Abstract
From spacecraft data, it is evident that electron pressure anisotropy develops in collisionless plasmas. This is in contrast to the results of theoretical investigations, which suggest this anisotropy should be limited. Common for such theoretical studies is that the effects of electron trapping are not included; simply speaking, electron trapping is a nonlinear effect and is, therefore, eliminated when utilizing the standard methods for linearizing the underlying kinetic equations. Here, we review our recent work on the anisotropy that develops when retaining the effects of electron trapping. A general analytic model is derived for the electron guiding center distribution of an expanding flux tube. The model is consistent with anisotropic distributions observed by spacecraft, and is applied as a fluid closure yielding anisotropic equations of state for the parallel and perpendicular components (relative to the local magnetic field direction) of the electron pressure. In the context of reconnection, the new closure accounts for the strong pressure anisotropy that develops in the reconnection regions. It is shown that for generic reconnection in a collisionless plasma nearly all thermal electrons are trapped, and dominate the properties of the electron fluid. A new numerical code is developed implementing the anisotropic closure within the standard twofluid framework. The code accurately reproduces the detailed structure of the reconnection region observed in fully kinetic simulations. These results emphasize the important role of pressure anisotropy for the reconnection process. In particular, for reconnection geometries characterized by small values of the normalized upstream electron pressure, , the pressure anisotropy becomes large with and strong parallel electric fields develop in conjunction with this anisotropy. The parallel electric fields can be sustained over large spatial scales and, therefore, become important for electron acceleration.
We gratefully acknowledge scientific contributions, help and support by several colleagues including Dr. M. Porkolab, Dr. A. Fasoli, Dr. W. Fox, Dr. N. Katz, Dr. O. Ohia, Dr. A. Vrublevskis, Dr. J. Ng, Dr. V. S. Lukin, Dr. H. Karimabadi, Dr. J. F. Drake, Dr. P. L. Pritchett, Dr. G. Lapenta, Dr. M. Øieroset, Dr. R. P. Lin, Dr. T. D. Phan, Dr. L.J. Chen, Dr. H. Ji, Dr. P. Cassak, Dr. F. I. Parra, Dr. A. Ram, Dr. D. Ernst, and Dr. J. Wright. The work at MIT was funded in part by DOE Grant Nos. DEFG0206ER54878 and ER55099, NASA grant NNX10AL11G, an NSF CAREER grant 0844620, and by the NASA Heliophysics Theory Program at LANL. Simulations were carried out using LANL institutional computing resources, the Pleiades computer at NASA, and Hopper at NERSC.
I. INTRODUCTION
II. ELECTRON PRESSURE ANISOTROPY IN LABORATORY AND SPACE PLASMAS
A. Reconnection experiments on Versatile Toroidal Facility (VTF)
B. Pressure anisotropy observed by spacecraft
III. GENERAL MODEL FOR THE ELECTRON DISTRIBUTION FUNCTION
A. Formulation of the problem
B. Solution for the passing electrons
C. Properties of the acceleration potential,
D. Solution for trapped electrons
E. Combined solution
F. Validation of the model
G. Local trapping
IV. THE EQUATIONS OF STATE
A. Closing the hierarchy of fluid equations
B. Limit of low density and high magnetic field
C. Limit of high density and low magnetic field
D. General case
V. FLUID MODELING WITH THE NEW CLOSURE
VI. APPLICABILITY OF THE EQUATIONS OF STATE
VII. ANTIPARALLEL RECONNECTION
A. Effect of pitch angle diffusion
B. Velocity space distributions near the X line
C. Scaling laws from momentum balance
VIII. APPLICATIONS OF THE EQUATIONS OF STATE
A. Asymmetric reconnection
B. Magnetic island coalescence
C. Electron energization in low β plasmas
IX. SUMMARY AND CONCLUSION
Key Topics
 Magnetic fields
 46.0
 Magnetic anisotropy
 36.0
 Magnetic reconnection
 34.0
 Equations of state
 27.0
 Electric fields
 23.0
Figures
(a) Illustration of the VTF open cusp experimental configuration. (b)Guiding center trajectory of a particle in the linear magnetic cusp. (c)Experimentally measured inplane electrostatic potential during magnetic reconnection in VTF. (d) Theoretically calculated pressure anisotropy during slow reconnection in a linear magnetic cusp.
(a) Illustration of the VTF open cusp experimental configuration. (b)Guiding center trajectory of a particle in the linear magnetic cusp. (c)Experimentally measured inplane electrostatic potential during magnetic reconnection in VTF. (d) Theoretically calculated pressure anisotropy during slow reconnection in a linear magnetic cusp.
(a) Gyrophase averaged electron distribution, , measured by the Wind spacecraft on 1 April 1999 during an encounter with a reconnection region in the deep magnetotail. (b) Distribution from a numerical model that reproduces the Wind observations. The match can only be obtained with the assumption that a parallel acceleration potential of 1 kV is present at the location of the measurement. (c) Example of the acceleration potential computed from the electric and magnetic configuration of kinetic simulation. The overlaid trapped electron trajectory is typical for the reconnection region.
(a) Gyrophase averaged electron distribution, , measured by the Wind spacecraft on 1 April 1999 during an encounter with a reconnection region in the deep magnetotail. (b) Distribution from a numerical model that reproduces the Wind observations. The match can only be obtained with the assumption that a parallel acceleration potential of 1 kV is present at the location of the measurement. (c) Example of the acceleration potential computed from the electric and magnetic configuration of kinetic simulation. The overlaid trapped electron trajectory is typical for the reconnection region.
Illustration of a flux tube expanding in time while being fed by “half Maxwellian” distributions from both ends of the tube. The distribution shown below the tube is representative for the electrons inside the tube, where the green area represents the contributions in velocity space from trapped electrons. Trapping can result from a local minimum in B or be caused by electric fields parallel to the flux tube as associated with a parallel acceleration potential, .
Illustration of a flux tube expanding in time while being fed by “half Maxwellian” distributions from both ends of the tube. The distribution shown below the tube is representative for the electrons inside the tube, where the green area represents the contributions in velocity space from trapped electrons. Trapping can result from a local minimum in B or be caused by electric fields parallel to the flux tube as associated with a parallel acceleration potential, .
Contour plots of the distribution in Eq. (16) , for . In (a), is evaluated with , whereas in (b) . The gray areas characterized by straight contours correspond to the trapped regions in velocity space with boundaries expressed in Eq. (17) .
Color contours of the electron distribution measured by Wind, also displayed above in Fig. 2(a) . The overlaid black contours represent in Eq. (16) , evaluated with and .
PIC simulation results: (a) Outofplane current density Jy , (b) magnetic field strength B with points used in Fig. 9 , where (white) and (black), and (c) plasma density n. Dashed lines represent inplane magnetic field lines. Simulation electron distribution functions with theoretical level lines superimposed along the cut right of the X line at the locations indicated in (a) ((d) , (e) , (f) , and (g) ).
PIC simulation results: (a) Outofplane current density Jy , (b) magnetic field strength B with points used in Fig. 9 , where (white) and (black), and (c) plasma density n. Dashed lines represent inplane magnetic field lines. Simulation electron distribution functions with theoretical level lines superimposed along the cut right of the X line at the locations indicated in (a) ((d) , (e) , (f) , and (g) ).
Possible profiles of as a function of the position l along the flux tube. The model in Eq. (15) is valid for the profiles in (a) with no local minimum in and (b) with one local minimum. For cases like that in (c) local trapping can occur, which causes changes to the distribution function not accounted for by Eq. (15) .
Possible profiles of as a function of the position l along the flux tube. The model in Eq. (15) is valid for the profiles in (a) with no local minimum in and (b) with one local minimum. For cases like that in (c) local trapping can occur, which causes changes to the distribution function not accounted for by Eq. (15) .
(a) The acceleration potential, , evaluated as a function of n for . The dashed line represents the Boltzmann scaling . (b) The parallel pressure, , evaluated as a function of n for . The straight dashed line represents the Boltzmann scaling p = nT. (c) The perpendicular pressure, , evaluated as a function of n for . The Boltzmann scaling , coincides with for .
(a) The acceleration potential, , evaluated as a function of n for . The dashed line represents the Boltzmann scaling . (b) The parallel pressure, , evaluated as a function of n for . The straight dashed line represents the Boltzmann scaling p = nT. (c) The perpendicular pressure, , evaluated as a function of n for . The Boltzmann scaling , coincides with for .
Comparison of the analytical equations of state in Eqs. (30) and (31) against PIC data from points marked in Fig. 6(b) .
Comparison of the analytical equations of state in Eqs. (30) and (31) against PIC data from points marked in Fig. 6(b) .
Contours of out of plane current density as observed in (a) a fluid simulation with isotropic pressure, (b) a fluid simulation applying our new anisotropic equations of state, and (c) a fully kinetic simulation.
Contours of out of plane current density as observed in (a) a fluid simulation with isotropic pressure, (b) a fluid simulation applying our new anisotropic equations of state, and (c) a fully kinetic simulation.
(a) Classification of simulation runs for in space of mass ratio and guide field . The red lines in the symbols represent the observed elongated electron current channels. (b) Classification of simulation runs for in space of electron beta , and mass ratio . (c) Outofplane current, Jy , in a fluid simulation with anisotropic electron pressure carried out with and .
(a) Classification of simulation runs for in space of mass ratio and guide field . The red lines in the symbols represent the observed elongated electron current channels. (b) Classification of simulation runs for in space of electron beta , and mass ratio . (c) Outofplane current, Jy , in a fluid simulation with anisotropic electron pressure carried out with and .
(a) Electron anisotropic pressure ratio from a kinetic simulation of antiparallel reconnection (see Ref. 45 for more details). (b) Typical trapped electron orbit, which passes repeatedly through the region of weak magnetic field (blue areas) in the outflow.
(a) Electron anisotropic pressure ratio from a kinetic simulation of antiparallel reconnection (see Ref. 45 for more details). (b) Typical trapped electron orbit, which passes repeatedly through the region of weak magnetic field (blue areas) in the outflow.
Electron distribution within neutral sheet. (a) Isosurface of the distribution at X line. The different colors correspond to the number of times the electrons are reflected in the layer. (b) Color plot is inplane electric field Ez , with contours of inplane projection of magnetic field lines. Electron orbits are shown traced back in time from the X line and characterized by 0 (red), 1 (blue), and 2 (magenta) reflections. The black ×symbols identify where the value of f for each trajectory is obtained using a relativistic version of Eq. (16) as the boundary condition.
Electron distribution within neutral sheet. (a) Isosurface of the distribution at X line. The different colors correspond to the number of times the electrons are reflected in the layer. (b) Color plot is inplane electric field Ez , with contours of inplane projection of magnetic field lines. Electron orbits are shown traced back in time from the X line and characterized by 0 (red), 1 (blue), and 2 (magenta) reflections. The black ×symbols identify where the value of f for each trajectory is obtained using a relativistic version of Eq. (16) as the boundary condition.
(a) B 2 and the difference predicted by the equations of state and directly from the PIC codes as functions of z along a cut through the X line. (b)(d) Predicted dependence on the upstream electron beta of various quantities along with PIC simulation results. (b) Characteristic Hall magnetic field strength BH normalized to upstream reconnecting field. (c) Maximum pressure ratio . (d) Maximum upstream acceleration potential normalized to electron temperature, .
(a) B 2 and the difference predicted by the equations of state and directly from the PIC codes as functions of z along a cut through the X line. (b)(d) Predicted dependence on the upstream electron beta of various quantities along with PIC simulation results. (b) Characteristic Hall magnetic field strength BH normalized to upstream reconnecting field. (c) Maximum pressure ratio . (d) Maximum upstream acceleration potential normalized to electron temperature, .
Contours of and from a kinetic simulations of asymmetric antiparallel reconnection. 60 The lefthand inflow region is characterized by , whereas for the right hand inflow region .
Contours of and from a kinetic simulations of asymmetric antiparallel reconnection. 60 The lefthand inflow region is characterized by , whereas for the right hand inflow region .
The (a) acceleration potential and (b) parallel electron pressure near the X line formed by two merging plasmoids.
The (a) acceleration potential and (b) parallel electron pressure near the X line formed by two merging plasmoids.
(Top panel) Contours of constant in a kinetic simulation of antiparallel reconnection. (Middle panels) Electron distributions for the points marked in the top panel. The magenta lines indicate the trapped passing boundaries. (Bottom panels) Electron distributions recorded by Cluster 1 spacecraft in a separator crossing. The magenta dots indicate the locations of measurements in velocity space. The simulated distributions in the middle panels qualitatively match these experimental distributions.
(Top panel) Contours of constant in a kinetic simulation of antiparallel reconnection. (Middle panels) Electron distributions for the points marked in the top panel. The magenta lines indicate the trapped passing boundaries. (Bottom panels) Electron distributions recorded by Cluster 1 spacecraft in a separator crossing. The magenta dots indicate the locations of measurements in velocity space. The simulated distributions in the middle panels qualitatively match these experimental distributions.
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