerner, Kusse and Hammer2009; Mitrofanov et al. Reference Mitrofanov, Krauz, Kubes, Scholz, Paduch and Zielinska2014; Huba Reference Huba2016) ( ${\sim}10^{24}~\text{m}^{-3}$ ) MHD-scale (Caballero Bendixsen et al. Reference Caballero Bendixsen, Bott-Suzuki, Cordaro, Krishnan, Chapman, Coleman and Chittenden2016) capacitively coupled plasma featuring an azimuthal magnetic field within its toroidal plasma sheath (Lindberg & Jacobsen Reference Lindberg and Jacobsen1961; Guillory et al. Reference Guillory, Rose, Lerner, Kusse and Hammer2009; Krishnan & Thompson Reference Krishnan and Thompson2010). Most DPFs are a few centimetres in both length and in radius and are small enough to be able to be operated inside the inner volume of a conducting coil of ${\lesssim}1~\text{m}^{3}$ . It is feasible that a dedicated experiment comprised of a conducting coil null point field and a central DPF, having the coil’s central spine axis aligned with the DPF axial direction (figure 4), could mimic the TSR magnetic field dynamics of figure 3.

Figure 4.

Generalized COMSOL Multiphysics model geometry of DPF (interior black contours) immersed in null point field (red vectors) produced by two conducting loops carrying equal current in opposite directions. The operation of the DPF apparatus within this field is the principle of operation. The optimized apparatus is anticipated to fit within ${\lesssim}1~\text{m}^{3}$ .

We have calculated magnetic field solutions for both the theoretical topology leading to TSR (Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011) and the described laboratory TSR system. The theoretical magnetic null point field is characterized by (Parnell et al. Reference Parnell, Smith, Neukirch and Priest1996),

(2.1) $$\begin{eqnarray}\boldsymbol{B}=B_{0}[x+{\textstyle \frac{1}{2}}(q-j_{\Vert })y,{\textstyle \frac{1}{2}}(q+j_{\Vert })x+py,j_{\bot }y-(p+1)z].\end{eqnarray}$$

The circular magnetic field perturbation used by Pontin et al. (Reference Pontin, Al-Hachami and Galsgaard2011) is a polar coordinate field in the fan plane of the form,

(2.2) $$\begin{eqnarray}B_{\unicode[STIX]{x1D703}}=B_{p}\exp \left[-\left(\frac{r-r_{0}}{a}\right)^{2}-\left(\frac{z-z_{0}}{b}\right)^{2}\right].\end{eqnarray}$$

A purely radial null with $p=1$ , $q=j_{\Vert }=j_{\bot }=0$ and $B_{0}=3~\text{T}$ will produce a magnetic field of the order of $10^{-1}~\text{T}$ approximately ${\sim}3~\text{cm}$ from the null region. A circular perturbation with $r_{0}=0.02~\text{m}$ , $a=0.01~\text{m}$ and $B_{p}=12~\text{T}$ , corresponding to centimetre-scale geometries, will produce a ring of flux of the order of $10^{-1}~\text{T}$ at a radial distance of $2~\text{cm}$ from the null. These combined fields are shown in figure 5(a) and depict the dynamics in figure 3.

For the coil–DPF system, a first-order approximation to a radial magnetic null point field was produced by solving the Biot–Savart law for two axially aligned, circular current carrying loops of radius 10 cm, offset from one another by 20 cm (figure 4). To produce a central positive null point similar to (2.1) and figure 5(a), each loop will have oppositely running current of magnitude 100 kA. This ensures a magnetic field strength of ${\sim}10^{-1}~\text{T}$ close to the null (see § 2). These first-order currents are obviously high, but can be mitigated to ${\lesssim}10^{2}$ . A using multi-layered helical conducting coil technology that produce fields of ${\gtrsim}1~\text{T}$ (Meinke et al. Reference Meinke, Senti and Stelzer1997; Goodzeit et al. Reference Goodzeit, Meinke and Ball2005). The initial magnetic curl of a DPF, leading to the formation of a flat-top toroidal plasma sheath, is created via a rapidly changing electric flux across coaxial conductors. We have solved for a general DPF field using the Ampère–Maxwell relation $\int \boldsymbol{B}\boldsymbol{\cdot }\text{d}\boldsymbol{l}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D716}_{0}(\text{d}/\text{d}t)\oint \boldsymbol{E}\boldsymbol{\cdot }\text{d}\boldsymbol{a}$ and a discharging capacitor bank $Q(t)=CV\text{e}^{-t/RC}$ as,

(2.3) $$\begin{eqnarray}B_{\unicode[STIX]{x1D703}}(r,t)\text{[T]}=\frac{-\unicode[STIX]{x1D707}_{0}V}{2\unicode[STIX]{x03C0}Rr}\text{e}^{-t/RC},\end{eqnarray}$$

where $V$ is the voltage, $R$ is the resistance, $C$ is the capacitance, $r$ is the radial distance and $t$ is the time elapsed since the discharge. We have adopted the values of $V=30~\text{kV}$ , $R=1~\text{m}\unicode[STIX]{x1D6FA}$ , and $C=12.5~\unicode[STIX]{x03BC}\text{F}$ which are representative values for similar experiments (e.g. Heo & Park Reference Heo and Park2002, Bures et al. Reference Bures, Krishnan, Madden and Blobner2010, El-Aragi Reference El-Aragi2010, Abdou et al. Reference Abdou, Ismail, Mohamed, Lee, Saw and Verma2012, Bures, Krishnan & James Reference Bures, Krishnan and James2012, Caballero Bendixsen et al. Reference Caballero Bendixsen, Bott-Suzuki, Cordaro, Krishnan, Chapman, Coleman and Chittenden2016). Time $t\approx 8\times 10^{-8}~\text{s}$ corresponds to an idealized capacitor bank rise time (Lerner, Krupakar Murali & Haboub Reference Lerner, Krupakar Murali and Haboub2011) with a magnetic field of $10^{-1}~\text{T}$ located 1 cm from the null. Figure 5(b) shows the superposition of the two-coil null point field and the azimuthal DPF perturbation field of (2.3). The similarities between the parametric and first-order hardware fields are obvious, i.e. both perturbations are able to deform the fan plane of the initial potential fields in a circular fashion. Therefore, it can be anticipated that, in the first order, a conducting coil–DPF combination can mimic the magnetic field dynamics leading to TSR (figure 3). In operational DPF systems, this magnetic curl is maintained and expanded radially toward the boundaries of the outer electrodes by the axial cascade of the toroidal plasma sheath following the snowplough model (Lindberg & Jacobsen Reference Lindberg and Jacobsen1961; Hart Reference Hart1964; Willenborg & Hendricks Reference Willenborg and Hendricks1976; Lee Reference Lee1983; Ziemer & Choueiri Reference Ziemer and Choueiri2001; Tang, Adams & Rusnak Reference Tang, Adams and Rusnak2010). The $\boldsymbol{J}\times \boldsymbol{B}$ force due to the interaction between the collisionally excited plasma current (Krishnan & Thompson Reference Krishnan and Thompson2010) and the magnetic curl of (2.3) induces the ${\sim}10~\unicode[STIX]{x03BC}\text{s}~\text{cm}^{-1}$ axial propagation of the sheath (Lee & Serban Reference Lee and Serban1996) in which the in situ flat-top plasma for reconnection resides.

Figure 5.

(a) Theoretical parametrization of radial potential null point field (Parnell et al. Reference Parnell, Smith, Neukirch and Priest1996) with applied circular magnetic field perturbation (Pontin et al. Reference Pontin, Al-Hachami and Galsgaard2011). (b) Circular DPF field immersed in radial potential null point field produced from current carrying loops. Both fields are topologically similar to figure 3.

3 TSR parameterization

While the qualitative topologies of the conducting coil–DPF combination mimic the magnetic field dynamics that is known to induce the TSR mode, it must be demonstrated that the quantitative plasma parameters of such a system can lead to the onset of reconnection. In either astrophysical or laboratory plasma environments, reconnection occurs within thin localized regions much smaller than the global length scale (current sheets). Within these localized regions, the frozen-in flux condition is broken and non-ideal effects, such as diffusion or particle de-magnetization, can begin to dominate and force field line connectivity changes through the plasma.

Here, we consider the quantitative plasma parameters for the centimetre-scale, capacitively coupled TSR experiment described in § 2 in both collisional resistive MHD (Gonzalez & Parker Reference Gonzalez and Parker2016) and collisionless two-fluid, ion–electron kinetic contexts (Büchner Reference Büchner1999). General plasma parameters such as the electron–ion collision frequency $\unicode[STIX]{x1D708}_{c}$ are derived from the Naval Research Laboratory (NRL) Plasma Formulary (Huba Reference Huba2016). Calculated parameters such as electron plasma frequency $\unicode[STIX]{x1D714}_{pe}$ are compared to the Formulary for verification. Table 1 shows a compilation of the requisite resistive MHD and kinetic parameters along with their formulas.

Table 1.

Plasma parameters calculated for the investigated TSR regime.

3.1 Resistive MHD

Resistive MHD first relies on a global frozen-in flux condition of length scale $L$ . These ideal MHD conditions are satisfied by a large Lundquist number ( $S\gg 1$ ) of the form,

(3.1) $$\begin{eqnarray}S=\frac{v_{A}L}{\unicode[STIX]{x1D702}}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}v_{A}L=\frac{\unicode[STIX]{x1D707}_{0}v_{A}Ln_{e}e^{2}}{m_{e}\unicode[STIX]{x1D708}_{c}},\end{eqnarray}$$

where $v_{A}$ is the local Alfvén velocity, $L$ is the length scale of the system, $\unicode[STIX]{x1D702}$ is the magnetic diffusivity $(\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E})^{-1}$ , $\unicode[STIX]{x1D707}_{0}$ is the vacuum permeability, $\unicode[STIX]{x1D70E}$ is the conductivity, $n_{e}$ is the electron number density, $e$ is the electric charge, $m_{e}$ is the electron mass and $\unicode[STIX]{x1D708}_{c}$ is the electron–ion collision frequency. Second, an abrupt change of plasma conditions results in a localized breakdown of ideal MHD (Kivelson & Russell Reference Kivelson and Russell1995; Kallenrode Reference Kallenrode2004). It is within this region where the limit $S\lesssim 1$ is reached that plasma diffusion across field lines is permitted and localized current sheets will form. The $S\sim 1$ limit of (3.1) predicts a current sheet thickness $l_{cs}$ of order (Kivelson & Russell Reference Kivelson and Russell1995),

(3.2) $$\begin{eqnarray}l_{cs}\sim \frac{1}{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}v_{A}}\sim \frac{m_{e}\unicode[STIX]{x1D708}_{c}}{\unicode[STIX]{x1D707}_{0}v_{A}n_{e}e^{2}}.\end{eqnarray}$$

Though a breakdown of ideal MHD must occur within the region of current sheet formation, as required for plasma advection across field lines, outside the current layer the ideal MHD approximation should remain valid.

In the MHD portion of table 1, the initial conducting plasma state is shown by $S\sim 10^{2}$ . The fully resistive plasma condition ( $S\sim 1$ ) via field line diffusion is expected to occur on small spatial scales of current sheet widths of the order of $10^{-5}~\text{m}$ . Such effects are predicted to occur on the diffusion time scale,

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{D}=\frac{l_{cs}^{2}}{\unicode[STIX]{x1D702}}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}l_{cs}^{2}=\frac{\unicode[STIX]{x1D707}_{0}n_{e}e^{2}l_{cs}^{2}}{m_{e}\unicode[STIX]{x1D708}_{c}}.\end{eqnarray}$$

The TSR $\unicode[STIX]{x1D70F}_{D}$ time scale of $10^{-8}~\text{s}$ , over which collisional instabilities dissipate, should be equivalent to a few Alfvén times $\unicode[STIX]{x1D70F}_{A}=l_{cs}/v_{A}$ that is a signature of 2-D X-line reconnection (Imshennik & Syrovatskiǐ Reference Imshennik and Syrovatskiǐ1967; Parnell et al. Reference Parnell, Neukirch, Smith and Priest1997). Such behaviour will be investigated further in § 4. These values in the collisional regime now must be compared to corresponding length and time scales predicted by collisionless kinetic theory to see how they differ, and to eventually determine what effects should be expected from the TSR set-up described in § 2.

3.2 Kinetics

MHD equations characterize the behaviour of plasma and magnetic fields to a high degree (Gonzalez & Parker Reference Gonzalez and Parker2016). However, in plasma environments where currents and the Hall MHD term ( $\boldsymbol{J}\times \boldsymbol{B}$ ) become important for determining the electric field and reconnection rate, e.g. the solar corona, a two-fluid, ion–electron kinetic model may be more appropriate. In this regime, collisionless reconnection occurs when the frozen-in condition is broken on the inertial length scales of electron motion due to anomalous resistivity. Such kinetic effects have been demonstrated in plasma simulations where the TSR mode dominates (Olshevsky et al. Reference Olshevsky, Divin, Eriksson, Markidis and Lapenta2015).

While the frozen-in condition was determined via resistive MHD equations, the breakdown must occur on inertial scales of the electron skin depth (Büchner Reference Büchner1999; Stenzel et al. Reference Stenzel, Urrutia, Griskey and Strohmaier2001),

(3.4) $$\begin{eqnarray}d_{e}=c\sqrt{\frac{\unicode[STIX]{x1D716}_{0}m_{e}}{n_{e}e^{2}}},\end{eqnarray}$$

where $c$ is the speed of light, $\unicode[STIX]{x1D716}_{0}$ is the permittivity of free space and $e$ is the electric charge. Table 1 shows that kinetic effects in this set-up would occur on scales an order of magnitude smaller than resistive MHD ( $d_{e}\sim 10^{-1}l_{cs}$ ).

Reconnection time scales due to kinetic effects should be a few electron gyro-periods $\unicode[STIX]{x1D70F}_{ce}=2\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}_{ce}^{-1}$ . Bulk plasma wave perturbations propagating at the Alfvén speed heat the electrons, effectively de-magnetizing them and breaking the frozen-in condition. Unfrozen electrons are free to induce changes in magnetic connectivity, and thus reconnection. Such effects must occur within a few electron gyro-periods (see § 4) so that electron energization is not damped away (Büchner Reference Büchner1999). It is easily seen that $\unicode[STIX]{x1D70F}_{ce}$ is substantially faster than diffusion time scales for each respective experiment by at least two orders of magnitude. These results suggest that we expect to see fast reconnection in the kinetic regime of this TSR experiment, and that turbulent dissipation will occur after non-ideal effects induce reconnection over many $\unicode[STIX]{x1D70F}_{ce}$ (Olshevsky et al. Reference Olshevsky, Divin, Eriksson, Markidis and Lapenta2015).

Figure 6.

Simulations of spine axis component current density evolution of (4.1) with $p=2.0$ , $j_{\Vert }=1.0$ , $B_{0}=10^{-1}~\text{T}$ and $n=10^{24}~\text{m}^{-3}$ , with deuterium plasma. (a) Resistive MHD scale with $l=l_{cs}=\unicode[STIX]{x1D716}=10^{-5}$ . The current amplification (left axis) begins to increase significantly after one $\unicode[STIX]{x1D70F}_{A}$ (vertical dotted line), well after many electron gyro-periods ( ${>}10\unicode[STIX]{x1D70F}_{ce}$ ). The behaviour exceeds the defined limit of valid behaviour $\unicode[STIX]{x1D716}\text{e}^{\unicode[STIX]{x1D714}t}j_{\Vert }<1$ (right axis) after $7.4\times 10^{-8}~\text{s}$ ( ${\sim}11\unicode[STIX]{x1D70F}_{A}$ , ${\sim}200\unicode[STIX]{x1D70F}_{ce}$ ). The $5\unicode[STIX]{x1D70F}_{A}$ threshold is also noted by the vertical solid line. (b) Kinetic scale with $l=d_{e}=\unicode[STIX]{x1D716}=10^{-6}$ . The current amplification (left axis) begins to increase significantly after one $\unicode[STIX]{x1D70F}_{ce}$ (vertical dotted line) and one $\unicode[STIX]{x1D70F}_{A}$ (vertical dotted line), and exceeds the defined limit of valid behaviour $\unicode[STIX]{x1D716}\text{e}^{\unicode[STIX]{x1D714}t}j_{\Vert }<1$ (right axis) after $8.9\times 10^{-9}~\text{s}$ ( ${\sim}13\unicode[STIX]{x1D70F}_{A}$ , ${\sim}25\unicode[STIX]{x1D70F}_{ce}$ ). The $10\unicode[STIX]{x1D70F}_{ce}$ threshold is also noted by the vertical dot–dashed line and is notably close in time to the $5\unicode[STIX]{x1D70F}_{A}$ threshold (vertical solid line).

4 Numerical analyses of TSR

To further elucidate the anticipated behaviour of the investigated TSR experimental regime, and to gauge the relative importance of resistive MHD versus kinetics, we have completed numerical analyses of the solutions to the linearized, low- $\unicode[STIX]{x1D6FD}$ , ideal MHD equations due to perturbations of a potential null point field (Parnell et al. Reference Parnell, Neukirch, Smith and Priest1997). To study the localized nature of current sheet formation and the time evolution of null point field line structuring, the formulation of Parnell et al. (Reference Parnell, Smith, Neukirch and Priest1996) was expanded to derive a model for a perturbation that induces a component of current in the $z$ -direction, corresponding to the spine axis of a null point (Parnell et al. Reference Parnell, Neukirch, Smith and Priest1997). The authors expressed this current density in the form,

(4.1) $$\begin{eqnarray}\boldsymbol{J}=B_{0}\unicode[STIX]{x1D716}\text{e}^{\unicode[STIX]{x1D714}t}(0,0,j_{\Vert })/l\unicode[STIX]{x1D707}_{0},\end{eqnarray}$$

where $B_{0}$ is the initial magnetic field, $\unicode[STIX]{x1D716}\ll 1$ , $l$ is the characteristic length over which reconnection occurs and current sheets form (Dungey Reference Dungey1953) (i.e. $l\sim l_{cs}$ for resistive MHD, and $l\sim d_{e}$ for kinetics), and the growth rate $\unicode[STIX]{x1D714}$ is,

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D714}=|p-1|v_{A}/l.\end{eqnarray}$$

Importantly, the temporal behaviour of (4.1) is only valid provided that $\unicode[STIX]{x1D716}\text{e}^{\unicode[STIX]{x1D714}t}j_{\Vert }<1$ (Parnell et al. Reference Parnell, Neukirch, Smith and Priest1997).

We have solved (4.1) and (4.2) for current sheets on both resistive MHD and kinetic scales of $l_{cs}$ and $d_{e}$ , respectively. Our magnetic field matrix parametrization following Parnell et al. (Reference Parnell, Smith, Neukirch and Priest1996) uses $p=2$ , identical to the TSR simulations of Pontin et al. (Reference Pontin, Al-Hachami and Galsgaard2011), and with current along the spine $j_{\Vert }=1$ and $q=j_{\bot }=0$ . This numerical parametrization satisfies the constraints of the matrix formulation of Parnell et al. (Reference Parnell, Smith, Neukirch and Priest1996). Other plasma parameters are taken from the TSR regime in table 1. In each case, the parameter $\unicode[STIX]{x1D716}\ll 1$ is chosen to be of order $l$ .

Figure 6(a) depicts equation (4.1) solutions for the spine axis component of the current density for the resistive MHD length scale of a current sheet width $l_{cs}=10^{-5}~\text{m}$ . The collapse of the null point and maximum current amplification occurs after $7.4\times 10^{-8}~\text{s}$ , or ${\sim}11\unicode[STIX]{x1D70F}_{A}$ and ${\sim}200\unicode[STIX]{x1D70F}_{ce}$ . Such Alfvén time scales are consistent with the MHD collapse of 2-D X-line diffusion regions (Imshennik & Syrovatskiǐ Reference Imshennik and Syrovatskiǐ1967; Parnell et al. Reference Parnell, Neukirch, Smith and Priest1997). However, the electron gyro-periods of $1\unicode[STIX]{x1D70F}_{ce}$ and $10\unicode[STIX]{x1D70F}_{ce}$ are also shown, and these both occur well prior to even $1\unicode[STIX]{x1D70F}_{A}$ . Therefore, the energization of electrons on diffusive time scales at these length scales will only break the frozen-in condition very slowly, if at all, since it is possible that electron energization may be damped away before resistive effects, and thus reconnection, can take over. Since these equations are based on ideal MHD, the current densities reached toward the end of this behaviour ${\gtrsim}10~\text{kA}~\text{cm}^{-2}$ can be considered an upper limit of any true laboratory plasma behaviour in our system. That is, the unique difference to other set-ups is the use of a dense deuterium plasma that will introduce additional resistivity to current amplification processes.

Figure 6(b) depicts identical equation (4.1) solutions for the spine axis component of the current density for the kinetic length scale of a current sheet width equal to the electron skin depth $d_{e}=10^{-6}~\text{m}$ . The collapse of the null point and its resulting current amplification occurs after $8.9\times 10^{-9}~\text{s}$ , or ${\sim}13\unicode[STIX]{x1D70F}_{A}$ and ${\sim}25\unicode[STIX]{x1D70F}_{ce}$ . Even on a smaller length scale, the Alfvén time scales are still consistent with MHD collapse. Fortunately, in comparison to the resistive MHD-scale solution, both the $1\unicode[STIX]{x1D70F}_{ce}$ : $1\unicode[STIX]{x1D70F}_{A}$ and $10\unicode[STIX]{x1D70F}_{ce}$ : $5\unicode[STIX]{x1D70F}_{A}$ milestones occur close together on kinetic scales. Therefore, kinetic effects are considered to be important in the considered TSR plasma regime, as electron energization corresponds to Alfvén wave perturbations, and fast reconnection shorter than the time scales predicted by resistive diffusion can occur. Such a kinetic approach is further justified by the ratio of the electron gyrofrequency to the electron plasma frequency $\unicode[STIX]{x1D714}_{ce}/\unicode[STIX]{x1D714}_{ce}$ in table 1. Since $\unicode[STIX]{x1D714}_{pe}\gg \unicode[STIX]{x1D714}_{ce}$ , current sheets of order $d_{e}$ are anticipated to be able to form within the TSR dense plasma regime.

As a final note on these predicted time scales, the described TSR experiment featuring a rigidly controlled isolated null point is obviously an open geometry along the spine axis that will not feature separators connecting pairs of nulls. This open geometry will not confine plasma well. However, plasma and current sheet evolution are the features we are aiming to control, not confinement. Thus, the predicted rapid time scales of reconnection, likely faster than $\unicode[STIX]{x1D70F}_{D}$ , may be difficult to diagnose from the first approximations presented here. We anticipate our analyses presented herein (e.g. table 1) to be a springboard for future analyses that will further constrain the presented experimental geometry.