Generalized Magnetohydrostatic Equilibria

This note shows the existence of a generalized class of magnetohydrostatic elliptic equilibria of a perfectly conducting fluid in systems with cylindrical topology for which the surfaces of constant pressure have an elliptic transverse cross section with the eccentricity given by an arbitrary function of the axial distance. Further, it is shown that a change in the parameters characterizing the solution leads to a generalized class of hyperbolic equilibria for which the surfaces of constant pressure have a hyperbolic transverse cross section with the eccentricity given by an arbitrary function of the axial distance. Whereas the first class of equilibria is of interest in a plasma-confinement system, the second class of equilibria is of interest in a magnetic-field-line reconnection system. One has for a magnetohydrostatic equilibrium of a perfectly conducting fluid Vp = J X B, (1) V X B = J, (2) V B = 0, (3) where p is the pressure of the fluid, J is the current density, and B is the magnetic field. Woolley [1] showed that Eqs. (l)-(3) yield a three-dimensional family of elliptic equilibria for which the surfaces of constant pressure have an elliptic transverse cross section with the eccentricity an arbitrary function of the axial distance z. These equilibria are of interest in a plasma-confinement system. The purpose of this note is first to show the existence of a more general class of elliptic equilibria than the one established by Woolley [1]. Further, it is shown that a change in the parameters characterizing the solution leads to a family of hyperbolic equilibria for which the surfaces of constant pressure have a hyperbolic transverse cross section with the eccentricity an arbitrary function of the axial distance z. These equilibria are of interest in a magnetic-field-line reconnection system 'Received April 12, 1985. ©1986 Brown University 487 488 BHIMSEN K. SHIVAMOGGI and represent a three-dimensional generalization of the solution given by Habbal and Tuan [2]. Magnetic field reconnection is the process by which magnetic field lines that are initially distinct link up, thereby lowering the potential energy of the magnetic field. In a system of rectangular Cartesian coordinates (x, y, z), let us prescribe for the magnetic field Bx=^[yf(z)], By = JP2[xh(z)], B: = 0, (4) which identically satisfies Eq. (3). Here f(z) and h(z) are arbitrary functions of z. A similar prescription was made by Lin [3] and Shivamoggi and Uberoi [4] in constructing exact solutions to the equations of magnetohydrodynamics of a dissipative fluid. Note that the prescription for the magnetic field given by Woolley [1] is a special case of (4). Using (4), Eq. (2) gives for the current density, Jx = -je;xh'; jv = ye;yf', j. = je{h ,y?;f, (5) where primes denote differentiation with respect to the argument. Using (4), Eqs. (1) and (2) give p = u(x, y) je2 + ye2). (6) Here co(.x, y) represents the total pressure—hydrodynamic plus magnetic. Using (4)-(6), Eq. (1) gives du/dx = ye2 Jfj, du/dv = (7) from which it is clear that one requires ■^i[yf(z)}-^2[xh(z)] =Jr(x,y). (8) Further, from (7) the integrability conditions for the existence of o>(jc, y) are Jf2 2 = JT, 2, (9a) or Jf,"/2 /yex = 3tr{'h 2/3f2 = (C (z), (9b) where k(z) is an arbitrary function of z. Let us differentiate the first equation in (9b) with respect to y, so that there follows /2.*y" = (io) Differentiating the first equation in (9b) with respect to z, one obtains iff 'je{' + yf2f 'yej= K yf 'ye; + k 'yex. (11) Using Eq. (10), Eq. (11) becomes iff'yec = K'yex, (12) which is consistent with the first equation in (9b). Similar deduction follows for the second equation in (9b). It is important to note from (12) that the present generalization exists only if k = k(z), and it degenerates to Woolley's [1] solution if k = const. GENERALIZED MAGNETOHYDROSTATIC EQUILIBRIA 489 There exist two distinct classes of solutions according as k + 0 or k = 0. For k # 0, one obtains from (7) and (9b) u(x, y) = /k + const. (13) Using (13), (6) becomes p=p0 + + Jf22), k # 0, (14) p0 being an arbitrary constant. Corresponding to k = 0, (9) gives Jfi = Gy/(z), Jf2 = Hxh(z), (15) where G and H are arbitrary constants. Using (15), (8) gives ]jf(z)h(z) = const. (16) First let us choose this constant to be imaginary, say iC, where C is real. This choice leads to a solution that is relevant to confining-field equilibria because in this solution the pressure p decreases monotonically as one moves away from the axis in a transverse plane z = const. Using (7), (15), and (16), (6) becomes P=P o 2 C2H . ^ 2JC2G , \ + i Uf/ + —+ 11 (17) " Gf2 I 1 \ Hh7 For the case G = H = 1, (17) becomes p = Po~ K2[i + p2(z)][-^2/p2(z) +y2](18) where p(z) =f{z)/c. Note that (18) is identical to the case obtained by Woolley [1] for k = 0. The intersection of the surfaces of constant pressure with a transverse plane z = const gives a family of nested ellipses x2/a2 + y2/b2 = 1 (19) within the bounding ellipse corresponding to p = 0 (assuming that p = 0 describes the boundary of the plasma). Here a2( ) = 2(p0-p)p2(z) 2(p0-p) () C2[l+P2(z)]' MZ) C2\\ + p2(z)] <20) The eccentricity a of the loci p = const in a given plane z = const is given by b2 a2 = 1 p2(z) b2 + a2 1 + p2(z) On the other hand, if we choose the constant in (16) to be real, say A, (6) becomes Gf2 I \ Hh 1 ' = = (21)