A&A 507, 2933 (2009)
Simple analytical examples of boundary driven evolution of a twodimensional magnetohydrostatic equilibrium
J. J. Aly
AIM  Unité Mixte de Recherche CEA  CNRS  Université Paris VII  UMR n 7158, Centre d'Études de Saclay, 91191 GifsurYvette Cedex, France
Received 20 July 2009 / Accepted 1 September 2009
Abstract
Aims. We construct families of timesequences of xinvariant
magnetostatic equilibria which describe ideal quasistatic evolutions
driven by stationary shearing motions imposed on a boundary. The change
in the thermal pressure of the plasma is determined by imposing either
an adiabatic, or an isothermal, or an isobaric, prescription.
Methods. We start from a well known family of linear
forcefree fields, on which we effect simple transforms.
Results. In either case, the magnetic field and the
pressure are
expressed analytically as functions of space and time. The field is
found to suffer an indefinite expansion, with a decrease to zero of the
pressure in the adiabatic and isothermal cases, and to eventually open.
Moreover, the configurations forming any sequence are shown to be
linearly stable with respect to xinvariant
perturbations.
Key words: magnetohydrodynamics (MHD)  Sun: magnetic fields  Sun: corona  Sun: coronal mass ejections (CMEs)
1 Introduction
Solar eruptive phenomena occurring in very elongated structures present in active regions have often been studied by using a simplified 2 D model. In the latter, the corona is represented by a halfspace containing a magnetized low beta highly conducting plasma with properties independent of the xcoordinate. The magnetic field has an arcade topology, and it is imposed to evolve quasistatically through a sequence of forcefree configurations as a result of slow shearing motions imposed to its footpoints on the ``photospheric'' boundary . Energy thus gets stored in the field, and one looks for the possibility of reaching some critical state beyond which a catastrophic release of a part of that energy becomes unavoidable. Analytical studies (Aly 1994,1990,1985) have shown that one of the most significant feature of such an evolution in the ideal MHD case is an indefinite expansion of the field leading asymptotically (for ) to its partial or full opening, with the formation of an infinitely thin currentsheet, a transition by reconnection to a lower energy state becoming however energetically favorable at some stage if resistivity is introduced in the model. Numerical simulations based on both dynamical and static schemes (Choe & Lee 1996; Amari et al. 1996) have lead to similar conclusions, and have also provided valuable descriptions of the nonideal reconnection phase.
Although the forcefree assumption appears to be relevant for representing slowly evolving structures lying in the midcorona, where the plasma beta has effectively a low value, it may be much less justified to use it for studying structures extending up to the upper corona. In the latter region, pressure and gravity forces are no longer fully negligible, and it has even been suggested by several authors that they may play a crucial role in the triggering of eruptive events (e.g., Shibasaki 2001; Low & Smith 1993). As yet these forces have been taken into account in some 2 D numerical simulations (Finn & Chen 1990; Choe & Lee 1996; Zwingmann 1987), but not much seems to have been done from an analytical point of view (see, however, Aly 1994, in which a particular example is discussed). This has lead us to undertake a general study of the evolution of a non forcefree 2 D equilibrium. As a first step, we have looked for exact solutions of the quasistatic evolution problem including the effects of the thermal pressure (gravity is still neglected). The aim of this paper is to present families of such solutions which seem to have not been noticed before, in spite of the fact that they can be obtained by effecting some simple transforms on a well known sequence of linear forcefree solutions (described, e.g., in Priest & Forbes 1990).
The paper is organized as follows. We first state precisely (Sect. 2) the general evolutionary problem in which we are interested. Thus we explain (Sect. 3) our method for transforming a sequence of forcefree fields describing an evolution driven by a stationary shearing velocity field into a nonforcefree sequence having the same property, with the pressure of the plasma evolving according to either an adiabatic prescription, or an isothermal one, or an isobaric one. This method works if the original sequence satisfies a peculiar condition, which is shown in Sect. 4 to be fulfilled by the sequence of linear forcefree fields alluded to above. We can thus obtain new sequences of equilibria, whose properties are studied in Sect. 5. Our results are summarized and discussed in Sect. 6.
2 Statement of the general problem
2.1 Assumptions
We use hereafter Cartesian coordinates (x,y,z). For , we define D, S, and to be, respectively, the domain , its lower plane boundary , and its crosssection . We assume that D contains a magnetized perfectly conducting plasma whose properties are left invariant by the translations parallel to the xaxis. We denote as and p(y,z,t), respectively, the magnetic field and the plasma thermal pressure at time t. Initially (at t=0) the system is in a given state of equilibrium, with the pressure force being balanced by the Lorentz force (the gravitational force is neglected here). The total energy (magnetic+thermal) per unit of xlength is finite, and is taken to have an arcade topology and no shear ( ). In the case , we also require that , i.e., the field does not thread the lateral boundaries, taken otherwise to be perfectly conducting.
For , a velocity field is applied to the magnetic footpoints on S (also taken to be perfectly conducting), with v(y) decreasing fast enough at infinity in the case for the energy input rate to keep a finite value. Consequently the system is driven into an evolution which is assumed to be quasistatic. Of course, we also need to fix a rule determining the behavior of the pressure. Hereafter, we shall use in turn the following prescriptions: (i) adiabatic prescription (Finn & Chen 1990; Choe & Lee 1996): the mass and the entropy contents in any flux tube are conserved, which amounts to consider the ``photosphere'' S as a wall impeding any exchange of matter and heat between the ``corona'' D and the subphotospheric region, while assuming that there are no heat sources or sinks in D. (ii) Isothermal prescription: the mass in any flux tube is conserved, while the temperature T keeps a constant externally fixed value. Then mass transfer through S is forbidden, while energy may be exchanged with a heat reservoir (thermostat). (iii) Isobaric prescription (Zwingmann 1987): the pressure keeps its initial value on each magnetic line. In that case, the photosphere is considered as a reservoir regulating p by allowing plasma to flow into or out the corona. There does not seem to be yet an agreement on which one of these assumptions may be the most realistic for describing the corona (see the discussion in Schindler 2006). Actually, it seems likely that to obtain a definitive answer to that question it will be necessary to introduce a more global model in which both the corona and the subphotospheric layers (and then the exchange between them) are taken into account. Meanwhile, studying the consequences of our three different assumptions and comparing them is certainly a profitable exercice.
2.2 Equations
We use the standard representation
in which the field is expressed in terms of a ``toroidal'' function B_{x}(y,z,t) and a ``poloidal'' flux function A(y,z,t). For A to be uniquely defined, we require
(we can impose Eq. (2) because , this condition resulting from the fact that the lateral walls are perfectly conducting and the assumption ). The level contours of A(y,z,t) in are the field lines of (just note that ).
Equilibrium at each time t requires B_{x} and p to be of the form
and A to satisfy the GradShafranov equation
where a dot denotes a derivative with respect to A at constant time (see, e.g., Schindler 2006, p. 78). Moreover, we need to have at each time t (finite energy condition)
The initial equilibrium is given, and it satisfies . Its lines have an arcade topology, i.e., they connect two points of S by just bridging above the socalled polarity inversion line (along which B_{z}=0). Without any loss of generality, we assume that all the lines emerge from the right side of the latter, which implies that A(y,z,t)>0. As a direct consequence of the frozenin law and of the velocity field on S being directed along , the flux function on S stays invariant,
and the lines in keep their initial topology,
It may be worth recalling that this means that is obtained from by a continuous deformation keeping fixed the footpoints on the lower boundary of .
Finally, we have to write equations prescribing how B_{x}(a,t)
and p(a,t)
change with time. For that we introduce the function
where , and its derivative
represents the area of the subdomain of comprised between the line and the yaxis, and then is the area of the domain comprised between the poloidal lines and . To get an equation for B_{x}(a,t), we consider a line of admitting as its projection onto , and define its shear X(a,t) to be the difference between the xcoordinates of its left and right footpoints, respectively. Then we have on the one hand (just use the equation determining a magnetic line and Eqs. (4) and (11)), and on the other hand , where denote the ycoordinates of the left () and right (+) footpoints (we use here the fact that X(a,t) is created by the stationary velocity field imposed on S). Therefore B_{x} has to obey the nonlocal equation
The pressure function p(a,t) is obviously given by
where p(a,0) is its given initial value, and we select for an isobaric evolution, for an isothermal evolution, and for an adiabatic evolution, with being the adiabatic index of the gas. For theoretical purpose, the latter will be taken here as an arbitrary parameter.
To summarize, our problem  referred to as EvPb hereafter  consists to determine a sequence satisfying Eqs. (2)/(3)(9) and (12), (13), when we are given an initial unsheared configuration (A,B_{x}=0,p)(y,z,0), a velocity profile v(y) (and then a shear function ), and a value of . In the case where , EvPb reduces to the forcefree evolutionary problem studied in the papers quoted in the introduction.
3 A tentative method to get solutions to EvPb having p 0 from a forcefree solution
We assume that we know a forcefree solution of EvPb  i.e., a solution for which the pressure vanishes , and we choose a time t_{0}>0. Then we set for
(14)  
(15)  
(16) 
where an overline indicates that a quantity is associated to , and is a given number either equal to 0, or equal to 1, or larger than 1. The timesequence that we obtain that way is a sequence of arcade equilibria as and have the same flux function, , and the same righthand side of the GradShafranov Eq. (6) ( ). It starts at the new initial time t_{0} from the nonpotential unsheared equilibrium
(17) 
and the thermal pressure evolves according to Eq. (13).
The shear X(a,t)
of the new field satisfies the first equality in Eq. (12) and then
where we have used the relation . If we now suppose that the factorization
holds true for some and some functions and f_{t0}(t), with f_{t0}(t_{0})=0 and f_{t0}(t) increasing, we can adopt t'=f_{t0}(t) as a new time (as we consider a quasistatic evolution, there is no dynamics involved and time just plays the role of a labeling parameter for the configurations), and we get after substituting f_{t0}^{1}(t') for t in a sequence which is clearly another solution to EvPb.
Our ``method'' for constructing a solution to EvPb having should now appear clearly. It just consists to look for forcefree solutions whose function X^{2} defined by Eq. (18) possesses the factorization property (19) for some value of , and to apply to it the transform explained above. Obviously the method works for any if we take (in that case the pressure p just evolves as did the magnetic pressure , and we can take and ). That it works for other values of may appear a priori dubious. However, it turns out to be the case, as shown in the next section.
4 An example of application of the method
4.1 A forcefree solution to EvPb
We start with a well known construction which may be found, e.g., in Priest & Forbes (1990). We first introduce the sequence of linear arcade forcefree fields defined in (with ) by
where B_{0}>0 is a constant, , and is a parameter ranging in ]k,k[. A short calculation shows that the differential area function (see Eq. (11)) associated to is given by
where . Using the first equality in Eq. (12), we thus obtain for the shear of a line of on which
(23) 
Next we set for
where T is some fixed number, and we define a timesequence of arcade forcefree fields by
(25) 
For , we have
and the shear function writes
is just of the form occuring in Eq. (12) and then appears to be the solution of the EvPb defined by the initial arcade potential field , and the stationnary velocity field
which is easily checked to produce the shear profile .
4.2 The new solutions
A simple look at Eq. (26) shows that the ratio (with t and t_{0} being arbitrary positive numbers) is independent of a, and that the term inside the bracket in the righthand side of Eq. (18) is always positive. Then the field has the factorization property (19) for all the values of , and we can apply the method of the previous section to produce solutions to EvPb with . Actually, it will prove more convenient here to start directly from the sequence .
We thus fix a number , and introduce the new time sequence of equilibria, , , by setting
where is an unknown function satisfying . This sequence starts from the unsheared equilibrium defined by
(32) 
with the latter having a beta at the origin given by
Using Eqs. (12), (22), and (30) we get the relation
(34) 
for the shear X(a,t). The latter can be put into the form by choosing to be of the form (27)  the shearing velocity field thus keeping the form (28), with T having the sign of B_{x}  and by requiring to be a solution of the equation
Simple considerations show that Eq. (35) has a unique solution for , with the latter satisfying
Then increases with t (actually from up to k^{2}) and with , while it decreases with .
In many cases, it is possible to give in closed form:
 For the isobaric law, ,
we have
 For the isothermal law, ,
we can write
 For the adiabatic law, ,
a closed form solution is possible for the particular values
of
for which Eq. (35)
reduces to an algebraic equation of degree no larger than 5 (
).
For instance, we obtain in the simplest case where
We shall refrain here from writing the quite heavy formulas for the other values of as they are not very illuminating.
5 Some properties of the new solutions
5.1 Expansion of the poloidal structure
It results immediately from Eq. (21) that the
maximum height, Z(a,t),
reached at time t by the line
is given by
(40) 
When t increases from 0 to infinity, increases from up to k^{2}, and then Z(a,t) increases monotonically up to infinity, with an asymptotically constant speed. Then shearing the footpoints leads to a continuous expansion of the poloidal structure and to its eventual opening. Moreover, it results from Eq. (36) that increasing leads to an increase of Z(a,t) while increasing leads to a decrease of that quantity. This behavior is illustrated in Fig. 1 where we have plotted Z(a,t)/L for a=a_{0}/2, , and such that (see Eq. (33)).
Figure 1: Evolution of Z(a_{0}/2,t)/L as a function of t/T for a) and b). In both a) and b), we have plotted ( from the bottom to the top) the forcefree case ( , black curve), and the three cases (green), (red), and (blue). 

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The open equilibrium which is asymptotically approached when is 1D for , being given by
(41)  
(42) 
and 1 D for , in which case
(43)  
(44) 
In these final states, the total pressure P (magnetic+thermal) is uniform ( ), but no current sheet is present (this is a peculiar feature due to their 1/1 D character).
5.2 Energy
Let us denote by W_{p}, W_{x}, and W_{i}, respectively, the poloidal magnetic energy (energy of ), the toroidal magnetic energy (energy of B_{x}), and the plasma internal energy (of course ``energy'' means here ``energy per unit of xlength''). Then we get after a short calculation
(45)  
(46)  
(47) 
and the total energy W=W_{p}+W_{x}+W_{i} is given by
(48) 
As it is easily checked, W_{p}, W_{x}, and W, increase monotonically with time in any case, and they all tend to infinity when , while W_{i} increases up to infinity in the isobaric case (), keeps a constant value in the isothermal case (), and decreases to zero in the adiabatic case ().
5.3 Linear stability
Consider any one of the configurations forming a solution to
EvPb,
and submit it  the driving boundary motions being
frozen  to an arbitrary small 2
D displacement field .
We require
to satisfy the conditions
(49) 
which express the fact that the boundary of D is rigid and perfectly conducting. Then we claim that the configuration is ideally linearly stable with respect to this perturbation. This statement is an immediate consequence of a general result reported in Schindler (2006, p. 216). According to the latter, a 2 D configuration in is stable if there does exist a fixed direction such that keeps the same sign in the whole domain. This is clearly the case here, with , as we have .
6 Conclusion
Up to now the analytical problem of the boundary driven quasistatic evolution of an xinvariant magnetostatic equilibrium occupying either a halfspace or a vertical slice of it has been essentially considered in the case where the magnetic field is forcefree. In this paper, we have proposed a method which allows to construct examples in which the thermal pressure of the plasma is taken into account. The method amounts to transform a solution of a forcefree EvPb into a solution of a nonforcefree EvPb, with p evolving according to some prescribed law, either adiabatic, or isothermal, or isobaric. For the method to apply, the initial forcefree solution has however to obey a strong peculiar constraint. The latter has been checked to be satisfied by a well known linear forcefree solution which has been already used by many authors, and we have explicitly constructed from it solutions to EvPb driven by the same boundary velocity field, under each one of the three prescriptions for p(t) recalled above. In the case where the constraint is not satisfied, the transform may be still effected, but it leads to evolutionary sequences which are driven by boundary motions whose velocity profile evolves in time and becomes in any case dependent of the imposed pressureprescription. Unfortunately, the transform from a forcefree state to a nonforcefree one which is at the base of the method is specific to the class of translationinvariant equilibria, for which the thermal pressure and the magnetic pressure associated to the component of the field along the direction of invariance intervene on the same footing in the GradShafranov equation. This property is no longer true for either the axisymmetric or the helical equilibria, and a fortiori for the 3D ones.
It should be noted that many authors have already provided examples of conversion of some particular magnetostatic equilibrium into a new one as a result of the application of an adequate mathematical transform (e.g., Lites et al. 1995; Low 1982; Aly 2009). For instance, Low 1982 has shown that any one of the specific unsheared 2D equilibria we have used in Sect. 4.2 as possible initial states of a quasistatic evolution, can be transformed into a 3D laminar equilibrium submitted to a uniform gravitational field. The new configuration consists of discrete, finitethickness flux tubes embedded in an isothermal fieldfree atmosphere, and it turns out to be linearly stable with respect to 3D perturbations. There is however something new in the present paper: the transform method has been applied not to a single equilibrium at a time, but to a whole evolutionnary sequence driven by stationary boundary motions, with another sequence of the same type being eventually obtained.
One of the weakness of the solutions we have presented is of course the presence of the lateral walls which impose to the field an artificial confinement, and it is necessary to study in details the more interesting case where the evolution takes place in the whole halfspace (). Preliminary results on that problem show that  as it can be a priori expected  the pressure does not change qualitatively the evolution when either the adiabatic or the isothermal prescription is adopted. In the isobaric case, however, the evolution cannot go on for ever. There is a critical time at which a global nonequilibrium phenomenon (Aly 1993) develops: there is no longer an equilibrium compatible with the constraints imposed to the system, and a dynamical evolution has to start in, which may be guessed to lead to an opening of the field. In spite of that, we feel that the new solutions are quite useful. They give explicit examples in which it is possible to evaluate quantitatively how a forcefree solution is changed when the effects of the thermal pressure of the plasma are introduced, they give an interesting test for checking the accurateness of the various exact estimates (generally in the form of upper and lower bounds on some physical quantities) that we shall present in our forthcoming paper on the general problem, and finally they may be used as test cases for numerical MHD codes of evolution.
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All Figures
Figure 1: Evolution of Z(a_{0}/2,t)/L as a function of t/T for a) and b). In both a) and b), we have plotted ( from the bottom to the top) the forcefree case ( , black curve), and the three cases (green), (red), and (blue). 

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In the text 