Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations.

*(English)*Zbl 1205.86001
Oxford Lecture Series in Mathematics and its Applications 32. Oxford: Clarendon Press (ISBN 0-19-857133-X/hbk). xi, 250 p. (2006).

Publisher’s description: Aimed at graduate students, researchers and academics in mathematics, engineering, oceanography, meteorology and mechanics, this text provides a detailed introduction to the physical theory of rotating fluids, a significant part of geophysical fluid dynamics. The text is divided into four parts, with the first part providing the physical background of the geophysical models to be analysed. Part II is devoted to a self contained proof of the existence of weak (or strong) solutions to the incompressible Navier-Stokes equations.

Part III deals with the rapidly rotating Navier-Stokes equations, first in the whole space, where dispersion effects are considered. The case where the domain has periodic boundary conditions is then analysed, and finally rotating Navier-Stokes equations between two plates are studied, both in the case of periodic horizontal coordinates and those in \(\mathbb R^2\).

In Part IV the stability of Ekman boundary layers, and boundary layer effects in magnetohydrodynamics and quasigeostrophic equations are discussed. The boundary layers which appear near vertical walls are presented and formally linked with the classical Prandlt equations. Finally spherical layers are introduced, whose study is completely open.

Part III deals with the rapidly rotating Navier-Stokes equations, first in the whole space, where dispersion effects are considered. The case where the domain has periodic boundary conditions is then analysed, and finally rotating Navier-Stokes equations between two plates are studied, both in the case of periodic horizontal coordinates and those in \(\mathbb R^2\).

In Part IV the stability of Ekman boundary layers, and boundary layer effects in magnetohydrodynamics and quasigeostrophic equations are discussed. The boundary layers which appear near vertical walls are presented and formally linked with the classical Prandlt equations. Finally spherical layers are introduced, whose study is completely open.

##### MSC:

86-02 | Research exposition (monographs, survey articles) pertaining to geophysics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76U05 | General theory of rotating fluids |

35Q30 | Navier-Stokes equations |

76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

86A05 | Hydrology, hydrography, oceanography |

86A10 | Meteorology and atmospheric physics |