Shallow Flows: Research Presented at the International Symposium on Shallow Flows, Delft, Netherlands, 2003

Cover......Page 1
Shallow Flows......Page 4
Table of contents......Page 6
Foreword......Page 12
Introduction......Page 14
1  DEFINITION......Page 16
2  OCCURENCES OF SHALLOW FLOWS......Page 17
3.1  Genemtion of 2DCS......Page 18
3.2  Growth of 2DCS......Page 21
4  METHODS OF INVESTIGATION......Page 22
REFERENCES......Page 24
Shallow jets, wakes and mixing layers......Page 26
INTRODUCTION......Page 28
2.1  The elliptic instability......Page 29
2.2  The hyperbolic instability......Page 30
3.1 The Zigzag instability in strongly stratified flow without rotation......Page 31
4.1  Experimental setup......Page 32
4.2  The state diagram......Page 33
RÉFÉRENCES......Page 34
2  EXPERIMENTAL SETUP......Page 36
3  PHYSICAL PARAMETERS......Page 37
4.1  Growth of the instability......Page 38
4.2  Velocity as a function of time......Page 39
5.1  Thickness of the current and of the interface......Page 40
5.3  Phase and growth rate of instability......Page 41
REFERENCES......Page 42
1 INTRODUCTION......Page 44
3.1  Governing stability equations......Page 45
3.2  Shallow wake parameterization......Page 46
3.3  Mapped Chebyshev polynomial method......Page 47
4.2  Laboratory experiments......Page 48
5  DISCUSSION AND CONCLUSIONS......Page 49
REFERENCES......Page 51
1  INTRODUCTION......Page 52
2  SHALLOW WATER EQUATIONS & 2DCS......Page 53
3  LINEAR STABILITY ANALYSIS......Page 54
4  GINZBURG-LANDAU EQUATION AS A MODEL OF NONLINEAR PHENOMENA......Page 55
5  DERIVATION OF THE GINZBURGLANDAU EQUATION FOR SHALLOW WATER FLOWS......Page 56
6  APPLICATION OF THE GL MODEL TO 2DCS......Page 57
REFERENCES......Page 58
1.1.1  Conceptual aspects......Page 60
1.2  Shallow flows......Page 61
2.1 Shallow wakes......Page 62
3  VORTEX FORMATION IN SHALLOW FLOWS: CONTROL......Page 64
ACKNOWLEDGMENTS......Page 67
REFERENCES......Page 68
1  INTRODUCTION......Page 70
2.2  Particle Image Velocimetry......Page 71
3.2  Large scale observations......Page 72
4.2  Velocity fields......Page 73
6  CONCLUSION......Page 74
REFERENCES......Page 75
1  INTRODUCTION......Page 76
2.1  Properties of the dipole......Page 77
2.4  Dipole trajectories for a sinusoidal forcing......Page 78
4  LABORATORY RESULTS......Page 79
5  GENERATION OF VORTICES IN OCEANOGRAPHIC SITUATIONS......Page 80
5.2 Other flushing processes......Page 82
REFERENCES......Page 83
2  EXPERIMENTS......Page 84
3.1  Volume of turbulent fluid......Page 85
3.2  Spreading rates......Page 86
4  STEADY MEAN FLOW......Page 87
4.2  Maximum dye concentration......Page 88
5.2  Constant coefficient of the entrainment......Page 89
5.3  Friction number......Page 90
REFERENCES......Page 91
2  EXPERIMENTAL SETUP......Page 92
3.1  Observations......Page 93
3.3  Geometrical rules for the single-vortex pattern......Page 94
5  NUMERICAL SIMULATION......Page 96
6  CONCLUSION......Page 98
REFERENCES......Page 99
1  INTRODUCTION......Page 100
2  THEORY OF QUASI-2D TURBULENCE IN SHALLOW FLUID LAYERS......Page 101
3  DNS OF 2D TURBULENCE WITH RAYLEIGH DAMPING......Page 102
4  EXPERIMENTS OF QUASI-2D TURBULENCE WITH VARYING FLUID DEPTHS......Page 104
5  DISCUSSION AND CONCLUSION......Page 106
REFERENCES......Page 107
1  INTRODUCTION......Page 108
3  EXPERIMENTAL SETUP......Page 109
4  STATISTICAL ANALYSIS......Page 110
6  AMPLIFICATION FUNCTION......Page 112
7  DISCUSSION AND CONCLUSIONS......Page 113
REFERENCES......Page 114
2  WHOLE-FIELD MEASUREMENT TECHNIQUES IN SHALLOW FLOW......Page 116
3.1  Procedure......Page 117
3.3  Small-scale turbulence modeling......Page 118
4.1  Phase-resolved transport for different models......Page 120
4.3  Secondary motion in LCS......Page 121
REFERENCES......Page 123
2  EXPERIMENTAL APPARATUS......Page 124
3  THEORY DISCUSSION AND FIRST RESULTS......Page 125
4  CONCLUSIONS......Page 127
REFERENCES......Page 128
2  METHODOLOGY......Page 130
3 RESULTS......Page 131
4 DISCUSSION......Page 135
REFERENCES......Page 137
Shallow stratified and rotating flows in ocean and atmosphere......Page 138
1  INTRODUCTION......Page 140
2  STRATIFIED AND ROTATING TURBULENCE......Page 143
3  TURBULENT JETS IN STRATIFIED FLUIDS......Page 145
4  WAKES IN STRATIFIED FLUIDS......Page 146
REFERENCES......Page 147
1  INTRODUCTION......Page 150
2.2  Numerical model......Page 151
3.1  Numerical results......Page 152
3.2  Laboratory results......Page 153
3.3  Interpretation......Page 154
REFERENCES......Page 155
2  GENERATION OF DISTURBANCES BY A HORIZONTAL STRIP MOVING ALONG THE HORIZONTAL PLANE......Page 156
3  EXPERIMENTAL SETUP......Page 158
4  MAIN EXPERIMENTAL RESULTS......Page 159
ACKNOWLEDGEMENTS......Page 160
REFERENCES......Page 161
1 INTRODUCTION......Page 162
2  SHALLOW LAYER MODEL FOR IDEALISED MESOSCALE FLOWS IN STABLE LAYERS......Page 163
3  GENERAL INTERPRETATION OF MODEL EQUATIONS......Page 164
3.2  Transverse and downwind effects......Page 165
4  COMPARISON BETWEEN SHALLOW LAYER MODEL AND NUMERICAL MESOSCALE RESULTS......Page 166
4.3  Comparison with numerical mesoscale simulation of flow with change in surface roughness at 12 and 2km resolution......Page 167
5  CONCLUSIONS......Page 168
REFERENCES......Page 169
3 RESULTS......Page 170
4  COMPARISON WITH OBSERVATIONS......Page 171
7  CONCLUSIONS......Page 173
REFERENCES......Page 174
1  INTRODUCTION......Page 176
2  TURBULENT HEAT FLUX......Page 177
3  THE CONVECTIVE-WIND INSTABILITY......Page 179
4  THE CONVECTIVE-SHEAR INSTABILITY......Page 180
5  DISCUSSION......Page 181
REFERENCES......Page 183
2.1  Flume set-up......Page 184
2.3  Flow observables......Page 185
4.1  Exchange flow......Page 186
4.2  Near-bed brackish layer......Page 188
REFERENCES......Page 190
1  INTRODUCTION......Page 192
2.1  The diffusive/hydraulic nature of exchange flows......Page 193
2.3  Effects of unsteadiness......Page 194
3  NUMERICAL SIMULATIONS......Page 195
REFERENCES......Page 198
1  INTRODUCTION......Page 200
2  THEORY......Page 201
3  COMPARISON OF RESULTS AGAINST LINEAR THEORY......Page 203
4  TRAPPED INTERNAL WAVES AND TURBULENCE......Page 204
5  CONCLUSION......Page 206
REFERENCES......Page 207
2  MATHEMATICAL APPARATUS......Page 208
3  FIELD DATA......Page 210
4  NUMERICAL DECOMPOSITION......Page 211
REFERENCES......Page 212
1  INTRODUCTION......Page 214
2.2  Sensitivity to parameter variations......Page 215
REFERENCES......Page 219
1  INTRODUCTION......Page 220
2.1  The hydrodynamic problem......Page 221
2.2  The morphodynamic problem......Page 222
4  RESULTS OF THE EXPERIMENTS......Page 223
4.2  Morphodynamic evolution of the inlet region......Page 224
5  CONCLUDING REMARKS......Page 227
REFERENCES......Page 228
1  INTRODUCTION......Page 230
2  MATHEMATICAL MODEL......Page 231
3  MEASUREMENTS IN SITU......Page 232
4  DATA COMPARISON......Page 234
5  CONCLUSIONS......Page 235
REFERENCES......Page 236
2.1  Field observations......Page 238
2.2  Conceptual modelling......Page 239
3  RESULTS......Page 240
4  DISCUSSION......Page 241
REFERENCES......Page 242
River and channel flows......Page 244
2.1  Groins......Page 246
2.3  Flow and sediment transport near structures......Page 247
2.5  Bifurcations......Page 248
3.1  Turbidity maximum......Page 249
3.4  Flow and transport on marshlands......Page 250
5.1  Wave-driven nearshore currents......Page 251
5.4  River mouths and tidal inlets......Page 252
6.2  Dredged spoil dispersal......Page 253
6.3  3-D bedforms......Page 254
REFERENCES......Page 255
1  INTRODUCTION......Page 258
2  TWO DIMENSIONAL MODEL......Page 259
3.1  Boundaries and wetting/drying fronts......Page 260
4  FLOOD SIMULATION......Page 261
REFERENCES......Page 263
1  INTRODUCTION......Page 264
2.2  Hydraulic conditions......Page 265
3.2  Instantaneous velocities......Page 266
3.4  Mean velocity distribution in the spanwise direction......Page 268
3.5  Schematized modeling for unsteadiness process of horizontal eddies......Page 270
REFERENCES......Page 271
2  REPRESENTATION OF COMPLEX TOPOGRAPHY IN 3D APPLICATIONS OF CFD......Page 272
3.1  Porosity treatment and modification of drag terms......Page 273
3.2  Boundary conditions......Page 274
3.5  Model geometry and boundary conditions......Page 275
4.2  Results......Page 276
5  FLOW FIELDS OVER GRAVELLY SURFACES......Page 277
REFERENCES......Page 278
1  INTRODUCTION......Page 280
3  DERIVATION OF A STABILITY PARAMETER FOR VSL......Page 282
4.2  Lateral shear layers......Page 284
REFERENCES......Page 286
1  INTRODUCTION AND BACKGROUND......Page 288
3  METHODS......Page 289
4  RESULTS AND DISCUSSION......Page 290
REFERENCES......Page 295
1  INTRODUCTION......Page 298
3  COMPUTATION RESULTS AND DISCUSSION......Page 299
4.1  Fluctuation phenomena on longitudinal ridges......Page 300
4.2  Method for identifying coherent structures......Page 302
REFERENCES......Page 304
2  MATHEMATICAL FORMULATION......Page 306
4.2  Mean velocity distribution......Page 307
4.4  Reynolds stress......Page 308
4.5  Two-point correlation’s and streak-spacing......Page 309
5  CONCLUDING REMARKS......Page 310
REFERENCES......Page 311
2.1  Governing equation and discretization......Page 312
2.3  Computational mesh......Page 313
3.2  Rectangular duct computations......Page 314
3.3  Two-dimensional flow on natural river bed......Page 316
REFERENCES......Page 317
1  INTRODUCTION......Page 318
REFERENCES......Page 319
1  INTRODUCTION......Page 320
2.2.1  Closure submodel at the centerline, neglecting inertia effects......Page 321
2.2.2  Extension over the width to cover the entire cross-section and inclusion of inertia effects......Page 322
3  THE SIMULATED EXPERIMENTS......Page 323
4.1  Water surface elevation......Page 324
4.3  Downstream velocity......Page 325
5  DISCUSSION......Page 326
REFERENCES......Page 327
1  INTRODUCTION......Page 330
3  FLOW IN SHALLOW HARBOURS......Page 331
4  RIVER MEANDERS......Page 333
5  CONCLUSIONS......Page 335
REFERENCES......Page 336
2  STUDY SITE AND FIELD MEASUREMENTS......Page 338
3  COMPUTATIONAL METHODS......Page 339
3.2  Grid independence and validation......Page 340
4.2  B37: large inner-bank recirculation......Page 341
5  DISCUSSION......Page 343
REFERENCES......Page 344
2  EXPERIMENTAL METHODS......Page 346
3.1  Bed form......Page 347
3.3  Flow structure and bed form......Page 348
REFERENCES......Page 351
1  INTRODUCTION......Page 352
2.2 Governing equations......Page 353
2.4  Calculation method of channel-changing stage......Page 354
2.4.1  Stage A......Page 355
3.2  Time-dependent secondary currents......Page 356
3.4  Turbulence characteristics......Page 357
4  CONCLUSIONS......Page 358
REFERENCES......Page 359
2  PREVIOUS WORKS......Page 360
3.2  Velocity and shear-stress distribution......Page 361
3.3  Periodical structures analysis, LCA 10 case, station C......Page 362
3.4  Vortex growth, LCA 10 case......Page 363
3.5  General analysis......Page 364
4.3  Wave-length comparisons......Page 365
REFERENCES......Page 367
1  INTRODUCTION......Page 368
3  WATER SURFACE OSCILLATION......Page 369
4  TIME AVERAGED FLOW VELOCITY......Page 370
5  CONCLUSIONS......Page 373
2.1  A moving boundary fitted coordinate (MBFC)......Page 374
2.3  Numerical model......Page 376
4  RESULTS AND DISCUSSION......Page 377
REFERENCES......Page 379
1  INTRODUCTION......Page 382
2.2 The discrete model......Page 383
3  EXPERIMENTAL SET-UP......Page 384
4  THE FEATURE TRACKING ALGORITHM......Page 385
5 RESULTS......Page 386
REFERENCES......Page 390
1  INTRODUCTION......Page 392
2  FORMULATION OF THE PROBLEM AND NUMERICAL PROCEDURE......Page 393
3  NUMERICAL SIMULATIONS AND RESULTS......Page 394
4  COMPARISON WITH EXPERIMENTAL OBSERVATION......Page 397
5  CONCLUDING REMARKS......Page 398
REFERENCES......Page 399
1  INTRODUCTION......Page 400
2.1  Sediment transport......Page 401
3.2  Boundary conditions......Page 402
4.2  Verification......Page 403
5.1  Bar migration......Page 404
5.2  Bar height......Page 405
5.3  Bar wavelength......Page 406
6  CONCLUSIONS......Page 407
REFERENCES......Page 408
Transport processes in shallow flows......Page 410
1 INTRODUCTION......Page 412
2  VORTICES IN THE MARINE ECOSYSTEM......Page 413
3  PASSIVE TRACERS IN VORTEX-DOMINATED FLOWS......Page 414
4  DYNAMICS OF REACTIVE TRACERS......Page 415
5  A SEMI-LAGRANGIAN APPROACH TO MARINE ECOSYSTEM DYNAMICS......Page 416
6  COHERENT VORTICES AND PRIMARY PRODUCTIVITY......Page 418
7  VORTICAL SHELTERS......Page 422
REFERENCES......Page 424
1  INTRODUCTION......Page 426
3  LIF MEASUREMENTS......Page 427
5  ADV AND UCM MEASUREMENTS......Page 428
6  RESULTS AND DISCUSSIONS......Page 429
REFERENCES......Page 432
1 INTRODUCTION......Page 434
3.1 Flow behaviour at different water depths......Page 435
3.2.2  Interaction with free surface......Page 437
3.3  Burst-like large-scale structures......Page 438
4  CONCLUSION......Page 440
REFERENCES......Page 441
1  INTRODUCTION......Page 442
2  FORMULATION OF THE PROBLEM: THE FLOW MODEL......Page 443
3.1  Numerical tests......Page 445
4  3D RESULTS AND DISCUSSION......Page 446
REFERENCES......Page 449
2  GOVERNING EQUATIONS......Page 452
3  NUMERICAL SOLVER......Page 453
3.3  Unsplit MUSCL-Hancock method......Page 454
3.6  Boundary conditions......Page 455
4.1  Pure diffusion......Page 456
4.3  Pure advection......Page 457
4.4  Wind-induced advection in a circular shallow lake......Page 458
5  CONCLUSIONS......Page 459
REFERENCES......Page 460
1  INTRODUCTION......Page 462
3  HAIRPIN PACKETS ORGANIZATION......Page 463
4  STRUCTURE OF SHEAR STRESS IN THE OUTER REGION......Page 465
5  TKE BUDGET IN THE OUTER REGION: THE ROLE OF ELONGATED SHEAR LAYERS......Page 466
6  IMPLICATIONS FOR GAS TRANSFER AND CONCLUSIONS......Page 468
REFERENCES......Page 469
1  INTRODUCTION......Page 470
3.1  Mean velocity and temperature......Page 471
3.3  Reynolds stress and heat flux......Page 473
3.5  Heat transfer at a gas-liquid interface......Page 474
REFERENCES......Page 475
2  HYDRAULIC MODEL......Page 476
4  LABORATORY MEASUREMENTS......Page 477
5  HYDRO-MORPHODYNAMIC COMPUTATIONS......Page 478
REFERENCES......Page 480
2  THEORETICAL BACKGROUND......Page 482
3  SKEWNESS COEFFICIENT......Page 484
4  RESULTS AND DISCUSSION......Page 485
REFERENCES......Page 489
1  INTRODUCTION......Page 490
2  METHOD AND VALIDATION......Page 491
3.2  Channel flow with groin fields......Page 494
3.3  Groin fields with different exchange behavior......Page 496
REFERENCES......Page 497
2  STUDY SITES......Page 498
3  METHODS AND METHODOLOGICAL PROBLEMS......Page 499
4  WHAT CAN WE DO WITH IMPERFECT MEASUREMENTS?......Page 500
6  DISCUSSION......Page 501
REFERENCES......Page 502
1  INTRODUCTION......Page 504
2  GENERAL CONCEPT......Page 505
3  LINEAR-LINEAR DIAGRAM OF SEDIMENT TRANSPORT......Page 506
4  PERSPECTIVES......Page 507
REFERENCES......Page 508
1  INTRODUCTION......Page 510
2.4  Oxygen microprofiles at the lake bottom......Page 511
4.1  Dynamics of BBL turbulence and DBL fluxes......Page 512
4.2  The scaling of δDEL......Page 513
REFERENCES......Page 514
1  INTRODUCTION......Page 516
2.3  Bed load and suspension sample collection......Page 517
3  EXPERIMENTAL RESULTS......Page 518
5  SUSPENSION CONCENTRATION DISTRIBUTION......Page 519
7  SUMMARY AND CONCLUSIONS......Page 520
REFERENCES......Page 522
2  THEORETICAL ANALYSIS FOR SECONDARY CURRENTS......Page 524
5  THEORETICAL ANALYSIS......Page 526
7  CONCLUSION......Page 528
REFERENCES......Page 529
Numerical modeling and turbulence closure techniques for shallow flows......Page 530
1  INTRODUCTION......Page 532
2.2  Free-surface movement......Page 533
3.2  Flow over a ramp......Page 535
3.4  Passage of a solitary wave over a submerged obstacle......Page 536
3.5  Flow about a surface-penetrating cylinder......Page 537
ACKNOWLEDGEMENTS......Page 538
REFERENCES......Page 539
1.3  Solution procedure......Page 540
2.5  Row-scaling......Page 541
3.2  Three-dimensional shallow-water equations......Page 542
4.4  Condition number and row-scaling......Page 543
4.6  Machine precision......Page 544
4.9  Truncation error......Page 545
REFERENCES......Page 546
1  INTRODUCTION......Page 548
2.2  Governing equations and numerical solutions......Page 549
3.1  The laboratory data......Page 550
3.2  Model set up......Page 551
4  COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS......Page 552
REFERENCES......Page 553
1  INTRODUCTION......Page 554
2  MATHEMATICAL FORMULATION......Page 555
4  RESULTS......Page 556
5  DISCUSSION AND CONCLUSIONS......Page 559
REFERENCES......Page 560
2 MODEL DESCRIPTION......Page 562
2.2.1 Space discretization......Page 563
2.2.4  Iterative methods......Page 564
3.1.1  Water surface profile......Page 565
3.1.2  Curved open channel......Page 566
3.1.3  Large reservoir......Page 567
3.3  Inundation risk and flood extension......Page 568
REFERENCES......Page 569
1.2  Survey of physical processes......Page 572
1.3  Considerations for numerical simulations......Page 573
2.2  Model set-up for a proper HLES......Page 574
2.5  Field measurements dedicated to flow separation......Page 575
3.1  Existing condition......Page 576
4  CONCLUSIONS......Page 577
REFERENCES......Page 578
1.1  Three-dimensional les......Page 580
1.2  Depth-average les......Page 581
2 3D-LES OF OPEN CHANNEL FLOW......Page 582
3  BACK SCATTER MODEL......Page 583
4  SHALLOW MIXING LAYER......Page 585
5  CONCLUSIONS......Page 586
REFERENCES......Page 587
1  INTRODUCTION......Page 588
2  DESIGN CRITERIA FOR AN SGS MODEL IN SHALLOW FLOWS WITH COMPLICATED GEOMETRY......Page 589
3  SHALLOW-WATER SGS MODEL......Page 590
4  SPATIAL AND TEMPORAL FILTERING......Page 591
5  ANALYSIS OF THE SGS MODEL......Page 592
REFERENCES......Page 593
APPENDIX......Page 594
2  GOVERNING EQUATIONS......Page 596
3  DISCRETIZATION METHOD......Page 597
5.1  Flow boundary conditions......Page 598
6.3.1  Inviscid frictionless flow......Page 599
6.3.3  Bed & wall friction......Page 600
7  APPLICATION OF THE MODEL......Page 601
7.4.1  Frictionless flow......Page 602
8  CONCLUSIONS......Page 603
REFERENCES......Page 604
1  INTRODUCTION......Page 606
3.1  Discontinuous formulation for the advective part......Page 607
3.2  Continuous formulation for the dispersive part......Page 608
4.1  Linear wave......Page 609
4.2  Equatorial Rossby waves......Page 610
REFERENCES......Page 612
2  SHALLOW WATER EQUATIONS......Page 614
3  FINITE ELEMENT APPROXIMATION......Page 615
4  DRYING AND WETTING......Page 616
6  VALIDATION AND RESULTS......Page 617
7  CONCLUSION AND DISCUSSION......Page 620
REFERENCES......Page 621
2.2  Finite element formulation......Page 622
3  INTRODUCTION OF KALMAN FILTER......Page 623
5.2.1  Finite element model......Page 624
5.3  Effect of Kalman filter on removal of noise......Page 626
REFERENCES......Page 627
2.1  The Boussinesq model......Page 628
2.2.3  Wetting/drying front......Page 629
3.1  Breaking and non-breaking solitary wave run-up over a plane beach......Page 630
3.2  Focused waves over a flat, frictionless bed......Page 632
3.3  Focused wave run-up over a plane beach at the UKCRF......Page 633
4  CONCLUSIONS......Page 634
REFERENCES......Page 635
2  SETTING OF COORDINATE SYSTEM......Page 636
3 KINETIC BOUNDARY CONDITION AT A WATER SURFACE IN A GENERALIZED CURVILINEAR COORDINATE......Page 637
4.2  Depth averaged momentum equation......Page 638
5.2  Calculation of metrics......Page 639
5.4 Water surface profile analysis......Page 640
6  CONCLUSIONS......Page 641
2.1  Property of overland flow......Page 642
2.3  Summary of the traveling time formulas......Page 643
3.3  Set-up of the numerical tests......Page 644
3.5  Simulation results......Page 645
3.7  New formulae to estimate Tc for flat terrain......Page 646
REFERENCES......Page 648
1 INTRODUCTION......Page 650
3.1  Derivation of a velocity distribution including accelerating/decelerating effects......Page 651
3.2  Procedure for calculation of velocity distribution......Page 653
4  COMPARISON BETWEEN EXPERIMENTS AND SIMPLE MODEL......Page 655
REFERENCES......Page 656
1 INTRODUCTION......Page 658
3  EXPERIMENTAL SETUP......Page 659
4 NUMERICAL MODEL SET UP......Page 660
5  COMPARISON OF RESULTS......Page 661
6  DISCUSSION OTHER CASES......Page 662
REFERENCES......Page 663
1  INTRODUCTION......Page 664
2.1  Basic equations......Page 665
2.5  Hydraulic conditions and computational domain......Page 666
3.1  Time-mean flow features......Page 667
3.2.1  Effects of inclination angle on unsteady flow features......Page 668
3.2.2  Period of the oscillation......Page 670
4  CONCLUDING REMARKS......Page 671
REFERENCES......Page 672
1  INTRODUCTION......Page 674
2  PHYSICAL AND NUMERICAL MODEL......Page 675
3.1  ‘Deep case’ vector plots......Page 676
3.2  ‘Shallow case’ vector plots......Page 677
3.3  Flux evolution......Page 678
4  TURBULENCE MODELLING......Page 680
REFERENCES......Page 681
Author index......Page 684