Linear Toroidal‐Inertial Waves on A Differentially Rotating Sphere with Application to Helioseismology: Modeling, Forward and Inverse Problems

ABSTRACT This paper develops a mathematical framework for interpreting observations of solar inertial waves in an idealized setting. Under the assumption of purely toroidal linear waves on the sphere, the stream function of the flow satisfies a fourth‐order scalar equation. We prove well‐posedness of wave solutions under explicit conditions on differential rotation. Moreover, we study the inverse problem of simultaneously reconstructing viscosity and differential rotation parameters from either complete or partial surface data. We establish convergence guarantee of iterative regularization methods by verifying the tangential cone condition, and prove local unique identifiability of the unknown parameters. Numerical experiments with Nesterov–Landweber iteration confirm reconstruction robustness across different observation strategies and noise levels.