A necessary condition for the stability of a class of three-dimensional laminated equilibria

A necessary condition is derived for the ideal magnetohydrodynamic stability of a class of three-dimensional laminated isothermal equilibria presented by Chou, Low, & Bhattacharjee (1993). The condition is derived by the use of a particular destabilizing trial function known as a 'ballooning mode.' This choice of trial function allows the necessary condition to be found by the solution of four coupled first-order ordinary differential equations that may be integrated along individual field lines. Here the boundary conditions model the line-tying effect of the dense photosphere. The necessary condition for stability together with the sufficient condition derived by Chou gives a bound on marginal stability, the tightness of the bound depending on the parameter regime. For the case of penumbral loops, considered by Chou, an approximate lower and upper bound on the critical value of the plasma beta for this model can be found. The upper bound on Beta, however, is smaller than that determined from observations.