Averaging over fast gravity waves for geophysical flows with arbitary

Embid, Pedro

doi:10.1080/03605309608821200

Cited by 167 publications

(209 citation statements)

References 12 publications

Supporting

Mentioning

191

Contrasting

Unclassified

Order By: Relevance

“…Therefore the quasigeostrophic component of the resonant equation (3.25) completely decouples. This fact was proved in BMN [6,11] by direct computation and also proved by Embid and Majda [26] using Ertel's theorem.…”

Section: For Real U(x) and ρ(X)mentioning

confidence: 81%

“…It follows that for fixed η, θ 2 , θ 3 , Π(k, −k − n, −n) is a polynomial of degree at most eight in k 3 . The leading power in k 8 3 is: 26) where m was eliminated via m = −k − n. If this leading term (4.26) is not zero, there are at most eight k 3 satisfying Π(k, −m − n, −n) = 0 for given k 1 , k 2 , n; this holds provided that…”

Section: Lemma 42 (Lemma On Restricted Convolutions) Let χ(K M N)mentioning

confidence: 96%

“…Clearly, when represented in Fourier modes, the operatorB in the right-hand side of (3.25) has coefficientsQ i1i2i3 kmn in (3.20) and the limit resonant equations are 26) whereÃ n =ν(n)ň 2 and the summation is over resonant terms. In equations (3.26)F i3 n =F QG,n for i 3 = 0 andF i3 n in the appropriate component ofF AG,n for i 3 = 1, 2.…”

Section: For Real U(x) and ρ(X)mentioning

confidence: 99%

“…This mathematical approach in the context of geophysical flows was initiated in BMN [4,6,10,11], Mahalov and Marcus [40]. In the context of symmetric hyperbolic systems (including weakly nonlinear geometric optics), related singular oscillating limits have been investigated by Joly, Metivier and Rauch [31,34], Schochet [45] (where the weakly compressible limit of 3D Euler equations was considered), Grenier [30], Embid and Majda [26]. For general "primitive" equations of geophysics, following methods for hyperbolic Euler-type systems of Bardos [17], Bourgeois and Beale [19] have demonstrated convergence to quasi-geostrophic limit equations for special classes of initial data (of infinite codimension).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2000

View full text Add to dashboard Cite

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.

show abstract

Section: For Real U(x) and ρ(X)mentioning

confidence: 81%

Section: Lemma 42 (Lemma On Restricted Convolutions) Let χ(K M N)mentioning

confidence: 96%

Section: For Real U(x) and ρ(X)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

2000

View full text Add to dashboard Cite

show abstract

“…In [7], he considered strong solutions and showed that the oscillating part goes to zero, whereas the quasi-geostrophic part goes to the unique global solution of the 3D quasi-geostrophic system whose initial data is the quasi-geostrophic part of the initial data. As in [21], [24] and [14], the methods used in [7] are given by the study of spectral properties of the matrix −L + 1 ε PA, where P denotes the Leray projection of L 2 onto the subspace of divergence free vector fields.…”

Section: Introductionmentioning

confidence: 99%