Gallagher, Isabelle Semi-classical analysis of oceanic flows. (English) Zbl 1348.35278 Lin, Fanghua (ed.) et al., Lectures on the analysis of nonlinear partial differential equations. Part 3. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-267-1/hbk). Morningside Lectures in Mathematics 3, 145-175 (2013). Summary: In these lecture notes we present results of C. Cheverry et al. [Duke Math. J. 161, No. 5, 845–892 (2012; Zbl 1244.35147)] and the author et al. [in: Nonlinear partial differential equations. The Abel symposium 2010. Proceedings of the Abel symposium, Oslo, Norway, September 28–October 2, 2010. Berlin: Springer. 231–254 (2012; Zbl 1252.35266)] in which the propagation of waves is studied, in a model describing the movement of oceans in large geographical zones. We consider a shallow water flow which is subject to strong rotation and linearized around an inhomogeneous stationary profile, and we prove that the underlying systems of PDEs can be diagonalized microlocally: the three linear propagators thus constructed correspond to particular types of waves, namely two Poincaré and one Rossby wave. We show how Mourre estimates allow to obtain the dispersion of Poincaré waves; in the case when the stationary profile is zonal we prove by ODE techniques that for initial data microlocalized in some codimension one set, Rossby waves are trapped for all times.For the entire collection see [Zbl 1342.35008]. MSC: 35Q86 PDEs in connection with geophysics 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 86A05 Hydrology, hydrography, oceanography 76B65 Rossby waves (MSC2010) 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35B44 Blow-up in context of PDEs Keywords:semi-classical analysis; microlocal analysis; Mourre estimates; geophysical flows Citations:Zbl 1244.35147; Zbl 1252.35266 PDFBibTeX XMLCite \textit{I. Gallagher}, Morningside Lect. Math. 3, 145--175 (2013; Zbl 1348.35278)