Eigenmode approach to coronal magnetic structure with differential solar rotation

Abstract

Diffusion of field line foot points in the photosphere couples with latitude‐dependent solar rotation to define an eigenvalue problem, such that coronal magnetic structures derived from certain specific linear combinations of spherical harmonic functions are found to rotate rigidly about the Sun. Such eigenmodes and their corresponding complex eigenfrequencies are readily identified by diagonalizing progressively larger matrix representations of the eigenvalue problem. For azimuthal harmonic number m = 1, the eigenvalue with the least negative imaginary part (≈ −0.6 year−1 for footpoint diffusion coefficient D⊥ = 600 km2s−1 and otherwise roughly proportional to D⊥1/2) corresponds to an eigenmode whose main component is the dipole (n = 1) moment perpendicular to the Sun's rotation axis. Associated values of Re ω11 correspond to heliomagnetic rotation at 99.4% of the Sun's equatorial rate for D⊥ = 600 km2s−1 and otherwise to a retrograde deviation roughly proportional to D⊥1/2 from the Sun's equatorial rotation rate. This eigenmode corresponds to the almost rigidly rotating coronal structure found by Wang et al. [1988] in their numerical simulations of the coronal magnetic field. The associated heliospheric current sheet rotates almost rigidly about the Sun despite the anchorage of adjacent field lines in a differentially rotating photosphere. This particular eigenmode also corresponds most nearly to the nonaxial part of the tilted dipole in the heliospheric model of Fisk [1996], whereby differential solar rotation leads to a latitudinal circulation of field lines through the heliosphere and thus to large‐scale heliospheric convection.