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Linear and nonlinear filtering for a two-layer quasi-geostrophic ocean model

Lander Besabe, Michele Girfoglio, Annalisa Quaini, Gianluigi Rozza

Abstract

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).

Linear and nonlinear filtering for a two-layer quasi-geostrophic ocean model

Abstract

Although the two-layer quasi-geostrophic equations (2QGE) are a simplified model for the dynamics of a stratified, wind-driven ocean, their numerical simulation is still plagued by the need for high resolution to capture the full spectrum of turbulent scales. Since such high resolution would lead to unreasonable computational times, it is typical to resort to coarse low-resolution meshes combined with the so-called eddy viscosity parameterization to account for the diffusion mechanisms that are not captured due to mesh under-resolution. We propose to enable the use of further coarsened meshes by adding a (linear or nonlinear) differential low-pass to the 2QGE, without changing the eddy viscosity coefficient. While the linear filter introduces constant (additional) artificial viscosity everywhere in the domain, the nonlinear filter relies on an indicator function to determine where and how much artificial viscosity is needed. Through several numerical results for a double-gyre wind forcing benchmark, we show that with the nonlinear filter we obtain accurate results with very coarse meshes, thereby drastically reducing the computational time (speed up ranging from 30 to 300).
Paper Structure (13 sections, 24 equations, 13 figures, 10 tables)

This paper contains 13 sections, 24 equations, 13 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Two-layer Quasi-Geostrophic Equations
  4. Low-pass differential spatial filter
  5. The discretized problem
  6. Time discretization
  7. Space discretization
  8. Numerical results
  9. Validation of our solver for the 2QGE
  10. Assessment of the linear and nonlinear filtering
  11. Numerical results for case 1
  12. Numerical results for case 2
  13. Conclusions and future perspective

Figures (13)

  • Figure 1: Validation: absolute error of the stream functions $\psi_1$ (top row) and $\psi_2$ (bottom row) for $Re = 10$ (first column), $Re = 100$ (second column), and $Re = 1000$ (third column). In each case, $Ro = 1$. The results were obtained with mesh size $1/256$.
  • Figure 2: Validation: absolute error of the stream functions $\psi_1$ (top row) and $\psi_2$ (bottom row) for $Ro = 0.1$ (first column), $Ro = 0.01$ (second column), and $Ro = 0.001$ (third column). In each case, $Re = 1$. The results were obtained with mesh size $h=1/256$.
  • Figure 3: Case 1: potential vorticity of the top and bottom layers at time $t=20,21$ computed by the DNS.
  • Figure 4: Case 1: Time-averaged stream function of the top layer $\widetilde{\psi}_1$ computed by DNS (first column) and 2QGE with no filtering (second column), 2QG-$\alpha$ (third column), and 2QG-NL-$\alpha$ (fourth column) with different coarse meshes. Note that the color bar may differ from one panel to the other.
  • Figure 5: Case 1: Time-averaged stream function of the bottom layer $\widetilde{\psi}_2$ computed by DNS (first column) and 2QGE with no filtering (second column), 2QG-$\alpha$ (third column), and 2QG-NL-$\alpha$ (fourth column) with different coarse meshes. Note that the color bar may differ from one panel to the other.
  • ...and 8 more figures