Eigenmodes of three-dimensional magnetic arcades in the Sun's corona

We develop a model of coronal-loop oscillations that treats the observed bright loops as an integral part of a larger three-dimensional (3D) magnetic structure comprised of the entire magnetic arcade. We demonstrate that magnetic arcades within the solar corona can trap MHD fast waves in a 3D waveguide. This is accomplished through the construction of a cylindrically symmetric model of a magnetic arcade with a potential magnetic field. For a magnetically dominated plasma, we derive a governing equation for MHD fast waves and from this equation we show that the magnetic arcade forms a 3D waveguide if the Alfvén speed increases monotonically beyond a fiducial radius. Both magnetic pressure and tension act as restoring forces, instead of just tension as is generally assumed in 1D models. Since magnetic pressure plays an important role, the eigenmodes involve propagation both parallel and transverse to the magnetic field. Using an analytic solution, we derive the specific eigenfrequencies and eigenfunctions for an arcade possessing a discontinuous density profile. The discontinuity separates a diffuse cylindrical cavity and an overlying shell of denser plasma that corresponds to the bright loops. We emphasize that all of the eigenfunctions have a discontinuous axial velocity at the density interface; hence, the interface can give rise to the Kelvin–Helmholtz instability. Further, we find that all modes have elliptical polarization with the degree of polarization changing with height. However, depending on the line of sight, only one polarization may be clearly visible