# Discontinuous Galerkin discretization for two-equation turbulence closure model

## TL;DR

This paper introduces a Discontinuous Galerkin discretization for two-equation turbulence models in ocean simulations, ensuring stability and positivity, and validates it through benchmarks and real-world river plume modeling.

## Contribution

It presents a novel DG discretization method for turbulence closure models that maintains stability and positivity in ocean modeling applications.

## Key findings

- Stable simulation of vertical mixing in stratified conditions
- Accurate river plume characteristics captured
- Validated with benchmarks and estuary simulation

## Abstract

Accurate representation of vertical turbulent fluxes is crucial for numerical ocean modelling, both in global and coastal applications. The state-of-the-art approach is to use two-equation turbulence closure models which introduces two dynamic equations to the system. Solving these equations numerically, however, is challenging due to the strict requirement of positivity of the turbulent quantities (e.g., turbulence kinetic energy and its dissipation rate), and the non-linear source terms that may render the numerical system unstable. In this paper, we present a Discontinuous Galerkin (DG) finite element discretization of the Generic Length Scale (GLS) equations designed to be incorporated in a DG coastal ocean model, Thetis. To ensure numerical stability, the function space for turbulent quantities must be chosen carefully. In this work, we propose to use zeroth degree elements for the turbulent quantities and linear discontinuous elements for the tracers and velocity. The spatial discretization is completed with a positivity preserving semi-implicit time integration scheme. We validate the implementation with standard turbulence closure model benchmarks and an idealized estuary simulation. Finally, we use the full three-dimensional model to simulate the Columbia River plume. The results confirm that the coupled model generates realistic vertical mixing, and remains stable under strongly stratified conditions and strong tidal forcing. River plume characteristics are well captured.