Numerical study of a regularized barotropic vorticity model of geophysical flow

Abstract

We study a Crank-Nicolson in time, finite element in space, numerical scheme for a Bardina regularization of the barotropic vorticity (BV) model. We derive the regularized model from the simplified Bardina model in primitive variables, present a numerical algorithm for it, and prove the algorithm is unconditionally stable with respect to the timestep size and optimally convergent in both space and time. Numerical experiments are provided that verify the theoretical convergence rates, and also that test the model/scheme on a benchmark double-gyre wind forcing experiment. For the latter test, we find the proposed model/scheme gives a good coarse mesh approximation to the highly resolved direct numerical simulation of the BV model, and compares favorably to related regularization model results.

Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

“…Equations (12) and (13) are supplemented by boundary conditions, such as ψ = ∂ψ ∂n = 0 on ∂Ω. More details regarding the parameters and nondimensionalization of the QGE are given in, e.g., [8,[30][31][32][33]. Note that the velocity can be recovered from the streamfunction according to the following formula:…”

On a Computational Advantageous Voigt Regularization for Geophysical Flows

“…The proposed algorithm was implemented in the software FreeFem++ 3 (see [7] for details on the implementation). Table 1 presents the convergence rates estimated using the analytical solution ψ = exp − 2π 2 Ro δ M L 3 sin(πx) sin(πy) (see [7]). In this table, the estimated convergence rates corroborate the theoretical convergence rates described in Theorem 3.1.…”

“…In this table, the estimated convergence rates corroborate the theoretical convergence rates described in Theorem 3.1. In the second test we evaluate the BV-Voigt model in the traditional Double Gyre Wind Forcing benchmark for Ro = 0.0016 and ( δM /L) 3 = 0.02 (for more details, see [7]). In this experiment, the solution of BV model is very sensitive to the mesh resolution.…”

“…Our goal in this paper is to study a Voigt regularization of the BV model, which has significant computational advantages compared to other regularizations such as BV-α [6] and BV-Bardina models [7]. BV-Voigt is a regularization of the BV model based on the Navier-Stokes-Voigt (NSV) equations [5], an incompressible viscoelastic model which is a smooth regularization of Navier-Stokes equations [4].…”

“…Using the proof of Theorem 1 presented in [7] with α = 0, it remains only to bound the extrapolation error caused by the linearization of the nonlinear term, which can be easily bound using Taylor series and the following terms…”