Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Abstract

Abstract. Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D "primitive" Navier-Stokes equations for strong stratification (large stratification parameter N). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear "2 1 2 dimensional" limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα, α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D "primitive" Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems.Mathematics Subject Classification. 76D05, 76D50, 76U05, 86A10.

Exponential Smallness of Inertia–Gravity Wave Generation at Small Rossby Number

“…The most standard of these is averaging, whereby (2.2)-(2.3) are averaged over the fast O( −1 ) time scale to leave only slow equations for s and for slowly-varying amplitudes of the fast variables f. Averaging has been applied only recently in the geophysical context, however (e.g. Majda & Embid 1998, Babin et al 2000, Wirosoetisno et al 2002, and with rather theoretical motivations. Instead, what has been extensively employed, is the idea of balance, which seeks to filer out fast inertia-gravity waves completely, mainly on the ground that these are weak in most of the atmosphere and oceans as well as poorly constrained by observations.…”

Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

“…In the above results, the Coriolis forcing term, with rotation parameter R, did not play any role in proving the global regularity. This is contrary to the cases of the three-dimensional fast rotating Euler, Navier-Stokes and Boussinesq equations by [2,3,4,5] where the authors take full advantage of the absence of resonances between the fast rotation and the nonlinear advection. This absence of resonances at the limit of fast rotation leads to strong dispersion and averaging mechanism (at the limit of fast rotation for large values R depending on the size of the initial data) that weakens the nonlinear effects and hence allows for establishing the global regularity result in the viscous Navier-Stokes case, and prolongs the life-space of the solution in the Euler case (see also [12,15] and references therein; in addition, see [1] for simple examples demonstrating the above mechanism).…”

“…For results concerning the short time existence and uniqueness of the inviscid primitive equations see, for example, [6,18,23,27] and references therein. Notably, it is unknown of whether the rotation term in the inviscid primitive equations, in particular for large values of R, plays a stabilizing mechanism by preventing the formation of singularity as in the case of Burgers equations [1,21], or by extending the life of span of the solution and postponing the blowup as in the case of the three-dimensional Euler equations [2,3,4,5,12,15]; this is a subject of ongoing and future research.…”

Slow Manifolds and Invariant Sets of the Primitive Equations

“…3 The nonhydrostatic case has been shown, for sufficiently strong rotation, to have a unique solution for all time by Babin et al (2000). 4 We stress that such bounds, having to account for all possible worst-case scenarios, usually do not give any meaningful estimate on how large the solution actually is; unhelpfully, ''estimate'' means ''bound'' in the mathematical literature.…”

“…D 3 21 q, has been used as an approximation to the primitive equations, (1) or its variants. When « is small and the initial conditions for the full primitive equations (1) are at (or near) geostrophic balance, it has been proven that, subject to some smoothness conditions, the solution of (12) is a good approximation to the solution of (1) over a time of order 1; see Bourgeois and Beale (1994) and Babin et al (2000) for the inviscid case in the Boussinesq PE, and Temam and Wirosoetisno (2007) for (1). Physically, this is not surprising, the more interesting question being whether and how the solution of (1) comes near geostrophic balance to begin with.…”