Numerical investigation of the regularity of the pressure for the primitive equations of the ocean Original Research Article

In this work we analyze the regularity of the pressure of the primitive equations (PE) of the ocean by numerical simulation and analysis. This model makes use of the hydrostatic pressure assumption. Several authors have shown that the hydrostatic hypothesis limits the regularity of the surface pressure to LD3/2(ω), where the weight D is the depth of the domain, and ω is the surface domain. Nevertheless, the same L2(ω) regularity as in the Navier–Stokes case can be proved when the domain has a talus. We address in this paper the question whether this gap of regularity is also observable in the numerical approximation of the primitive equations. Specifically, for the numerical solver of the primitive equations considered, we prove convergence of the surface pressure in LD3/2(ω) and in L2(ω′) for subdomains ω′⊂ω with talus. This lack of regularity near the border is confirmed by numerical tests, where we observe the formation of an infinite normal derivative in ∂ω of the surface pressure as the size of the talus vanishes. Moreover, we show that the LD3/2(ω) regularity obtained in the theory for domains without talus is not far from being optimal.