On the feasibility of self-similar solutions of the MHD equations near a magnetic null point

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Highlights

  • A classical ansatz for the MHD equations near a magnetic null point is studied.

  • The transmission conditions to connect this with an outer bounded solution are detailed.

  • It is found that the outer total pressure may act to blow up the original geometry.

  • Even when it does not, instability is generic in this type of problems.

Abstract

Hyperbolic neutral points of plane magnetic fields are the main feature in simple models of magnetic reconnection. There is some discussion on the stability of the magnetic geometry as depending on the boundary conditions. We consider a classical family of self-similar solutions to the magnetohydrodynamic equations in a bounded neighborhood of the neutral point and connect them with well behaved solutions at infinity. It is found that even in the case when there is no blow up to the whole configuration, the magnetic topology is generically unstable.

Introduction

Hyperbolic critical points of the magnetic field in two-dimensional magnetohydrodynamics have a long history as models for studying the mechanism of magnetic reconnection, but they are also relevant in other important phenomena in Astrophysics. A recent example is the conversion between types of hydromagnetic waves taking place near magnetic null points [1], [2]. Let us review briefly the main points of the fast reconnection problem: Sweet and Parker [3], [4] had studied the rate of energy conversion in the collision of two masses of plasma provided with opposite magnetic fields. This rate turns out to be much smaller than the one observed in several phenomena, such as solar flares. The first proposal to remedy this was made by Petschek [5], [6]; central to his hypothesis was the possibility of keeping short the current sheet where opposing field lines meet, so that the X-form of the field lines would not degenerate. The original model was later generalized to a whole family of solutions [7], [8]. However, D. Biskamp [9], [10], [11], in a series of numerical experiments, showed that the current sheet tends to expand and the rate of reconnection to decrease, concluding that the Petschek model was not feasible. Other researchers, notably E.R. Priest and his collaborators, argued that the model consistency depends on the appropriate boundary conditions, and that the essence of process remains a valid explanation in several instances [12], [13], [14], [15], [16]. Other authors remain skeptical of the presence in nature of this process; it seems that at constant resistivity, Petschek’s slow shocks do not occur [17], [18], [19]. All those arguments depend in an essential way in what happens at the sheet, but even taking into account only the global geometry of these X-type configurations, the opinion that field lines tend to flatten out seems very extended [20], [21]. The real process of fast magnetic reconnection is sure to involve a far more complex description than the one of single-fluid magnetohydrodynamics; at least the Hall current and possibly electron inertia seem necessary [22], [23], [24]. We will focus on the dynamics of the problem, i.e. if a reasonable boundary condition may keep the geometry of the field lines. There exists a family of self-similar solutions that have proved useful to study basic features of magnetic reconnection [25], [26], [27], as well as the effect of additional forcings such as the Hall current [28], [29], [30]. These functional forms of magnetic field and velocity, however, cannot hold in the whole plane because both quantities behave linearly and thus grow indefinitely at infinity, which is unphysical. We have assumed that these solutions connect outside a bounded domain with other solutions, presumably well behaved at infinity, so that the Rankine–Hugoniot relations must hold at the interface. It will be shown that the outer total pressure acts as a forcing on the inner domain, and that even when one avoids a blow up of the velocity, the X-type topology of the magnetic field lines is generically unstable. Therefore these self-similar solutions may only be acceptable as a short time description of the behavior of the flow.

Section snippets

Evolution equations and boundary conditions

We start from the equations of ideal MHD as applied to an incompressible fluid: vt+vv=p+J×B,Bt=×(v×B),v=B=0, where v represents the fluid velocity, B the magnetic field, J=×B the current density and p the kinetic pressure. (1) is the momentum equation and (2) the induction one. We assume that both v and B lie in the (x,y) plane and depend only on these variables. Then the solenoidal character of v and B in simply connected domains translates into the existence of a streamfunction ϕ

Evolution of the magnetic topology

Let us call the outer total pressure p¯(t,x,y), so that p¯ must have the form given in (19) in Γ, but now it is considered an external term which is determined by the outer flow and field. As stated, this means that it is enough to consider a point different from 0, (x0(t),y0(t))Γ(t). Thus 12(x0(t)2y0(t)2)γ̇(t)12(x0(t)2+y0(t)2)γ(t)22α0β0(x0(t)2+y0(t)2)=p¯(t,x0(t),y0(t)). There exists the singular case, given by the lines x=±y; if x0(t)2=y0(t)2, then γ(t)2=4α0β02p¯(t,x0(t),y0(t))x0(t)2+y

Conclusions

The evolution of an hyperbolic neutral point in two dimensional magnetohydrodynamics possesses some interest as the geometry of choice in simple models of magnetic reconnection, as well as in some other phenomena such as mode conversion. While in the classical Sweet–Parker configuration the X formed by the magnetic field line tends to flatten and the current sheet to extend laterally, in a series of models pioneered by Petschek the X remains narrow and the current sheet short. This has been

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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