The linear oscillations of a semi-infinite polytropic atmosphere threaded by a uniform vertical magnetic field are studied in the ideal dissipationless limit. For a prescribed real horizontal wavenumber, k, there are an infinite number of discrete complex eigenfrequencies ωn - iΓn, n = 0, 1, 2, ..., corresponding to oscillations which decay in time due to "radiative damping" by quasi-Alfvénic slow-modes (s-modes) propagating along the magnetic field lines in the direction of increasing fluid density. This damping becomes particularly weak in the two extreme limits of the horizontal wavelength 2π/k being much greater than, or much less than, the thickness L of the highly magnetized (magnetic pressure ≳ gas pressure) surface boundary layer of the equilibrium atmosphere. For kL ≪ 1 the real part of the complex eigenfreguencies varies with the horizontal wavenumber according to the familiar p-mode dispersion relation ω ∼ √gk, where g is the constant gravitational acceleration that pertains to unmagnetized polytropic atmospheres. In the opposite limit, kL ≫ 1, a different dispersion relation, ω ∼ k√gL, obtains. The s-mode damping of the atmospheric oscillations is maximal when kL ≃ 1, where the damping time can be of order a few oscillation periods. Surprisingly, for an infinite set of discrete horizontal wavenumbers, one of these complex eigenfrequencies is purely real, indicating the presence of undamped trapped normal modes of oscillation of the magnetized polytrope. We refer to the set of all such undamped modes as trapped π-modes, and following the usual conventions, we designate a particular mode by πn,n+ν where n (= 0, 1, 2, ...) counts the number of nodes in the horizontal displacement eigenfunction, while n + ν (ν = 1, 2, 3, ...) counts the number of nodes in the vertical displacement eigenfunction. The real oscillation frequencies of these trapped modes organize into a sequence of bands which correspond to distinct values of ν, with the fine-scale frequency splitting within each band arising from the variations with n. The fundamental ν = 1 band is characterized by oscillation periods of order π√L/g (for a complete adiabatically stratified polytrope with a ratio of specific heats γ = 5/3). Although this special set of trapped modes is not complete in the sense that their superposition does not provide the most general solution of an initial-value problem, they are the characteristic modes of oscillation of the atmosphere and thus will dominate the solution of the initial-value problem after the initial transients have decayed. This magnetized atmosphere is too simplified to provide quantitative predictions about oscillatory motions in sunspot umbrae, or the coupling of these oscillations to external p-mode forcing, but the qualitative nature of these findings should carry over to these more realistic magnetic field configurations. In particular, we would suggest that the trapped π-modes of a realistic sunspot atmosphere must play a role in the internal sunspot oscillatory motions analogous to that played by their p-mode counterparts in the oscillations of the surrounding unmagnetized quiet Sun. Some preliminary encouragement follows by noting that for typical solar surface values g = 274 m s-2 and L = 600 km, the periods of the ν = 1 band given are of order 3 minutes, which is a dominant period of the oscillations observed within sunspot umbrae.
- Sun: magnetic fields
- Sun: oscillations