Space missions have revealed much about the global structure and dynamics of the Sun and planets within the solar system. But similar objects that lie beyond the range of being visited by a spacecraft, remain best described as point sources of light. Often, a one-dimensional (function of radius) model does wonders in explaining their first-order observational signatures, but is rarely sufficient to even try to paint a complete picture. In reality, they are complex, nonlinear, fluid objects that exhibit intriguing stationary and time-variable behaviour in three dimensions! Thus, it is important to model them as such. Unfortunately, with increasing complexity of these models, they not only become super-expensive, but also less intuitive. This is where reduced-order modelling strikes a balance, elegantly sitting between simplified and convoluted models. In this talk, I will show how reduced-order theory, anchored in linearized analysis and benchmarked against nonlinear simulations, can be leveraged to uncover hidden mysteries in stellar and exoplanetary systems.

I shall go over two scenarios pertaining to rotating hydrodynamic spheres. The first concerns the interior circulation of stars, specifically the problem of outward angular momentum transport, contributions to which by internal flows are typically neglected in 1-D evolutionary models. Are Sun-like stars in internal rotational steady state or still evolving? What role do compositional gradients play in this process? We use a primitive equation framework from atmospheric sciences to answer these questions. The second shifts focus to the surface (atmospheric) circulation of exoplanets, specifically due to variable irradiation from their hosts or to their orbital configuration. Using a shallow water climate model, I identify novel climate regimes including oscillating hotspots on tidally locked planets, and resonance states on asynchronous planets like Earth with permanent hot and cold spots. For both systems, I will present analytical scaling laws, limits demarcating linear from nonlinear behaviour, and implications for follow-up science.