Note: Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Part 1 Mathematical tools -- M1 Algebra of vectors -- M1.1 Basic concepts and definitions -- M1.2 Reference frames -- M1.3 Vector multiplication -- M1.3.1 The scalar product of two vectors -- M1.3.2 The vector product of two vectors -- M1.3.3 The dyadic representation, the general product of two vectors -- M1.3.4 The scalar triple product -- M1.3.5 The vectorial triple product -- M1.3.6 The scalar product of a vector with a dyadic -- M1.3.7 Products involving four vectors -- M1.4 Reciprocal coordinate systems -- M1.5 Vector representations -- M1.6 Products of vectors in general coordinate systems -- M1.7 Problems -- M2 Vector functions -- M2.1 Basic definitions and operations -- M2.2 Special dyadics -- M2.2.1 The conjugate dyadic -- M2.2.2 The symmetric dyadic -- M2.2.3 The antisymmetric or skew-symmetric dyadic -- M2.2.4 The adjoint dyadic -- M2.2.5 The reciprocal dyadic -- M2.3 Principal-axis transformation of symmetric tensors -- M2.4 Invariants of a dyadic -- M2.4.1 The first scalar of a dyadic -- M2.4.2 The vector of a dyadic -- M2.4.3 The second scalar of a dyadic -- M2.4.4 The third scalar of a dyadic -- M2.5 Tensor algebra -- M2.6 Problems -- M3 Differential relations -- M3.1 Differentiation of extensive functions -- M3.2 The Hamilton operator in generalized coordinate systems -- M3.3 The spatial derivative of the basis vectors -- M3.4 Differential invariants in generalized coordinate systems -- M3.4.1 The first scalar or the divergence of the local dyadic… -- M3.4.2 The Laplacian of a scalar field function -- M3.4.3 The vector of the local dyadic… -- M3.5 Additional applications -- M3.6 Problems -- M4 Coordinate transformations -- M4.1 Transformation relations of time-independent coordinate systems -- M4.1.1 Introduction. , M4.1.2 Transformation of basis vectors and coordinate differentials -- M4.1.3 Transformation of vectors and dyadics -- M4.2 Transformation relations of time-dependent coordinate systems -- M4.2.1 The addition theorem of the velocities -- M4.2.2 Orthogonal q systems -- M4.2.3 The generalized vertical coordinate -- M4.3 Problems -- M5 The method of covariant differentiation -- M5.1 Spatial differentiation of vectors and dyadics -- M5.2 Time differentiation of vectors and dyadics -- M5.3 The local dyadic of v -- M5.4 Problems -- M6 Integral operations -- M6.1 Curves, surfaces, and volumes in the general q system -- M6.2 Line integrals, surface integrals, and volume integrals -- M6.3 Integral theorems -- M6.3.1 Stokes' integral theorem -- M6.3.2 Gauss' divergence theorem -- M6.4 Fluid lines, surfaces, and volumes -- M6.5 Time differentiation fluid integrals -- M6.5.1 Time differentiation of fluid line integrals -- M6.5.2 Time differentiation of fluid surface integrals -- M6.5.3 Time differentiation of fluid volume integrals -- M6.6 The general form of the budget equation -- M6.6.1 The budget equation for the partial masses of atmospheric air -- M6.6.2 The first law of thermodynamics -- M6.7 Gauss' theorem and the Dirac delta function -- M6.8 Solution of Poisson's differential equation -- M6.9 Appendix: Remarks on Euclidian and Riemannian spaces -- M6.10 Problems -- M7 Introduction to the concepts of nonlinear dynamics -- M7.1 One-dimensional flow -- M7.1.1 Fixed points and stability -- M7.1.2 Bifurcation -- M7.1.2.1 Saddle-node bifurcation -- M7.1.2.2 Transcritical bifurcation -- M7.1.2.3 Pitchfork bifurcation -- M7.2 Two-dimensional flow -- M7.2.1 Linear stability analysis -- M7.2.2 Classification of linear systems -- M7.2.3 Two-dimensional nonlinear systems -- M7.2.4 Limit cycles -- M7.2.5 Hopf bifurcation -- M7.2.6 The Liapunov function. , M7.2.7 Fractal dimensions -- Part 2 Dynamics of the atmosphere -- 1 The laws of atmospheric motion -- 1.1 The equation of absolute motion -- 1.2 The energy budget in the absolute reference system -- 1.3 The geographical coordinate system -- 1.3.1 Operations involving the rotational velocity vOmega -- 1.3.1.1 The divergence of v in the geographical coordinate system -- 1.3.1.2 Rotation and the vector gradient of vOmega -- 1.3.2 The centrifugal potential -- 1.3.3 The budget operator -- 1.4 The equation of relative motion -- 1.5 The energy budget of the general relative system -- 1.6 The decomposition of the equation of motion -- 1.7 Problems -- 2 Scale analysis -- 2.1 An outline of the method -- 2.2 Practical formulation of the dimensionless flow numbers -- 2.3 Scale analysis of large-scale frictionless motion -- 2.4 The geostrophic wind and the Euler wind -- 2.5 The equation of motion on a tangential plane -- 2.6 Problems -- 3 The material and the local description of flow -- 3.1 The description of Lagrange -- 3.2 Lagrange's version of the continuity equation -- 3.2.1 Preliminaries -- 3.2.2 The mass-conservation equation in the Lagrangian form -- 3.3 An example of the use of Lagrangian coordinates -- 3.3.1 General remarks -- 3.3.2 The thermo-hydrodynamic equations -- 3.3.3 Difference approximations -- 3.3.4 Initial values and boundary conditions -- 3.3.4.1 Approximate determination of… -- 3.3.4.2 The interpolated velocity…on the trajectory -- 3.3.5 The numerical stability condition -- 3.4 The local description of Euler -- 3.5 Transformation from the Eulerian to the Lagrangian system -- 3.6 Problems -- 4 Atmospheric flow fields -- 4.1 The velocity dyadic -- 4.1.1 The three-dimensional velocity dyadic -- 4.1.2 The two-dimensional velocity dyadic -- 4.2 The deformation of the continuum -- 4.2.1 The representation of the wind field. , 4.2.2 Flow patterns and stability -- 4.3 Individual changes with time of geometric fluid configurations -- 4.3.1 The relative change of the material line element -- 4.3.2 The directional change of the material line element -- 4.3.3 The change in volume of a rectangular fluid box -- 4.3.4 Two-dimensional examples -- 4.4 Problems -- 5 The Navier-Stokes stress tensor -- 5.1 The general stress tensor -- 5.1.1 Volume forces -- 5.1.2 Surface forces -- 5.2 Equilibrium conditions in the stress field -- 5.3 Symmetry of the stress tensor -- 5.4 The frictional stress tensor and the deformation dyadic -- 5.5 Problems -- 6 The Helmholtz theorem -- 6.1 The three-dimensional Helmholtz theorem -- 6.2 The two-dimensional Helmholtz theorem -- 6.3 Problems -- 7 Kinematics of two-dimensional flow -- 7.1 Atmospheric flow fields -- 7.2 Two-dimensional streamlines and normals -- 7.2.1 Two-dimensional streamlines -- 7.2.2 Construction of normals -- 7.3 Streamlines in a drifting coordinate system -- 7.4 Problems -- 8 Natural coordinates -- 8.1 Introduction -- 8.2 Differential definitions of the coordinate lines -- 8.3 Metric relationships -- 8.4 Blaton's equation -- 8.5 Individual and local time derivatives of the velocity -- 8.6 Differential invariants -- 8.6.1 The horizontal divergence of the velocity -- 8.6.2 Vorticity or the vertical component of… -- 8.6.3 The Jacobian operator and the Laplacian -- 8.7 The equation of motion for frictionless horizontal flow -- 8.8 The gradient wind relation -- 8.9 Problems -- 9 Boundary surfaces and boundary conditions -- 9.1 Introduction -- 9.2 Differential operations at discontinuity surfaces -- 9.3 Particle invariance at boundary surfaces, displacement velocities -- 9.4 The kinematic boundary-surface condition -- 9.4.1 External boundary surfaces -- 9.4.2 Internal boundary surfaces. , 9.4.3 The generalized vertical velocity at boundary surfaces -- 9.5 The dynamic boundary-surface condition -- 9.6 The zeroth-order discontinuity surface -- 9.6.1 The inclination of the zeroth-order DS -- 9.6.2 A discontinuity surface of zeroth-order in the geostrophic wind field -- 9.7 An example of a first-order discontinuity surface -- 9.8 Problems -- 10 Circulation and vorticity theorems -- 10.1 Ertel's form of the continuity equation -- 10.2 The baroclinic Weber transformation -- 10.3 The baroclinic Ertel-Rossby invariant -- 10.4 Circulation and vorticity theorems for frictionless baroclinic flow -- 10.4.1 A general baroclinic vortex theorem -- 10.4.2 Ertel's vortex theorem -- 10.4.3 Ertel's conservation theorem, potential vorticity -- 10.4.4 The general vorticity theorem -- 10.4.5 Rossby's formulation of the potential vorticity -- 10.4.6 Helmholtz's baroclinic vortex theorem -- 10.4.7 Thomson's and Bjerkness' baroclinic circulation theorems -- 10.4.8 Interpretation of Bjerkness' circulation theorem -- 10.5 Circulation and vorticity theorems for frictionless barotropic flow -- 10.5.1 The barotropic Ertel-Rossby invariant -- 10.5.2 Barotropic vortex theorems of Ertel, Helmholtz, and Thomson -- 10.5.3 Vortex lines and vortex tubes -- 10.5.4 The vorticity theorem for the barotropic atmosphere -- 10.6 Problems -- 11 Turbulent systems -- 11.1 Simple averages and fluctuations -- 11.2 Weighted averages and fluctuations -- 11.3 Averaging the individual time derivative and the budget operator -- 11.4 Integral means -- 11.5 Budget equations of the turbulent system -- 11.6 The energy budget of the turbulent system -- 11.7 Diagnostic and prognostic equations of turbulent systems -- 11.8 Production of entropy in the microturbulent system -- 11.8.1 Scalar fluxes -- 11.8.2 Vectorial fluxes -- 11.8.3 The scalar phenomenological equations. , 11.8.4 The vectorial phenomenological equations.

Note: Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- 1.1 The atmospheric radiation field -- 1.2 The mean global radiation budget of the Earth -- 1.3 Solar-terrestrial relations -- 1.3.1 The equation of time -- 1.3.2 Geographical coordinates and the solar position -- 1.3.3 Long-term variations of the Earth's orbital parameters -- 1.4 Basic definitions of radiative quantities -- 1.5 The net radiative flux density vector -- 1.6 The interaction of radiation with matter -- 1.6.1 Absorption -- 1.6.2 Scattering -- 1.6.3 Emission -- 1.7 Problems -- 2 The radiative transfer equation -- 2.1 Eulerian derivation of the radiative transfer equation -- 2.1.1 The exchange of photons -- 2.1.2 The absorption of photons -- 2.1.3 The scattering of photons -- 2.1.4 The emission rate -- 2.1.5 The budget equation of the photon distribution function -- 2.2 The direct-diffuse splitting of the radiance field -- 2.3 The radiatively induced temperature change -- 2.4 The radiative transfer equation for a horizontally homogeneous atmosphere -- 2.5 Splitting of the radiance field into upwelling and downwelling radiation -- 2.6 The solution of the radiative transfer equation for a horizontally homogeneous atmosphere -- 2.6.1 The scattering atmosphere -- 2.6.2 The nonscattering atmosphere -- 2.7 Radiative flux densities and heating rates -- 2.7.1 The scattering atmosphere -- 2.7.2 The nonscattering atmosphere -- 2.8 Appendix -- 2.8.1 Local thermodynamic equilibrium -- 2.9 Problems -- 3 Principles of invariance -- 3.1 Definitions of the scattering and transmission functions -- 3.2 Diffuse reflection in a semi-infinite atmosphere -- 3.3 Chandrasekhar's four statements of the principles of invariance -- 3.4 The inclusion of surface reflection -- 3.5 Diffuse reflection and transmission for isotropic scattering -- 3.6 Problems. , 4 Quasi-exact solution methods for the radiative transfer equation -- 4.1 The matrix operator method -- 4.1.1 Derivation of the addition theorems -- 4.1.2 The optical properties of a homogeneous elementary layer -- 4.1.3 The doubling algorithm -- 4.1.4 Inhomogeneous atmospheres -- 4.2 The successive order of scattering method -- 4.3 The discrete ordinate method -- 4.4 The spherical harmonics method -- 4.5 The finite difference method -- 4.5.1 Vertical discretization in the finite difference method -- 4.5.2 Treatment of the boundary conditions -- 4.5.3 Computation of mean radiances and flux densities -- 4.6 The Monte Carlo method -- 4.6.1 Determination of photon paths -- 4.6.2 Treatment of absorption -- 4.7 Appendix -- 4.7.1 The reflection matrix at the ground -- 4.8 Problems -- 5 Radiative perturbation theory -- 5.1 Adjoint formulation of the radiative transfer equation -- 5.2 Boundary conditions -- 5.2.1 Vacuum boundary conditions -- 5.2.2 Boundary conditions for a reflecting surface -- 5.2.3 Inclusion of surface reflection in the formulation of the radiances -- 5.3 Radiative effects -- Example I -- Example II -- Example III -- 5.4 Perturbation theory for radiative effects -- 5.4.1 Basic perturbation theory -- 5.4.2 An alternative formulation of the radiative effect -- 5.4.3 Evaluation of the perturbation integral -- 5.5 Appendix -- 5.5.1 Linear operator and its adjoint -- 5.5.2 Superposition formula for the inclusion of Lambertian surface reflection -- 5.6 Problems -- 6 Two-stream methods for the solution of the radiative transfer equation -- 6.1 Delta-scaling of the phase function -- 6.2 The two-stream radiative transfer equation -- 6.3 Different versions of two-stream methods -- 6.3.1 Two-stream method with hemispheric isotropy -- 6.3.2 The Eddington approximation -- 6.3.3 Discrete ordinates formalism -- 6.3.4 Practical improved flux method. , 6.4 Analytical solution of the two-stream methods for homogeneous layer -- 6.5 Approximate treatment of scattering in the infrared spectral region -- 6.6 Approximations for partial cloud cover -- 6.6.1 Partial cloud cover with random overlap -- 6.6.2 Partial cloud cover with maximum overlap -- 6.7 The classical emissivity approximation -- 6.8 Radiation charts -- 6.9 Radiative equilibrium -- 6.10 Problems -- 7 Transmission in individual spectral lines and in bands of lines -- 7.1 The shape of single spectral lines -- 7.1.1 The Lorentz line -- 7.1.2 The thermal Doppler line -- 7.1.3 The Voigt profile -- 7.2 Band models -- 7.2.1 Mean absorption in a single Lorentz line -- 7.2.2 Band model for nonoverlapping lines -- 7.2.3 Random band models -- Goody's exponential model -- Godson's inverse power model -- The Malkmus model -- 7.2.4 Elsasser's regular model -- 7.2.5 The Schnaidt model -- 7.3 The fitting of transmission functions -- 7.3.1 Exponential sum-fitting of transmissions -- 7.3.2 The k-distribution method -- Basic illustration of the k-distribution method -- Algorithm for computing the k-distribution -- The cumulative k-distribution -- 7.3.3 The correlated k-distribution method -- 7.3.4 The k-distribution method for special situations -- Two overlapping gases -- Gray absorption coefficient -- Regular band of nonoverlapping rectangular lines -- Regular band of triangular lines -- Regular band of nonoverlapping Lorentz lines -- Single scattering properties for inhomogeneous atmospheres -- 7.4 Transmission in inhomogeneous atmospheres -- 7.4.1 One-parameter scaling -- 7.4.2 The two-parameter scaling technique of Curtis and Godson -- 7.5 Results -- 7.6 Appendix -- 7.6.1 Maxwell's velocity distribution and the mean molecular velocity -- 7.6.2 Original derivation of the Ladenburg and Reiche function. , 7.6.3 The Mittag-Leffler theorem and the Elsasser model -- Step 1 -- Step 2 -- 7.7 Problems -- 8 Absorption by gases -- 8.1 Introduction -- 8.2 Molecular vibrations -- 8.2.1 Two coupled harmonic oscillators -- 8.2.2 Review of physical principles -- 8.2.3 Linear triatomic molecules -- 8.2.4 Nonlinear triatomic molecules -- 8.3 Some basic principles from quantum mechanics -- 8.3.1 Stationary and coherent states -- (i) Stationary states -- (ii) Coherent states -- 8.3.2 The Schrödinger equation -- 8.3.3 Hamilton operator for a charged particle in an electromagnetic field -- 8.3.4 The interaction Hamiltonian -- 8.3.5 Computation of transition probabilities -- 8.3.6 Einstein transition probabilities -- 8.3.7 Line intensities -- 8.4 Vibrations and rotations of molecules -- 8.4.1 The harmonic oscillator -- 8.4.2 Vibration of diatomic molecules -- 8.4.3 Vibration of polyatomic molecules -- 8.4.4 Rotation of diatomic molecules -- 8.4.5 Vibration-rotation of diatomic molecules -- 8.5 Matrix elements, selection rules and line intensities -- 8.5.1 The harmonic oscillator -- 8.5.2 The rigid rotator -- 8.6 Influence of thermal distribution of quantum states on line intensities -- 8.7 Rotational energy levels of polyatomic molecules -- 8.7.1 Linear molecules -- 8.7.2 Symmetric top molecules -- 8.7.3 Spherical top molecules -- 8.7.4 Asymmetric top molecules -- 8.8 Appendix -- 8.8.1 The Hamilton function -- 8.8.2 Macroscopic fields and Maxwell's equations -- 8.9 Problems -- 9 Light scattering theory for spheres -- 9.1 Introduction -- 9.2 Maxwell's equations -- 9.3 Boundary conditions -- 9.4 The solution of the wave equation -- 9.4.1 Solution of the scalar wave equation in spherical coordinates -- 9.4.2 Solution of the vector wave equation in spherical coordinates -- 9.5 Mie's scattering problem -- 9.5.1 The incoming wave -- 9.5.2 The scattered and the interior waves. , 9.5.3 Rayleigh scattering -- 9.6 Material characteristics and derived directional quantities -- 9.6.1 Extinction, scattering and absorption coefficients -- 9.6.2 The scattering function and the scattering phase function -- 9.7 Selected results from Mie theory -- 9.8 Solar heating and infrared cooling rates in cloud layers -- 9.9 Problems -- 10 Effects of polarization in radiative transfer -- 10.1 Description of elliptic, linear and circular polarization -- 10.1.1 Linear polarization -- 10.1.2 Circular polarization -- 10.2 The Stokes parameters -- 10.3 The scattering matrix -- 10.3.1 Representation of the electric vector in the scattering plane -- 10.3.2 Transformation of the Stokes vector -- 10.4 The vector form of the radiative transfer equation -- 10.5 Problems -- 11 Remote sensing applications of radiative transfer -- 11.1 Introduction -- 11.2 Remote sensing based on short- and long-wave radiation -- 11.2.1 Methods based on the extinction of solar radiation -- Example: determination of the aerosol and ozone optical depth -- 11.2.2 Methods based on thermal emission -- (1) Nadir-looking instrument -- (2) Ground-based observations -- 11.3 Inversion of the temperature profile -- 11.3.1 Direct linear inversion -- 11.3.2 Linear inversion with constraints -- 11.3.3 Chahine's relaxation method -- 11.3.4 Smith's iterative inversion method -- 11.4 Radiative perturbation theory and ozone profile retrieval -- 11.5 Appendix -- 11.5.1 Example for an ill-posed inversion problem -- 11.6 Problems -- 12 Influence of clouds on the climate of the Earth -- 12.1 Cloud forcing -- 12.2 Cloud feedback in climate models -- 12.2.1 Cloud feedback in response to doubling atmospheric CO2 -- 12.2.2 Other trace gases -- 12.2.3 Liquid water and cloud microphysics feedback -- 12.2.4 Climatic impact due to aerosols -- 12.3 Problems -- Answers to problems -- Chapter 1 -- Chapter 2. , Chapter 3.