A potential energy conserving finite element method for turbulent variable density flow: application to glacier-fjord circulation

TL;DR

This paper presents a novel finite element method for turbulent variable density flow that conserves energy and improves turbulence modeling, demonstrated through glacier-fjord circulation simulations.

Contribution

It introduces a new energy-conserving finite element discretization with a symmetric viscosity operator for stratified turbulence, validated on glacier-fjord models.

Findings

Enhanced energy conservation in simulations.

Better turbulence resolution with less artificial diffusion.

Accurate modeling of glacier-fjord circulation.

Abstract

We introduce a continuous Galerkin finite element discretization of the non-hydrostatic Boussinesq approximation of the Navier-Stokes equations, suitable for various applications such as coastal ocean dynamics and ice-ocean interactions, among others. In particular, we introduce a consistent modification of the gravity force term which enhances conservation properties for Galerkin methods without strictly enforcing the divergence-free condition. We show that this modification results in a sharp energy estimate, including both kinetic and potential energy. Additionally, we propose a new, symmetric, tensor-based viscosity operator that is especially suitable for modeling turbulence in stratified flow. The viscosity coefficients are constructed using a residual-based shock-capturing method and the method conserves angular momentum and dissipates kinetic energy. We validate our proposed method through numerical tests and use it to model the ocean circulation and basal melting beneath the ice tongue of the Ryder Glacier and the adjacent Sherard Osborn fjord in two dimensions on a fully unstructured mesh. Our results compare favorably with a standard numerical ocean model, showing better resolved turbulent flow features and reduced artificial diffusion.