Renormalization group method applied to the primitive equations

Abstract

In this article we study the limit, as the Rossby number e goes to zero, of the primitive equations of the atmosphere and the ocean. From the mathematical viewpoint we study the averaging of a penalization problem displaying oscillations generated by an antisymmetric operator and by the presence of two time scales.

Existence and regularity results for the primitive equations in two space dimensions

“…and, since this is needed in [9], we assume space periodicity with period M, that is, all functions are taken to satisfy…”

“…We first show how we can express the diagnostic variables w and p in terms of the prognostic variables u, v and ρ. For each U = (u, v, ρ) ∈ V we can determine uniquely w = w(U ) from (1.1d), 9) since w(x, 0) = 0, w being odd in z. Furthermore, writing that w(x, −L 3 /2, t) = w(x, L 3 /2, t), we also have…”

“…The objective of this article is to derive various results of existence and regularity of solutions for the Primitive Equations of the ocean (PEs) in two space dimensions. These results, besides their intrinsic interest, are needed in [9] which is another motivation of this work.We consider the PEs in their nondimensional form (see Section 5) : …”

“…We mention here the similar works of Bresch, Kazhikhov and Lemoine [2] and of Ziane [13], who consider different boundary conditions and do not consider the higher regularity results needed in [9]; see also [11]. For the non-dimensional form of the PEs, we refer here for example to [4], [8], and [12] but a substantial amount of literature is available on this subject.…”

“…The motivations for considering periodic boundary conditions is that there are needed in studies on homogeneous turbulence of the atmosphere and also for the study of the renormalized equations considered in [9].…”