Self oscillations of a forced inhomogeneous hydromagnetic cavity

We study the nonlinear development of MHD waves in a one dimensional cavity stratified by gravity and embedded in background, homogeneous, vertical magnetic field which produces an anisotropic viscosity. The plasma is compressible and its temperature is homogeneous over most of the cavity, rising to large values in a narrow 'transition zone' near its upper boundary. Disturbances at the cavity's boundaries are assumed to be 'quasi resonant' with the fundamental mode of the discrete Alfvén spectrum. The linear behaviour of the cavity is given by a detailed computation of the Alfvén and sound eigenmodes and eigenfrequencies. The non linear behaviour of the fundamental Alfvén mode is studied by Galerkin and multiple scale analyses which reduce the full MHD equations to a two dimensional, driven, dissipative dynamical system. Self oscillations of the Alfvén wave amplitude set in via a Hopf bifurcation as the background magnetic field is varied past a critical value (B_cr ≃ 1.5 G). In solar chromospheric conditions the period of these oscillations lies in the hour range.