Anmerkung: 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.