Zdunkowski W., Bott A. Dynamics of the atmosphere: A course in theoretical meteorology

Cambridge University Press, 2003. — 793 p.Dynamics of the Atmosphere is a textbook with numerous exercises and solutions, written for senior undergraduate and graduate students of meteorology and related sciences. It may also be used as a reference source by professional meteorologists and researchers in atmospheric science. In order to encourage the reader to follow the mathematical developments in detail, the derivations are complete and leave out only the most elementary steps. The book consists of two parts, the first presenting the mathematical tools needed for a thorough understanding of the second part. Mathematical topics include a summary of the methods of vector and tensor analysis in generalized coordinates; an accessible presentation of the method of covariant differentiation; and a brief introduction to nonlinear dynamics. These mathematical tools are used later in the book to tackle such problems as the fields of motion over different types of terrain, and problems of predictability. The second part of the book begins with the derivation of the equation describing the atmospheric motion on the rotating earth, followed by several chapters that consider the kinematics of the atmosphere and introduce vorticity and circulation theorems. Weather patterns can be considered as superpositions of waves of many wavelengths, and the authors therefore present a discussion of wave motion in the atmosphere, including the barotropic model and some Rossby physics.Achapter on inertial and dynamic stability is presented and the component form of the equation of motion is derived in the general covariant, contravariant, and physical coordinate forms. The subsequent three chapters are devoted to turbulent systems in the atmosphere and their implications for weather-prediction equations. At the end of the book newer methods of weather prediction, such as the spectral technique and the stochastic dynamic method, are introduced in order to demonstrate their potential for extending the forecasting range as computers become increasingly powerful.Preface
Mathematical tools
Algebra of vectors
Basic concepts and definitions
Reference frames
Vector multiplication
Reciprocal coordinate systems
Vector representations
Products of vectors in general coordinate systems
Vector functions
Basic definitions and operations
Special dyadics
Principal-axis transformation of symmetric tensors
Invariants of a dyadic
Tensor algebra
Differential relations
Differentiation of extensive functions
The Hamilton operator in generalized coordinate systems
The spatial derivative of the basis vectors
Differential invariants in generalized coordinate systems
Additional applications
Coordinate transformations
Transformation relations of time-independent coordinate systems
Transformation relations of time-depende
inate systems
The method of covariant differentiation
Spatial differentiation of vectors and dyadics
Time differentiation of vectors and dyadics
The local dyadic of vP
Integral operations
Curves, surfaces, and volumes in the general qi system
Line integrals, surface integrals, and volume integrals
Integral theorems
Fluid lines, surfaces, and volumes
Time differentiation of fluid integrals
The general form of the budget equation
Gauss’ theorem and the Dirac delta function
Solution of Poisson’s differential equation
Appendix: Remarks on Euclidian and Rieman spaces
Introduction to the concepts of nonlinear dynamics
One-dimensional flow
Two-dimensional flow
Dynamics of the atmosphere
The laws of atmospheric motion
The equation of absolute motion
The energy budget in the absolute reference system
The geographical coordinate system
The equation of relative motion
The energy budget of the general relative system
The decomposition of the equation of motion
Scale analysis
An outline of the method
Practical formulation of the dimensionless numbers
Scale analysis of large-scale frictionless motion
The geostrophic wind and the Euler wind
The equation of motion on a tangential plane
The material and the local description of flow
The description of Lagrange
Lagrange’s version of the continuity equation
An example of the use of Lagrangian coordinates
The local description of Euler
Transformation from the Eulerian to the Lagrangian system
Atmospheric flow fields
The velocity dyadic
The deformation of the continuum
Individual changes with time of geometric fluid configurations
The Navier–Stokes stress tensor
The general stress tensor
Equilibrium conditions in the stress field
Symmetry of the stress tensor
The frictional stress tensor and the deformation dyadic
The Helmholtz theorem
The three-dimensional Helmholtz theorem
The two-dimensional Helmholtz theorem
Kinematics of two-dimensional flow
Atmospheric flow fields
Two-dimensional streamlines and normals
Streamlines in a drifting coordinate system
Natural coordinatesDifferential definitions of the coordinate lines
Metric relationships
Blaton’s equation
Individual and local time derivatives of the velocity
Differential invariants
The equation of motion for frictionless horizontal flow
The gradient wind relation
Boundary surfaces and boundary conditionsDifferential operations at discontinuity surfaces
Particle invariance at boundary surfaces, displacement velocities
The kinematic boundary-surface condition
The dynamic boundary-surface condition
The zeroth-order discontinuity surface
An example of a first-order discontinuity surface
Circulation and vorticity theorems
Ertel’s form of the continuity equation
The baroclinic Weber transformation
The baroclinic Ertel–Rossby invariant
Circulation and vorticity theorems for frictionless baroclinic flow
Circulation and vorticity theorems for frictionless barotropic flow
Turbulent systems
Simple averages and fluctuations
Weighted averages and fluctuations
Averaging the individual time derivative and the budget operator
Integral means
Budget equations of the turbulent system
The energy budget of the turbulent system
Diagnostic and prognostic equations of turbulent systems
Production of entropy in the microturbulent system
An excursion into spectral turbulence theory
Fourier Representation of the continuity equation and the equation of motion
The budget equation for the amplitude of the kinetic energy
Isotropic conditions, the transition to the continuous wavenumber space
The Heisenberg spectrum
Relations for the Heisenberg exchange coefficient
A prognostic equation for the exchange coefficient
Concluding remarks on closure procedures
The atmospheric boundary layerPrandtl-layer theory
The Monin–Obukhov similarity theory of the neutral Prandtl layer
The Monin–Obukhov similarity theory of the diabatic Prandtl layer
Application of the Prandtl-layer theory in numerical prognostic models
The fluxes, the dissipation of energy, and the exchange coefficients
The interface condition at the earth’s surface
The Ekman layer – the classical approach
The composite Ekman layer
Ekman pumping
Appendix A: Dimensional analysis
Appendix B: The mixing length
Wave motion in the atmosphere
The representation of waves
The group velocity
Perturbation theory
Pure sound waves
Sound waves and gravity waves
Lamb waves
Lee waves
Propagation of energy
External gravity waves
Internal gravity waves
Nonlinear waves in the atmosphere
e barotropic model
The basic assumptions of the barotropic model
The unfiltered barotropic prediction model
The filtered barotropic model
Barotropic instability
The mechanism of barotropic development
Appendix
Rossby waves
One- and two-dimensional Rossby waves
Three-dimensional Rossby waves
Normal-mode considerations
Energy transport by Rossby waves
The influence of friction on the stationary Rossby wave
Barotropic equatorial waves
The principle of geostrophic adjustment
Appendix
ertial and dynamic stability
Inertial motion in a horizontally homogeneous pressure field
Inertial motion in a homogeneous geostrophic wind field
Inertial motion in a geostrophic shear wind field
Derivation of the stability criteria in the geostrophic wind field
Sectorial stability and instability
Sectorial stability for normal atmospheric conditions
Sectorial stability and instability with permanent adaptation
The equation of motion in general coordinate systemsThe covariant equation of motion in general coordinate systems
The contravariant equation of motion in general coordinate systems
The equation of motion in orthogonal coordinate systems
Lagrange’s equation of motion
Hamilton’s equation of motion
Appendix
The geographical coordinate system
The equation of motion
Application of Lagrange’s equation of motion
The first metric simplification
The coordinate simplification
The continuity equation
The stereographic coordinate system
The stereographic projection
Metric forms in stereographic coordinates
The absolute kinetic energy in stereographic coordinates
The equation of motion in the stereographic Cartesian coordinates
The equation of motion in stereographic cylindrical coordinates
The continuity equation
The equation of motion on the tangential plane
The equation of motion in Lagrangian enumereation coordinates
Orography-following coordinate systems
The metric of the η system
The equation of motion in the η system
The continuity equation in the η system
The stereographic system with a generalized vertical coordinate
The ξ transformation and resulting equations
The pressure system
The solution scheme using the pressure system
The solution to a simplified prediction problem
The solution scheme with a normalized pressure coordinate
The solution scheme with potential temperature as vertical coordinate
A quasi-geostrophic baroclinic modelThe first law of thermodynamics in various forms
The vorticity and the divergence equation
The first and second filter conditions
The geostrophic approximation of the heat equation
The geostrophic approximation of the vorticity equation
The ω equation
The Philipps approximation of the ageostrophic component of the horizontal wind
Applications of the Philipps wind
A two-level prognostic model, baroclinic instabilityThe mathematical development of the two-level model
The Phillips quasi-geostrophic two-level circulation model
Baroclinic instability
excursion concerning numerical procedures
Numerical stability of the one-dimensional advection equation
Application of forward-in-time and central-in-space difference quotients
A practical method for the elimination of the weak instability
The implicit method
The aliasing error and nonlinear instability
Modeling of atmospheric flow by spectral techniquesThe basic equations
Horizontal discretization
Predictability
Derivation and discussion of the Lorenz equations
The effect of uncertainties in the initial conditions
Limitations of deterministic predictability of the atmosphere
Basic equations of the approximate stochastic dynamic method
Answers to Problems
List of frequently used symbols
References and bibliography