Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

Abstract

Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems.However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models. IntroductionNumerical modeling of the coastal ocean is important for many environmental and industrial applications. Typical scenarios include modeling circulation at regional scales, coupled river-estuary-plume systems, river networks, lagoons, and harbors.Length scales range from some tens of meters in rivers and embayments to tens of kilometers in the coastal ocean; water depth ranges from less than a meter to kilometer scale at the shelf break. The time scales of the relevant processes range from minutes to hours, yet typical simulations span weeks or even decades. The dynamics are highly non-linear, characterized by local smallscale features such as fronts and density gradients, internal waves, and baroclinic eddies. These physical characteristics imply that coastal ocean modeling is intrinsically multi-scale, which imposes several technical challenges.Most coastal ocean models solve the hydrostatic Navier-Stokes equations under the Boussinesq approximation -a valid approximation for mesoscale and sub-mesoscale (1 km) processes. Small-scale processes (< 100 m) are, however, inherently three-dimensional where non-hydrostatic effects can be important, especially in areas with pronounced density structure and stratification (Marshall et al., 1997b;Mahadevan, 2006). Non-hydrostatic modeling requires very high horizontal mesh resolu-1 arXiv:1711.08552v2 [physics.ao-ph]

Discontinuous Galerkin discretization for two-equation turbulence closure model

“…The turbulence closure model was implemented in the Thetis three-dimensional circulation model (Kärnä et al, 2018). Thetis solves the hydrostatic equations with a semi-implicit DG finite element method.…”

“…Vertical diffusion is treated implicitly. The model formulation is described in detail in Kärnä et al (2018).…”

“…It is worth noting that the commonly-used Crank-Nicolson scheme is not L-stable and may lead to unphysical solutions. The simplest choice is the first order Backward Euler method (used in Kärnä et al (2018) for example). This scheme is, however, known to be dissipative.…”

“…The advection terms are discretized with standard DG upwind technique as implemented in the hydrodynamical model (Kärnä et al, 2018). The diffusion terms are treated with Symmetric Interior Penalty Galerkin method (SIPG; Epshteyn and Rivière, 2007) with a penalty factor σ = 1/L where L is the vertical element size (Kärnä et al, 2018).…”

“…We present the weak formulation of the gov-erning equations and a positivity preserving, semi-implicit time integration scheme. The model is implemented in the Thetis ocean model (Kärnä et al, 2018).…”