The 2.5-D analytical model of steady-state Hall magnetic reconnection

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007ja012852, 2008 The 2.5-D analytical model of steady-state Hall magnetic reconnection D. B. Korovinskiy, 1 V. S. Semenov, 1 N. V. Erkaev, 2,3 A. V. Divin, 1 and H. K. Biernat 4,5 Received 2 October 2007; revised 16 November 2007; accepted 14 December 2007; published 3 April [1] An analytical model of steady-state magnetic reconnection in a collisionless incompressible plasma is developed using the electron Hall MHD approximation. It is shown that the initial complicated system of equations may be split into a system of independent equations, and the solution of the problem is based on the solution of the Grad-Shafranov equation for a magnetic potential. This equation is found to be fundamental for the whole problem analysis. An electric field potential jump across the electron diffusion region and the separatrices is proved to be the necessary condition for steady-state reconnection. Besides of this fact, it is found that the protons in-plane motion obeys to Bernoulli law. The solution obtained demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities and the formation of Hall current systems and a magnetic field structure as expected. Citation: Korovinskiy, D. B., V. S. Semenov, N. V. Erkaev, A. V. Divin, and H. K. Biernat (2008), The 2.5-D analytical model of steady-state Hall magnetic reconnection, J. Geophys. Res., 113,, doi: /2007ja Introduction [2] The process of magnetic reconnection is a widely distributed Nature phenomenon and an actual scientific problem during the last decades. The first model of magnetic reconnection was developed by Sweet [1958] and Parker [1957] using the theory of resistive magnetohydrodynamics (MHD). They suggested a diffusion mechanism of magnetic energy conversion. Various attempts to alter the Sweet-Parker model were made, since it failed to explain the energy release rate observed during solar flares. In 1964, Petschek presented another model of magnetic reconnection [Petschek, 1964], in which standing slow shock waves supply the conversion of magnetic energy. This mechanism, being much faster than the Sweet-Parker scheme, explained the observed energy release rate. However, it was shown later that spatial nonuniform resistivity is required to support the Petschek scenario [e.g., Sato and Hayashi, 1979; Biskamp, 1986; Erkaev et al., 2000; Uzdensky and Kulsrud, 2000; Biskamp and Schwarz, 2001; Erkaev et al., 2002]. This anomalous resistivity in collisionless plasmas may be caused by microinstabilities, such as the lowerhybrid drift instability [Huba et al., 1977], the Buneman instability [Drake et al., 2003], the ion-acoustic instability [Kan, 1971; Coroniti, 1985], or others [Büchner and Daughton, 2007]. 1 State University of St. Petersburg, St. Petersburg, Russia. 2 Institute of Computational Modelling, Russian Academy of Sciences, Siberian Branch, Krasnoyarsk, Russia. 3 Siberian Federal University, Krasnoyarsk, Russia. 4 Space Research Institute, Austrian Academy of Sciences, Graz, Austria. 5 Institute of Physics, University of Graz, Graz, Austria. Copyright 2008 by the American Geophysical Union /08/2007JA012852$09.00 [3] There is another approach which leads to a selfconsistent model of fast reconnection and which does not require anomalous resistivity. This process is developed using MHD with the Hall effect invoking [Sonnerup, 1979; Terasawa, 1983; Hassam, 1984], which causes a Petschek-like configuration due to the generation of dispersive waves [Mandt et al., 1994; Rogers et al., 2001]. It turns out that the contribution of this effect appears close to the X-line, at length scales in order of the proton inertial length (skin depth), which is defined as l p = c/w p, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c is the speed of light, w p = 4pne 2 =m p is the proton plasma frequency, and m p is the proton mass. Inside this region, the Hall effect decouples the proton(ion) and electron motion. One of the consequences of this decoupling is a demagnetization of protons, tearing them off from the magnetic field lines, while the magnetic field stays frozen into the electron fluid. Electrons, in their turn, demagnetize much closer to the X-line, inside the so-called electron diffusion region (EDR), because of inertia or nongyrotropic pressure [Vasyliunas, 1975]. A scaling analysis gives d l e, where d is the thickness of the EDR, l e = pc/w ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e is the electron inertial length (skin depth), and w e = 4pne 2 =m e is the electron plasma frequency. Considerable efforts were dedicated to the mechanism of electron demagnetization studies [e.g., Lyons and Pridemore- Brown, 1990; Cai et al., 1994; Biskamp et al., 1997; Horiuchi and Sato, 1997; Shay et al., 1998; Kuznetsova et al., 1998; Hesse and Winske, 1998]. However, in the GEM Challenge project [Birn et al., 2001], it was found that the Hall reconnection rate is insensitive to the dissipation mechanism. The rate of magnetic reconnection is nearly independent of the strength of dissipation in hybrid simulations with resistivity [Mandt et al., 1994], two-fluid simulations [Ma and Bhattacharjee, 1996; Biskamp et al., 1997], 1of13

2 and particle-in-cell (PIC) simulations [Shay and Drake, 1998; Hesse et al., 1999]. In addition, the Hall reconnection rate is independent on the magnitude of the electron mass [Shay and Drake, 1998; Hesse et al., 1999; Pritchett, 2001; Ricci et al., 2002] and the system size [Shay et al., 1999; Huba and Rudakov, 2004]. To summarize the results obtained by all types of numerical simulations [Birn et al., 2001], the essential feature of fast reconnection is the presence of the Hall effect, which provides the rate of the steady-state reconnection to be approximately a constant of the order of 0.1 [Shay et al., 1999; Huba and Rudakov, 2004]. However, this result is not incontrovertible. It has been found in studies of forced reconnection and double tearing mode reconnection that the reconnection rate has a dependence on the electron dissipation and the system size [Grasso et al., 1999; Wang et al., 2001; Porcelli et al., 2002; Bhattacharjee et al., 2005]. [4] Apart from the reconnection rate, the Hall effect leads to a distinctive current structure which produces an out-ofplane magnetic field with a quadrupolar configuration. These features were first pointed out by Sonnerup [1979]. Since then, this behavior has been studied carefully in numerical simulations and observed in the magnetosphere [e.g., Øieroset et al., 2001; Mozer et al., 2002; Scudder et al., 2002; Nagai et al., 2003; Runov et al., 2003; Nagai and Fujimoto, 2005; Alexeev et al., 2005] and in laboratory experiments [Ren et al., 2005; Cothran et al., 2005]. Thus, the Hall MHD (HMHD) approach proves to be a minimally sufficient description for magnetic reconnection process in collisionless plasmas. [5] Unfortunately, analytical studies of magnetic reconnection in the frame of HMHD meet considerable difficulties. The analytical problems arise due to the EDR, where the electron inertia and the pressure tensor anisotropy are the dominant physical effects causing electron demagnetization and magnetic field line disruption. But these effects are the subject to kinetic theory, not to HMHD. Numerical simulations point out how these difficulties can be overcome. As far as reconnection rate does not depend on the mechanism of dissipation, one can restrict the EDR contribution studies to the size of the EDR rather than its internal structure. Numerical simulations [Shay et al., 1998] confirmed the thickness of the EDR to be of the order of l e for antiparallel Hall reconnection. The length of the EDR was thought to be 10l e [Shay et al., 1999; Huba and Rudakov, 2004], but later results suggest that it expands to the edge of the proton dissipation region, 10l p [Daughton et al., 2006]. This contradiction seems to be solved in recent PIC simulations demonstrating that EDR develops a twoscale structure along the outflow direction. The length of the electron current layer is found to be sensitive to the proton/ electron mass ratio, approaching 0.6l p for the realistic electron mass value. In addition, an elongated outflow electron jet develops into the proton acceleration direction, and its length extends to tens of the proton skin depths [Shay et al., 2007]. [6] Actually, analytical studies of the problem may be simplified, moreover, by using the electron Hall MHD (EHMHD) approximation. In the nearest vicinity of the stagnation point, the proton velocities are small as compared to the electron velocities. Hence, one may consider the electric current in this region as the electron current only. This assumption is the matter of EHMHD. The applicability domain of this approach is of the order of the proton inertial length l p [Biskamp, 2000], but smaller than the HMHD region. Nevertheless, to study the problem using the EHMHD approximation proves to be of material significance. In particular, a quite detailed EHMHD-analysis of the problem has been performed by Uzdensky and Kulsrud [2006]. In the mentioned paper the authors have shown that with the time-stationarity and translational symmetry conditions, the magnetic and electron velocity fields can be expressed in terms of only a single one-dimensional function. This function is a magnetic potential of the in-plane magnetic field (poloidal flux function). Besides, the authors have found out that, neglecting the ion current, one has the Grad-Shafranov equation for this potential. Unfortunately, sufficient attention has not been payed to this result, in our view. Namely, the Grad-Shafranov equation for magnetic potential can be put in the foundation of a clear analytical model of self-consistent steady-state collisionless magnetic reconnection in an incompressible plasma. Moreover, it allows to obtain some information about the electric field behavior without solving any equations. This subject is the main content of the present paper. [7] The paper is organized as follows: In section 2 we formulate the problem using the EHMHD formalism. In section 3 we split the system of the original EHMHD equations and proceed to the Grad-Shafranov equation for a magnetic potential. In section 4 we obtain the analytical solution of this equation making use of the boundary layer approximation. Section 5 is dedicated to the discussion of the out-of-plane magnetic field, and of the electric potential and electric field behavior. The pattern of the numeric solution is presented in section 6. In the last section we summarize the results obtained. 2. Problem Formulation [8] We examine the problem of steady-state collisionless magnetic reconnection in the vicinity of the X-line, where we are permitted to apply the EHMHD approximation, i.e., at the scale of the proton skin depth. We avoid the description of the EDR internal processes and consider only its size, which we suppose to be of the order of l e in its cross section and hl p along it, where h is a coefficient of the order of 1. Outside the EDR the plasma is supposed to be nonresistive; furthermore, it is assumed to be quasi neutral and incompressible. [9] Our coordinate system is chosen as follows: the X-axis coincides with the magnetic field at its direction at infinity, the Y-axis is directed along the X-line, and the Z-axis is perpendicular to both of them. Here we present a 2.5-D model, meaning that the y-components are included in our consideration, but all quantities are assumed to be independent of y. [10] The two-fluid description of the problem is determined by the following equations, r V p r Vp ¼rpþne E þ 1 c V p B ; ð1þ 2of13

3 E þ 1 c V e B ¼ 0; rb ¼ 4pne c re ¼ 0; rb ¼ 0; rv p;e ¼ 0: ð2þ V p V e ; ð3þ Here, (1) is the equation of the proton motion, where p is the proton gas pressure, and V p is the proton bulk velocity; (2) is the Ohm law, where V e is the electron bulk velocity, (3) is the Ampère law, (4) is the Faraday law, (5) is the Gauss law, and (6) is the mass conservation law for each particle species. For the simplicity, we neglect the tensor of the electron pressure in the Ohm law (2). However, a gradient of the scalar electron pressure may be easily taken into consideration, as it is shown in the next section. [11] The system (1) (6) is appropriate for the Hall MHD problem formulation under the condition V V p, where V is the plasma bulk velocity. Next, we consider this system under the EHMHD approximation V p V e (but V p 6¼ 0), which yields ð4þ ð5þ ð6þ structure, quantity P upper (x) can be obtained, in principle, from the matching condition for external and internal solutions. We restricted our study to the internal region only; therefore, for simplicity, we assume P upper (x) =const and, consequently, P = const. [13] Note that system (1) (6) does not contain any information about the EDR. Nevertheless, its size may be taken into consideration, if we transform these equations to another form, which is subject to the next section. 3. Grad-Shafranov Equation [14] In this section we formulate our problem in terms of potentials. First, we introduce dimensionless quantities: the magnetic field strength ~B = B/B 0, the proton and electron bulk velocities ~V p,e = V p,e /V A, the electric field strength ~E = E/E A, the gas pressure ~p = p/p 0, and the length scales ~r = r/l p. Here p 0 = B 2 0 /4p, and r =(x, y, z). [15] Second, we introduce the electric potential F via ~E = ~r~f. Omitting the tildes, we rewrite equations (1) (6) for normalized quantities, bearing in mind the EHMHD approximation (7) (9), V p r Vp ¼rprF; ð11þ V e B ¼rF; rb ¼V e ; ð12þ ð13þ j ¼ ne V p V e ffineve ; ð7þ rb ¼ 0; ð14þ rb ¼ 4pne c 4pne V p V e ffi V e ; c ð8þ rv p;e ¼ 0: ð15þ E þ 1 c V p B ¼ 1 c V 1 p V e B ffi c V e B: In our steady-state two-dimensional case, the electric field E y must be a constant, according to Faraday s law (4). So, we define E y ¼ E A ; ð9þ ð10þ where e is the reconnection rate which we assume to be small, e 1, and E A = 1 c B 0V A is the Alfvén electric field. Here, B 0 is the magnetic field value at the upper boundary of the examined region (it will be defined more accurately below) and V A is the corresponding proton Alfvén velocity. [12] Finally, we assume the total pressure P = p +(1/8p) (B x 2 + B y 2 + B z 2 ) to be a constant in the region considered. The constancy of P along z axis is a straightforward consequence of the boundary layer approximation, so, generally speaking, we have here P = P(x). The distribution of P(x) is fixed by its distribution P upper (x) at the upper boundary of the EHMHD domain. Since the last one is embedded in the large-scale [16] Note that F has a linear dependence on y, =. Under this point of view, we can present F as a sum of two terms, F(x, y, z) =f(x, z) y. The gradient of the isotropic electron pressure p e can easily be taken into account by introducing an effective potential ~ f = f p e [Vasyliunas, 1975; Uzdensky and Kulsrud, 2006]. [17] Equation (11) may be split into two equations, for out-of-plane proton velocity V py and for in-plane velocity V p? =(V px, V pz ). For V py we have V p? r? Vpy ¼ : ð16þ Here and below,? denotes the x-z plane. For a given function V p? (x, z) equation (16) has a solution, which can be achieved by using the method of characteristics [e.g., Sobolev, 1989]. Z r ds V py ðþ¼ r jv p? j þ V pyðr 0 Þ; ð17þ r 0 where ds is an elementary displacement along the in-plane projection of the proton trajectory, and r 0 marks the initial point of integration. 3of13

4 [18] Equation for the in-plane proton velocity is V p? r? Vp? ¼r? p r? f: ð18þ Using the identity (V p? r? )V p? = 1 2 r 2? V p? V p? r? V p?, and applying the operator (V p? ) to equation (18), we come to the Bernoulli law for protons, 1 V p? r? 2 V p? 2 þ p þ f ¼ 0: [19] Integrating equation (19), we have ð19þ 1 2 V 2 p? þ p þ f ¼ const traj; ð20þ where const traj is a constant along the trajectory. Expressing the gas pressure through the total and magnetic pressures, we obtain finally, 1 2 V 2 p? 1 2 B2 þ f ¼ const traj : ð21þ Here B 2 = B x 2 + B z 2 + B y 2, and the constant total pressure P has been included in the const traj. [20] Following the commonly used way [Biskamp, 2000], we introduce a magnetic potential A(x, z) and an electron velocity stream function. A is defined by B? ¼r Ae y ; ð22þ where e y is the unit vector. [21] Thus, Ampère s law (13), being written in components, has the form point of integration, B y (r 0 ) = 0, and n is the quadrant number. By a somewhat other way, namely by using the volume-per-flux quantity, equation (24) and its solution (25) were obtained by Uzdensky and Kulsrud [2006]. [23] On the other hand, equation (24) reveals that the functions A and B y compose a couple of natural variables for our problem, instead of x and z. Using definitions (22) and (23), we can write Therefrom, we get da x Adx z Adz ¼ B z dx B x dz; db y x B y dx z B y dz ¼V ez dx þ V ex dz: dx ¼ 1 D V exda þ B x db y ; dz ¼ 1 D V ezda þ B z db y : Using these expressions, ¼ 1 þ 1 ¼ 1 ð V e? r? Þ; ¼ y þ 1 ¼ 1 ð B? r? Þ: ð27þ Now, we note that rf is orthogonal to the vectors V e and B, according to (12). This leads to two ðv e? r? Þf ¼V ¼ V ey; ð28þ V ex ; V ez : This means that the out-of-plane magnetic field B y is the stream function for the electron in-plane velocity [Biskamp, 2000], V e? ¼r B y e y : ð23þ So, curves B y = const are the electron in-plane trajectories, while curves A = const are the in-plane magnetic field lines. [22] Furthermore, from the y-component of the Ohm law (12) we obtain ¼ V ex B z V ez B x @z A; B ðx; zþ D; ð24þ where D is the Jacobian of the transformation of variables (x, z)! (A, B y ). For a given magnetic potential A(x, z) we can solve this equation analogously to (16). The solution is B y ðþ¼ r ð1þ nþ1 Z r r 0 ds jb? j ; ð25þ where ds is an elementary displacement along the in-plane projection of the magnetic field line, r 0 marks the ðb? r? Þf ¼B ¼ B y: ð29þ We rewrite these equations by using expressions (26) and (27), ¼ y ¼ B y : ð30þ ð31þ According to Ampère s law (13), we get V ey = D? A, where D? is the in-plane Laplace operator, D? 2 /@x 2 +@ 2 /@z 2. Then, equation (30) takes ¼ D?A: ð32þ Integrating (31), we obtain an equation for the potential f of the in-plane electric field, f ¼ 1 2 B2 y þ GA ð Þ; where G(A) is an unknown integration constant. ð33þ 4of13

5 [24] By using the equation (33), we arrive at two essential expressions. First, substituting (33) in (21), we see that the electric field fully compensates the magnetic pressure force, created by the out-of-plane magnetic field, 1 2 V p? B2 x þ B2 z þ GA ð Þ ¼ consttraj : ð34þ Second, substituting (33) in (32), we conclude that magnetic potential A must satisfy the Grad-Shafranov equation, D? A ¼ dgðaþ da : ð35þ This result appears to be evident for a number of years. For example, only one step separates equation (35) and equation (6.49) in the work of Biskamp [2000] (B rda = 0, in our notations). Equation (35) has also been obtained by Uzdensky and Kulsrud [2006], who have pointed out that this equation for the potential of the in-plane magnetic field is a straightforward consequence of neglecting the ion current. However, in the cited paper, the authors did not pay attention to the key role of this equation in the whole problem, which in the presented investigation is a main subject. [25] Finally, we arrive at the following system of equations: V ey D? A ¼ dgðaþ da ; ð36þ B y ðþ¼ r ð1þ nþ1 Z r r 0 f ¼ 1 2 B2 y þ GA ð Þ; ds jr? Aj ; 1 2 V p? jr?aj 2 þ GA ð Þ ¼ const traj ; ð37þ ð38þ ð39þ Z r ds V py ðþ¼ r jv p? j þ V pyðr 0 Þ; ð40þ r 0 r? V p? ¼ 0; ð41þ where (36) is the equation for the magnetic potential A, (37) is the equation for out-of-plane magnetic field B y, (38) is the equation for the electric potential f, (39) is the Bernoulli equation for the in-plane motion of protons, (40) is the equation for the out-of-plane proton velocity V py, and (41) is the continuity equation. Thus, the initial complicated system is split, and the solution of the problem is based on the solution of the Grad-Shafranov equation for the magnetic potential (36). 4. Boundary Layer Approximation [26] First, we note that the potential A, evidently, must keep a saddle-type configuration to provide the required magnetic field structure. Since A is defined with an accuracy Figure 1. G(A). Sketch of the behavior of V ey (A) dg/da and to a constant, we can fix it by the requirement A(0,0) = 0. Then, according to the definition of A in equation (22), its sign is positive in the outflow regions (OR) and negative in the inflow regions (IR). Therefore, A(x,z) = 0 at the separatrices, including the origin. [27] Furthermore, to solve the Grad-Shafranov equation (36), we have to specify the function on the right-hand side, dg/da. Physically, this function is the electron bulk velocity V ey, and this circumstance allows one to describe dg/da qualitatively. It is clear that electrons can be accelerated into the y-direction by the electric field E y inside the EDR, and partly by the electric field E z via the electric drift. Both sources are strongly localized in the z-direction (with a scale length of the order of l e ) near the curves A = 0, that is near the X-line and separatrices. Therefore, V ey (A) has a shape similar to a d-like function. It turns out that the concrete functional dependence is not really much important due to high localization of this function. Because of these reasons, we can figure out the likely behavior of the function dg(a)/da, which has a minimum at the origin that is at the point A=0,and tends to zero, while jaj increases (see Figure 1). The numeric value of the minimum can be estimated by using the Ampère law (13). Indeed, accordingly to this equation, we have maxjv ey j 1 ¼ l p =l e ¼ V Ae ; ð42þ l e =l p where V Ae is the electron Alfvén velocity, and l e /l p is the EDR width measured in proton skin depth units. By integrating function dg(a)/da, we obtain G(A) (see Figure 1). [28] Now we apply the scaling consideration to show that the region of values of G(A) is strictly limited by the boundary conditions. The scaling of the problem is defined by the inclination angle of the separatrices, which is in the order of e 1. Hence, the tangential components, x, B x, V x are in the order of 1, while the normal components, z, B z, V z are much less, namely in the order of. Therefore, we use the boundary layer approximation to simplify the Grad-Shafranov equation (36). Neglecting the x-derivative in equation (36) and integrating it over z, weobtain 2 ¼ GA ð ÞþCx ðþ; ð43þ 5of13

6 where C(x) is a integration constant. = B x (x,z), and as far as B x (x,0) = 0 due to the symmetry condition, C(x) is defined as Cx ðþ¼gax; ½ ð 0ÞŠ: ð44þ This function has a simple physical meaning. Considering equation (43) at the separatrices, where G(A) = 0, we conclude that Cx ðþ¼ 1 2; 2 Bsep ð45þ where the index sep indicates the value at the separatrix. [29] Let us introduce a function B x 0 (x) which we define as B 0 x x ðþlim x B xðx; zþ: ð46þ z!1 In the boundary layer approximation, we treat the upper boundary of the considered region as infinity. With this definition, equation (43) yields Cx ðþ¼ B0 x ðþlim x GAx; ½ ð zþš: ð47þ z!1 Since at the origin quantity C(0) = 0 according to (44), equation (47) gives lim GA0; ½ ð zþš ¼ 1 2 z!1 2 B0 x ðþ: 0 ð48þ [30] In the IR, the potential A(x, z) is a monotonic decreasing function of z for a fixed x (and a monotonic increasing function of x for a fixed z in OR). The function G(A) is monotonic as well. Therefore, lim GAx; ½ ð zþš lim GA0; ½ ð zþš ¼ 1 2 z!1 z!1 2 B0 x ð0þ: ð49þ [31] Now we define the normalization constant B 0 as B x 0 (0). With this normalization, the inequality (49) gives (see Figure 1) max jgðaþj ¼ 1 2 : ð50þ Equations (43) and (44) allow to calculate the potential A(x, z), if the boundary condition is fixed as a function B z (x,0). Thus, an analytical solution of the Grad-Shafranov equation for the potential A (in the boundary layer approximation) is p Ax; ð zþ ¼ ffiffiffi Z z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 GAx; ½ ð z 0 ÞŠþCx ðþdz 0 ; 0 ð51þ Cx ðþ¼gax; ½ ð 0ÞŠ; ð52þ Ax; ð 0Þ ¼ Z x 0 B z ðx 0 ; 0Þdx 0 : ð53þ Here, the sign on the right-hand side of (51) alters at the separatrices: it is negative in the IR and positive in the OR. [32] There is another way to preset the boundary condition because function C(x) determines the field line flexion at the upper boundary. Indeed, according to expressions (47) and (49), we can write C(x) ffi 1/2((B x 0 ) 2 (x) (B x 0 ) 2 (0)). Therefore, by prescribing function B z (x,0), we determine function B x 0 (x) automatically, and vice versa. However, one of the basic features of the process, namely the electric potential behavior, can be obtained without solving any equations. 5. Electric Potential [33] Here we discuss the out-of-plane magnetic field B y and the drop of the electric potential f, which consists of two terms, G(A) and (1/2)B 2 y. When moving from IR to OR, the change in G(A) is equal to 1, according to expression (50) (see Figure 1). Moreover, the main part of this reduction must take place near the separatrices, including the origin, because dg(a)/da = V ey has a sharp minimum there. So, G(A) has an abrupt jump across the separatrices and the diffusion region. [34] Let us consider the contribution of the second term of f. According to equation (37), B y has a singularity at the separatrices in absence of the EDR. Using approximation G(A) =a A in the vicinity of the separatrices, where a is a positive constant, we obtain after some algebra the following representation for function B y at the separatrices, B sep y (x), B sep y ðþ¼b x EDR y þ ln x ; ð54þ x 0 where B y EDR is the small EDR contribution and x 0 is the EDR length, so that we assume here x > x 0. From this we see that the absence of EDR (x 0 = 0) makes reconnection impossible, because lim x0!0 B y = 1. This means that B y would create an infinite potential barrier at the electrons way if the EDR does not prevent this. This logarithmic singularity is very similar to the simplest solution discussed by Biskamp [2000]. On the other hand, the logarithmic singularity is weak, hence even a short-length EDR can essentially reduce the value of B y. Besides, the logarithmic term in (54) is multiplied by the small parameter. As a result, the main contribution to the electric potential f comes from the function G(A) rather than from the term (1/2)B y 2. Thus, the jump of the electric potential Df, which is responsible for the proton acceleration, is the jump of the function G(A), jdfj jd G(A)j 1, or in dimensional units, jdfj B2 0 4pne : ð55þ Note that this estimation is obtained as a straightforward consequence of the condition of the Grad-Shafranov equation solvability (equations (43) (50)). So, B 0 2 /(4p ne) is the threshold value of the electric potential jump, which is necessary for steady-state reconnection to proceed. [35] According to (20), one half of the energy is spent for the proton acceleration and the other half for the work against the gas pressure. As far as width of the separatrices is of order of l e, the electric field E z = jdfj/l e B 0 V Ae /c is 6of13

7 Figure 2. (top) Magnetic potential A(x, 0). (bottom) Electron velocity V ey as a function of the A potential. Thin horizontal and vertical lines mark the EDR border. much larger than the reconnection electric field E y, in qualitative agreement with numerical simulations [e.g., Shay et al., 1998], spacecraft observations [e.g., Mozer et al., 2002; Wygant et al., 2005], laboratory experiments [e.g., Matthaeus et al., 2005], and suggested by analytical theory [e.g., Mozer, 2006; Uzdensky and Kulsrud, 2006]. For example, for the Earth magnetotail conditions B = 10 nt and a plasma concentration n =1cm 3 the electric field strength will be je z j50 mv/m. 6. Numerical Solution [36] In order to demonstrate the efficiency of our model, we present a pattern of numerical solutions. The region under consideration is a rectangle, which length is limited by the EHMHD applicability condition, and its width is restricted by the validity of the boundary layer approximation. So, we set the half-length and half-width of the rectangle to be equal to 1 and 0.3, respectively. We choose a d-like function V ey (A) as follows: V ey ðaþ dgðaþ da ¼1 m p A 2 þ m 2 : ð56þ The parameter m controls the sharpness of V ey (A), i.e., the EDR size in the A-space, and the extreme value of V ey (A). According to our estimation (42), we set m = This value provides min V ey = V ey (0) = 1/pm = V Ae. By integration of V ey (A) we obtain GA ð Þ ¼ 1 p arctan A m : ð57þ Bearing in mind results of numeric simulations by Shay et al. [1999], we fix the boundary condition B z (x, 0) as a linear function, B z ðx; 0Þ ¼ x; ð58þ where is the reconnection rate. Its value is set equal to 0.1 to facilitate a comparison with numerical simulations. Then, we defined the EDR as a rectanglep with half-width d equal to the electron inertial length l e = ffiffiffiffiffiffiffiffiffiffiffiffiffi m e =m p and half-length hl p, where h = 0.5 in accordance with Shay s estimation [Shay et al., 2007]. This value fulfils the requirement that the effective acceleration of electrons in y direction takes place mostly inside the EDR. With the chosen and m as above, we have at the right boundary of the EDR: V ey (0.5, 0) = 1/3V ey (0, 0) (see Figure 2). [37] First, the magnetic potential A(x, z) is found by solving the Grad-Shafranov equation (36) in the boundary layer approximation, using the system of equations (51) (53). Curves A = const are in-plane magnetic force lines, they are presented in Figure 3. Then, the in-plane magnetic field B? = (B x, B z ) is calculated in accordance with definition (22). [38] Next, we evaluate the out-of-plane magnetic field B y by expression (37). The integration is performed in the backward direction, that is starting from the outer boundary of the box and moving along the force lines towards the axis (X in the OR, and Z in the IR). As far as the solution (37) is applicable out of the EDR only, the integration is terminated at the EDR boundary. In fact, staying in the frame of our theory, we cannot account the EDR contribution, but we assume it to be a small correction to the calculated B y, due to the symmetry and smoothness of B y (x, z). Therefore, we neglect this correction. Knowing B y, we calculate the electron velocities V ex and V ez according to (23), then, the electric potential f is evaluated from equation (38), and the inplane electric field E? =(E x, E z ) is found. [39] We use equation (39) to compute the proton trajectories. The boundary conditions at the upper boundary of the box are fixed as follows: V upper px = 0 and V upper pz = B x /B 2. This means that we assume protons to inflow vertically with the drift velocity. The continuity equation (41) allows to simplify the computational scheme considerably. Namely, we introduce a stream function z(x, z) for the proton velocity according to V p? ¼r ze y : ð59þ 7of13

8 Figure 3. Proton trajectories (thick) and magnetic field lines (thin). Then, we calculate z at the upper boundary, zðx; z 1 Þ ¼ zðþ¼ x Z x 0 Vpz upper ðx 0 Þdx 0 : ð60þ This definition gives z(0) = 0. Using equation (39), we also calculate const traj (x) at the upper boundary. It is easy to see, that const traj (0) = 0 as well. Knowing z(x) and const traj (x), we obtain the function const(z), which is equal to zero, evidently, at the point z = 0. By using this function and the boundary layer approximation jv pz jjv px j, equation (39) can be rearranged to the ¼ ffiffiffi r p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 constðzþga ð Þþ 1 2 B2 x þ B2 z : ð61þ The boundary condition is the const(z 0 ) value at the halfaxis x 0. Both half-axis, x 0 and z 0 belong to the one trajectory, so that z 0 = z(x,0) = z(0, z 1 ) = 0. Integrating equation (61) under the boundary condition const(z 0 ) = const(0) = 0, we calculate the stream function z(x, z), and then V p? is found. The curves z(x, z) =const give us the proton in-plane trajectories, which are presented in Figure Summary [40] In the previous sections we presented a 2.5-D analytical model of steady-state magnetic reconnection in a collisionless incompressible plasmas. Our investigations are restricted to the vicinity of the X-line at a length scale of the proton skin depth. Making use of scaling of the problem, we used the electron Hall MHD description of plasmas for developing the model and a boundary layer approximation for finding a solution. Here, we collect the essential features of the presented model. [41] The mathematical formulation of the model consists of equations (36) (41) under the assumption of constancy of the total pressure. The cornerstone of the model is the Grad-Shafranov equation (36) for the magnetic potential A of the in-plane magnetic field B?. The family of functions G(A) satisfying the physical requirements of the problem defines the family of the possible solutions. G(A) has a clear physical sense, namely, it is the main part of the electric potential f. From the other hand, the derivative dg(a)/da is an electron out-of-plane bulk velocity V ey. This velocity has an abrupt minimum in the center of EDR, in the origin, because inside the EDR the electric field E y accelerates demagnetized electrons unimpeded. Since the EDR is thin, we assume V ey (A) tobead-like function having a small but nonzero half-width. The sharpness of this function should be also smoothed due to the electric drift. By integrating V ey (A), we get G(A) (Figure 1), while the boundary conditions give us the range of its values. The limiting values of G(A) are determined by the magnetic pressure at the upper boundary of the investigated region (z = z 1 ). By normalizing the magnetic field by B 0 B x (0, z 1 ), we obtain the fact that G(A) takes values from 1/2 in the inflow region to 1/2 in the outflow region. So, the physical conditions of the problem define G(A) well enough for appropriate modeling. [42] The variation of G(A) defines the drop of the electric potential f from its maximum at the upper boundary to its minimum in the outflow region, jdfj B 0 2 /(4pne). This estimation is obtained as a straightforward consequence of the condition of the solvability of the Grad-Shafranov equation. Therefore, B 0 2 /(4pne) is a threshold value of the electric potential jump, which is necessary for steady-state reconnection to proceed. The most part of this drop is localized in a thin layer surrounding the separatrices, which scale is in order of the electron diffusion region width, i.e., in order of the electron skin depth l e. The thinness of this sheet leads to the presence of a strong electric field je z j jdfj/l e. Besides, it leads to a powerful acceleration of electrons into the X-line direction. As far as the EDR width 8of13

9 Figure 4. Out-of-plane magnetic field B y. The B y values are shown by color. is of the order of l e, jv ey j must reach up to electron Alfvén velocity V Ae, both at the origin, and at the separatrices (at the separatrices, including the origin, the magnetic potential A = 0, and max jv ey (A)j = jv ey (0)j). Thus, the electron out-ofplane current splits into thin wings mapping separatrices. This conclusion coincides with results of numerous numerical simulations by, e.g., Shay et al. [1999], Daughton et al. [2006], Schmitz and Grauer [2006], and Shay et al. [2007]. [43] The in-plane magnetic field B? defines the out-ofplane magnetic field B y (25), although solution (25) is valid outside EDR only, where the electrons are magnetized. This claims unequivocally that magnetic reconnection is impossible in absence of the EDR. In that case electrons are magnetized everywhere, including the origin, and we have B y =±1 at the separatrices because B? (0,0) = 0. Therefore, calculating B y, we stopped our computation at the EDR boundary. [44] Hence, a reasonable question arises about the influence of the EDR size on B y. The logarithmic representation of B y sep (54) claims that B y is relatively low-sensitive to the EDR length, so it depends on the in-plane field B? and the out-of-plane electric field E y mainly. Therefore, we assume the EDR contribution to B y to be small, since B y (0, 0) = 0 due to the symmetry condition. The B y structure is presented at Figure 4. Neglecting the proton currents led to the expected result, namely that the B y magnitude turned out to be overstated; we obtained the maxjb y j0.75, while the largest observed value is [Mozer et al., 2002; Alexeev et al., 2005]. Figure 5. Curves B y = const electron trajectories. The color bar shows B y values. 9of13

10 Figure 6. In-plane electron velocity V ex spatial distribution. The color bar shows V ex values. [45] In its turn, the magnetic field B y is a stream function for the electron in-plane bulk velocity, i.e., for the in-plane electric current in the EHMHD approximation. So, curves with B y = const are projections of the electron trajectories onto the plane y = 0. These trajectories are presented at Figure 5. One can see the electron trajectories are counterstreaming the separatrix. Together with the electron velocities distribution presented in Figures 6 and 7, this constitutes the picture of the Hall current structure; slow electrons more from one side of the separatrix into the inflow region (V e 1V Ap ), and fast ones (V e 10V Ap ) from the other side. [46] The electric potential f consists of two components, f = G(A) B y 2. In fact, the second term is small, since B y is close to zero everywhere, except of the thin sheet near the separatrices. Moreover, its maximum value is slightly shifted into the outflow region, in the weak in-plane magnetic field area, according to equation (25) (the magnitude of this shift depends on the EDR width d, and when d! 0 the maximum shifts toward the separatrix). Meanwhile, quantity G(A) is positive in the IR, it is equal to zero at the separatrices, and it is negative in the OR. So, in the thin sheet of large B y values, two components of f just cancel each other, and its shape just slightly smoothes (see Figure 8). Figure 7. In-plane electron velocity V ez spatial distribution. The color bar shows V ez values. 10 of 13

11 Figure 8. Electric potential f(x, z). The color bar shows f values. [47] At last, the proton in-plane motion obeys the Bernoulli law: the sum of the kinetic energy (V 2 px + V 2 pz ), gas pressure and electric field energy is constant along the proton in-plane trajectories (19). Curiously enough, the electric field fully compensates the magnetic pressure force, created by the out-of-plane magnetic field B y (34). [48] The protons are accelerated by the drop of the electric potential jdfj ffijdg(a)j ffib 2 0 /(4pne). One half of this energy is spent for their acceleration, the other half is for the work against the gas pressure (see Figures 8 and 9). [49] Summarizing, the Grad-Shafranov equation for the magnetic potential A is found to be a helpful instrument for Hall reconnection modeling. The function G(A) and the boundary condition of B z (x,0) (or B x (x, z 1 )) are the main degrees of freedom in the model developed. Though the model permits some arbitrariness in the choice of these functions (but strongly restricted arbitrariness!), in actual situations, this choice must provide a coordination of the solution in the EHMHD domain with the solution in the outer region. The matching of such solutions is an attractive Figure 9. Proton velocity V px spatial distribution. The color bar shows V px values. 11 of 13

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