Abstract
AbstractIt is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction/damping, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the lateral boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no-heat-flux physical boundary condition. Consequently, the second order parabolic heat equation is over-determined with two different boundary conditions. In a previous work we proposed one remedy to this problem by introducing a fourth-order artificial hyper-diffusion to the heat transport equation and proved global regularity for the proposed model. A shortcoming of this higher-oder diffusion is the loss of the maximum/minimum principle for the heat equation. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction/damping term for the vertical component of the velocity in the hydrostatic balance equation. In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics. That is, we show global existence, uniqueness and continuous dependence of the strong solutions on initial data for this model. Unlike the 3D viscous PG model, we are still unable to show the uniqueness of the weak solution. Notably, we also demonstrate in what sense the additional damping term, suggested by Salmon, annihilate the ill-posedness in the original system; consequently, it can be viewed as “regularizing” term that can possibly be used to regularize other related systems.
Highlights
The starting point in the derivation of the ocean circulation models is Boussinesq equations which are the Navier–Stokes equations with rotation and a heat transport equation
In a previous work we proposed one remedy to this problem by introducing a fourth-order arti cial hyper-di usion to the heat transport equation and proved global regularity for the proposed model
In this paper we prove the global, for all time and all initial data, well-posedness of strong solutions to the three-dimensional Salmon’s planetary geostrophic model of ocean dynamics
Summary
The starting point in the derivation of the ocean circulation models is Boussinesq equations which are the Navier–Stokes equations with rotation and a heat transport equation. Based on physical ground Samelson and Vallis [26] have argued that in closed ocean basin, with the no-normal- ow boundary conditions, this model can be solved only in restricted domains which are bounded away from the lateral boundary, ∂M × (−h, ) It cannot be utilized in the study of the large-scale circulation. Samelson and Vallis proposed in [26] various dissipative schemes to overcome these physical and numerical di culties They propose to add either a linear Rayleighlike drag/friction/damping or a conventional eddy viscosity to the horizontal components of the momentum equations, and a horizontal di usion in the thermodynamic equation (subject to no-heat- ux at the lateral boundary.) The planetary geostrophic (PG) model with conventional eddy viscosity has been studied mathematically in [4], [24], [25].
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