Nonlinear thermal instability in two dimensions

Nonlinear thermal disturbances are analyzed for a two-dimensional structure taking into account thermal conduction parallel to and perpendicular to the magnetic field, as well as heating and cooling effects. In general, small structures are linearly stable while larger ones are unstable. Heat conduction perpendicular to the field has a stabilizing effect and increases the maximum stable size of a structure. In many cases, the second-order growth rate is positive (enhancing heating but preventing cooling) for very large structures and is negative (opposite effect) otherwise. The perpendicular conduction causes a negative correction other than for the largest structures. This perpendicular conduction is particularly important for structures in the marginal linear state; strong cooling occurs in the absence of perpendicular conduction but if such conduction is included and is strong enough, catastrophic heating may occur. Perpendicular heat conduction is found to be most significant in long, thin, cool structures.