TOPICS ON ENVIRONMENTAL AND PHYSICAL GEODESY Compiled by Jose M. Redondo Dept. Fisica Aplicada UPC, Barcelona Tech. November 2014 Contents 1 Vector calculus identities 1 1.1 Operator notations ........................................... 1 1.1.1 Gradient ............................................ 1 1.1.2 Divergence .......................................... 1 1.1.3 Curl .............................................. 1 1.1.4 Laplacian ........................................... 1 1.1.5 Special notations ....................................... 1 1.2 Properties ............................................... 2 1.2.1 Distributive properties .................................... 2 1.2.2 Product rule for the gradient ................................. 2 1.2.3 Product of a scalar and a vector ................................ 2 1.2.4 Quotient rule ......................................... 2 1.2.5 Chain rule ........................................... 2 1.2.6 Vector dot product ...................................... 2 1.2.7 Vector cross product ..................................... 2 1.3 Second derivatives ........................................... 2 1.3.1 Curl of the gradient ...................................... 2 1.3.2 Divergence of the curl ..................................... 2 1.3.3 Divergence of the gradient .................................. 2 1.3.4 Curl of the curl ........................................ 3 1.4 Summary of important identities ................................... 3 1.4.1 Addition and multiplication .................................. 3 1.4.2 Differentiation ......................................... 3 1.4.3 Integration .......................................... 4 1.5 See also ................................................ 4 1.6 Notes and references ......................................... 4 1.7 Further reading ............................................ 4 2 Flow velocity 5 2.1 Definition ............................................... 5 2.2 Uses .................................................. 5 2.2.1 Steady flow .......................................... 5 2.2.2 Incompressible flow ...................................... 5 i ii CONTENTS 2.2.3 Irrotational flow ........................................ 5 2.2.4 Vorticity ............................................ 5 2.3 The velocity potential ......................................... 5 2.4 References ............................................... 6 3 Laminar flow 7 3.1 Relationship with the Reynolds number ................................ 7 3.2 Examples ............................................... 7 3.3 Laminar flow barriers ......................................... 8 3.4 See also ................................................ 8 3.5 References .............................................. 8 3.6 External links ............................................. 8 4 Chaos theory 9 4.1 Introduction .............................................. 9 4.2 Chaotic dynamics ........................................... 10 4.2.1 Sensitivity to initial conditions ................................ 10 4.2.2 Topological mixing ...................................... 11 4.2.3 Density of periodic orbits ................................... 11 4.2.4 Strange attractors ....................................... 11 4.2.5 Minimum complexity of a chaotic system .......................... 12 4.2.6 Jerk systems ......................................... 12 4.3 Spontaneous order ........................................... 13 4.4 History ................................................. 13 4.5 Distinguishing random from chaotic data ............................... 15 4.6 Applications .............................................. 15 4.6.1 Computer science ....................................... 16 4.6.2 Biology ............................................ 16 4.6.3 Other areas .......................................... 16 4.7 See also ................................................ 17 4.8 References ............................................... 17 4.9 Scientific literature ........................................... 20 4.9.1 Articles ............................................ 20 4.9.2 Textbooks ........................................... 20 4.9.3 Semitechnical and popular works ............................... 21 4.10 External links ............................................. 22 5 Fractal 23 5.1 Introduction .............................................. 24 5.2 History ................................................. 24 5.3 Characteristics ............................................ 26 5.4 Brownian motion ........................................... 27 CONTENTS iii 5.5 Common techniques for generating fractals .............................. 27 5.6 Simulated fractals ........................................... 27 5.7 Natural phenomena with fractal features ............................... 28 5.8 In creative works ........................................... 28 5.9 Applications in technology ...................................... 29 5.10 See also ................................................ 29 5.10.1 Fractal-generating programs ................................. 29 5.11 Notes ................................................. 29 5.12 References ............................................... 29 5.13 Further reading ............................................ 31 5.14 External links ............................................. 32 6 Wind wave 33 6.1 Wave formation ............................................ 34 6.2 Types of wind waves ......................................... 35 6.3 Wave shoaling and refraction ..................................... 37 6.4 Wave breaking ............................................. 37 6.5 Science of waves ............................................ 37 6.6 Wind wave models ........................................... 39 6.7 Seismic signals ............................................. 39 6.8 Internal waves ............................................. 39 6.9 See also ................................................ 39 6.10 Notes ................................................. 39 6.11 References ............................................... 40 6.11.1 Scientific ........................................... 40 6.11.2 Other ............................................. 41 6.12 External links ............................................. 41 7 Mixing (process engineering) 42 7.1 Mixing classification .......................................... 42 7.2 Liquid–liquid mixing ......................................... 42 7.2.1 Single-phase blending ..................................... 42 7.2.2 Multi-phase mixing ...................................... 43 7.3 Gas–gas mixing ............................................ 43 7.4 Solid–solid mixing ........................................... 43 7.4.1 Mixing mechanisms ...................................... 43 7.5 Liquid–solid mixing .......................................... 43 7.5.1 Solid suspension ........................................ 44 7.5.2 Solid deagglomeration .................................... 44 7.6 Liquid–gas mixing ........................................... 44 7.7 Gas–solid mixing ........................................... 44 7.8 Multiphase mixing ........................................... 44 iv CONTENTS 7.9 Constitutive equations ......................................... 45 7.10 Laboratory mixing ........................................... 45 7.11 Mixing in microfluidics ........................................ 46 7.12 Industrial mixing equipment ...................................... 46 7.12.1 Turbines ............................................ 46 7.12.2 Close-clearance mixers .................................... 46 7.12.3 High shear dispersers ..................................... 46 7.12.4 Static mixers ......................................... 47 7.13 See also ................................................ 47 7.14 References ............................................... 47 7.15 External links ............................................. 47 8 Liquid bubble 48 8.1 Common examples .......................................... 48 8.2 Physics and chemistry ......................................... 49 8.2.1 Appearance .......................................... 49 8.2.2 Applications .......................................... 49 8.2.3 Pulsation ........................................... 49 8.3 Physiology and medicine ....................................... 50 8.4 See also ................................................ 50 8.5 References .............................................. 50 8.6 External links ............................................. 50 9 Synthetic aperture radar 51 9.1 Functional principle .......................................... 51 9.1.1 Algorithm ........................................... 51 9.1.2 More complex operation ................................... 52 9.1.3 Polarimetry .......................................... 52 9.1.4 Interferometry ......................................... 53 9.1.5 Ultra-wideband SAR ..................................... 54 9.1.6 Doppler-beam sharpening ................................... 54 9.1.7 Chirped (pulse-compressed) radars .............................. 55 9.2 Typical operation ........................................... 55 9.3 Image appearance ........................................... 56 9.4 History ................................................. 57 9.4.1 Origin and early development (ca. 1950–1975) ........................ 58 9.4.2 Relationship