Suppression of Hall-Term Effects by Gyroviscous Cancellation in Steady Collisionless Magnetic Reconnection

0031-9007= The formation of an ion-dissipation region, in which motions of electrons and ions decouple and fast magnetic reconnection occurs, is demonstrated during a steady state of two-dimensional collisionless driven reconnection by means of full-particle simulations. The Hall-term effect is suppressed due to the gyroviscous cancellation at scales between the ion-skin depth and ion-meandering-orbit scale, and thus ions are tied to the magnetic field. The ion frozen-in constraint is strongly broken by nongyrotropic pressure tensor effects due to ion-meandering motion, and thus the ion-dissipation region is formed at scales below the ion-meandering-orbit scale. A similar process is observed in the formation of an electrondissipation region. These two dissipation regions are clearly observed in an out-of-plane current density profile.

Collisionless magnetic reconnection is a fundamental mechanism for the rapid release of magnetic energy in solar corona, high temperature tokamaks, magnetospheric substorms, and laboratory plasmas [1][2][3][4][5][6][7].Recent computer simulation studies have revealed that the rate of reconnection by collisionless processes is very large compared to those obtained by using resistive magnetohydrodynamics (MHD) [8].Rapid reconnection is demonstrated to occur when the motions of electrons and ions decouple in a narrow region around the reconnection point, in two-fluid simulations, in hybrid simulations, and in full-particle simulations [8][9][10][11][12][13][14][15][16].This decoupling is explained in terms of two-fluid effects and particle orbit effects, which are not included in the MHD model, and which form an iondissipation region.In this region the ion frozen-in condition is broken and the ions are unmagnetized, while the electrons are tied to the field [8][9][10][11][12][13][14][15][16].This difference between the ion flow and electron flow causes fast magnetic reconnection because it leads to the generation of fast electron flows in the ion-dissipation region.We call the region, in which the plasma's frozen-in condition is violated, the dissipation region, although the violation is caused by collisionless effects.The violation dominantly controls the formation of the dissipation region and of the current density profile.
When the electrons are tied to the magnetic field in the ion-dissipation region, we expect that the ion frozen-in constraint is broken at scales below the ion-skin depth, because where J n=d i v i ÿ v e , v i , v e , n, and d i c=! pi are the electric current, ion flow velocity, electron flow velocity, plasma density, and ion-skin depth, respectively.On the other hand, full-particle simulations show that the ion frozen-in condition is violated only within the ion-meandering-orbit scale in steady collisionless reconnection [17][18][19].Note that the meandering motion is a bouncing motion in a field reversal region.
In this Letter, we work out this discrepancy between the two spatial scales controlling the ion-dissipation region: the ion-skin depth d i suggested by the electron frozen-in condition and the ion-meandering-orbit scale l mi suggested by full-particle simulations.We show that the ion pressure tensor plays two crucial roles in the formation of the iondissipation region.One role is by the gyroviscous pressure [20,21], which suppresses the violation of the frozen-in condition due to the inertia effect, so that the ions are almost tied to the field within d i .The other role is played by the nongyrotropic pressure, which causes strong violation of the frozen-in condition within l mi .
We analyze a steady state of collisionless driven reconnection without any guide field in an open system with the mass ratio m i =m e 800 by using the two-dimensional electromagnetic particle-in-cell simulation code developed in our previous work [9,10,17,18].The simulation domain is a square open region in the xy plane.At the upstream boundary, the ions and electrons are tied to the magnetic field, and thus the plasma inflow is driven by E B drift due to an external electric field applied in the z direction [17].The applied field takes a uniform profile with a constant value E 0 when the system relaxes to a steady state [17,18].At the downstream boundary, the plasma can freely flow in or out [18].The initial condition is given by a one-dimensional Harris-type equilibrium as B x y B 0 tanhy=y h and Py B 2 0 =8 sech 2 y=y h .We set the time step to be !ce0 t 0:02 and use a 512 512 point grid and 25:6 10 6 particles.
Figure 1 shows the perspective view of the out-of-plane current density profile in the reconnection plane after the system relaxes to a steady state.The profile has a sharp peak at the center where reconnection occurs, and this point is called the X point.The current density profile along the inflow direction consists of two parts.One is a sharp peak and the other is a low shoulder.Figure 2  This scale corresponds to the location where the local ion Larmor radius i y v ti y=! ci y is equal to the scale length of the magnetic field i y L B y, where L B y B x =@B x =@y, and thus the ion dynamics is fully kinetic within l mi .Note that the local ion cyclotron frequency !ci y eBy=m i c p and the local ion thermal velocity v ti y T i y=m i p are evaluated by using the local magnetic field By and the local ion temperature T i y calculated from the particle velocities and particle positions.The sharp peak is formed through the electron dynamics.The current layer has no structure characterized by the ion-skin depth d i , which is evaluated by using the averaged density in the current layer.Note that two-fluid effects should be included within this scale, while a MHD description is valid beyond this scale.We call the region between d i and l mi the two-fluid region.
We focus on the formation of the dissipation region that enables fast magnetic reconnection.Figure 3 shows the spatial profiles of various terms measuring the violation of the frozen-in condition along the vertical line passing through the X point in the steady state.The out-of-plane electric field E z that is induced at the upstream boundary by the applied electric field E 0 ÿ0:04B 0 c spreads over the system and becomes uniform.The value of the electric field at the X point, y 0, is the reconnection electric field.Figure 3(a) shows that ÿv i B is almost the same as the electric field at the outside of the region indicated by ion scales, and thus the ions are tied to the field.The ion frozen-in condition is strongly broken within the ion-meandering-orbit scale l mi .On the other hand, the ions are almost tied to the field in the two-fluid region (l mi < y < d i ), and correspondingly the Hall term J B is small between d i and l mi and becomes large within l mi .Figure 3(b) shows the relation E z ÿv e Bj z holds ex-cept near the Xpoint.This implies that the electrons are still tied to the field in the ion-dissipation region where the ion frozen-in condition is broken.This electron frozen-in condition implies the violation of the ion frozen-in condition within d i as explained by making use of Eq. ( 1).We note that this electron frozen-in condition is consistent with the results of the Swarthmore spheromak experiment [7], which showed the electron inertia term is small in the region where the Hall term is large.
In order to investigate the violation mechanism of the frozen-in constraint in relation to the formation of the dissipation region, let us examine the nonideal terms in the momentum equations at steady state based on the particle simulation data, where P i and P e are the ion and electron pressure tensors, respectively.The normalizations are 3(c) and 3(d) show the spatial profiles of the outof-plane component of the terms in Eqs. ( 2) and (3).It is worth noting that the ion pressure tensor term r P i cancels out the ion inertia term v i rv i in the two-fluid region (l mi < y < d i ) where the ion inertia term is large and works toward breaking the frozen-in condition as shown in Fig. 3(c).Therefore, this cancellation leads to the suppression of the Hall-term effects and to the maintenance of the ion frozen-in constraint within d i .The ion frozen-in condition is strongly broken, and the v i B term is small in the full kinetic region within l mi where the pressure tensor term is dominant.This large pressure tensor term is originated from the nongyrotropic ionmeandering motion.The strong violation leads to the formation of the low shoulder of current density profile in shows that the electron pressure tensor term r P e and the electron inertia term v e rv e break the electron frozen-in constraint near the X point.The electron inertia term vanishes at the X point, while the electron pressure tensor term has a sharp peak, and it balances the electric field there.Thus, the reconnection electric field is generated by the nongyrotropic pressure tensor originating from the electron-meandering motion at the X point as suggested by Refs.[15,19].The electron pressure tensor seems to cancel out the electron inertia effect in the region between d e and l me .Unfortunately, the cancellation is not clear because the electron-skin depth d e is close to the electron-meandering-orbit scale l me in our simulation.A similar counterbalance is also demonstrated in the simulation of electron-positron plasmas [22].We summarize the violation of the frozen-in condition and the counterbalance in Table I.
Let us consider how the ion pressure tensor compensates the frozen-in condition for the violation due to the ion inertia effect within the ion-skin depth.This effect is generated by the gyroviscous cancellation [20,21], which is a common property of magnetized plasmas and is caused by the finite-Larmor-radius effect, n dv di dt r gv ?ÿr ?g ; (4) where d=dt @=@t v i r, v i v ki v EB v di , r gv ?r P ?ÿ r ?p ? , and subscripts jj and ?denote parallel and perpendicular components.In this equation v EB E B=B 2 , v di d i nB 2 B rp i , gv , p ? , and g are the E B flow velocity, ion diamagnetic flow velocity, gyroviscous tensor, perpendicular pressure, and a correction of scalar pressure [21].The out-of-plane component (the z component) of Eq. ( 4) at steady state is v i rv diz 1 n r Pj z 0, where Aj z stands for a z component of A and the relations r ?g j z r ?p ? j z 0 have been used because @=@z 0 in the two-dimensional system.Thus, Eq. ( 4) implies that the z component of the ion inertia term balances the pressure tensor term v i rv iz v i rv diz ÿ 1 n r Pj z between d i and l mi because the ion flow velocity v iz is dominated by the diamagnetic flow velocity v diz for l mi < y < d i as shown in Fig. 4. Therefore, the violation due to the ion inertia term is being counteracted by the gyrotropic pressure tensor term, and the Hall term is small,

TABLE I. Violation of frozen-in condition.
Violation of ion frozen-in condition In summary, we have worked out the discrepancy between the two spatial scales that can control an iondissipation region, where fast reconnection occurs, in steady collisionless reconnection.One is the ion-skin depth d i predicted from the fact that electrons are tied to the magnetic field in the ion-dissipation region because of Eq. ( 1).The other is the ion-meandering-orbit scale l mi , which is revealed from full-particle simulations.We found that gyroviscous cancellation occurs at scales between d i and l mi , and correspondingly the ion pressure tensor compensates the ion frozen-in condition for the violation due to the ion inertia effect.Therefore, the Hall term is small at scales between d i and l mi .The nongyrotropic pressure tensor effect due to the ion-meandering motion strongly violates the ion frozen-in condition within l mi , and it dominantly controls the width of the ion-dissipation region.Hence, there exists the current layer controlled by the ion-meandering-orbit scale and no spatial structure characterized by the ion-skin depth in the current density profile during the steady state of collisionless reconnection.This result is consistent with the observed current layer width of 0:4d i i in the magnetic reconnection experiment [5].Note that the difference between d i and l mi is not an order but a factor in our simulation.The difference should be large in low-plasmas.We also note that the ion-skin depth controls the current layer width when the reconnection is unsteady as shown in Refs.[12,13,19].
A similar counterbalance is observed in the violation of the electron-frozen-in condition.Our numerical results suggest that the electron pressure tensor term counteracts the violation due to the electron inertia within the electronskin depth, and thus the width of the electron-dissipation region is mainly controlled by the electron-meanderingorbit scale.Unfortunately, the counterbalance is not clear compared to that of the ions because the electron-skin depth is close to the electron-meandering-orbit scale in our simulations.The electron-dissipation region corresponds to the sharp peak in the current density profile, and thus we have the two-scale structure of the current layer as shown in Fig. 2.
We found that the cancellation occurs when the plasma is about one and expect that the cancellation is universal when plasma is less than one because the gyroviscous cancellation is a common property of magnetized plasmas.

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illus-PRL 95, 045003 (2005) P H Y S I C A L R E V I E W L E T T E 2005 The American Physical Society trates this two-scale structure in detail, where d0 is the Debye length.The low shoulder is controlled by the ionmeandering-orbit scale l mi , which satisfies l mi i l mi .

FIG. 2 .FIG. 1 (Fig. 2 .
FIG.2.The spatial profile of the out-of-plane current density along the vertical line passing the X point in the steady state.

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FIG. 3 (color).The spatial profile of each term in the out-of-plane component of the ion momentum equation and of the electron momentum equation along the vertical line passing the X point in the steady state.

FIG. 4 .
FIG. 4. The spatial profile of the out-of-plane ion flow velocity (solid line) and the out-of-plane ion diamagnetic flow velocity (dotted line) along the vertical line passing the X point in the steady state.