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A Physically Consistent Formulation of Lateral Friction in Shallow-Water Equation Ocean Models

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  • 1 Centre for Ocean–Atmospheric Prediction Studies, The Florida State University, Tallahassee, Florida
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Abstract

Dissipation in numerical ocean models has two purposes: to simulate processes in which the friction is physically relevant and to prevent numerical instability by suppressing accumulation of energy in the smallest resolved scales. This study shows that even for the latter case the form of the friction term should be chosen in a physically consistent way. Violation of fundamental physical principles reduces the fidelity of the numerical solution, even if the friction is small. Several forms of the lateral friction, commonly used in numerical ocean models, are discussed in the context of shallow-water equations with nonuniform layer thickness. It is shown that in a numerical model tuned for the minimal dissipation, the improper form of the friction term creates finite artificial vorticity sources that do not vanish with increased resolution, even if the viscous coefficient is reduced consistently with resolution. An alternative numerical implementation of the no-slip boundary conditions for an arbitrary coast line is considered. It was found that the quality of the numerical solution may be considerably improved by discretization of the viscous stress tensor in such a way that the numerical boundary scheme approximates not only the stress tensor to a certain order of accuracy but also simulates the truncation error of the numerical scheme used in the interior of the domain. This ensures error cancellation during subsequent use of the elements of the tensor in the discrete version of the momentum equations, allowing for approximation of them without decrease in the order of accuracy near the boundary.

Abstract

Dissipation in numerical ocean models has two purposes: to simulate processes in which the friction is physically relevant and to prevent numerical instability by suppressing accumulation of energy in the smallest resolved scales. This study shows that even for the latter case the form of the friction term should be chosen in a physically consistent way. Violation of fundamental physical principles reduces the fidelity of the numerical solution, even if the friction is small. Several forms of the lateral friction, commonly used in numerical ocean models, are discussed in the context of shallow-water equations with nonuniform layer thickness. It is shown that in a numerical model tuned for the minimal dissipation, the improper form of the friction term creates finite artificial vorticity sources that do not vanish with increased resolution, even if the viscous coefficient is reduced consistently with resolution. An alternative numerical implementation of the no-slip boundary conditions for an arbitrary coast line is considered. It was found that the quality of the numerical solution may be considerably improved by discretization of the viscous stress tensor in such a way that the numerical boundary scheme approximates not only the stress tensor to a certain order of accuracy but also simulates the truncation error of the numerical scheme used in the interior of the domain. This ensures error cancellation during subsequent use of the elements of the tensor in the discrete version of the momentum equations, allowing for approximation of them without decrease in the order of accuracy near the boundary.

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