Flux tube oscillations and coronal heating

Wave transmission in low beta magnetic flux tubes has, mathematically, the same structure as the propagation of electromagnetic waves in optical fibers. In both cases the problem is reducible to a single wave equation for the longitudinal component of the perturbed field along the fiber/tube axis. We derive this equation, solve the dispersion relation associated with it, and assign three wave numbers to each mode. In cylindrical coordinates (r, phi, z), for a given phi-wave number, the plane of the r- and z- wave numbers is divided into one "mode zone" in which each grid point is a possible mode of the system and one "forbidden zone" in which no mode may dwell. The cutoff line, the boundary of the two zones, is given both analytically and numerically. Next we introduce weak resistive and viscous dissipation to the system, solve for the decay time of each mode and for the densities of heat generation rates by each dissipative process. The two densities have identical spatial dependencies, but different magnitudes. The resistive heat rate is inversely proportional to the Lundquist number, S, and the viscous one to the Reynolds number, R. The time decay exponent is proportional to the sum (S-1 + R-1).