### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Fully Kinetic Simulation Results
- 3. Electron Momentum Balance
- 4. Predicted Scalings
- 5. Summary
- Acknowledgments
- References

[1] In situ observation of the Earth's magnetosphere has identified Hall magnetic fields as a key signature of collisionless magnetic reconnection. The inflow portion of the reconnection diffusion region is further characterized by strong electron pressure anisotropy. These two features are tightly linked in a quantitative model, which is verified using fully kinetic simulations. The model predicts the Hall field strength and the maximum electron pressure anisotropy as functions of the upstream ratio of electron fluid and magnetic pressures.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Fully Kinetic Simulation Results
- 3. Electron Momentum Balance
- 4. Predicted Scalings
- 5. Summary
- Acknowledgments
- References

[2] Magnetic reconnection allows an often violent reconfiguration of magnetic field lines within a plasma. It accompanies diverse phenomena including magnetic substorms in the Earth's magnetosphere, solar flares, coronal mass ejections, and sawtooth crashes and disruptions in tokamaks. In plasmas with negligible collisions, found in the magnetosphere and the solar wind, reconnection involves the decoupling of electrons and ions in the diffusion region. *Sonnerup* [1979] predicted that this leads to the formation of a quadrupolar Hall magnetic field centered on the reconnection region. The Hall field structure has now been observed in spacecraft data and numerical simulation, and it has been measured in laboratory experiments [*Øieroset et al.*, 2002; *Borg et al.*, 2005; *Drake et al.*, 2008; *Daughton et al.*, 2006; *Hesse et al.*, 2008; *Ren et al.*, 2005; *Brown et al.*, 2006]. Despite the fact that the Hall fields are now considered to be a key signature of collisionless reconnection in space data, there has heretofore been no quantitative theory for the strength of the Hall fields.

[3] Recent in situ observation of the Earth's magnetosphere reveals two more characteristic features of the diffusion region: *Chen et al.* [2008] found electron pressure anisotropy in the inflow plasma with *p*_{∥} > *p*_{⊥} (where directions are with respect to the magnetic field), and *Phan et al.* [2007] observed evidence for an electron outflow jet near the X-line similar to one observed in their simulations. The outflow electrons stream faster than the **E** × **B** speed, and the associated current produces the Hall field.

[4] In reconnection research, emphasis has typically been placed on the component of the electron momentum balance equation,

in the direction of the reconnection electric field. To study the Hall currents that develop with reconnection, however, it proves more fruitful to first consider what balances the −*ne***u**_{e} × **B** ≈ **J**_{⊥} × **B** force on the electrons. This force is of course perpendicular to both the magnetic field and the electron flow. We find that averaged over the electron jets **J**_{⊥} × **B** ≃ ∇ · **P**. Considering, for example, a pressure anisotropy with *p*_{∥} − *p*_{⊥} ≃ 4*nT*_{e} directly outside the jets of width ∼2*ρ*_{e} (typical of values observed in kinetic simulations), ∣∇ · **P**/(*ne*)∣ ≃ 4*nT*_{e}/(*ne* 2*ρ*_{e}) = *v*_{th,e}*B*. This term is much larger than the reconnection electric field, *E*_{rec} ≃ 0.1*V*_{A}*B*. The electron jets streaming near the thermal speed across the magnetic field are thus controlled by the electron pressure anisotropy. Based on analytical results and data from three PIC simulation runs (on two different codes), the electron pressure anisotropy and Hall fields and currents are found to scale with the upstream plasma conditions.

[6] In anti-parallel reconnection, the initial magnetic geometry contains a neutral sheet where the field vanishes. During the reconnection process, regions of very weak magnetic field persist around neutral points. In principle, this precludes the use of the above equations of state, which assume the particles are magnetized. We find, however, that the electron pressure anisotropy predicted by the equations of state just outside the neutral sheet, where the electrons are still magnetized, imposes useful relationships between the upstream electron pressure and the Hall magnetic fields.

### 2. Fully Kinetic Simulation Results

- Top of page
- Abstract
- 1. Introduction
- 2. Fully Kinetic Simulation Results
- 3. Electron Momentum Balance
- 4. Predicted Scalings
- 5. Summary
- Acknowledgments
- References

[7] We use three particle-in-cell (PIC) simulation runs to explore the application of the equations of state to anti-parallel reconnection. Figure 1 shows results from one of the simulations. The simulation is translationally symmetric in the *z*-direction, has a total domain of 2560 × 2560 cells = 400*d*_{e} × 400*d*_{e}, tracks roughly 2 × 10^{9} particles, and implements open boundary conditions for both the particles and the fields described by *Daughton et al.* [2006]. The initial state is a Harris neutral sheet with gradients in the *y* direction and is characterized by the following parameters: *m*_{i}/*m*_{e} = 400, *T*_{i}/*T*_{e} = 5, *ω*_{pe}/*ω*_{ce} = 2, and background density = 0.3 *n*_{0} (peak Harris density). Magnetic reconnection with a single X-line evolves from a small perturbation, and we consider a time with approximately steady-state reconnection. We also use results from two other similar simulation runs with the same mass ratio of *m*_{i}/*m*_{e} = 400. One was performed on the same code with open boundary conditions, but used a temperature ratio of *T*_{i}/*T*_{e} = 1, and the last was run on the code P3D.

[8] The density *n* is fairly uniform in the vicinity of the X-line, while the value of *B* becomes very low (Figures 1a and 1b). The quadrupolar out-of-plane Hall magnetic field *B*_{z} is shown in Figure 1c. We focus on the inner electron diffusion layer where strong electron currents *j*_{z} and *j*_{x} (Figures 1d and 1e) flow in a narrow channel. The electron jets flow at nearly the electron thermal speed (based on the upstream temperature), and are localized to a layer around 4*d*_{e} wide.

[9] As stated above, the equations of state apply to the inflow region where the electrons are magnetized. They agree with the PIC simulation to within ∼10% up to a layer a few *d*_{e} = *c*/*ω*_{pe} wide, where the full electron pressure tensor *P*_{ij} is used to define *p*_{⊥} = [*P*_{ij}(*δ*_{ij} − *b*_{i}*b*_{j})]. The pressure anisotropy in the inflow region is substantial: in the present simulation (which had the strongest anisotropy of the three simulations), it reaches nearly *p*_{∥}/*p*_{⊥} ∼ 7 (see Figure 3a). Figure 3b shows the PIC pressure anisotropy *p*_{∥}/*p*_{⊥} divided by the fluid model estimates to illustrate their agreement in the inflow region. In the outflow layer (the rectangular outlines in Figures 3a and 3c), where the equations of state are inapplicable, the pressure becomes nearly isotropic because the electrons are effectively pitch-angle scattered by passing through the very weak magnetic field. A typical trapped electron trajectory, as shown in Figure 2, crosses the region of weak magnetic field repeatedly as it moves through the outflow region. As noted by *Chen et al.* [2008], strong electron pressure anisotropy may therefore help identify the inflow region in space data.

### 3. Electron Momentum Balance

- Top of page
- Abstract
- 1. Introduction
- 2. Fully Kinetic Simulation Results
- 3. Electron Momentum Balance
- 4. Predicted Scalings
- 5. Summary
- Acknowledgments
- References

[10] The strong pressure anisotropy predicted by the equations of state contributes to electron momentum balance in the inner electron region. Here, the electrons carry almost all of the current (more than 90% throughout the region) and correspondingly nearly all of the **J** × **B** force exerted by the magnetic field on the plasma. We highlight the pressure anisotropy by writing steady-state electron momentum balance, assuming −*ne***u**_{e} = **J** = ∇ × **B**, in the form

where *F*_{j} contains the electric field, non-gyrotropic pressure, and inertia contributions. In the electron layer, the magnetic field lines are strongly curved and ∇_{i}*b*_{i}*b*_{j} is large. To leading order, its coefficient must be small, or (*p*_{∥} − *p*_{⊥} − *B*^{2}) ≈ 0. Physically, this means the magnetic tension force across the layer associated with the bent field lines, indicated schematically in Figure 3d, is largely balanced by the anisotropic electron pressure.

[11] We consider *x*-momentum balance for differentially narrow fluid elements extending ∼4*d*_{e} across the outflow jet (for example, the small shaded box in Figure 3c). The largest contributions come from *B*^{2}*b*_{x}*b*_{y} and (*p*_{∥} − *p*_{⊥})*b*_{x}*b*_{y} evaluated immediately outside the jet and are plotted in Figure 3e. A smaller contribution is shown from *x* gradients in the stress tensor integrated across the layer. The other electron terms that are not plotted are a similar size or smaller. These terms become important where the electron flows peak and then terminate roughly 40*d*_{e} downstream from the X-line. Finally, the integrated magnetic force on the ions is plotted. It is small, and we neglect it when we assume **J** = −*ne***u**_{e}. Thus, averaged over the layer,

and the volume-averaged perpendicular electron Hall currents are supported by the pressure anisotropy directly outside the jets, where **P** ≃ (*p*_{∥} − *p*_{⊥})**bb** + *p*_{⊥}**I** with *p*_{∥} ≫ *p*_{⊥}.

### 4. Predicted Scalings

- Top of page
- Abstract
- 1. Introduction
- 2. Fully Kinetic Simulation Results
- 3. Electron Momentum Balance
- 4. Predicted Scalings
- 5. Summary
- Acknowledgments
- References

[12] Together with the equations of state *p*_{∥}(*n*, *B*) and *p*_{⊥}(*n*, *B*), the main result from force balance considerations, *p*_{∥} − *p*_{⊥} ≈ *B*^{2} all along the electron layer, determines important Hall physics parameters. Figure 4 shows *p*_{∥} − *p*_{⊥} and *B*^{2} as functions of *y* along a typical cut 15*d*_{e} to the right of the X-line using both the simulation data and our equations of state. By solving *p*_{∥}(*n*, *B*) − *p*_{⊥}(*n*, *B*) = *B*^{2} (where the two dashed lines in Figure 4 intersect), we find the value of the magnetic field strength immediately outside the electron jet, which we denote by *B*_{H}.

[13] Taking the density as approximately uniform with *n* ≈ *n*_{∞}, we obtain *B*_{H} as a function only of the ratio of electron to magnetic pressure at the inflow boundary, *β*_{e∞} = 2*p*_{e∞}/*B*_{∞}^{2}. This relation is shown in Figure 4b. An approximate form good to 10% for *β*_{e∞} < 1 is

The above scaling is confirmed by the three PIC simulations of reconnecting current sheets with varying electron *β*_{e∞}. *B*_{H} is evaluated where the out-of-plane electron current reaches 40% of its maximum [roughly (2–4)*d*_{e} from the peak] and is marked in Figure 4b for three numerical studies. The middle simulation is from the code P3D using fully periodic boundary conditions [*Shay et al.*, 2007], and the other two runs were performed on the open boundary code [*Daughton et al.*, 2006].

[14] The momentum balance constraint *p*_{∥} − *p*_{⊥} = *B*^{2} applies everywhere along the length of the electron outflow jets (as seen in Figure 3d). For a given *β*_{e∞} and a roughly uniform density equal to its value in the ambient plasma, = *n*/*n*_{∞} = 1, the equations of state predict a unique value of *B*_{H} that satisfies the momentum balance condition. This implies another result consistent with the simulations: the magnetic field strength is nearly uniform along the current sheet. Although its magnitude is roughly constant and equal to the fixed value *B* = *B*_{H}, the magnetic field may rotate along the outflow. The component rotated out of the plane is the Hall field *B*_{z}. The value of *B*_{H} determined from our equations of state outside the current layer is therefore an upper bound for ∣*B*_{z}∣ in the inner diffusion region.

[15] Similarly, the equations of state provide an estimate for the maximum electron pressure ratio *p*_{∥}/*p*_{⊥}. As visible in Figure 4a, the equations of state break down slightly before *B* reaches the predicted value of *B*_{H}. We find empirically from the codes, however, that evaluating the equations of state *p*_{∥}(*n*, *B*) and *p*_{⊥}(*n*, *B*) at *n* ∼ *n*_{∞} and *B* = 1.25*B*_{H}, where the equations of state are valid, gives a good estimate for the maximum *p*_{∥}/*p*_{⊥}. The scaling plotted in Figure 4c is approximately

[16] As explained by *Le et al.* [2009], the enhanced parallel pressure predicted by the equations of state results largely from electron trapping and heating by a parallel electric field. The effect of the parallel electric field is parameterized by the acceleration potential defined by *Egedal et al.* [2009] as Φ_{∥}(**x**) = **E** · *d***l**, where the integral is taken along the magnetic field from the point **x** to the ambient plasma where **E** · **B** = 0. In Figure 4d, we plot the value of Φ_{∥} predicted by our model at the point of maximum upstream pressure anisotropy. Note that at low *β*_{e∞}, a large Φ_{∥} develops, scaling roughly as

and the majority of inflow electrons are trapped. Because *β*_{e} is typically low in Earth's magnetotail, electrical trapping in the inflow region is likely a crucial mechanism for creating the upstream electron pressure anisotropy with *p*_{∥} > *p*_{⊥} observed by both the Cluster and Wind spacecraft near reconnecting current sheets [*Chen et al.*, 2008; *Øieroset et al.*, 2002]. Our model, however, should be generalized for *β*_{e∞} < 0.01 to account for effects of a large Φ_{∥} > 10*T*_{e}/*e*. In particular, a modification is made to the underlying distribution function on which the equations of state are based [*Egedal et al.*, 2010].