Recent advances in numerical modeling of solar inertial modes

Inertial modes are low-frequency global oscillations in rotating fluids, restored by the Coriolis force. Long-term observations of horizontal flows on the solar surface have led to the detection of various inertial modes on the Sun, opening a new window into the structure and dynamics of the solar interior. Here, we focus on recent advances in the theoretical modeling of solar inertial modes within a purely hydrodynamic framework, leaving magnetohydrodynamic effects beyond the scope of this review. Because the low-frequency oscillations of a differentially rotating spherical shell depend sensitively on the rotation profile and the properties of the medium, they are primarily studied using numerical models. We first summarize how linear eigenmode analyses have enabled the identification and classification of different families of inertial modes observed on the Sun, and how these models reveal the strong diagnostic potential of inertial modes for probing poorly constrained parameters in the deep convection zone, such as turbulent viscosity and superadiabaticity. The amplitudes of linearly stable modes can be estimated from their response to stochastic forcing, whereas those of linearly unstable modes must be determined using nonlinear numerical modeling. We further discuss the possible dynamical roles of large-amplitude inertial modes in transporting heat and angular momentum in the Sun. By clarifying what has been established and what remains uncertain, this review provides a framework for interpreting current observations and guiding future theoretical developments in this rapidly evolving field.