The Linear Response of a Magnetic Flux Tube to Buffeting by External p-Modes. I

Abstract

The linear response of a thin vertical magnetic flux tube to buffeting by p-modes in the ambient atmosphere is examined with the aim of understanding the interaction of acoustic modes with sausage tube waves. The idealized case of an isothermal atmosphere is considered, which has the mathematical advantage that the differential equation for the vertical component of the Lagrangian displacement in the tube can be solved analytically. A lower boundary condition is employed that permits the tube wave to leak out through this boundary. This has the important consequence that the p-mode interaction with flux tubes does not exhibit a resonant behavior. The detailed behavior of the vertical displacement in the tube and its dependence on various parameters are examined. An equation for the wave energy in a thin flux tube is derived along with analytic expressions for the wave energy density and vertical energy flux. The variation of the tube response Xi (defined as the ratio of the total wave energy, integrated over the length of the tube, to the p-mode energy in the external atmosphere) is investigated for different values of the dimensionless external horizontal wavenumber Kx, mode order n, and beta , where beta is the ratio of the gas to magnetic pressure in the tube that, by assumption, is constant with depth. It is found that when n is small, the response of the tube increases gradually with Kx until reaching a maximum, and thereafter it drops very sharply. As n increases, the maximum shifts to lower values of Kx. For fixed values of Kx and beta , Xi increases with n and then falls off after reaching a maximum. A similar dependence of Xi on beta is found. Line widths of p-modes are also calculated, and their dependence on Kx and frequency is studied. Finally, an application of the results to the solar atmosphere is discussed, and the limitations of the model are pointed out.