Magnetic reconnection in three-dimensional quasi-helical pinches

Abstract

This paper deals with the phenomenology of magnetic reconnection during reversed-field pinch helical self-organization. Numerical results obtained by solving a three-dimensional nonlinear visco-resistive fluid model to describe the hot current-carrying plasma are summarized. Magnetic reconnection manifests itself during the plasma dynamics, interrupting the persistence of quasi-helical states. The main signatures of magnetic reconnection in reversed-field pinches are discussed: partial conversion of magnetic into kinetic energy, current sheet formation, steepening of plasma current profiles, locking of the angular phases between different Fourier components of the magnetic field. The latter is recognized as the three-dimensional trigger of the reconnection events. Then the paper deals with the temporal scales of the process: low visco-resistive dissipation in the model, corresponding to high plasma current in the experiments, results in longer characteristic time between reconnection events. Furthermore, it is confirmed that the scaling of the reconnection rate is compatible with a modified Sweet–Parker model. A discussion of magnetic reconnection during the 2D simplified tokamak internal kink mode evolution, showing the development of secondary tearing instabilities, is presented and the similarities with RFP evolution are highlighted.

Appendices

Appendix 1: On the reconnection rate and possible modeling approaches

In this appendix a long-standing problem in reconnection theory is briefly reviewed, i.e., the fast reconnection rates⁷ (proportional to the inverse Alfvén time) observed in astrophysical and laboratory systems like in the solar flares and in the tokamak sawtoothing instability. The quest for a solution to this problem helps in categorizing the possible strategies for modeling magnetic reconnection. A resistive diffusion process would give too slow reconnection rates, $\tau ^{-1} \propto \eta$ ($\eta$ is the plasma dimensionless resistivity), at low values of resistivity measured in astrophysical or laboratory plasmas. A modeling strategy is to describe magnetic reconnection in the magnetohydrodynamic (MHD) regime, and consider it as a phenomenon localized in the vicinity of the region where magnetic field lines break, while elsewhere magnetic field lines are frozen into the plasma (Parker 1957): Sweet and Parker consider a two-dimensional current sheet and an incompressible inflow of plasma in the reconnection region gives a faster rate $\tau ^{-1} \propto \eta ^{0.5}$, a result later modified including viscous effects (Park et al. 1984) (the “modified Sweet–Parker scaling” $\tau ^{-1} \propto \eta ^{0.5}(1+\nu /\eta )^{-0.25}$, with $\nu$ plasma scalar viscosity, discussed in Sect. 4.4).

Petschek proposed a scenario in which reconnection proceeds at faster rates, with $\tau ^{-1}\propto 1/\ln {\eta }$ (Petschek 1963), by assuming a shorter reconnection layer and that the plasma inflow was redirected outside the reconnection region by shock waves, converting magnetic energy into ions kinetic energy (see Zweibel and Yamada 2009; Biskamp 2000 for a good description of the Sweet–Parker and Petschek models).

Later, the occurrence of instabilities of the current sheet itself (the so-called plasmoids) has been studied, starting from Furth et al. (1963), Biskamp (1986) and continuing in the next decades (Loureiro et al. 2007; Bhattacharjee et al. 2009; Comisso and Grasso 2016; Günter et al. 2014; Loureiro and Uzdensky 2015; Comisso and Bhattacharjee 2016; Del Sarto and Ottaviani 2017; Stanier et al. 2019) discovering the possibility of reconnection rates independent of resistive effects $\tau ^{-1} \propto \eta ^0$.

Furthermore, it was also shown that collisionless effects could speed-up the reconnection rate to values compatible with observations (Ottaviani and Porcelli 1993; Biskamp et al. 1995; Grasso et al. 1999; Borgogno et al. 2005, 2017; Grasso et al. 2020). Other MHD effects have also been investigated, in particular the role of turbulence in influencing the current sheet diffusion (Matthaeus and Lamkin 1986; Lapenta 2008), or the study of Hall-MHD (Biskamp et al. 1995) or two-fluid (Kleva et al. 1995) or kinetic effects, in particular when the current sheet transverse dimension gets smaller than length scales related to the ion sound Larmor radius or the ion skin depth (Yamada et al. 2010).

Boozer (2018) claims that the most overlooked feature in modeling fast magnetic reconnection is three-dimensionality. Restricting the evolution to two-dimensional spaces obscures the ideal evolution of magnetic fields lines, and in particular their exponential sensitivity to parallel electric fields which can give Alfvén speed reconnection with elementary magnetohydrodynamics modeling.

Appendix 2: Helical flux function $\chi$

The computation of the helical flux function $\chi (\mathbf {r})$, discussed in Sect. 1 and used in Sect. 3.1, proceeds as follows. Let us consider a magnetic field $\mathbf {B}(\mathbf {r})$ in a three-dimensional space with a curvilinear coordinate system whose coordinates are labeled as $u^i=(u^1,u^2,u^3)$. Let us suppose that the system has a symmetry, i.e., $\partial / \partial u^3=0$. The magnetic field can be written using the vector potential $\mathbf {A}(\mathbf {r})$ as:

$$\begin{aligned} \mathbf {B}=\nabla \times \mathbf {A} = \frac{\varepsilon ^{ijk}}{J}\frac{\partial A_j}{\partial u^i}\mathbf {e}_k, \end{aligned}$$

(7)

where $\varepsilon ^{ijk}$ represents the Levi-Civita tensor, J the jacobian of the coordinate transformation, $A_j$ the covariant component of the vector potential and $\mathbf {e}_k$ is the covariant basis vector. Imposing the relation $\mathbf {B} \cdot \nabla \chi =0$ choosing a gauge $A_1=0$ and remembering that $\partial / \partial u^3=0$ it is found that

$$\begin{aligned} \frac{\partial A_3}{\partial u^2}\frac{\partial \chi }{\partial u^1}-\frac{\partial A_3}{\partial u^1}\frac{\partial \chi }{\partial u^2}=0, \end{aligned}$$

(8)

meaning that the equality is satisfied if

$$\begin{aligned} \chi =A_3=\mathbf {A} \cdot \mathbf {e}_3. \end{aligned}$$

(9)

In helical geometry the helical flux function is given by the third covariant component of the vector potential, that will be indicated as $A_{3\mathrm{h}}$. The next step is the determination of the helical flux function in cylindrical geometry, using the transformation law of vector between different geometries. Indicating with $A_{i\mathrm{h}}$ the covariant components of the vector potential in helical geometry and with $A_{i\mathrm{c}}$ the covariant components of the vector potential in cylindrical geometry the transformation law can be written as (D’Haeseleer et al. 1991) $A_{i\mathrm{c}}=A_{i\mathrm{h}}\frac{\partial u^{i\mathrm{h}}}{\partial u^{i\mathrm{c}}}$. The helical geometry is defined by the three coordinates $u^{i\mathrm{h}}=(\rho ,u,l)$, while the cylindrical one is defined by $u^{i\mathrm{c}}=(r,\theta ,z)= m\theta +\frac{n}{R_0} z$. The quantity $\rho$ represents a radial coordinate, u represents a helical angle and the third coordinate l is ignorable in the case of helical symmetry under analysis. The coordinates of the helical geometry can be written in terms of the cylindrical coordinates as $\rho =r$, $u = m\theta +\frac{n}{R_0} z$ and $l = z$. Other choices are possible, in particular for the ignorable coordinate.

This identification allows transforming the helical flux function in helical geometry to cylindrical geometry.

Simple calculations show that $\chi =A_{3\mathrm{h}}=A_{3\mathrm{c}}-\frac{n}{mR_0}A_{2\mathrm{c}}$, that, in terms of the physical cylindrical components of the vector potential $(\tilde{A}_\mathrm{r},\tilde{A}_{\theta },\tilde{A}_z)$ the helical flux function (multiplied by m) is given by:

$$\begin{aligned} \chi =m\tilde{A}_z-\frac{n}{R_0}r\tilde{A}_{\theta }. \end{aligned}$$

(10)