Abstract
This paper focuses on the energy conservation properties of a hydrostatic, Boussinesq, coastal ocean model using a classic finite difference method. It is shown that the leapfrog time-stepping scheme, combined with the sigma-coordinate formalism and the motions of the free surface, prevents the momentum advection from exactly conserving energy. Because of the leapfrog scheme, the discrete form of the kinetic energy depends on the product of velocities at odd and even time steps and thus appears to be possibly negative when high-frequency modes develop. Besides, the study of the energy balance clarifies the numerical choices made for the computation of mixing processes. The time-splitting technique used to reduce the computation costs associated to the resolution of surface waves leads to the well-known external and internal mode equations. We show that these equations do not conserve energy if the coupling of these two modes is forward in time. Even if non-linear terms are negligible, this shortcoming can be significant regarding the pressure gradient term ‘frozen’ over a baroclinic time step. An alternative energy-conserving time-splitting technique is proposed in this paper. Discussion and conclusions are conducted in the light of a set of numerical experiments dedicated to surface and internal gravity waves.
Highlights
The purpose of this study is to evaluate and enforce the energy conservation properties of the sigma-coordinate free-surface ocean model Symphonie, described in Auclair et al (2000)
The numerical schemes of an ocean circulation model are considered in details in this work, from the point of view of the global conservation of energy
It is shown that some energy conservation properties are lost due to the time-splitting method used to limit computational costs associated to the resolution of surface waves
Summary
The purpose of this study is to evaluate and enforce the energy conservation properties of the sigma-coordinate free-surface ocean model Symphonie, described in Auclair et al (2000). Concerning mixing terms, the expected consistency with the turbulent closure scheme is, to our knowledge, rather rarely mentioned in the literature (see, Burchard, 2002) This particular point is addressed in this paper. The pressure gradient force (PGF) is probably the most important term because of the large exchange between kinetic and potential energies through vertical advection of density. In oceanic models using terrain following coordinates, PGF schemes are sensitive to truncation errors because of potentially steep bathymetric slopes, in coastal areas (Haney, 1991) This issue motivated several studies concerning ways to improve the accuracy of numerical schemes. The present paper focuses on aspects rarely studied in the past: lack of energy conservation due to leapfrog temporal schemes, including free-surface motion, Asselin filter and time splitting of internal and external modes. We present a numerical experiment of internal wave generation over a submarine ridge, after which we present our conclusions
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