About the Authors |
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xv | |
Preface |
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xix | |
Acknowledgments |
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xxi | |
List of Acronyms |
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xxiii | |
Definitions and Notation |
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xxv | |
Introduction and Scope |
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xxix | |
Part I Background |
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3 | (13) |
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1.1 Typical Motions and Scales |
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4 | (4) |
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8 | (8) |
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8 | (2) |
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10 | (2) |
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12 | (1) |
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13 | (1) |
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1.2.5 Aggregation and Identity |
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14 | (2) |
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2 Drifters and Their Numerical Simulation |
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16 | (22) |
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16 | (4) |
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20 | (5) |
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20 | (4) |
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24 | (1) |
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24 | (1) |
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25 | (4) |
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2.3.1 Basic Lagrangian Model |
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25 | (1) |
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26 | (3) |
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2.4 Model Validation with Drifters |
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29 | (6) |
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31 | (4) |
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35 | (3) |
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2.5.1 Drifter Assimilation |
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37 | (1) |
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3 Probability and Statistics - A Primer |
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38 | (78) |
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3.1 Basics - Random Numbers |
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38 | (9) |
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3.1.1 Continuous Distributions: f and F |
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38 | (1) |
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3.1.2 Properties: Survival, Hazard Rate |
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39 | (2) |
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3.1.3 Properties: Mean, Variance, Moments |
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41 | (2) |
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3.1.4 Properties: Median, Mode, Quartile |
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43 | (1) |
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3.1.5 Properties: Other Means |
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44 | (1) |
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44 | (1) |
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3.1.7 Discrete Distributions: Pi and Fj |
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45 | (2) |
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3.2 Some Common Distributions |
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47 | (17) |
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3.2.1 Continuous Distributions |
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47 | (11) |
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3.2.2 Discrete Distributions |
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58 | (5) |
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3.2.3 Importance of G, U, B, Pois |
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63 | (1) |
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3.2.4 The Central Limit Theorem |
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63 | (1) |
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3.3 Generating Random Numbers |
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64 | (6) |
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64 | (3) |
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3.3.2 Some Specific Deviates |
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67 | (3) |
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70 | (12) |
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71 | (1) |
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72 | (4) |
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76 | (6) |
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82 | (1) |
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82 | (17) |
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83 | (3) |
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3.5.2 Correlation and Autocorrelation |
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86 | (1) |
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3.5.3 Autocorrelated Time Series |
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86 | (6) |
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3.5.3.1 Separation-Based Covariance and Correlogram |
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89 | (2) |
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3.5.3.2 Correlogram and Impulse Response |
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91 | (1) |
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3.5.4 Autocorrelated Eulerian Fields |
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92 | (3) |
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3.5.5 Generating Covariance |
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95 | (3) |
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3.5.6 Summary - Covariance |
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98 | (1) |
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99 | (15) |
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3.6.1 Individual Residence Time |
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100 | (1) |
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3.6.2 Aggregate Properties: Relaxation of Initial Condition |
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100 | (2) |
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102 | (1) |
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103 | (1) |
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3.6.5 Summary - Steady State |
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104 | (1) |
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105 | (2) |
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107 | (1) |
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107 | (1) |
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3.6.9 Example - Branch Point |
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108 | (1) |
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3.6.10 A Network of Boxes |
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108 | (5) |
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110 | (1) |
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111 | (2) |
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3.6.11 Closing Ideas - Particles in Boxes |
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113 | (1) |
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114 | (1) |
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115 | (1) |
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4 Dispersion by Random Walk |
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116 | (55) |
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4.1 Introduction: Discrete Drunken Walk |
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117 | (2) |
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119 | (2) |
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4.2.1 Resolved and Subgrid Motion |
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119 | (1) |
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120 | (1) |
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4.3 The ARO Model - Uncorrelated Random Walk and Simple Diffusion |
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121 | (12) |
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4.3.1 The Displacement Process |
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121 | (1) |
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4.3.2 Correspondence to Diffusion |
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122 | (2) |
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124 | (1) |
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4.3.4 Inhomogeneous Diffusion |
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125 | (1) |
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4.3.5 Anisotropic Diffusion |
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126 | (2) |
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4.3.6 Shear and Convergence |
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128 | (3) |
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4.3.7 Metrics of Resolution |
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131 | (1) |
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4.3.8 Stepsize and the Need for Autocorrelation |
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132 | (1) |
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4.4 The AR1 Model - Autocorrelated Velocity |
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133 | (11) |
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4.4.1 AR1: Continuous Form and Its Discretization |
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134 | (2) |
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4.4.2 AR1: Discrete Canonical Form |
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136 | (3) |
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139 | (3) |
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4.4.4 Some Summary Observations about the AR1 Model |
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142 | (2) |
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4.5 The AR2 Model - Autocorrelated Acceleration |
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144 | (12) |
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4.5.1 Discrete Canonical Forms |
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144 | (4) |
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148 | (5) |
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4.5.3 AR2 Displacement and Link to Diffusion |
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153 | (3) |
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4.6 The AR1-s Model - Spinning Walk |
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156 | (9) |
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4.6.1 AR1-s Complex Velocity |
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157 | (2) |
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159 | (2) |
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161 | (1) |
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162 | (3) |
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4.7 Summary - Four Random Walk Models |
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165 | (4) |
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166 | (1) |
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167 | (1) |
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167 | (2) |
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169 | (1) |
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4.8 Concluding Remarks - Random Motion |
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169 | (2) |
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5 Boundary Conditions, Boundary Layers, Sources |
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171 | (23) |
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5.1 Boundary Layers: Continuum and Discretization |
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171 | (2) |
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5.2 Discretized Boundaries |
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173 | (4) |
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5.2.1 Particle Motion and Change |
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173 | (1) |
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5.2.2 States and Transitions |
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174 | (1) |
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175 | (1) |
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5.2.4 Basic Needs for Boundary Particles |
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176 | (1) |
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177 | (1) |
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177 | (2) |
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5.4 A Repellant Boundary Layer |
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179 | (4) |
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179 | (1) |
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180 | (1) |
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181 | (1) |
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181 | (2) |
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183 | (8) |
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183 | (3) |
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186 | (2) |
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188 | (3) |
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5.6 Beyond the Boundary - Exogenous Sources |
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191 | (1) |
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192 | (2) |
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194 | (33) |
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6.1 Reynolds Stresses and the Gradient Flux Relation |
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194 | (3) |
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6.1.1 The Gradient Flux Relation |
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196 | (1) |
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197 | (7) |
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197 | (2) |
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6.2.2 Turbulent Kinetic Energy |
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199 | (3) |
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6.2.3 Turbulent Length Scale |
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202 | (2) |
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6.3 Vertical Closure Examples |
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204 | (17) |
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6.3.1 Level 2.5 Formulation |
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204 | (4) |
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6.3.1.1 Governing Equations |
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205 | (1) |
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6.3.1.2 Vertical Boundary Conditions |
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206 | (2) |
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6.3.2 Level 2.0 Formulation |
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208 | (1) |
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209 | (2) |
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6.3.4 Steady-State Point Model, Level 2 Closure |
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211 | (9) |
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212 | (2) |
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214 | (1) |
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6.3.4.3 Rotation and Wind - The Ekman Layer |
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215 | (5) |
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6.3.5 Some Implementations |
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220 | (1) |
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221 | (2) |
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223 | (4) |
Part II Elements |
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7 Meshes: Interpolation, Navigation, and Fields |
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227 | (44) |
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227 | (14) |
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227 | (1) |
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227 | (1) |
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228 | (5) |
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233 | (2) |
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7.1.5 Spherical-Polar Coordinates |
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235 | (1) |
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7.1.6 Horizontal Interpolation |
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235 | (3) |
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7.1.6.1 Higher Order Interpolation |
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237 | (1) |
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238 | (1) |
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7.1.8 Location and Navigation |
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239 | (2) |
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7.1.8.1 Global and Local Coordinates |
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239 | (1) |
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7.1.8.2 Is a Particle in an Element? |
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239 | (1) |
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7.1.8.3 Motion within an Element |
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239 | (1) |
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7.1.8.4 When Does a Particle Leave an Element? |
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240 | (1) |
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7.2 Vertical Discretization |
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241 | (4) |
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7.2.1 Separation of Variables |
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241 | (1) |
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7.2.2 Special Functions and Global z-Interpolation |
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242 | (1) |
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7.2.3 Piecewise Local z-Interpolation |
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242 | (2) |
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7.2.4 Vertical Interpolation |
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244 | (1) |
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7.3 3-D Location, Interpolation, Navigation |
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245 | (4) |
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7.3.1 Interpolation on a Single Element |
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246 | (1) |
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7.3.2 Is a Particle in an Element? |
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247 | (1) |
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7.3.3 Motion within an Element |
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247 | (1) |
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7.3.4 When Does a Particle Leave an Element? |
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248 | (1) |
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7.3.5 Summary: An Overlay of Horizontal Meshes |
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249 | (1) |
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249 | (15) |
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7.4.1 The Union of Elements |
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249 | (1) |
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7.4.2 Essential Data Structures |
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250 | (3) |
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253 | (1) |
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7.4.4 Subsidiary Data Structures |
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253 | (4) |
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7.4.5 The Galerkin Projection |
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257 | (2) |
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259 | (2) |
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7.4.7 Some Mesh Generation Packages |
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261 | (1) |
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262 | (1) |
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263 | (1) |
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7.5 Quadrilateral Elements |
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264 | (7) |
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7.5.1 Local Coordinates and Interpolation |
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265 | (2) |
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267 | (2) |
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269 | (2) |
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271 | (26) |
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271 | (1) |
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8.2 Scattering among Elements |
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272 | (1) |
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8.3 Scattering within an Element |
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273 | (6) |
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274 | (5) |
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8.3.1.1 Uniform Distribution on a Triangle |
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275 | (1) |
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8.3.1.2 Linear Distribution on a Triangle |
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276 | (1) |
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276 | (3) |
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279 | (1) |
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8.4 Projections: The Density of a Set of Particles |
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279 | (18) |
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8.4.1 Simple Fixed Mesh Projections |
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281 | (3) |
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8.4.2 The Least Squares Projection |
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284 | (3) |
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287 | (1) |
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8.4.4 Example: Particles on a Mesh |
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287 | (1) |
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8.4.5 Convergence - The Small Δx Problem |
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288 | (2) |
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290 | (7) |
Part III Applications |
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9 Noncohesive Sediment - Dense Particles |
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297 | (40) |
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297 | (1) |
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9.2 Three States: P, M, B |
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298 | (2) |
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300 | (1) |
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301 | (2) |
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9.5 Bottom Boundary Layer |
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303 | (1) |
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303 | (6) |
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9.6.1 The Threshold of Motion - The Shields Parameter |
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304 | (1) |
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305 | (2) |
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9.6.3 Initial Vertical Position - Entrainment Lift |
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307 | (2) |
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9.7 Vertical Motion: The Rouse Number and ze |
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309 | (1) |
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310 | (3) |
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313 | (5) |
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318 | (1) |
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319 | (3) |
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9.12 Theoretical Extensions |
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322 | (1) |
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9.13 A Simple Particle Model |
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322 | (1) |
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323 | (12) |
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9.14.1 Bed-Load - Suspended Load Transport |
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324 | (1) |
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9.14.2 Bed-Load Particle Velocity |
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325 | (1) |
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9.14.3 Suspended Load Particle Velocity |
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326 | (1) |
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9.14.4 Sediment Dispersion Coefficients |
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327 | (1) |
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328 | (1) |
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328 | (7) |
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335 | (2) |
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10 Oil - Chemically Active Particles |
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337 | (52) |
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337 | (5) |
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338 | (1) |
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339 | (1) |
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340 | (1) |
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10.1.4 Subsurface Releases |
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341 | (1) |
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342 | (2) |
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10.2.1 Surface and Subsurface Parcels |
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343 | (1) |
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10.2.2 Generic Model Needs |
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344 | (1) |
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344 | (10) |
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345 | (9) |
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10.4 Weathering Processes |
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354 | (12) |
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354 | (3) |
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357 | (1) |
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10.4.3 Density and Viscosity |
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358 | (1) |
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10.4.4 A Simple Weathering Model |
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359 | (4) |
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363 | (3) |
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366 | (4) |
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10.5.1 Stokes Drift and Surface Velocity |
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366 | (2) |
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10.5.2 Random Walk Models |
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368 | (1) |
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10.5.3 Droplet Rise Velocity |
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369 | (1) |
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10.5.4 Droplet Size Distribution |
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370 | (1) |
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10.6 Density and Crowding |
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370 | (3) |
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371 | (1) |
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371 | (2) |
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373 | (11) |
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10.7.1 Surface Source - Entrainment |
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374 | (1) |
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375 | (4) |
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10.7.3 Subsurface Source - Blowout |
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379 | (2) |
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381 | (3) |
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384 | (4) |
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384 | (1) |
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10.8.2 Subsurface Releases |
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384 | (1) |
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10.8.3 The Deepwater Horizon Incident |
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385 | (3) |
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388 | (1) |
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11 Individual-Based Models - Biotic Particles |
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389 | (66) |
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389 | (1) |
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11.2 Diversity in the Cohort |
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389 | (2) |
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11.3 Individual-Based States |
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391 | (1) |
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11.3.1 Mixing and Aggregation |
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391 | (1) |
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391 | (1) |
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11.3.3 Eulerian and Lagrangian Quantities |
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392 | (1) |
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11.3.4 Continuous and Logical States; State Transitions |
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392 | (1) |
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392 | (6) |
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11.4.1 Growth Rate Distribution - Analytic Example |
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393 | (1) |
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11.4.2 Growth Rate Distribution: Simulation |
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394 | (2) |
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11.4.3 Multiple Equilibria |
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396 | (1) |
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396 | (2) |
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398 | (8) |
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11.5.1 Simulations: Individual-Based Mortality |
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399 | (2) |
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11.5.2 Individual Survivorship Probability: Geometric Distribution |
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401 | (2) |
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11.5.3 Lifetime Distribution |
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403 | (1) |
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11.5.4 Cohort Abundance: Binomial Distribution |
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404 | (1) |
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405 | (1) |
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11.5.6 Example: Growth, Mortality, Motion |
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406 | (1) |
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406 | (12) |
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11.6.1 Example: Three Stages |
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408 | (3) |
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11.6.2 Stage Completion Rate, Residence Time - Constant α |
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411 | (1) |
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11.6.3 The General Case - Variable α |
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412 | (2) |
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11.6.4 Recap: Stage Completion |
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414 | (4) |
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418 | (2) |
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11.8 Motion - Advection + Random Walk |
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420 | (1) |
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421 | (5) |
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11.9.1 Example Formulations |
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422 | (4) |
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426 | (1) |
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427 | (17) |
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427 | (7) |
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428 | (3) |
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431 | (2) |
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11.11.1.3 Recent Summaries - Larval Fish |
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433 | (1) |
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434 | (4) |
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438 | (5) |
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11.11.3.1 The Gulf of Maine |
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438 | (3) |
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441 | (2) |
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11.11.4 Summary - Plankton IBMs |
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443 | (1) |
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11.12 Population Connectivity |
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444 | (3) |
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11.13 Summary - Individual-Based Modeling |
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447 | |
Part IV Appendices |
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455 | (3) |
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458 | (3) |
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461 | (2) |
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D Rates and Rate Limiters |
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463 | (5) |
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468 | (2) |
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F Covariance Matrix for Shear and Convergence |
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470 | (2) |
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G Summary - ARn Recursions |
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472 | (2) |
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H Distribution Properties for Linear Triangles |
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474 | (5) |
Bibliography |
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479 | (28) |
Index |
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507 | |