Fast magnetic reconnection due to anisotropic electron pressure

Magnetic reconnection allows for large-scale conversion of magnetic energy into kinetic energy and heat by changing magnetic topology. It occurs in a wide range of environments, such as solar eruptions, planetary magnetospheres, fusion devices, and astrophysical settings.¹ One key unsolved problem is what determines the rate that reconnection proceeds.^2,3

In simplified two-dimensional (2D) systems often employed in simulations, the reconnection rate E is determined by the aspect ratio of the current sheet, but it is not understood what controls its length. In collisional plasmas, current layers are elongated,^4,5 so collisional reconnection is relatively slow. For less collisional 2D systems, elongated layers break and produce secondary islands,^6–9 giving normalized reconnection rates of 0.01.^9–11 However, this is ten times slower than the fastest rates seen in simulations.^12–17 

The GEM Challenge¹⁸ showed that the Hall term, when active, is sufficient to produce short current layers with $E≃0.1$⁠. The interpretation of this is still under debate.^19–25 One can ask—do other mechanisms limit the length of current layers that could help explain what causes fast reconnection?

In this Letter, we report that fast reconnection can be caused by electron pressure anisotropy using the Chew-Goldberger-Low (CGL) equations of state²⁶ in the generalized Ohm's law. This has not been seen previously because (a) most simulations use no out-of-plane (guide) magnetic field, but this mechanism requires one and (b) previous fluid simulations included pressure anisotropies only in the momentum equation, which does not produce fast reconnection.^27–29 This result is distinguished from known results that off-diagonal elements of the electron pressure tensor balance the reconnection electric field at the reconnection site^30–32 and agyrotropies contribute near the reconnection site.^33,34 Neither effect is present for the CGL equations because the pressure tensor is gyrotropic. As with the Hall effect, gyrotropic pressure does not break the frozen-in condition.³⁵ Nonetheless, it plays a crucial role in allowing fast reconnection in this regime.

Gyrotropic pressures are different parallel p_|| and perpendicular $p⊥$ to the magnetic field.³⁶ The CGL, or double adiabatic, equations of state²⁶ follow rigorously from kinetic theory in the ideal limit (no heat conduction) with strong enough magnetic fields so particles are magnetized. Previous studies treated gyrotropic pressures in tearing instabilities.^37–42 Electron pressure anisotropies have garnered interest lately since they are self-generated by reconnecting magnetic field lines.⁴³ The resulting equations of state⁴⁴ are valid for guide fields no stronger than the reconnecting magnetic field.

Simulations are carried out using the two-fluid code F3D (Ref. 45) modified to include gyrotropic pressures. The code updates the continuity equation, momentum equation, Faraday's law, and pressure equations. The electric field E is given by the generalized Ohm's law

where v is velocity, B is magnetic field, J is current density, n is number density, e is proton charge, $pe$ is the electron pressure tensor, η is resistivity, m_e is electron mass, and each term on the right can be turned off, including the Hall term $J×B/nec$⁠. The momentum equation is

where $ϵ=1−4π(p||−p⊥)/B2, I$ is the identity tensor, ρ is mass density, and p is total (electron plus ion) pressure.

When pressure anisotropies are used, we employ the CGL equations, equivalent to $pσ||B2/ρ3$ and $pσ⊥/ρB$ being constants²⁶ for species σ. We write them as evolution equations [see Eqs. (17) and (18) in Ref. 46] with $E·J$ omitted for simplicity. The numerical implementation is benchmarked using Alfvén waves and the firehose and mirror instabilities. For isotropic plasmas, $pσ/ρ5/3$ = constant.

All quantities are normalized: magnetic fields to the reconnecting magnetic field B₀, densities to the value ρ₀ far from the current sheet, velocities to the Alfvén speed $cA=B0/(4πρ0)1/2$⁠, lengths to the ion inertial length $di=c/ωpi$⁠, electric fields to $cAB0/c$⁠, resistivities to $4πcAdi/c2$⁠, and pressures to $B02/4π$⁠. The simulation size is $Lx×Ly=204.8×102.4$ in a doubly periodic domain with 4096 × 2048 cells. This system is large enough that boundaries do not play a role; a steady state prevails for an extended time.

The initial reconnecting magnetic field is $Bx(y)=tanh[(y+Ly/4)/0.5]−tanh[(y−Ly/4)/0.5]−1$⁠. Unless otherwise stated, the guide field is large at 5 and increases at the current sheet to balance total pressure. The density ρ = 1 and pressure $pσ=5$ are initially uniform (⁠$pσ⊥=pσ||=5$ for anisotropic). When electron pressure is evolved, ions are cold, and vice versa.

All simulations use $me=mi/25$ unless otherwise stated, which is acceptable because E is insensitive to m_e (Ref. 47) and length scales for the ions (⁠$cs/Ωci≃0.7$⁠) and electrons (⁠$de=c/ωpe=0.2$⁠) are sufficiently separated.⁴⁸ The resistivity is 0.005, chosen so that if reconnection is Sweet-Parker-like, the layer thickness is $(ηLx/4)1/2≃0.5$ which makes it marginal against secondary islands.⁷ Reconnection is seeded using a coherent magnetic perturbation of amplitude 0.014. Initial random velocity perturbations of amplitude 0.04 break symmetry. The equations employ fourth order diffusion with coefficient $D4=2.5×10−5$ to damp noise at the grid.

Benchmark simulations using two-fluid simulations (with the Hall term, electron inertia, and isotropic electron pressure) reveal a well-known open exhaust, as shown by the out-of-plane current density J_z in Fig. 1(a). Various simulations are then performed without the Hall term. When the CGL equations are used on the electrons (which we call eCGL), an open exhaust occurs (panel (b)). Panel (c) is for the same system but with m_e = 0, showing that electron inertia does not cause the open exhaust. Panel (d) is when the CGL equations are used on the ions (which we call iCGL). The current sheet is elongated like in Sweet-Parker reconnection. To further identify the key physics, a simulation of an unphysical system is tested: the electron pressure anisotropy is included in the momentum equation [Eq. (2)] but not in generalized Ohm's law [Eq. (1)]. The result is an elongated current sheet (panel (e)). The three with open exhausts are fast, $E≃0.06−0.1$⁠, while the elongated sheets give the Sweet-Parker rate of 0.01. The thickness of the current sheets in (a) and (b) are near 0.2, showing that layers in eCGL go down to d_e as in two-fluid reconnection. In contrast, the layer thickness for (d) and (e) is 0.525 and 0.6 (the Sweet-Parker thickness). The conclusion is clear: the eCGL equations give rise to fast reconnection even with no Hall term, and the key physics is the electron pressure gradient in generalized Ohm's law.

The physics when electron pressure anisotropies dominate bears similarities to Hall reconnection with a guide field.⁴⁹ The z component of Eq. (1) in terms of the flux function ψ defined as $Ez=(1/c)∂ψ/∂t$ is

With the Hall term present, the left side reveals magnetic flux is convected by electrons, so electrons carrying the current drag the reconnecting magnetic field out of the plane.⁵⁰ This produces a quadrupole out-of-plane magnetic field B_z,⁵¹ shown for the two-fluid simulation in Fig. 2(a). With a strong guide field, the gas pressure (not shown) develops a quadrupole with opposite polarity to maintain total pressure balance.⁴⁹ 

Without the Hall term, Eq. (3) implies magnetic flux is convected by ions.⁵² The magnetic field is dragged out of the plane by ions, giving a B_z quadrupole with opposite polarity as in Hall reconnection, displayed for the eCGL simulation in Fig. 2(b). (An instability is visible in the exhaust. The system is not firehose or mirror unstable; it is likely a drift instability.) To balance total pressure, $pe⊥$ develops a quadrupole of opposite polarity, displayed in Fig. 2(c). The density (not shown) develops a quadrupole like that of $pe⊥$⁠. This requires a parallel electric field pointing from low density to high, which comes from a parallel electron pressure with high p_e|| in regions of low $pe⊥$ so p_e|| has a quadrupole of opposite polarity as $pe⊥$⁠, shown in Fig. 2(d). Thus, an electron pressure anisotropy is self-generated. It contributes to the reconnection electric field as $Epe,z=−(1/ne)(∇·pe)z=−(1/ne)(B·∇)[(pe⊥−pe||)Bz/B2]$⁠, plotted as the solid line in a vertical cut through the X-line in Fig. 2(e). For comparison, the dashed line shows the Hall electric field E_Hz in the two-fluid simulation and the dashed-dotted line shows the resistive electric field E_η in the iCGL simulation. The structure of E_pe,z is similar to the known E_Hz profile in two-fluid reconnection. Note that eCGL with m_e = 0 also has quadrupoles, but iCGL with slow reconnection does not.

The guide field is key to the physics. If it is too large, the ion Larmor radius falls below electron or resistive scales which prevent fast reconnection, analogous to Hall reconnection.⁵³ If it is too small, the pressure change due to the B_z quadrupole is small, so the effect in the previous paragraph is negligible. We quantify this by finding when E_pe,z is dominated by other contributions to Ohm's law, which for the present simulations is the resistive term. A scaling analysis gives $Epe,z/Eη∼0.1βe0(Bg/Brec)/(2ηc2/4πcAdi)$⁠, where 0.1 is E for fast reconnection, $βe0=8πpe0/(Brec2+Bg2)$ is the electron plasma beta, and B_rec and B_g are the reconnecting and guide field strengths. This ratio is small if B_g is sufficiently big or small. We confirm this in simulations with $pe⊥=pe||=1$⁠; the predicted range is $0.05≪Bg≪20$⁠. (Formally, the CGL model is invalid for small B_g, so this tests fundamental physics independent of the appropriateness of the CGL model.) We find reconnection is Sweet-Parker-like for B_g = 0 and 0.1, has a short current layer with $E≃0.03$ for a transitional guide field B_g = 0.25, is fast with $E≃0.05$ for B_g = 0.5, 5, and 7.5, and is again Sweet-Parker-like for B_g = 15. These results agree with the prediction.

This system yields an interesting way to study the cause of fast reconnection. It was proposed that reconnection is fast if linear perturbations to a homogeneous equilibrium permit dispersive waves with faster phase speeds at smaller scales,⁵⁴ such as the whistler or kinetic Alfvén wave in Hall-MHD. This has been controversial because reconnecting fields are not homogeneous.

We present the linear theory of a plasma with pressure anisotropies, the Hall term, and electron inertia. Rather than using CGL, we generalize by taking ion and electron pressures to be arbitrary functions of ρ and B, i.e., $pσ⊥=pσ⊥(ρ,B)$ and $pσ||=pσ||(ρ,B)$⁠. This captures adiabatic, CGL, and Egedal's equations of state.^43,44 The dispersion relation relating the frequency ω to the wavevector k is

where

Here, $D=1+k2de2, pB=B2/8π$ is the magnetic pressure, $ϵ̃=1−4π(p||0−p⊥0)/B02$ is the equilibrium anisotropy parameter for the total pressure, $ϵ̃e$ is similarly defined for the electrons, $∂(A1,A2)/∂(x,y)=(∂A1/∂x)(∂A2/∂y)−(∂A1/∂y)(∂A2/∂x)$ is a Poisson bracket-type operator, and the 0 subscript denotes equilibrium quantities. This reduces to known results in the limits of anisotropic-MHD with the CGL equations (⁠$dσ→0,pσ⊥/ρB=$ constant, $pσ||B2/ρ3$ = constant)⁵⁵ and isotropic two-fluid (⁠$pσ||=pσ⊥$⁠).⁵⁴ 

We find pressure anisotropies introduce dispersive waves even when the Hall term is absent. All terms in Eq. (4) with $di2$ give dispersive waves. For the high ω, high k with $ϵ̃=ϵ̃e=1$ limit, $ω2≃C4≃(k||2cA2di2/D2){k2−k⊥2[∂(pe||−pe⊥)/∂pB]0}$⁠. The k² term comes from the Hall term and is the standard whistler wave, while the $k⊥2$ term is a whistler-like wave coming from the electron pressure anisotropy in generalized Ohm's law. For eCGL, this is, $ω2=(3/2)β0k∥2k⊥2cA2di2$⁠, where $β0=p0/pB$⁠. Similarly, following Ref. 54, the intermediate frequency range gives $ω2≃C2/C4$⁠. In the $k⊥≫k||$⁠, short wavelength, cold ion limit, this yields $ω2=(k||2k2cA2di2/D2){(∂pe||/∂ρ)0−[∂(pe⊥,pe||)/∂(ρ,pB)]0}/[cA2/D+(cA2/D)(∂pe⊥/∂pB)0+(∂pe⊥/∂ρ)0]$⁠. In the low β₀ limit, the first term gives the standard kinetic Alfvén wave, while the second is a kinetic Alfvén-type wave arising from the pressure anisotropy. In eCGL, this wave has $ω2=(5/2)β0k∥2k2di2(p0/ρ0)$⁠.

There are many ways to test the dispersive wave model. For cold ions, dispersive waves from anisotropies persist. However, they vanish identically for cold electrons. The dispersive wave model predicts fast reconnection for eCGL but slow reconnection for iCGL, consistent with our simulations. Equation (4) implies there are dispersive waves without the Hall term when there is an equilibrium pressure anisotropy, independent of the equations of state, consistent with previous studies.^40,56

Interestingly, when Egedal's equations of state^43,44 are employed in simulations without the Hall term, reconnection is Sweet-Parker-like (J. Egedal, private communication). Thus, simply having an electron pressure anisotropy is insufficient to cause fast reconnection; the pressure anisotropy must have a particular form. Fluid modeling of other equations of state could provide insight about what physically sets the length of the current layer.

We now show that electron pressure anisotropies can dominate the Hall term in real systems. First, we have performed particle-in-cell simulations with parameters similar to the fluid simulations, confirming that the CGL model (⁠$p||∝ρ3/B2$ and $p⊥∝ρB$⁠) is reasonably reproduced (plots not shown). Also, electron pressure anisotropies dominate the Hall term for the parameters of the simulations in Fig. 1. Fig. 2(f) shows a simulation of eCGL with the Hall term; the contribution to the reconnection electric field of the pressure anisotropy (solid line) dominates the Hall term (dashed line).

We suspect electron pressure anisotropy dominates when dispersive wave terms due to the anisotropy dominate the standard whistler and kinetic Alfvén waves in Eq. (4). In C₂, the first term with $di2$ gives the standard kinetic Alfvén wave. The second term with $di2$ is the most important term arising from the electron pressure anisotropy (by a factor of β, which is small for many systems of interest). In the $ϵ̃e=ϵ̃=1$ limit, a simple calculation reveals that the electron pressure contribution of the kinetic Alfven wave is completely cancelled by part of the electron pressure anisotropy. This implies that it always dominates the Hall term when T_e > T_i with low β. Therefore, when T_e > T_i, the anisotropy is the dominant mechanism for the entire parameter regime previously thought to be the kinetic Alfvén regime of reconnection⁵⁴—low β, high in-plane β based on B_rec, and strong guide field (but not strong enough to make the ion Larmor radius smaller than d_e). Physically, T_i needs to be smaller than T_e because if it is large enough, it can dominate the electron pressure effect discussed here.

There are physical systems where reconnection in this parameter regime could occur. The solar wind and some tokamaks are low β where significant guide fields are expected and T_e > T_i is possible.

We gratefully acknowledge the support by NSF grant AGS-0953463 (PAC) and NASA grants NNX10AN08A (PAC), NNX14AC78G (JFD), NNX14AF42G (JFD), and NNX11AD69G (MAS). This research used computational resources at the National Energy Research Scientific Computing Center and NASA Advanced Supercomputing. We thank M. Kuznetsova, J. Egedal, and C. Salem for helpful conversations.