A simple mechanical model for resonance absorption: The Alfven resonance

We consider resonance absorption of magnetohydrodynamic waves, and the Alfven resonance layer in particular. We show that the dissipative layer can be modeled as a simple mechanical system consisting of a few harmonic oscillators which are coupled by friction. The mechanical model reproduces known results for the externally driven system in steady state, such as the structure of the dissipative layer, the ''waves'' of heating which propagate across the layer, and the fact that the total heating is independent of time. The total work done on the oscillators by the driver is always positive; the external driver sees the total system as a single damped oscillator driven exactly at resonance; Nonetheless, some of the oscillators return energy back to the driver. The total kinetic energy of all the oscillators and the total potential energy are nearly independent of time, because the integrals, across the dissipative layer, of the square of the velocity and the square of the displacement, are truly constants in time. Waves of kinetic and potential energy propagate across the system in the same sense as the waves of heating. We also investigate an initial value problem in which the driver is turned on at t = 0. There is no single number representing the time required for the dissipative layer to reach a steady state. The waves of heating which are found in the steady state are also present in the buildup phase. However, if the driver is turned off after the system has reached a steady state, then the waves of heating are less obvious. We consider the effects of a nonlinear frictional coupling between the oscillators, designed to mimic the effects of Kelvin-Helmholtz instabilities. The nonlinear coupling has surprisingly little effect on the system. The total steady state heating rate is the same as in the linear system; even with nonlinear dissipation, the dissipative layer adjusts itself to absorb a predetermined amount of energy being pumped in by the external driver. The waves of heating which are found in the linear system are still present. We find no evidence of chaotic behavior.