Stability analysis of split-explicit free surface ocean models: Implication of the depth-independent barotropic mode approximation

Abstract

The evolution of the oceanic free-surface is responsible for the propagation of fast surface gravity waves, which approximatively propagate at speed √ gH (with g the gravity and H the local water depth). In the deep ocean, this phase speed is roughly two orders of magnitude faster than the fastest internal gravity waves. The very strong stability constraint imposed by those fast surface waves on the time-step of numerical models is handled using a mode splitting between slow (internal / baroclinic) and fast (external / barotropic) motions to allow the possibility to adopt specific numerical treatments in each component. The barotropic mode is traditionally approximated by the vertically integrated flow because it has only slight vertical variations. However the implications of this assumption on the stability of the splitting are not well documented. In this paper, we describe a stability analysis of the mode splitting technique based on an eigenvector decomposition using the true (depth-dependent) barotropic mode. This allows us to quantify the amount of dissipation required to stabilize the approximative splitting. We show that, to achieve stable integrations, the dissipation usually applied through averaging filters can be drastically reduced when incorporated at the level of the barotropic time stepping. The benefits are illustrated by numerical experiments. In addition, the formulation of a new mode splitting algorithm using the depth-dependent barotropic mode is introduced.

A Scalable Semi‐Implicit Barotropic Mode Solver for the MPAS‐Ocean

“…It is worth noting that we performed an additional simulation (ES-DE, a gray dash-dot line in Fig. 6) using the sigma vertical coordinate and the second-order averaging filter to match configurations in Demange et al (2019). Our result now closer, albeit still slightly more dissipated than their result (see the middle panel of their Fig.…”

“…10), meaning that our model configurations for this test case and the split-explicit subcycling scheme implemented in MPAS-O are working properly. The more dissipated barotropic energy in our result compared to Demange et al (2019) would result from differences in the model discretizations and numerical schemes used.…”

“…In one-dimensional tests consisting of a simple cosine wave test and a surface gravity wave test, we focus on the accuracy of each time stepping method. The two-dimensional (x-z slice domain) internal tide test case (Marsaleix et al, 2008;Demange et al, 2019) is more challenging than the simple one-dimensional test cases and is evaluated to investigate the barotropic energy dissipation. We include a fourth-order non-subcycled (unsplit) Runge-Kutta (RK) method already implemented in MPAS-O as a reference solution for the oneand two-dimensional test cases.…”

“…The internal tide test case is a two-dimensional (x-z slice) test case with a needle-like bathymetry that generates a nonlinear internal tide (Marsaleix et al, 2008;Demange et al 2019).…”

“…The splitting methods (ES and SI with Δ = 10 s) contain numerical dissipation relative to the UE method even though they are also second-order accurate (Dukowicz and Smith, 1994;Shchepetkin and McWilliams, 2005;Higdon, 2008). With increasing time step size, the total barotropic energy for the split time stepping methods decreases rapidly (Marsaleix et al, 2008;Demange et al, 2019). Trends for both the semi-implicit and explicit-subcycling barotropic mode solvers are similar, although the semi-implicit solver is slightly more dissipative for Δ = 240 s and 1,080 s. Judging from a tiny amount of the scaled barotropic mechanical energy for Δ = 1,080 s at day 12 8.82 × 10 E‹ for ES and 5.82 × 10 E‹ for SI), it seems the fluid simulated by the splitting methods is reaching steady state.…”