On the steady state of nonlinear quasiresonant Alfven oscillations in one-dimensional magnetic cavity

We study the steady state of nonlinear, small-amplitude, quasiresonant Alfv en oscillations in a homogeneous dissipative hydromagnetic cavity which is forced by the shear motion of its boundaries. It is shown that, even in the case of strong nonlinearity, these oscillations can be represented, to lea ding order, by a sum of two solutions in the form of oppositely propagating waves with permanent shapes. An infinite set of nonlinear equations for th e Fourier coefficients of these solutions is derived which, in general, adm its multiple solutions, depending on the re-scaled total Reynolds number, R , and mistuning, Delta, between the frequency of the boundary forcing and t he first Alfven eigenmode of the cavity. Two types of solutions are found. On the one hand, low-modal solutions set in over the entire parameter range studied, which can be represented, with a remarkable accuracy, by very few Fourier modes even at very large R. For a fixed a the time-averaged energy , epsilon, that can be stored in the cavity is saturated, as R increases, t o a value which is approximately proportional to epsilon(2), epsilon(3) << 1 being the Alfven Mach number of the boundary motions. The time-averaged a bsorbed power (the Poynting flux S) scales as 1/R. For suitable values of R and a catastrophic transitions occur between these solutions, in which the average power released scales as R, provided R < epsilon(-1/2). The second type of solutions sets in for a narrow window of Delta and develops large gradients (shocks) which need to be represented by many Fourier modes. For a fixed a the build-up of these gradients takes place starting from a low-m odal solution in either a continuous way, by increasing R, or in a sudden c atastrophic way as R becomes smaller than a critical value. Both epsilon an d S are saturated as R is increased. It is suggested that both types of sol utions can explain bright events in the solar atmosphere.