The oscillations of a slowly rotating star have long been classified into spheroidal and toroidal modes. The spheroidal modes include the well-known 5-min acoustic modes used in helioseismology. Here we report observations of the Sun's toroidal modes, for which the restoring force is the Coriolis force and whose periods are on the order of the solar rotation period. By comparing the observations with the normal modes of a differentially rotating spherical shell, we are able to identify many of the observed modes. These are the high-latitude inertial modes, the critical-latitude inertial modes, and the equatorial Rossby modes. In the model, the high-latitude and critical-latitude modes have maximum kinetic energy density at the base of the convection zone, and the high-latitude modes are baroclinically unstable due to the latitudinal entropy gradient. As a first application of inertial-mode helioseismology, we constrain the superadiabaticity and the turbulent viscosity in the deep convection zone.
Context. In a previous paper, we studied the effect of latitudinal rotation on solar equatorial Rossby modes in the β-plane approximation. Since then, a rich spectrum of inertial modes has been observed on the Sun, which is not limited to the equatorial Rossby modes and includes high-latitude modes.
Aims. Here we extend the computation of toroidal modes in 2D to spherical geometry using realistic solar differential rotation and including viscous damping. The aim is to compare the computed mode spectra with the observations and to study mode stability.
Methods. At a fixed radius, we solved the eigenvalue problem numerically using a spherical harmonics decomposition of the velocity stream function.
Results. Due to the presence of viscous critical layers, the spectrum consists of four different families: Rossby modes, high-latitude modes, critical-latitude modes, and strongly damped modes. For each longitudinal wavenumber m ≤ 3, up to three Rossby-like modes are present on the sphere, in contrast to the equatorial β plane where only the equatorial Rossby mode is present. The least damped modes in the model have eigenfrequencies and eigenfunctions that resemble the observed modes; the comparison improves when the radius is taken in the lower half of the convection zone. For radii above 0.75 R⊙ and Ekman numbers E < 10−4, at least one mode is unstable. For either m = 1 or m = 2, up to two Rossby modes (one symmetric and one antisymmetric) are unstable when the radial dependence of the Ekman number follows a quenched diffusivity model (E ≈ 2 × 10−5 at the base of the convection zone). For m = 3, up to two Rossby modes can be unstable, including the equatorial Rossby mode.
Conclusions. Although the 2D model discussed here is highly simplified, the spectrum of toroidal modes appears to include many of the observed solar inertial modes. The self-excited modes in the model have frequencies close to those of the observed modes with the largest amplitudes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.