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Mathematical problems of the dynamics of incompressible fluid on a rotating sphere. (English) Zbl 1391.76003

Cham: Springer (ISBN 978-3-319-65411-9/hbk; 978-3-319-65412-6/ebook). xii, 239 p. (2017).
The book consists of eight chapters. The author discusses some mathematical problems and numerical calculations of two-dimensional dynamics of a liquid on a rotating sphere. Only viscous and ideal incompressible liquids are considered. The second chapter is devoted to spherical harmonics, geographical maps, Hilbert spaces on a sphere, and some estimations. The governing equations of the dynamics of an ideal fluid on a rotating sphere, used in Chapters 3 and 4, are as follows: \[ \begin{aligned} & \Delta\frac{\partial}{\partial t}\Psi+J(\Psi,\Delta\Psi+2\mu)=-\sigma\Delta\Psi+\nu(-\Delta\Psi)^{s+1}+F,\\ & \Delta\Psi(\mu,\lambda,t=0)=\Delta\Psi_0.\\ & \vec u=\vec n\times\nabla\Psi,\quad \nabla\vec u=0,\,\Delta\Psi(\mu,\lambda)=\frac{\partial}{\partial\mu}(1-\mu^2)\frac{\partial}{\partial\mu}+\frac{1}{1-\mu^2}\frac{\partial^2}{\partial\lambda^2}.\end{aligned} \] Here, \(\vec n\) is the unit outward normal to the sphere at a point \(x(\mu,\lambda)\), \(\mu,\lambda\) are geographic coordinates, \(J(\Psi,\Delta\Psi)\) takes into account the advective effect, \(J(\Psi,2\mu)\) includes the sphere rotation, \(\sigma\Delta\Psi\) describes the influence of the planetary boundary layer, and \(\nu(-\Delta\Psi)^{s+1}\) takes into account the turbulent viscosity. In Chapter 3, fundamental approaches to the solvability of vorticity equations on a sphere are presented: the solvability of a stationary vorticity equation, asymptotic behavior of solutions, dimension of the vorticity equation attractor. Several problems of dynamics of modons are considered in the fourth chapter.
In the fifth chapter, the Rossby-Haurwitz wave stability is discussed, whereas Chapter 6 studies the stability of solitary waves (modons) and Wu waves. In the seventh chapter, the stability of shear and zonal flows is reviewed using linear and nonlinear approaches. The nonlinear stability analyses are based on the first Lyapunov method. Here, the reader can find the kinetic energy evolution, and perturbations of a zonal jet. Numerical studies of linear stability are presented in Chapter 8.
On balance, the book contains a deep analysis of mathematical problems of two-dimensional dynamics of an ideal liquid on a rotating sphere and some numerical calculations of the related problems. However, a comparison of solutions discussed in the work with the data of observations in the Earth’s atmosphere would be desirable. This book may be useful for scientists, graduate students, and for all interested in the numerical calculations of dynamics of a liquid on a rotating sphere.

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
76U05 General theory of rotating fluids
76D33 Waves for incompressible viscous fluids
76E20 Stability and instability of geophysical and astrophysical flows
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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