Regularity "in Large" for the 3D Salmon's Planetary Geostrophic Model of Ocean Dynamics

TL;DR

This paper proves the global well-posedness of Salmon's 3D planetary geostrophic ocean model, addressing previous issues of ill-posedness by establishing existence, uniqueness, and stability of solutions for all initial data.

Contribution

It demonstrates the global well-posedness of Salmon's model, providing a rigorous mathematical foundation for its use in ocean dynamics simulations.

Findings

Proved global existence of strong solutions.

Established uniqueness and continuous dependence on initial data.

Addressed ill-posedness by introducing a Rayleigh-like friction term.

Abstract

It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, 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 literal 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. Another remedy for this problem was suggested by R. Salmon by introducing an additional Rayleigh-like friction 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.