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Boundary conditions and plane magnetic reconnection
Fluid Dynamics Research 39 (2007) 447 – 456
Boundary conditions and plane magnetic reconnection Manuel Núñez∗ Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain Received 8 February 2005; received in revised form 4 December 2006; accepted 5 December 2006 Communicated by M.-E. Brachet
Abstract It is well known that several classical models of plane magnetic reconnection turn out to be more or less efficient according to the choosing of the boundary conditions. We prove analytically that one important datum, the variation of the area of the reconnection region, may be found, except by resistive effects, directly in terms of the flux and stream functions at the boundary. If we include the Hall effect in our model, the vertical component of the field at the boundary is also necessary. Since this area is a measure of the capacity of the system to keep the magnetic nozzle open, this provides a rigorous quantitative measure of the essential role of boundary conditions on the efficiency of the process. © 2007 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 52.35.Vd; 52.30.Cv; 96.60.Rd Keywords: Reconnection region; Current sheet; Flux and stream functions
1. Introduction Magnetic reconnection is one of the most important astrophysical processes because of its relevance in the behavior of the magnetospheres of stars and planets, being e.g. responsible of such spectacular phenomena as solar flares. Its theoretical study began with Sweet (1958) and Parker (1957, 1963), who analyzed the collision of two plasma masses possessing opposite magnetic fields. The resulting ∗ Tel.: +34 983 423924; fax: +34 983 423013.
E-mail address: [email protected] 0169-5983/$32.00 © 2007 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.12.001
448
M. Núñez / Fluid Dynamics Research 39 (2007) 447 – 456
Fig. 1. Reconnection at magnetic neutral lines.
Fig. 2. Throttling of the plasma flow.
configuration is a familiar one (Fig. 1): the plasma approaches the neutral sheet where magnetic field lines reconnect, and escapes laterally. The occurrence and properties of this type of geometries in a quasi-static two-dimensional situation was studied in depth by Syrovatskii (1971), Imshennik and Syrovatskii (1967), Syrovatskii and Bulanov (1981). The process is physically sound, but it produces a conversion rate from magnetic to kinetic energy far lower than the one actually found in many astrophysical events, actually explosive. Moreover, the tendency of this configuration is to lengthen the magnetic nozzle and therefore to slow the outflow of plasma (Fig. 2), thus diminishing the efficiency of the energy conversion: it is often said that the plasma is throttled. The first attempt to overcome this problem was due to Petschek (1964), who tried to keep the current sheet short by postulating the existence of magnetoacoustic shocks taking the plasma rapidly away from the neutral, or current sheet. Petschek’s mechanism was later developed and generalized (Axford, 1967; Sonnerup, 1970; Yeh and Axford, 1970; Vasyliunas, 1975). However, the stability of the Petschek configuration was questioned by numerical experiments who failed to keep the current sheet short (Biskamp, 1986, 1993); it later became clear that an appropriate election of boundary conditions could stabilize it and yield fast reconnection rates (Yan et al., 1992; Priest and Forbes, 1992, 2000). The preferred explanation today argues that ions and electrons decouple near the current sheet: this allows the presence of dispersive waves known as whistlers, which help in taking the plasma away from the sheet and thus avoid the throttling of the process. Hence two-fluid MHD equations are necessary to model the plasma at least in the vicinity of the current sheet, yielding correct results in several instances (Drake et al.,
M. Núñez / Fluid Dynamics Research 39 (2007) 447 – 456
449
1997; Shay et al., 2001; Rogers et al., 2001; Biskamp, 2000; Nakamura and Fujimoto, 2006; Nakamura et al., 2006). A simpler, but often satisfactory intermediate step is to include the Hall term in the induction equation, and this is the course we will follow. Finally, we must mention that the effect of turbulence on magnetic reconnection is highly relevant and may destroy the neat image of the current sheet (Matthaeus and Lamkin, 1985, 1986). Our objective is far simpler: since boundary conditions have been proved decisive in the efficiency of reconnection, we wish to find if at least some features of its evolution may be proved to depend directly on them without resource to the integration of the equations. In particular, we will consider the area of the region bounded by two fixed magnetic field lines: if these evolve as in Fig. 2, the area will decrease in time as they tend to close on the current sheet. The addition of the Hall term makes the MHD system intrinsically three-dimensional: however, solutions may be found depending only on two space variables (Craig and Watson, 2003), and this is the case we will consider. We will prove that, except for the diffusive effects, the knowledge of the stream and magnetic flux functions at the boundary, together with the vertical component of the magnetic field, determine completely the area of this region. This knowledge may best be associated with the setting of Dirichlet boundary conditions on the domain boundary for the velocity and the magnetic field. If and are known at time zero and both ∇ and ∇ are known for all time at the boundary, then also and are known for all time. In fact, the full set of Dirichlet conditions is not really necessary to determine the flux functions: it is enough to know the tangential components of velocity and magnetic field at the boundary. This is important because, although the full MHD system is parabolic and Dirichlet conditions are admissible, in the collisionless (ideal) case often studied complete Dirichlet conditions overdetermine the system, while tangential Dirichlet conditions do not.
2. Mathematical setting of the problem Although our arguments will apply to very general geometries, for the sake of concretion we will study a rectangle U centered at the current sheet (see Fig. 3): the thick curves will be magnetic field lines, and their meaning will be explained shortly. As in Fig. 1, the flow approaches the sheet from above and below, and leaves horizontally. Q2
Q1 y
P2
P1
Γ1 Γ0
Ψ=0
x
P3
P4 Q3
Q4
Fig. 3. The reconnection region.
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We must be aware that this is a quasi static picture and that there is no guarantee that this geometric structure will be maintained in a dynamic situation. However, the symmetries of velocity and magnetic field will hold if the original state as well as the boundary conditions keep them. These are: the magnetic field is antisymmetric with respect to both coordinate axes, whereas the velocity is symmetric. Hence field lines are symmetric (but have opposite senses) with respect to both axes, and streamlines are also symmetric, but have the same sense. The induction equation, governing the evolution of the magnetic field, is jB jt
= B − u · ∇B + B · ∇u − h∇ × (J × B)
∇ · B = ∇ · u = 0.
(1)
u represents the (ion) velocity, B the magnetic field, J = ∇ × B the current density, the resistivity and h( 0) the Hall constant, taken as zero in the pure MHD setting. In the collisionless case is taken as zero. As stated before, all the magnitudes will be taken as depending only on the two space variables (x, y), so that both field lines and streamlines lie within cylindrical surfaces whose projection on the (x, y) plane are the level surfaces of the respective flux functions and : u = (y , −x , W ), B = (y , −x , Z). From now on, we will consider only the projection on the (x, y) plane and talk about lines instead of surfaces. Any field of the form A = (g, h, ), where hx − gy = Z is a vector potential for B: B = ∇ × A. and are determined uniquely up to additive constants (depending on time). “Uncurling” the induction equation we find the equation for A: jA jt
= A + u × B − h × (J × B) + ∇ ,
(2)
where is a potential function. Since A is determined up to the addition of any gradient, we may choose the gauge to get ∇ =0, but this does not allow one to fix boundary conditions for A; the problem of choosing an adequate gauge is always delicate. However, since we are only interested in the third component of A, we may reason as follows: since B does not depend on z, we may take also A independent of z for all time. Then (2) implies ∇ = ∇ (t, x, y), i.e. (t, x, y, z) = (t)z + 1 (t, x, y), and therefore jz = (t). Thus the third component of (2) yields j jt
= + (u × B)3 − h(J × B)3 + (t).
(3)
Let us take in the current sheet as = 0: since we intend to study the evolution of the region comprised between two flux lines = i , it is convenient to anchor the central line to avoid spurious effects due to a time-dependent . Analogously, we fix the flow flux function by giving the value = 0 at the union of
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the coordinate axes. Since B = 0 there, (3) yields (t) = −(t, x, 0),
for any point (x, 0) of the current sheet. Since J = (Zy , −Zx , −), (t) = J3 (t, x, 0, 0),
i.e. the independent term is the value of the vertical current at any point of the current sheet. We intend to analyze the evolution in time of the region V dotted in Fig. 3, comprised between two flux lines 0 : = 0 < 0, 1 : = 1 > 0, in order to see how the region where reconnection takes place grows or decreases in area. The flux line = 0, the only one where branching initially occurs, is formed by a central portion (the current sheet) where the field vanishes, and two lateral separatrices. There are several ways to study the variation of a region under an evolution equation. One of the simplest (although involving some use of differentiation of generalized functions) consists in writing the equation satisfied by the characteristic function of this region (whose value is one within it and zero outside). To this end, we first consider any real differentiable function of a single variable. Since j jt
f () = f ()
j jt
,
(3) implies that f () satisfies jf () jt
= f () + f ()(u × B)3 − hf ()(J × B)3 + f ().
(4)
Let us study the advective and Hall terms. Since the third component of u × B is x y − y x , we have f ()(u × B)3 = f ()(x y − y x ) = (∇ × (∇(f ◦ ))) · e3 , where e3 = (0, 0, 1). We have denoted here by f ◦ the composition of both functions, to emphasize the fact that the gradient ∇ must be applied to the composition, and not only to f . Analogously, the third component of −h(J × B) is h(Zy x − Zx y ), and therefore −hf ()(J × B)3 = h(∇ × (Z∇(f ◦ ))) · e3 . Integrating (4) in U and using Stokes’ theorem, j f () dx dy = f () dx dy + f ()∇ · dr jt U jU U +h Zf ()∇ · dr + Uf () dx dy. jU
(5)
In order to analyze the evolution of the size of the relevant subdomains of U , we will need to take nonsmooth f ’s: explicitly we will take f to be the characteristic function (0 ,1 ) (whose value is one within the interval (0 , 1 ) and zero outside). In order to interpret (5) for this function, we will approximate f by smooth functions fn and analyze the limit of the associated identities. Not surprisingly, the value of f will turn out to be the differential in the sense of distributions of f .
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3. Area of the reconnection region Let us approximate (0 ,1 ) pointwise by a sequence of smooth bounded functions fn with support contained in (0 − 1/n, 1 + 1/n). Any of the classical convergence theorems shows that fn () dx dy → (0 ,1 ) () dx dy = m(U ∩ (0 < < 1 )) = m(V ), (6) U
U
that is, the limit is the area of the region V , limited as stated by the flux lines 0 and 1 within U . fn () tends to zero both within the interval (0 , 1 ) and outside it, but the jump at the endpoints implies that for any smooth function g with compact support contained in (a, b), with (0 , 1 ) ⊂ (a, b), a
b
fn ()g() d = −
b a
fn ()g () d → g(0 ) − g(1 ),
(7)
by a simple integration by parts. This, of course, is the same as saying that the limit of fn in the sense of distributions is the difference of Dirac measures 0 − 1 ,
which is the differential of f in the sense of distributions. In order to apply this to the limits of the integrals, we will consider in some detail the term f () dx dy. U
The problem is that while (7) provides the limit value of the integral in of fn g, we have a double integral which we need to reduce to an iterated one of the form d (some function). In general this involves a complicated change of variables, but in our case we know that fn vanishes outside a small neighborhood of 0 and 1 . There we may take orthogonal coordinates so that i becomes effectively one of the coordinate axes; a simple application of Fubini’s theorem will yield the appropriate form of the double integral. Thus, let us fix one i ∈ {0, 1}. For n0 large enough, the set i −1/n0 < < i + 1/n0 forms a neighborhood V0 of the flux curve i : = i within U for which there exist orthogonal trajectories = const. to the flux ones, so that (, ) is an orthogonal system of coordinates there. This may fail for arbitrary open sets or when i branches and ceases to be a one-dimensional manifold, but we may assume that this does not happen for i and a small neighborhood of it. Then j(x, y) d d , fn () dx dy = fn () j ( ,
) V0 W0 where W0 is the counterimage of V0 by the transform (x, y) → (, )(x, y): it has the form (i − 1/n0 , i + 1/n0 ) × (0, L), although the -length L varies somewhat with (since we are confined
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to U ). Since ∇ , ∇ are orthogonal, the jacobian is (|∇ ||∇ |)−1 , and we are left with 0 +1/n0 L d
fn () dx dy = fn () d |∇ ||∇ | 0 −1/n0 0 V0 1 +1/n0 L d . + fn () d |∇ ||∇ | 1 −1/n0 0 As explained before, provided all the relevant functions are continuous, the limit of this when n → ∞ is L L 1 0 d − d , |∇ ||∇ | |∇ ||∇ | 0 0 and, since d /|∇ | = ds (arc length parameter), we are left with the limit ds − ds. 0 |∇ | 1 |∇ | Since = −J3 , this term may be written as J3 J3 ds − ds. 1 |∇ | 0 |∇ | As for the integral of the independent term f () (remember that depends only on the time), by using the same argument as before we find ds ds − f () dx dy = U 0 |∇ | 1 |∇ | ds ds − , = J3 (0) 0 |∇ | 1 |∇ | where, as stated previously, J3 (0) denotes the third component of the current density at any point of the current sheet (depending only on time). Therefore the sum of both terms is J3 − J3 (0) J3 − J3 (0) ds − ds. |∇ | |∇ | 1 0 Let us consider now the boundary integrals. Here the problem of converting the integrals to integrals in simplifies considerably, since at each separate quadrant, is a valid parameter for jU . Thus if r is a positive parametrization of jU , where the positive sense of jU corresponds to increasing (first and third quadrants), we have ∇ · r > 0, whereas when it corresponds to a decreasing (second and fourth quadrants), ∇ · r < 0. Thus, at each quadrant Cj ,
jU ∩Cj
gf n ()∇ ·
dr =
Ij
gf n () d,
where Ij is the interval described by , which includes 0 and 1 .
(8)
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Let Pj be the point where the flux line 0 intersects jU ∩ Cj , Qj the same for 1 (see Fig. 3). Since fn tends to 0 − 1 , the integral in (8) tends to g(Pj ) − g(Qj ). Adding all together and replacing g by the value + hZ occurring in (5), the contribution of the boundary terms is 4 4 (Z(Pj ) − Z(Qj )). ((Pj ) − (Qj )) + h j =1
j =1
Joining all the terms, we obtain the equation for the evolution of the area of V : J3 − J3 (0) J3 − J3 (0) j ds − ds m(V ) = |∇ | |∇ | jt 0 1 +
4 4 (Z(Pj ) − Z(Qj )). ((Pj ) − (Qj )) + h j =1
(9)
j =1
The last two terms are determined by the values of and Z at the boundary; to find the points Pi , Qj we need to know at jU . Only the resistive terms require the knowledge of certain quantities within U , that in principle must be found by integration of the relevant equations. In the classical quasi-static model of Syrovatskii (1971) the field is taken as current-free within U , except for the current sheet. There J3 (0) must be taken as the jump of y across the sheet, [y ] = [B1 ] = y (0, 0+) − y (0, 0−). Since in this case vanishes at i , the first two terms in the right-hand side of (9) become 1 1 ds [y ]. ds − 1 |∇ | 0 |∇ | We cannot guarantee that the current-free condition will last when considering an evolving state, so in general the whole integral in (9) should be included. Notice that |∇ | is precisely the size of the plane component of the magnetic field. Unless this becomes extremely low, for the small values of found in astrophysical plasmas its contribution is likely to be much smaller than the advective or Hall ones. Finally, recall that as asserted in the introduction, the knowledge of and at time zero, plus the knowledge of the plane tangential components of u, B and the value of Z for all time at jU determine directly the nonresistive terms in (9) for all time, without need of integrating the induction equation (1). 4. Conclusions The influence of boundary conditions upon magnetic reconnection processes is a highly relevant topic when analyzing the efficiency of different mechanisms. While in general there is very little that can be said a priori without integration in time of the appropriate MHD equations, some aspects are simple enough to admit a direct representation in terms of the boundary conditions. One of these is the area of the plane region V limited by two magnetic field lines. A decrease of this area could indicate a throttling of the magnetic nozzle and therefore a collapse of energy conversion. We find that the rate of variation of this area is determined exclusively by the values of the stream and flux functions in the boundary of the domain under consideration, plus the vertical component there of the magnetic field if we add the Hall term to the induction equation; in the presence of magnetic diffusivity, a term must be added involving certain line integrals upon the original field lines. Therefore, in the collisionless case, the knowledge of
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the tangential components of magnetic field and velocity upon the boundary of the domain determine completely the area of V without need of solving the MHD equations. It would be absurd to exaggerate the importance of this result: the area of V represents at best a partial indication of how reconnection evolves, and Dirichlet boundary conditions are not the only ones possible in reconnection modeling. In fact there is no universally accepted criterion for this: Biskamp (1986, 1993) admits any boundary condition that does not lead to shocks in the vicinity of the boundary, which leaves considerable freedom; other times the boundary conditions are set by the numerical integration method chosen (Yan et al., 1992). As Priest and Forbes (2000) explain, boundary conditions depend on the particular application and there is nothing particularly meaningful in fixing the stream and flux functions at the boundary. Still, the result obtained certainly shows the decisive influence of boundary conditions in determining essential features of the reconnection process.
Acknowledgments The author wishes to thank Dr. Luis Alberto Tristán for his help in handling the graphics of this paper. Partially supported by Junta de Castilla y León under contract VA003A05.
References Axford, W.I., 1967. Magnetic storm effects associated with the tail of the magnetosphere. Space Sci. Rev. 7, 149–157. Biskamp, D., 1986. Magnetic reconnection via current sheets. Phys. Fluids 29, 1520–1531. Biskamp, D., 1993. Nonlinear Magnetohydrodynamics. Cambridge University Press, Cambridge. Biskamp, D., 2000. Magnetic Reconnection in Plasmas. Cambridge University Press, Cambridge. Craig, I.J.D., Watson, P.G., 2003. Magnetic reconnection solutions based on a generalized Ohm’s law. Solar Phys. 214, 131–150. Drake, J.F., Biskamp, D., Zeiler, A., 1997. Breakup of the electron current layer during 3D collisionless magnetic reconnection. Geophys. Res. Lett. 24, 2921–2924. Imshennik, V.S., Syrovatskii, S.I., 1967. Two-dimensional flow of an ideally conducting gas in the vicinity of the zero line of a magnetic field. Sov. Phys. JETP 25 (4), 656–664. Matthaeus, W.H., Lamkin, S.L., 1985. Rapid magnetic reconnection caused by finite amplitude fluctuations. Phys. Fluids 28 (1), 303–307. Matthaeus, W.H., Lamkin, S.L., 1986. Turbulent magnetic reconnection. Phys. Fluids 29 (8), 2513–2534. Nakamura, T.K.M., Fujimoto, M., 2006. Magnetic reconnection within MHD-scale Kelvin–Helmhotlz vortices triggered by electron inertial effects. Adv. Space Res. 37, 522–526. Nakamura, T.K.M., Fujimoto, M., Otto, A., 2006. Magnetic reconnection induced by weak Kelvin–Helmholtz instability and the formation of the low-latitude boundary layer. Geophys. Res. Lett. 33 (4), L14106. Parker, E.N., 1957. Sweet’s mechanism for merging magnetic fields in conducting fluids. J. Geophys. Res. 62, 509–520. Parker, E.N., 1963. The solar flare phenomenon and theory of reconnection and annihilation of magnetic fields. Astrophys. J. Supp. 8, 177–211. Petschek, H.E., 1964. Magnetic field annihilation. in: Ness W.H. (Ed.), Physics of Solar Flares. NASA SP-50, Washington D.C. pp. 425–439. Priest, E.R., Forbes, T.G., 1992. Does magnetic reconnection exist? J. Geophys. Res. 97, 16757–16772. Priest, E.R., Forbes, T.G., 2000. Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press, Cambridge. Rogers, B.N., Denton, R.E., Drake, J.F., Shay, M.A., 2001. Role of dissipative waves in collisionless magnetic reconnection. Phys. Rev. Lett. 87(19), 195004 1-4.
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Shay, M.A., Drake, J.F., Rogers, B.N., Denton, R.E., 2001. Alfvénic collisionless magnetic reconnection and the Hall term. J. Geophys. Res. 106, 3759–3772. Sonnerup, B.U.O., 1970. Magnetic field reconnection in a highly conducting incompressible fluid. J. Plasma Phys. 4, 161–174. Sweet, P.A., 1958. The production of high-energy particles in solar flares. Nuovo Cimento 8 (Ser. X), 188–196. Syrovatskii, S.I., 1971. Formation of current sheets in a plasma with a frozen-in magnetic field. Sov. Phys. JETP 33 (5), 933–940. Syrovatskii, S.I., Bulanov, S.V., 1981. Pinch sheets and reconnection in astrophysics. Ann. Rev. Astron. Astrophys. 19, 163–227. Vasyliunas, V.M., 1975. Theoretical models of magnetic field line merging. Rev. Geophys. Space Phys. 13, 303–336. Yan, M., Lee, L.C., Priest, E.R., 1992. Fast magnetic reconnection with small shock angles. J. Geophys. Res. 97, 8277–8293. Yeh, T., Axford, W.I., 1970. On the reconnection of magnetic field lines in conducting fluids. J. Plasma Phys. 4, 207–229.
Boundary conditions and plane magnetic reconnection Manuel Núñez∗ Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain Received 8 February 2005; received in revised form 4 December 2006; accepted 5 December 2006 Communicated by M.-E. Brachet
Abstract It is well known that several classical models of plane magnetic reconnection turn out to be more or less efficient according to the choosing of the boundary conditions. We prove analytically that one important datum, the variation of the area of the reconnection region, may be found, except by resistive effects, directly in terms of the flux and stream functions at the boundary. If we include the Hall effect in our model, the vertical component of the field at the boundary is also necessary. Since this area is a measure of the capacity of the system to keep the magnetic nozzle open, this provides a rigorous quantitative measure of the essential role of boundary conditions on the efficiency of the process. © 2007 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. PACS: 52.35.Vd; 52.30.Cv; 96.60.Rd Keywords: Reconnection region; Current sheet; Flux and stream functions
1. Introduction Magnetic reconnection is one of the most important astrophysical processes because of its relevance in the behavior of the magnetospheres of stars and planets, being e.g. responsible of such spectacular phenomena as solar flares. Its theoretical study began with Sweet (1958) and Parker (1957, 1963), who analyzed the collision of two plasma masses possessing opposite magnetic fields. The resulting ∗ Tel.: +34 983 423924; fax: +34 983 423013.
E-mail address: [email protected] 0169-5983/$32.00 © 2007 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved. doi:10.1016/j.fluiddyn.2006.12.001
448
M. Núñez / Fluid Dynamics Research 39 (2007) 447 – 456
Fig. 1. Reconnection at magnetic neutral lines.
Fig. 2. Throttling of the plasma flow.
configuration is a familiar one (Fig. 1): the plasma approaches the neutral sheet where magnetic field lines reconnect, and escapes laterally. The occurrence and properties of this type of geometries in a quasi-static two-dimensional situation was studied in depth by Syrovatskii (1971), Imshennik and Syrovatskii (1967), Syrovatskii and Bulanov (1981). The process is physically sound, but it produces a conversion rate from magnetic to kinetic energy far lower than the one actually found in many astrophysical events, actually explosive. Moreover, the tendency of this configuration is to lengthen the magnetic nozzle and therefore to slow the outflow of plasma (Fig. 2), thus diminishing the efficiency of the energy conversion: it is often said that the plasma is throttled. The first attempt to overcome this problem was due to Petschek (1964), who tried to keep the current sheet short by postulating the existence of magnetoacoustic shocks taking the plasma rapidly away from the neutral, or current sheet. Petschek’s mechanism was later developed and generalized (Axford, 1967; Sonnerup, 1970; Yeh and Axford, 1970; Vasyliunas, 1975). However, the stability of the Petschek configuration was questioned by numerical experiments who failed to keep the current sheet short (Biskamp, 1986, 1993); it later became clear that an appropriate election of boundary conditions could stabilize it and yield fast reconnection rates (Yan et al., 1992; Priest and Forbes, 1992, 2000). The preferred explanation today argues that ions and electrons decouple near the current sheet: this allows the presence of dispersive waves known as whistlers, which help in taking the plasma away from the sheet and thus avoid the throttling of the process. Hence two-fluid MHD equations are necessary to model the plasma at least in the vicinity of the current sheet, yielding correct results in several instances (Drake et al.,
M. Núñez / Fluid Dynamics Research 39 (2007) 447 – 456
449
1997; Shay et al., 2001; Rogers et al., 2001; Biskamp, 2000; Nakamura and Fujimoto, 2006; Nakamura et al., 2006). A simpler, but often satisfactory intermediate step is to include the Hall term in the induction equation, and this is the course we will follow. Finally, we must mention that the effect of turbulence on magnetic reconnection is highly relevant and may destroy the neat image of the current sheet (Matthaeus and Lamkin, 1985, 1986). Our objective is far simpler: since boundary conditions have been proved decisive in the efficiency of reconnection, we wish to find if at least some features of its evolution may be proved to depend directly on them without resource to the integration of the equations. In particular, we will consider the area of the region bounded by two fixed magnetic field lines: if these evolve as in Fig. 2, the area will decrease in time as they tend to close on the current sheet. The addition of the Hall term makes the MHD system intrinsically three-dimensional: however, solutions may be found depending only on two space variables (Craig and Watson, 2003), and this is the case we will consider. We will prove that, except for the diffusive effects, the knowledge of the stream and magnetic flux functions at the boundary, together with the vertical component of the magnetic field, determine completely the area of this region. This knowledge may best be associated with the setting of Dirichlet boundary conditions on the domain boundary for the velocity and the magnetic field. If and are known at time zero and both ∇ and ∇ are known for all time at the boundary, then also and are known for all time. In fact, the full set of Dirichlet conditions is not really necessary to determine the flux functions: it is enough to know the tangential components of velocity and magnetic field at the boundary. This is important because, although the full MHD system is parabolic and Dirichlet conditions are admissible, in the collisionless (ideal) case often studied complete Dirichlet conditions overdetermine the system, while tangential Dirichlet conditions do not.
2. Mathematical setting of the problem Although our arguments will apply to very general geometries, for the sake of concretion we will study a rectangle U centered at the current sheet (see Fig. 3): the thick curves will be magnetic field lines, and their meaning will be explained shortly. As in Fig. 1, the flow approaches the sheet from above and below, and leaves horizontally. Q2
Q1 y
P2
P1
Γ1 Γ0
Ψ=0
x
P3
P4 Q3
Q4
Fig. 3. The reconnection region.
M. Núñez / Fluid Dynamics Research 39 (2007) 447 – 456
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We must be aware that this is a quasi static picture and that there is no guarantee that this geometric structure will be maintained in a dynamic situation. However, the symmetries of velocity and magnetic field will hold if the original state as well as the boundary conditions keep them. These are: the magnetic field is antisymmetric with respect to both coordinate axes, whereas the velocity is symmetric. Hence field lines are symmetric (but have opposite senses) with respect to both axes, and streamlines are also symmetric, but have the same sense. The induction equation, governing the evolution of the magnetic field, is jB jt
= B − u · ∇B + B · ∇u − h∇ × (J × B)
∇ · B = ∇ · u = 0.
(1)
u represents the (ion) velocity, B the magnetic field, J = ∇ × B the current density, the resistivity and h( 0) the Hall constant, taken as zero in the pure MHD setting. In the collisionless case is taken as zero. As stated before, all the magnitudes will be taken as depending only on the two space variables (x, y), so that both field lines and streamlines lie within cylindrical surfaces whose projection on the (x, y) plane are the level surfaces of the respective flux functions and : u = (y , −x , W ), B = (y , −x , Z). From now on, we will consider only the projection on the (x, y) plane and talk about lines instead of surfaces. Any field of the form A = (g, h, ), where hx − gy = Z is a vector potential for B: B = ∇ × A. and are determined uniquely up to additive constants (depending on time). “Uncurling” the induction equation we find the equation for A: jA jt
= A + u × B − h × (J × B) + ∇ ,
(2)
where is a potential function. Since A is determined up to the addition of any gradient, we may choose the gauge to get ∇ =0, but this does not allow one to fix boundary conditions for A; the problem of choosing an adequate gauge is always delicate. However, since we are only interested in the third component of A, we may reason as follows: since B does not depend on z, we may take also A independent of z for all time. Then (2) implies ∇ = ∇ (t, x, y), i.e. (t, x, y, z) = (t)z + 1 (t, x, y), and therefore jz = (t). Thus the third component of (2) yields j jt
= + (u × B)3 − h(J × B)3 + (t).
(3)
Let us take in the current sheet as = 0: since we intend to study the evolution of the region comprised between two flux lines = i , it is convenient to anchor the central line to avoid spurious effects due to a time-dependent . Analogously, we fix the flow flux function by giving the value = 0 at the union of
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the coordinate axes. Since B = 0 there, (3) yields (t) = −(t, x, 0),
for any point (x, 0) of the current sheet. Since J = (Zy , −Zx , −), (t) = J3 (t, x, 0, 0),
i.e. the independent term is the value of the vertical current at any point of the current sheet. We intend to analyze the evolution in time of the region V dotted in Fig. 3, comprised between two flux lines 0 : = 0 < 0, 1 : = 1 > 0, in order to see how the region where reconnection takes place grows or decreases in area. The flux line = 0, the only one where branching initially occurs, is formed by a central portion (the current sheet) where the field vanishes, and two lateral separatrices. There are several ways to study the variation of a region under an evolution equation. One of the simplest (although involving some use of differentiation of generalized functions) consists in writing the equation satisfied by the characteristic function of this region (whose value is one within it and zero outside). To this end, we first consider any real differentiable function of a single variable. Since j jt
f () = f ()
j jt
,
(3) implies that f () satisfies jf () jt
= f () + f ()(u × B)3 − hf ()(J × B)3 + f ().
(4)
Let us study the advective and Hall terms. Since the third component of u × B is x y − y x , we have f ()(u × B)3 = f ()(x y − y x ) = (∇ × (∇(f ◦ ))) · e3 , where e3 = (0, 0, 1). We have denoted here by f ◦ the composition of both functions, to emphasize the fact that the gradient ∇ must be applied to the composition, and not only to f . Analogously, the third component of −h(J × B) is h(Zy x − Zx y ), and therefore −hf ()(J × B)3 = h(∇ × (Z∇(f ◦ ))) · e3 . Integrating (4) in U and using Stokes’ theorem, j f () dx dy = f () dx dy + f ()∇ · dr jt U jU U +h Zf ()∇ · dr + Uf () dx dy. jU
(5)
In order to analyze the evolution of the size of the relevant subdomains of U , we will need to take nonsmooth f ’s: explicitly we will take f to be the characteristic function (0 ,1 ) (whose value is one within the interval (0 , 1 ) and zero outside). In order to interpret (5) for this function, we will approximate f by smooth functions fn and analyze the limit of the associated identities. Not surprisingly, the value of f will turn out to be the differential in the sense of distributions of f .
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3. Area of the reconnection region Let us approximate (0 ,1 ) pointwise by a sequence of smooth bounded functions fn with support contained in (0 − 1/n, 1 + 1/n). Any of the classical convergence theorems shows that fn () dx dy → (0 ,1 ) () dx dy = m(U ∩ (0 < < 1 )) = m(V ), (6) U
U
that is, the limit is the area of the region V , limited as stated by the flux lines 0 and 1 within U . fn () tends to zero both within the interval (0 , 1 ) and outside it, but the jump at the endpoints implies that for any smooth function g with compact support contained in (a, b), with (0 , 1 ) ⊂ (a, b), a
b
fn ()g() d = −
b a
fn ()g () d → g(0 ) − g(1 ),
(7)
by a simple integration by parts. This, of course, is the same as saying that the limit of fn in the sense of distributions is the difference of Dirac measures 0 − 1 ,
which is the differential of f in the sense of distributions. In order to apply this to the limits of the integrals, we will consider in some detail the term f () dx dy. U
The problem is that while (7) provides the limit value of the integral in of fn g, we have a double integral which we need to reduce to an iterated one of the form d (some function). In general this involves a complicated change of variables, but in our case we know that fn vanishes outside a small neighborhood of 0 and 1 . There we may take orthogonal coordinates so that i becomes effectively one of the coordinate axes; a simple application of Fubini’s theorem will yield the appropriate form of the double integral. Thus, let us fix one i ∈ {0, 1}. For n0 large enough, the set i −1/n0 < < i + 1/n0 forms a neighborhood V0 of the flux curve i : = i within U for which there exist orthogonal trajectories = const. to the flux ones, so that (, ) is an orthogonal system of coordinates there. This may fail for arbitrary open sets or when i branches and ceases to be a one-dimensional manifold, but we may assume that this does not happen for i and a small neighborhood of it. Then j(x, y) d d , fn () dx dy = fn () j ( ,
) V0 W0 where W0 is the counterimage of V0 by the transform (x, y) → (, )(x, y): it has the form (i − 1/n0 , i + 1/n0 ) × (0, L), although the -length L varies somewhat with (since we are confined
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to U ). Since ∇ , ∇ are orthogonal, the jacobian is (|∇ ||∇ |)−1 , and we are left with 0 +1/n0 L d
fn () dx dy = fn () d |∇ ||∇ | 0 −1/n0 0 V0 1 +1/n0 L d . + fn () d |∇ ||∇ | 1 −1/n0 0 As explained before, provided all the relevant functions are continuous, the limit of this when n → ∞ is L L 1 0 d − d , |∇ ||∇ | |∇ ||∇ | 0 0 and, since d /|∇ | = ds (arc length parameter), we are left with the limit ds − ds. 0 |∇ | 1 |∇ | Since = −J3 , this term may be written as J3 J3 ds − ds. 1 |∇ | 0 |∇ | As for the integral of the independent term f () (remember that depends only on the time), by using the same argument as before we find ds ds − f () dx dy = U 0 |∇ | 1 |∇ | ds ds − , = J3 (0) 0 |∇ | 1 |∇ | where, as stated previously, J3 (0) denotes the third component of the current density at any point of the current sheet (depending only on time). Therefore the sum of both terms is J3 − J3 (0) J3 − J3 (0) ds − ds. |∇ | |∇ | 1 0 Let us consider now the boundary integrals. Here the problem of converting the integrals to integrals in simplifies considerably, since at each separate quadrant, is a valid parameter for jU . Thus if r is a positive parametrization of jU , where the positive sense of jU corresponds to increasing (first and third quadrants), we have ∇ · r > 0, whereas when it corresponds to a decreasing (second and fourth quadrants), ∇ · r < 0. Thus, at each quadrant Cj ,
jU ∩Cj
gf n ()∇ ·
dr =
Ij
gf n () d,
where Ij is the interval described by , which includes 0 and 1 .
(8)
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Let Pj be the point where the flux line 0 intersects jU ∩ Cj , Qj the same for 1 (see Fig. 3). Since fn tends to 0 − 1 , the integral in (8) tends to g(Pj ) − g(Qj ). Adding all together and replacing g by the value + hZ occurring in (5), the contribution of the boundary terms is 4 4 (Z(Pj ) − Z(Qj )). ((Pj ) − (Qj )) + h j =1
j =1
Joining all the terms, we obtain the equation for the evolution of the area of V : J3 − J3 (0) J3 − J3 (0) j ds − ds m(V ) = |∇ | |∇ | jt 0 1 +
4 4 (Z(Pj ) − Z(Qj )). ((Pj ) − (Qj )) + h j =1
(9)
j =1
The last two terms are determined by the values of and Z at the boundary; to find the points Pi , Qj we need to know at jU . Only the resistive terms require the knowledge of certain quantities within U , that in principle must be found by integration of the relevant equations. In the classical quasi-static model of Syrovatskii (1971) the field is taken as current-free within U , except for the current sheet. There J3 (0) must be taken as the jump of y across the sheet, [y ] = [B1 ] = y (0, 0+) − y (0, 0−). Since in this case vanishes at i , the first two terms in the right-hand side of (9) become 1 1 ds [y ]. ds − 1 |∇ | 0 |∇ | We cannot guarantee that the current-free condition will last when considering an evolving state, so in general the whole integral in (9) should be included. Notice that |∇ | is precisely the size of the plane component of the magnetic field. Unless this becomes extremely low, for the small values of found in astrophysical plasmas its contribution is likely to be much smaller than the advective or Hall ones. Finally, recall that as asserted in the introduction, the knowledge of and at time zero, plus the knowledge of the plane tangential components of u, B and the value of Z for all time at jU determine directly the nonresistive terms in (9) for all time, without need of integrating the induction equation (1). 4. Conclusions The influence of boundary conditions upon magnetic reconnection processes is a highly relevant topic when analyzing the efficiency of different mechanisms. While in general there is very little that can be said a priori without integration in time of the appropriate MHD equations, some aspects are simple enough to admit a direct representation in terms of the boundary conditions. One of these is the area of the plane region V limited by two magnetic field lines. A decrease of this area could indicate a throttling of the magnetic nozzle and therefore a collapse of energy conversion. We find that the rate of variation of this area is determined exclusively by the values of the stream and flux functions in the boundary of the domain under consideration, plus the vertical component there of the magnetic field if we add the Hall term to the induction equation; in the presence of magnetic diffusivity, a term must be added involving certain line integrals upon the original field lines. Therefore, in the collisionless case, the knowledge of
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the tangential components of magnetic field and velocity upon the boundary of the domain determine completely the area of V without need of solving the MHD equations. It would be absurd to exaggerate the importance of this result: the area of V represents at best a partial indication of how reconnection evolves, and Dirichlet boundary conditions are not the only ones possible in reconnection modeling. In fact there is no universally accepted criterion for this: Biskamp (1986, 1993) admits any boundary condition that does not lead to shocks in the vicinity of the boundary, which leaves considerable freedom; other times the boundary conditions are set by the numerical integration method chosen (Yan et al., 1992). As Priest and Forbes (2000) explain, boundary conditions depend on the particular application and there is nothing particularly meaningful in fixing the stream and flux functions at the boundary. Still, the result obtained certainly shows the decisive influence of boundary conditions in determining essential features of the reconnection process.
Acknowledgments The author wishes to thank Dr. Luis Alberto Tristán for his help in handling the graphics of this paper. Partially supported by Junta de Castilla y León under contract VA003A05.
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