Part I. Atmospheric Dynamical Models.
Chapter 1. Introduction. The introduction provides an overview on atmospheric dynamics, and on numerical weather prediction (NWP), together with its history. It contains comments on the usage of the spectral technique in NWP.
Chapter 2. Governing Atmospheric Dynamics. The fundamental equations governing atmospheric flows, relevant for the Earth's atmosphere, are laid out. Consideration is given to the rotational and gravitational restoring forces. The global energy balance is illustrated in its essence.
Chapter 3. The Primitive Equations. The closed set of hydrostatic primitive equations (HPEs) is derived in detail for the vertical σ-coordinate. The HPEs are reformulated in spherical geometry in their vorticity-divergence representation, together with continuous energy considerations. The HPE potential-vorticity equation is considered in vectorial form.
Chapter 4. The Shallow-Water Model. The shallow-water equations (SWEs) are developed as a barotropic subset of the HPEs. The vorticity-divergence representation of the SWEs is considered, together with specific analytical balanced solutions and energetic considerations.
Chapter 5. The Barotropic Vorticity Equation. The barotropic vorticity equation (BVE) is introduced as a subset of the SWEs, representing the simplest geophysically relevant dynamical model. The analytical form of Rossby-Haurwitz wave solutions to the BVE on the sphere is discussed.
Chapter 6. Balanced Flow. The concept of balanced quasigestrophic (QG) flow together with the impact of the QG approximation is quantitatively discussed within the SWEs. Further, conditions for hydrodynamic instability in barotropic and baroclinic flows are recorded. The concept of normal mode (NM) analysis is discussed within the framework of the HPEs.
Part II. Spectral Numerical Models.
Chapter 7. The Spectral Method. The essence of the spectral method in its representing unknown functions in terms of expansion coefficients on known basis functions is explained. Illustrations are given for the linear and nonlinear advection equation. Spectral discrete energy conservation implied by use of the Galerkin technique is demonstrated. The phenomenon of aliasing is discussed quantitatively. The duality of the spectral and physical representations of functions is described through explicit formulation of the continuous transforms between both of these representations. The transform method is discussed in general, as well as in its specific application within the spectral integration of the BVE. The spectral construction of isotropic correlations in spherical geometry is considered.
Chapter 8. Vertical Discretization. The specifics of the vertical discretization of the HPEs, as used within PEAK, are outlined, in terms of the Lorenz vertical arrangement of variables. The energetic consistency of this discretization is demonstrated.
Chapter 9. Time Integration. The semi-implicit time integration technique, as used within PEAK, is fully discussed. The starting point of the discussion is given by the separation of processes related to linear gravity-wave activity from the full HPE dynamics. The leap-frog scheme is illustrated.
Chapter 10. Code Structure of PEAK. The structure of the code of the HPE model PEAK model is discussed, by giving consideration to how the computational flow is controlled within PEAK, as well as by describing all essential subroutines in detail. The chapter contains a table listing a set of nineteen PEAK reference experiments that may easily be carried out by readers on a personal computer.
Chapter 11. Experimentation with PEAK. Following specific guidelines on setting up the exectuable PEAK code, this chapter is concerned with the validation of the baroclinic PEAK configuration, primarily against the benchmark test suite suggested by [PST04].
Chapter 12. Barotropic PEAK Configurations. The identification and execution of barotropic variants of PEAK is described. Published reference solutions, such as [WDH+92], [GSP04], and [GSP06] serve to validate these variants of PEAK.
Part III. Appendices.
Appendix A. Tensor Analysis. This appendix contains all details on geometrical and dynamical aspects required for implementing the HPEs for numerical use as appropriate for PEAK.
Appendix B. Spectral Basis Functions. A brief, but comprehensive summary of the associated Legendre functions is presented, together with theory relevant for their efficient and stable numerical evaluation.
Appendix C. The PEAK Model Code. This appendix contains printed listings of all code referred to in this book, and referenced in detail on the code page.
Sunday, 03 March 2024