On the Propagation of Oceanic Waves Driven by a Strong Macroscopic Flow

Abstract

In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around an inhomogeneous (nonzonal) stationary profile. This extends the study (Cheverry et al. in Semiclassical and spectral analysis of oceanic waves, Duke Math. J., accepted), where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting.

Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.

Appendix: A Comparison Result

For the sake of completeness, we state here the result which shows the stability of the propagation under an O(ε ^∞) error on the propagator. This result has been used in the proof of the diagonalization when comparing A and T _±, T _R , and in the proof of dispersion when comparing T _± and $T^{0}_{\pm}$.

Proposition 5.3

Let A _ε and $\tilde{A}_{\varepsilon}$ be two pseudo-differential operators such that

• iA _ε is hermitian in L ²(R ^d),

• $A_{\varepsilon}-\tilde{A}_{\varepsilon}=O(\varepsilon ^{\infty})$ microlocally on Ω⊂R ^2d.

Let $\tilde{\varphi}$ be a solution to

$$i\partial _t\tilde{\varphi}+\tilde{A}_\varepsilon \tilde{\varphi}=0$$

microlocalized in Ω, and φ be the solution to

$$i\partial _t \varphi+ A_\varepsilon \varphi=0$$

with the same initial data. Then, for all N∈N,

$$\sup_{ t\leq\varepsilon ^{-N}} \| \varphi(t)-\tilde{\varphi}(t)\|_{L^2(\mathbf{R}^d)} =O(\varepsilon ^\infty).$$

Proof

The proof is based on a simple energy inequality and is completely straightforward. We have This leads to

$$\| \varphi(t)-\tilde{\varphi}(t)\|^2_{L^2(\mathbf{R}^d)}=O(\varepsilon ^\infty) t,$$

which concludes the proof. □