New Measure of the Dissipation Region in Collisionless Magnetic Reconnection
A new measure to identify a small-scale dissipation region in collisionless magnetic reconnection is proposed. The energy transfer from the electromagnetic field to plasmas in the electron’s rest frame is formulated as a Lorentz-invariant scalar quantity. The measure is tested by two-dimensional particle-in-cell simulations in typical configurations: symmetric and asymmetric reconnection, with and without the guide field. The innermost region surrounding the reconnection site is accurately located in all cases. We further discuss implications for nonideal MHD dissipation.
pacs:52.35.Vd, 94.30.cp, 95.30.Qd, 52.27.Ny
Magnetic reconnection Birn & Priest (2007) is a fundamental process in many plasma systems, ranging from laboratory and solar-terrestrial environments to extreme astrophysical settings. The violation of the ideal condition, , is essential to allow the magnetic flux transport across the reconnection point. The critical “diffusion region” (DR) where the ideal condition is violated is of strong interest for understanding the key mechanism of reconnection. In collisionless plasmas, since ions decouple first from the magnetic fields, it is thought that the DR consists of an ion-scale outer region and an electron-scale inner region.
In two-dimensional (2D) reconnection problems in the - plane, a popular criterion to identify the innermost “electron diffusion region” (EDR) is the out-of-plane component of the electron nonideal condition, , where
and the electron pressure tensor. In particular, it is known that the divergence of the pressure tensor sustains a finite at the reconnection point, arising from local electron dynamics Hesse et al. (1999, 2011).
Recent large-scale particle-in-cell (PIC) simulations have shed light on the electron-scale structures around the reconnection site. Earlier investigations Daughton et al. (2006); Fujimoto (2006) found that the EDR identified by Daughton et al. (2006) or the out-of-plane electron velocity Fujimoto (2006) extends toward the outflow directions. Previous research has suggested that the EDR has a two-scale substructure: the inner EDR of and the outer EDR of with a super-Alfvénic electron jet Karimabadi et al. (2007); Shay et al. (2007). Satellite observations found similar signatures far downstream of the reconnection site Phan et al. (2007). The roles of these EDRs are still under debate; however, there is a growing consensus that only the inner EDR or a similar small-scale region should control the reconnection rate Shay et al. (2007); Klimas et al. (2008); Hesse et al. (2008). Importantly, it was recently argued that the outer EDR is non- or only weakly dissipative, because the super-Alfvénic jet and condition stem from projections of the diamagnetic electron current in a suitably rotated frame Hesse et al. (2008).
Meanwhile, a serious question has been raised by numerical investigations on asymmetric reconnection, whose two inflow regions have different properties such as in reconnection at the magnetopause Pritchett & Mozer (2009a); Mozer & Pritchett (2010). It was found that various quantities including fail to locate the reconnection site in asymmetric reconnection, especially in the presence of an out-of-plane guide field Pritchett & Mozer (2009a). Considering the debate on inner or outer EDRs and the puzzling results in asymmetric reconnection, it does not seem that is a good identifier of the critical region.
In this Letter, we propose a new measure to identify a small, physically significant region surrounding the reconnection site. We construct our measure based on the following three theoretical requirements. First, we are guided by the notion that dissipation should be related to nonideal energy conversion. Second, we desire a scalar quantity. If we use a specific component of a vector, we have to choose an appropriately rotated frame Hesse et al. (2008). Using a scalar quantity instead, we do not need to find the right rotation in a complicated magnetic geometry. Third, it should be insensitive to the relative motion between the observer and the reconnection site. For example, the reconnection site can retreat away Oka et al. (2008), or, for example, the entire reconnection system may flap over a satellite due to the magnetospheric motion.
Our strategy is as follows. We choose a frame that can be uniquely specified by the observer. Among several candidates, we choose the rest frame of an electron’s bulk motion because it would be the best one to characterize electron-scale structures. Next we consider the energy transfer from the field to plasmas in this frame, which is a scalar quantity. We then expand it with observer-frame quantities. The obtained measure meets all three requirements. It is a Lorentz invariant (frame-independent scalar) and is related to the nonideal energy transfer.
We follow the spacelike convention (,+,+,+). Let us start from the electromagnetic tensor ,
Using a 4-velocity , where is the Lorentz factor , we obtain a 4-vector of the rest-frame electric field Anile (1989),
Here the prime sign denotes the properties in the rest frame of an arbitrary motion , and is the inverse Lorentz transformation from the moving frame. The components of and are given by,
We also use the 4-current , where is the charge density. The current can be split into the conduction current and the convection current, a projection of the motion of the non-neutral frame Møller (1972),
such that is purely spacelike.
Let us define a dissipation measure , the energy conversion rate in the moving frame. The contraction of the covariant and contravariant vectors gives us a Lorentz-invariant scalar,
Choosing the frame of electron bulk motion (the number density’s flow), we obtain the electron-frame dissipation measure,
In the nonrelativistic limit, one can simplify Eq. (7) by setting . One can confirm this by multiplying and .
In ion-electron plasmas, since , where is the proper density, we obtain the following relation between the electron-frame and ion-frame measures,
Such a symmetric relation is reasonable, as ions are the current carrier in the electron’s frame and vice versa. If ions consist of multiple species, , where denotes ion species and is the charge number.
To see how our measure characterizes the reconnection region, we have carried out 2D nonrelativistic PIC simulations. The length, time, and velocity are normalized by the ion inertial length , the ion cyclotron frequency , and the ion Alfvén speed , respectively. The mass ratio is , and the electron-ion temperature ratio is . Periodic () and conductive wall () boundaries are used. Four runs (-) are carried out. Runs 1 and 2 employ a Harris-like configuration, and . The domain of is resolved by cells. particles are used. The speed of light is . In run 2, we impose a uniform guide-field . Runs 3 and 4 employ asymmetric configuration. Since no kinetic equilibrium is known, we employ the following fluid equilibrium proposed by Ref. Pritchett (2008), and . Across the current sheet, magnetic fields and the density vary from and to and . The domain of is resolved by grid points. particles are used. The speed of light is . In run 4, a guide-field is added. In all runs, reconnection is triggered by a small flux perturbation.
The panels in Fig. 1 present the popular measure and the electron-frame dissipation in run 1 in the well-developed stage. They are normalized by and , respectively. Another option is to employ the upstream normalization Karimabadi et al. (2007); Shay et al. (2007) or its hybrid extension for asymmetric cases Cassak & Shay (2007), but these are beyond the scope of this paper. All quantities are averaged over to remove noise. In Fig. 1(a), one can recognize a positive region near the reconnection site and a negative channel which extends to the outflow direction. They correspond to the inner and outer EDRs Karimabadi et al. (2007); Shay et al. (2007). On the other hand, Fig. 1b gives a different picture. There is a positive region near the reconnection site, indicating that the strong energy transfer occurs there. Hereafter we call it “dissipation region” of . At the reconnection point, a main contributor to is . In this case, the charge term is responsible for of the total value, due to significant charge separation, . As we move to the outflow direction, is gradually replaced by . This makes the dissipation region longer than the inner EDR. Based on the scale height, the aspect ratio of the dissipation region, , is similar to the universal reconnection rate of . Outside of minor fluctuations, there are no significant structures in the downstream region ().
We have also studied the ion-frame dissipation in this case. The spatial profile closely resembles to that of , as indicated by Eq. (8) for such a quasineutral plasma ().
The panels in Fig. 2 show the dissipation measure in three other runs. One can see that it excellently identifies compact regions about the reconnection sites in all cases. In the symmetric guide-field run [Fig. 2(a)], the dissipation region is tilted slightly anticlockwise. This is associated with the electron cavities with a parallel electric field along one pairs of separatrices Pritchett & Coroniti (2004). In the asymmetric cases, excellently works even in the most challenging case with a guide-field Pritchett & Mozer (2009a). The field reversal lines (), shown by the dashed lines, are located inside our dissipation regions. The peak amplitude of is high in the guide field cases, as the strong electron current is confined. This deserves further investigation, because the kinetic dissipation mechanism is different in guide-field cases Hesse et al. (2011). There are charge-separated regions along the separatrices in the guide-field cases and on the boundary with the upper inflow regions in the asymmetric cases. Such charge-separation effects are included in via the last term in Eq. (7), which usually improves the identification of the dissipation region.
Let us focus on the dissipation region in runs 1 and 3. The panels in Fig. 3 show the composition of the reconnection electric field along the inflow lines, based on Eq. (1). The shadows indicate the rescaled amplitude of the dissipation measure .
In the symmetric case [run 1; Fig. 3(a)], is balanced by the electron pressure tensor term Hesse et al. (1999) at the center and the bulk inertial term in surrounding regions. The dissipation region is located in a narrow region on local electron inertial scale, -. There, the bouncing electrons carry a strong electron current, and is intense accordingly. We also note that the electron ideal condition is weakly violated on the larger scale of the local ion inertial length, -. Such an outer structure is related to ion’s decoupled motion, and we will see a clear two-scale structure at sufficiently large times Ishizawa et al. (2004).
In asymmetric reconnection, it is known that the field reversal () and the flow stagnation points () are usually not collocated Cassak & Shay (2007); Pritchett (2008); Cassak & Shay (2009). In our run [Fig. 3(b)], the electron flow stagnation point is located on the upper side () while the field reversal is on the lower side (). Therefore the motional electric field even becomes negative around the dissipation region. Importantly, remains positive there and it resonates with a main current . Interestingly, also contributes to in the lower side (). On the other hand, even though is an order of magnitude larger than and on the upper side () Pritchett & Mozer (2009a), it does not contribute to because the vertical current is negligible. Physically, such a strong is overemphasized by the diamagnetic drift where the density gradient is strong. Regarding , we find that the ideal condition is violated in the lower-side upstream of . The pressure tensor terms, both and , are not negligible there. We find that they stem from the gyrotropic electron pressure tensor in the upstream region. This tells us that these effects are due to the drift motion of gyrating electrons. We also notice that the time derivative term () is a key contributor on the upper side (), because the reconnection site moves upward very slowly. Even when the structure is stationary in a frame , the time derivative is not always zero in the observer frame. The fact renders the analysis more difficult and it provides another motivation for using a frame-independent measure.
Let us discuss the relevance for the MHD energy budget. For simplicity we limit our discussion to the nonrelativistic regime. Defining an MHD velocity , we find
The total energy transfer can be decomposed to
The first term stands for the work done by the Lorentz force on the MHD fluids. This operates in the ideal MHD also. The second term is the work done by the Coulomb force, also interpreted as the energy transfer by the convection current. These two disappear in the MHD frame. The last term is responsible for the nonideal energy transfer. In a quasineutral plasma , the positive () plays the same role as an irreversible dissipation given by in the resistive MHD. One can see that positive regions are enhanced and localized around the reconnection sites in Figs. 1(b) and 2.
From the kinetic viewpoint, gyrating particles undergo various drift motions such as , diamagnetic, and curvature drifts. When non- drifts appear, the bulk flow no longer comoves with the field lines, and therefore the ideal condition is no longer a useful concept. Drift motions lead to the electromagnetic energy dissipation, if and only if they involve nonideal energy conversion. For example, we do not recognize a significant nonideal energy transfer in the outer EDR in Fig. 1(b), where due to the diamagnetic effect Hesse et al. (2008). On the other hand, around the reconnection sites, nongyrotropic or field-aligned electrons carry intense currents and then they enhance the nonideal energy transfer.
In summary, we have proposed an electron-frame dissipation measure [Eq. (7)] to identify a physically relevant, small-scale region surrounding the reconnection point. We have demonstrated that it works excellently in typical configurations. Furthermore, we identified its relation to nonideal MHD dissipation, which is essential to the reconnection problem.
Our finding will benefit several research fields. One is the satellite observation of reconnection. NASA is preparing the Magnetospheric Multiscale (MMS) mission to observe the electron-scale structures in near-Earth reconnection sites. By using Eq. (7) with , one can identify the dissipation region regardless of the motion and the orientation of the reconnection site. Another example is relativistic astrophysics. Owning to a growing attention to reconnection, numerical modeling of relativistic reconnection has been growing in importance Zenitani & Hoshino (2005), but basic properties are much less known than in the nonrelativistic case. Our Lorentz-invariant measure can be readily applied to the relativistic dissipation region problem Hesse & Zenitani (2007). Further numerical work is desirable to further test our measure in three dimensions for these systems.
Acknowledgements.One of the authors (S.Z.) acknowledges support from JSPS Postdoctoral Fellowships for Research Abroad. This work was supported by NASA’s MMS mission.
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