Numerical modeling of ocean physics is essential for multiple applications. However, the large range of scales and interactions involved in ocean dynamics make numerical modeling challenging and expensive. Many regional ocean models resort to a hydrostatic (HS) approximation that reduces the computational burden. However, a challenge is to capture and study ocean phenomena involving complex dynamics over a wider range of scales and processes, from regional to small scales (e.g., thousands of kilometers to meters), resolving submesocales, nonlinear internal waves, subduction, and overturning where they occur. Many such local dynamics require non-hydrostatic (NHS) ocean models. The main computational cost for NHS models arises from solving a globally coupled elliptic PDE for the NHS pressure. Our main research thrust is to optimally reduce these costs so that the NHS dynamics are resolved where needed.

We start from a high-order hybridizable discontinuous Galerkin (HDG) finite element NHS ocean solver, which is well suited for multidynamics systems. We present a new adaptive algorithm to decompose a domain into NHS and HS dynamics subdomains and solve their corresponding equations, thereby reducing the cost associated with the NHS pressure solution step. The NHS/HS subdomains are adapted based on new numerical NHS estimators, such that NHS dynamics is used only where needed. Since the choice of boundary condition imposed on the internal boundaries between subdomains is crucial to maintain accuracy, we explore and compare different choices. To evaluate the computational costs and accuracy of the adaptive NHS-HS solver, we first complete several analyses using internal solitary waves (e.g. Vitousek and Fringer 2011). We then complete more realistic NHS-HS simulations of Rayleigh-Taylor instability-driven subduction events by nesting with our MSEAS realistic and operational data-assimilative HS ocean modeling system.