An analytical model of “Electron-Only” magnetic reconnection rates

To highlight key features critical to the rate determination, we carry out 2D PIC simulations of magnetic reconnection in plasmas of realistic proton-to-electron mass ratio m_i/m_e = 1836. We employ the setup of case A by Pyakurel et al.⁶ that has a guide field B_g = −8B_x0, where B_x0 is the reconnecting component. The ion β_i = 3.54 and electron β_e = 0.35. These are chosen based on the parameters of the MMS electron-only event², but with five different system sizes, L_x × L_z = 1.28d_i × 2.56d_i, 2.56d_i × 2.56d_i, 3.84d_i × 3.84d_i, 5.12d_i × 5.12d_i, and 7.68d_i × 7.68d_i. Details of the simulations setup are in the “Methods” section. The units used in the presentation include the ion cyclotron time \({\Omega }_{{{\rm{ci}}}}^{-1}\equiv {({{\rm{e}}}{B}_{x0}/{m}_{{{\rm{i}}}}c)}^{-1}\), the in-plane ion Alfvén speed \({V}_{{{\rm{Ai}}}0}={B}_{x0}/{(4\pi {n}_{0}{m}_{{{\rm{i}}}})}^{1/2}\) based on the upstream density n₀, and the ion inertial length \({d}_{{{\rm{i}}}}\equiv c/{(4\pi {n}_{0}{{{\rm{e}}}}^{2}/{m}_{{{\rm{i}}}})}^{1/2}\).

Character of “electron-only” reconnection

PIC simulations capture electron-only reconnection when the domain size is small enough. Figure 1 shows the essential features in the L_x = 2.56d_i case. The electron outflow speed V_ex (Fig. 1a) indicates active transport of reconnected magnetic flux. Unlike in ion-coupled standard reconnection, it is evident that ion outflows V_ix do not develop in Fig. 1b. Interestingly, electron-only reconnection has a higher reconnection rate than the standard reconnection rate of \({{\mathcal{O}}}(0.1)\)^31,32,33,34, as shown in Fig. 1e. This is somewhat expected because magnetic flux transport is now not limited by the ion Alfvén speed, as in the ion-coupled reconnection, but by the faster electron Alfvén speed since ions are not magnetized/coupled within the small domain. Naively, if the estimate of the typical EDR aspect ratio  ~0.1 times the ratio of the electron Alfvén speed \({V}_{{{\rm{Ae}}}0}={B}_{x0}/{(4\pi {n}_{0}{m}_{{{\rm{e}}}})}^{1/2}\) and the ion Alfvén speed V_Ai0 is used, we get the normalized reconnection rate

where E_R is the reconnection electric field. Note that, throughout this paper, the subscript “0” is reserved for upstream asymptotic values. This R value, however, is too high compared to the simulation results, as shown in Fig. 1e. The rate only gets closer to unity \({{\mathcal{O}}}(1)\), and a scaling law has not been developed yet.

To address this issue, one key observation is that the limiting speed is actually much lower than the asymptotic electron Alfvén speed V_Ae0. Figure 1c shows cuts of the x-direction electron flow velocity V_ex in blue, ion flow velocity V_ix in red, and the E × B drift velocity in black along the midplane (z = 0). Electrons reach a peak outflow speed V_ex,peak ≃ 0.15V_Ae0 when they exit the EDR (the red box in Fig. 1b). This V_ex,peak value (also shown as the purple horizontal line in Fig. 1d) is, instead, close to the electron Alfvén speed based on the local B_x at the EDR-scale in the nonlinear stage; this can be seen by comparing it with the blue line in Fig. 1d near the edge of the red shaded vertical band of d_e-scale. We will denote this relation by \({V}_{{{\rm{e}}}x,{{\rm{peak}}}} \sim {V}_{{{\rm{Ae}}}}\equiv {B}_{x{{\rm{e}}}}/{(4\pi n{m}_{{{\rm{e}}}})}^{1/2}\).

Farther downstream in Fig. 1c, V_ex plateaus to a super ion Alfvénic value of 1.7V_Ai0 that is only 4% of the asymptotic electron Alfvén speed V_Ae0. This critical speed limits the flux transport. The time evolution of the electron outflow velocity V_ex cuts (Fig. 2b), demonstrates the development of the plateauing of V_ex after the reconnection rate also reaches its plateau (Fig. 1e). Similar V_ex plateaus (of different values) also develop in other four simulations of different system sizes, as shown in rest panels of Fig. 2. Note that the plateau in the smallest system (L_x = 1.28d_i) in Fig. 2a is less clear due to the back-pressure that will be discussed later. Overall, it is expected that a lower flux transport speed leads to a reconnection rate lower than the estimation in Eq. (1). We will denote this limiting speed as \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\), which is, the electron outflow speed at a distance L₀ downstream of the X-line. Farther downstream of this location, the exhaust opening angle quickly decreases to 0, as marked in Fig. 1b.

The limiting speed of the flux transport

The first goal is to derive this limiting speed \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\). We start from the electron momentum equation in the steady state

The term on the left-hand side (LHS) is the electron flow inertia. The terms on the right-hand side (RHS) are the magnetic tension force, magnetic pressure gradient force, electric force, and the divergence of the electron pressure, respectively. Note that the ion flow velocity ∣V_i∣ ≪ electron velocity ∣V_e∣ condition (i.e., ions do not carry the electric current J) and Ampère’s law were used to turn the Lorentz force—eV_e × B/c ≃ J × B/(nc) into the two magnetic forces in Eq. (2). Balancing the electron flow inertia with the magnetic tension B ⋅ ∇ B/4π will lead to an electron jet moving at the electron Alfvén speed. However, the jet can be slowed down by other terms on the RHS, especially the in-plane electric field E. One important source is the Hall electric field E_Hall = J × B/enc that arises from the separation of the lighter electron flows from the much heavier ion flows. E_Hall acts to slow down electrons and speed up ions to self-regulate itself³⁵; thus, we expect E_x pointing in the same direction as the outflows that slow down the electron jet^36,37.

To quantify this phenomenon, we take the “finite-difference approximation” of Eq. (2) at point “1” in Fig. 3a. In the x-direction, the momentum equation reads

where the targeted quantity V_ex3 is V_ex at point “3”, etc. Being similar to the analysis from Fig. 1c of Liu et al.³⁸, this equation, moreover, includes the in-plane electric field critical to the acceleration of electron outflows within the Hall region. This approach allows one to derive the algebraic relation between key quantities while considering the magnetic geometry of the system^35,38,39,40. Here, we ignored the electron pressure gradient and the \({B}_{y}^{2}\) gradient along path 2–3. These are justified since ΔP_exx and \(\Delta ({B}_{y}^{2})/8\pi\) are relatively small^37,41 compared to \({B}_{x0}^{2}/8\pi\) (∝ tension) in Fig. 3b.

To estimate E_x1, we analyze the steady-state Faraday’s law \(\oint {{\bf{E}}}\cdot d\overrightarrow{\ell }=0\) and the original momentum equation along the closed loop (2-3-4-5-2) in Fig. 3a. Unlike path 2–3, the flow inertia ∣nm_eV_e ⋅ ∇ V_e∣ along the integral path 3-4-5-2 is negligible compared to ∣B ⋅ ∇ B/4π − ∇ B²/8π∣ = ∣J × B/c∣ ≃ ∣enV_e × B/c∣, so we can write

Term vanishes since V_ey = 0 at the upstream; term vanishes because V_ex = 0 along the inflow symmetry line. Terms and roughly cancel each other because \(\int{V}_{{\rm{e}}y}{B}_{x}dz\propto \int{J}_{y}{B}_{x}dz\propto \int({\partial }_{z}{B}_{x}){B}_{x}dz=\Delta ({B}_{x}^{2})/2\), which is \({B}_{x0}^{2}/2\) for the 3–4 and \(-{B}_{x0}^{2}/2\) for the 5–2 integral paths. This equation can then be approximated as

The LHS used the fact that E_x increases monotonically from 0 at the X-line to point “3.” The first integral on the RHS holds because the outflow V_ex is narrowly confined within the separatrices. In the next step, we further approximate \(\int_{3}^{6}{B}_{y}dz\simeq [({B}_{y6}+{B}_{y3})/2]{\delta }_{0}\). And, the last integral \(\int_{4}^{5}{V}_{{{\rm{e}}}z}dx\simeq \int_{3}^{4}{V}_{{{\rm{e}}}x}dz\simeq {V}_{{{\rm{e}}}x3}{\delta }_{0}\), since the particle fluxes going through sides 2–3 and 2–5 are negligible due to the symmetry shown in Fig. 3a and incompressibility is used. With the upstream B_y0 ≃ B_y3 as in Fig. 3a, we can then combine the two terms on the RHS to derive

Here the last equality used Ampère’s law (B_y6 − B_y3)/δ₀ ≃ (4π/c)neV_ex3. We note that the electric field E_x1 is basically determined by the convection of the Hall magnetic quadrupole field (i.e., B_y6 − B_y3) and \({\int}_{23452}{{\bf{E}}}\cdot d\overrightarrow{\ell }=0\), as illustrated in Fig. 4a.

While this model mimics the characteristics of the electron current system of an idealized exhaust, it does not consider the effect of the closed boundary, which can be significant in a small system. In particular, the high ion pressure originating from the initial current sheet will accumulate into the plasmoid at a fixed location. With nearly immobile ions, where nm_iV_i ⋅ ∇ V_i is negligible compared to other forces in the ion momentum equation, enE ≃ ∇ P_i^37,41, as illustrated in Fig. 4b. In the small system size limit, one would expect that enE_x1 ≃ (P_i3 − P_i2)/L₀ can be easily of the order of \({B}_{x0}^{2}/(8\pi {L}_{0})\) due to the build-up of pressure within the plasmoid and the depletion of the pressure component xx at the X-line³⁵, as shown by the central dip in the ΔP_ixx (green) curve of Fig. 3b.

Hence, we will impose a reasonable condition where the sum of the plasma and magnetic pressures completely cancels the magnetic tension in the L_x → 0 limit. This can be done by including this ion back pressure into the full E_x1 using a function f(L_x),

We choose f(L_x) = sech(L_x/Δ_f) so that, for L_x ≫ Δ_f then f → 0, corresponding to Fig. 4a. For L_x ≪ Δ_f then f → 1, where the outflow is shut off and the ion pressure gradient dominates, as in Fig. 4b. The length scale Δ_f will later be determined to be Δ_f = 1.28d_i, and the f-profile is shown in Fig. 5b. The ion–electron interaction is primarily mediated by the electric field within the Hall region. Hence, it seems appropriate to heuristically include the effect of ion back pressure into the electric field estimation.

Plugging Eq. (7) back to Eq. (3), and realizing B_z1 ≃ B_z7 ≃ (δ₀/L₀)B_x7, the separatrix slope S_lope ≃ δ₀/L₀, B_z3 ≃ 2B_z1, B_x7 ≃ B_x6/2, and B_x6 ≃ B_x0 from the magnetic field line geometry (see the flux function contour in Fig. 1a), we obtain the limiting speed

A critical feature in Eq. (8) is \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\propto {\delta }_{0}^{-1}\), which provides a faster jet in a narrower exhaust. Without the corrections gathered within the square root, if δ₀ → d_e then \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\to {V}_{{{\rm{Ae}}}0}\) (i.e., also true for δ₀ ≪ d_e when the electron inertial effect \({({d}_{{{\rm{e}}}}/{\delta }_{0})}^{2}\) within the square root is retained). This is responsible for the faster flux transport speed at sub-d_i-scales, but it transitions to the ion Alfvén speed when δ₀ → d_i, because \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\to {V}_{{{\rm{Ai}}}0}\), as in ion-coupled standard reconnection. In the limit δ₀ ≫ d_i, one needs to consider the full two-fluid equations [e.g. ref. ⁴²], coupling ions back to the scale larger than the typical ion diffusion region (IDR) size. The resulting \({V}_{{{\rm{out}}},{{\rm{e}}}}{| }_{{L}_{0}}\) remains ion Alfvénic [e.g. ref. ⁴³].

This scale-dependent velocity is the dispersive property discussed in the idea of Whistler/Kinetic Alfvén wave (KAW)-mediated reconnection^{28,29,30,42,44}, but here we also include the reduction by the back pressure (parameterized by f) within a small system. The flow is stopped when f → 1 in Eq. (8), corresponding to the limit L_x ≪ Δ_f where the total pressure gradient completely cancels the tension force in Eq. (3). Finally, the outflow speed is also reduced with a larger opening angle (S_lope↑).

Geometry and reconnection rates

This limiting speed not only determines how fast magnetic flux is convected into the outflow exhaust but also the upstream magnetic geometry and, thus, the strength of the reconnecting magnetic field immediately upstream of the EDR. All together, one can derive the electron-only reconnection rate.

We closely follow the approach in ref. ³⁵ to estimate the magnetic field strength B_xe immediately upstream of the EDR of size 2L_e × 2δ_e, as marked by the red box in Fig. 1b and δ_e ~ d_e. One can write

where L₀ and δ₀ are the exhaust length and half-width. Other relevant quantities are annotated in Fig. 1b. For instance, V_in,e is the electron inflow speed at z = δ_e while \({V}_{{{\rm{in}}},{{\rm{e}}}}{| }_{{\delta }_{0}}\) is the value at z = δ₀. The first equality of Eq. (9) used the frozen-in condition upstream of the EDR. The second equality holds because of the incompressibility and V_ex,peak ≃ V_Ae. The third equality approximates the separatrix as a straight line to simplify the geometry. The fourth and fifth equalities used similar arguments to the quantities at the edge of the larger L₀ × δ₀ box. Finally, in the 2D steady-state, E_y is uniform. Thus, the equality between the first and the last terms gives,

An important difference from Liu et al.³⁵ is that B_xi in their Eq. (5) is now replaced by B_x0, since the entire system is within the IDR.

Liu et al.³⁵ further estimated the depletion of the pressure component along the inflow direction, caused by the vanishing energy conversion J ⋅ E_Hall = J ⋅ (J × B/nec) = 0; note that E_Hall dominates within the IDR and this pressure depletion provides the localization mechanism necessary for fast reconnection. One can then use force balance along the inflow direction to relate B_xe to the separatrix slope S_lope³⁵. In the case where the guide field at the X-line does not change much from its upstream value, like B_y2 in Fig. 3, we get

The only difference is again that B_xi in Eq. (9) of Liu et al.³⁵ is now replaced by B_x0. In order to get the full solution from Eqs. (8), (10), and (11), one still needs to relate δ₀ to S_lope. We approximate

as it is reasonable to expect 2L₀ to be on the order of the system size L_x, as in Fig. 1b. We can then equate Eqs. (8) and (10) and solve for S_lope numerically.

Once S_lope is determined, we can estimate the normalized reconnection rate,

The last equality used B_z3/B_x0 ≃ B_z6/B_x6 ≃ S_lope. In Fig. 5a, the prediction of R as a function of L_x without including the back pressure effect (i.e., f = 0) is shown as the green dashed curve, while the prediction with nonzero f(L_x) (given in Fig. 5b) is shown as the black solid curve. In a similar format, the limiting speed (Eq. (8)) is shown in Fig. 5c, while the more pronounced peak electron jet speed \({V}_{{{\rm{e}}}x,{{\rm{peak}}}}\simeq {V}_{{{\rm{Ae}}}}=({B}_{x{{\rm{e}}}}/{B}_{x0}){({m}_{{{\rm{i}}}}/{m}_{{{\rm{e}}}})}^{1/2}{V}_{{{\rm{Ai}}}0}\) is shown in Fig. 5d. The estimated exhaust width (Eq. (12)) is shown in Fig. 5e. Simulation results are plotted as orange symbols, whose values can be read off from Figs. 1e and 2.

Overall, the green dashed curves already work reasonably well for 2.56d_i≤ L_x≤ 10d_i cases, but they overestimate quantities in the L_x = 1.28d_i case. For this reason, we set the length scale Δ_f = 1.28d_i in f(L_x) to parametrize the back pressure effect that suppresses the outflow and rate. This corrects the predictions, and the resulting black solid curves capture the scaling of these key quantities in Fig. 5a, c, d, e; the quantitative agreements are within a factor of 2. Importantly, the rate (R) is now bounded by a value \(\simeq {{\mathcal{O}}}(1)\), addressing the key question that motivates this work.