Spectral numerical weather prediction models.

*(English)*Zbl 1239.86001
Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611971-98-9/pbk; 978-1-611971-99-6/ebook). xxv, 498 p. (2012).

This monograph presents numerical weather prediction using spectral methods for solving the governing equations. A specific example for the implementation, the Primitive Equation Atmospheric Research Model Kernel (PEAK), is provided. The book is organized into three parts. First the governing equations of the atmosphere dynamics are described in the form of a model hierarchy of the hydrostatic primitive equations. Spectral approximations of these sets of partial differential equations are presented in part two. Finally, the third part contains three appendices summarizing tensor analysis, spectral basis functions and the PEAK model code in Fortran.

After a basic introduction, part one presents the principles and governing equations of atmospheric dynamics from a physical and meteorological point of view. A hierarchy of continuous atmospheric models is given which are then considered in part two for numerical discretization. Starting with the primitive equations and conservation principles, the shallow water model is presented. It describes the motion in a shallow layer of incompressible fluid subject to rotation and gravity and it is currently used as intermediate model for geophysical flows in the atmosphere and the ocean. Then the barotropic vorticity equation is introduced in spherical geometry and the concept of balanced flows is provided.

Part two introduces the spectral method for solving partial differential equations encountered in geophysical flows. Practical aspects of Galerkin and collocation discretizations are discussed as well as the problem of dealiasing and different time integration techniques. As example the one-dimensional Burgers equation is studied in some detail. Aspects of the vertical discretization are also addressed. Three chapters are then dedicated directly to the PEAK model. The code structure is presented together with its numerical implementation. Numerical experiments with the PEAK model including some validation are also shown. In addition barotropic PEAK configurations are presented. At the end of each chapter some exercises are proposed.

The appendices recall some basics of tensor analysis including spherical coordinates. Spectral basis functions, i.e., the Legendre polynomials, which are the building blocks of the spherical harmonics, are described and finally a listing of the Fortran PEAK code is printed. A bibliography with numerous references concludes this volume.

The book is accessible for readers with a modest mathematical background and the presentation avoids some mathematical rigor compared to numerical books on spectral methods. Many shortcuts in the text require some accustoming and make the text sometimes difficult to read.

After a basic introduction, part one presents the principles and governing equations of atmospheric dynamics from a physical and meteorological point of view. A hierarchy of continuous atmospheric models is given which are then considered in part two for numerical discretization. Starting with the primitive equations and conservation principles, the shallow water model is presented. It describes the motion in a shallow layer of incompressible fluid subject to rotation and gravity and it is currently used as intermediate model for geophysical flows in the atmosphere and the ocean. Then the barotropic vorticity equation is introduced in spherical geometry and the concept of balanced flows is provided.

Part two introduces the spectral method for solving partial differential equations encountered in geophysical flows. Practical aspects of Galerkin and collocation discretizations are discussed as well as the problem of dealiasing and different time integration techniques. As example the one-dimensional Burgers equation is studied in some detail. Aspects of the vertical discretization are also addressed. Three chapters are then dedicated directly to the PEAK model. The code structure is presented together with its numerical implementation. Numerical experiments with the PEAK model including some validation are also shown. In addition barotropic PEAK configurations are presented. At the end of each chapter some exercises are proposed.

The appendices recall some basics of tensor analysis including spherical coordinates. Spectral basis functions, i.e., the Legendre polynomials, which are the building blocks of the spherical harmonics, are described and finally a listing of the Fortran PEAK code is printed. A bibliography with numerous references concludes this volume.

The book is accessible for readers with a modest mathematical background and the presentation avoids some mathematical rigor compared to numerical books on spectral methods. Many shortcuts in the text require some accustoming and make the text sometimes difficult to read.

Reviewer: Kai Schneider (Marseille)

##### MSC:

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

86A10 | Meteorology and atmospheric physics |

65Z99 | Applications to the sciences |

65Z05 | Applications to the sciences |

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |