Application of the MHD energy principle to magnetostatic atmospheres [report]

E.G. Zweibel
1984 unpublished
We apply the MHD energy principle to .the stability of a magnetized atmosphere which is bounded below by much denser fluid, as is the solar corona. Me treat the two fluids as ideal; the approximation which is consistent with the energy principle, and use the dynamical conditions that must hold at a fluid-fluid interface to show that if vertical displacements of... the lower boundary are permitted, then the lower atmosphere must be perturbed as well. However, displacements which do not perturb
more » ... ch do not perturb the coronal boundary can be properly treated as isolated perturbations of the corona alone. DISCLAIMER This report lis prepared su in account of work sponsored by an agency of the United States Government. Neither the United Slates Government nor any agency thereof, nor any of tbeir employees, makes any warranty, express or implied, or assumes any legal liability or responsi bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use woukt not infringe privately owned rights. Refer ence herein to any specific commercial product, process, or service hy trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom mendation, or favoring by the United Stales Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily slate or reflect Ujose of the United States Government or any agency thereof. liSTSIBUTfOH OF THIS B3CMMFHT IS IKffiKTED t" 2 I. INTRODUCTION Studies of the equilibrium and stability of magnetized plasma in a gravitational field ore important 1 -to many areas ot astrophysics, including solar and stellar physics (reviewed by Priest, 1982, and Rosner et al., 1984). The stability of structures in the solar corona is relevant to understanding the onset of eruptive activity, as well as the necessary conditions for equilibrium. Because even the simplest models of coronal features are sufficiently inhomogeneous that solving the full mode problem is very difficult, many studies of coronal MHD stability have used the energy principle method of Bernstein et al. (1958; hereafter BFKK) to determine stability without calculating the modes themselves. Coronal magnetic fieldlines are thought to be connected to the lower solar atmosphere (chromosphere and photosphere), which is much denser than the corona, and Ultimately to extend into the solar interior. Rather than considering the stability of the composite system consisting of hot gas and cooler underlying material, most studies of coronal MHD stability have imposed a boundary at the coronal base and have treated the lower atmosphere only through its influence on the boundary conditions! Several different assumptions about the boundary conditions on £|, the component of the fluid displacement * parallel to the magnetic field S, have been made in the literature. Schindler et al. (1983) chose t * Or as if the photosphere ware a rigid boundary. Einaudi and van Hoven (1981) imposed parity constraints on €g that allow 5 B jt Q, ifc, 0 d (19S4a,b) did not explicitly restrict £j at all. In this paper, we discuss the influence of the photospheric boundary condition on stability by assuming that both the upper and lower atmosphere are ideal fluids. It is clear that material in the solar atmosphere does not I 3 always behave adiabatically. Radiative processes, thermal conduction, a.id some form of heating all play rolea in the structure. Haas flow la often present. The two-fluid treatment in this paper is an idealization, but it is an idealization which is consistent with the MHD energy principle, and should be a good approxination as long as the HHD tine scales are rapid conpared to • the tine scale on which mass exchange occurs between the two fluids. , equilibria with flows, and nonadiabatic perturbations, cannot be studied with the ideal HHD energy principle. we find that for coronal stability problems in which gravitational stratification is included, the effect of nonaero 5| is to force the lower atmosphere to be perturbed. This arises in a natural way from the conditions at the boundary between the two fluids. In Sec. II, we use the HHD energy principle to demonstrate the existence of a surface integral and a perturbation of the lower atmosphere when ; ( is not zero at the boundary. In Sec. Ill, we discuss the effect of the boundary terms on various results in the literature. Section IV is a discussion together with conclusions. 8 from both fluids. Proceeding similarly to BFKK (see also Roberts, 1967), we rewrite the surface integral in Eq. (11) using the boundary conditions satisfied by ?. Bote that even if we use the extended energy principle, so that we allow trial functions in the volume integral fiw F which do not satisfy the boundary conditions (6), we must evaluate 6w g assuming that these conditions are satisfied. This has not always been done in the literature. According to B}« (4),-n x ( is continuous at the interface. Using Eqs. (5) and <6), Ap and $ + t * is" are continuous as well, as is X * n. These results enable us to write 2«W S = -<£ d 2 *{(n.£) [1.V(P + B 2 /2)] -(n-S) (l-VB>f} Using the equation of mechanical equilibrium, this Lecomes 2«W S = i dM(n«^ {[B'VB)'t * P?'g] -(n-B) (f.VB>& or 2fitt s = -£ d 2 x{(n't)(t'|)p* [nx(|x|).VB]4} Evidently, the second term involves only tangential derivatives of i at the interface. But the tangential derivatives of B are continuous; this term is therefore zero. If we take g * yg and let the boundary lie in the x-z plane, tnen fiw-takes the final form 2«W S -/ dxdzCygfpjj -Pj (12) 9 where Pj and P u denote the densities in the lower and upper atmosphere, respectively. Equation (12) is exactly what is expected when one considers the Rayleigh-Taylor instability between two media of different density (e.g., Chandrasekhar, 1961). Since PJJ » P u in the problem considered here, the 3 irfaee term is positive. Ihus we have shown that for nonzero g, the presence of flow across the unperturbed fluid boundary (nonzero ? ( ) tends to be stabilizing. The surface term given in Bq. (12) arises naturally from the dynamics of the problem, and must be included in any evaluation of ' ? e -y/H ) 2 ] (15) Here, J is the current density, H is the thermal scale height, and 3/a s • 1/B S • V is the derivative along a field line. To derive stability criteria. Hood assumes that the perturbations are isothermal (T = 1) and minimizes with respect to 3£ z /3z and 7 •(te~y />Ii ). This results in the conditions v • t = ^ , (16) He then takes the limit of infinitely large wave number in the z direction; k z + °°. Since condition [17) requires that the product k a S z be finite, the nagnetic tension term (B'VE ) z in 5w, which corresponds to bending the field lines out of their equilibrium plane, becomes negligible. (See GLlman 1979, Asseo et al. 1980, Zweibei 1981). Thus, 6"w is reduced to the form 2«W > /^.{(^IjMvA.V^)-i£ -£]}*,* , (,B> B B exists if the parameter 20H which measures the current exceeds a certain threshold. Hia solutions for A 1 are ev rather than odd, functions of x. According to the arguments above, the stability boundary for isolated coronal modes should be at a larger value of the current parameter than that found by Hood. The stability boundary for these modes can be found by solving the Euler equation for the integral (18) with A 1 • 0 at the end and apex of a field line. within the framework of the present analysis. Hood has found a sufficient, but not necessary, stability condition. IV. DISCUSSION AND CONCLUSIONS In this paper, we have considered the lower boundary condition for coronal MHD stability problems. These systems are characterized by magnetic field lines which connect the corona to a much denser, underlying atmosphere. Tt;eir ideal MHD stability has b ,ed using the SFKK energy principle. The lower atmosphere has simply been modeled 93 a rigid, conducting wall in some previous treatments (e.g., Schindler e_t _al. 1983) on which the fluid displacement f vanishes. Other studies have allowed a nonvanishing fluid displacement parallel to the magnetic field at the lower boundary, but required the perpendicular components ?j_ to vanish. If displacement of the lower boundary is permitted, there are two basic ways in which this can be interpreted physically. One approach is to consider the boundary as a contact surface between two ideal fluids of different temperatures and densities. We pursued this approach here. The alternative is to consider tne lower boundary as a source or sink of mass. This is only possible if heat gain or loss mechanisms are available to effect a. phase change between the two fluids. Although the latter viewpoint certainly has 1 7 elements of physical real.'.sm, the ideal HHD energy principle is inapplicable to such systems. We reviewed the boundary conditions which apply to a fluid-fluid interface and pointed out that these boundary conditions lead to surface integrals in the perturbed potential energy Sw which represent Mv work done at the interface between the fluids. These terns £<Bj. 12)] can be written in a form which involves the density contrast between the two fluids, and is the same term one derives in an analysis of the Rayleigh Taylor instability at an interface between unraagnetized fluids, since the lower atmosphere is much denser than the upper atmosphere, the surface term is strongly stabilizing. It vanishes when the displacement normal to the boundary vanishes. We also found that &* FC cannot be ignored for displacements with f * n?*0. thus, there are two types of displacements; the isolated coronal modes, with f • n = o, for which 8w s and 6w FC can legitimately be set equal to zero, and the displacements with t * n j 0. For the latter, the stability problem consists of jointly minimizing *W FC , 4w FL , and 5w s . If the volume term SWpg alone is minimized, as was done by Hood (1984a,b) , the result can be used to give a sufficient, but not necessary, condition for stability of the isolated coronal modes. We showed that Hood's minimization of 5w FC for a particular set of equilibria (Hood 1984b) led to nonvanishing ? • n. The imposition of E • n =» 0 on the boundary requires that Hood's trial functions A 1 have odd parity. We would argue, therefore, that some of the equilibria that Hood predicted are unstable are actually stable, according to the two-fluid analysis. The rigid boundary condition with ? = 0 has a. vanishing surface term. + This assumes that the rigid boundary condition with 5=0 leads to a selfconsistent problem in which Sw^, alone is minimized. This seema to be the 18 simplest approach to treating the corona as an isolated system. The full problem, involving thermal exchange and dynamical forcing by motions of the fieldiine endpoints, will have to be explored by other methods than the MHD energy principle, ACKNOWLEDGMENTS It is a pleasure to acknowledge discussion with P, J. Cargill,
doi:10.2172/6302305 fatcat:ymfxd6gt5fedxmqvitpnq727v4