 | Hutter K. Physics of Lakes / K.Hutter, Y.Wang, I.P.Chubarenko. Vol.2: Lakes as Oscillators. - Berlin; Heidelberg: Springer, 2011. - xxxiv, 646 p.: ill. - (Advances in geophysical and environmental mechanics and mathematics; 2). - Incl. bibl. ref. and indexes. - ISBN 978-3-642-19111-4; ISSN 1866-8348
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11 The Role of the Earth's Rotation: Fundamentals -
Rotation and Stratification Influenced Dynamics ............. 1
11.1 Estimations by Dimensional Reasoning .................. 1
11.1.1 Tributary Affected Advection .................. 2
11.1.2 Wind-Induced Circulation ...................... 3
11.1.3 Barotropic and Baroclinic Wave Dynamics ....... 4
11.2 Rotation Influenced Shallow Water Waves ............... 6
11.3 A Brief Classification of Rossby Waves ............... 13
11.4 Plane Linear Waves in a Rotating Stratified Fluid .... 19
11.4.1 Waves in a Linearly Stratified Rotating
Unbounded Boussinesq Fluid ................... 20
11.4.2 Waves in a Stably Stratified Shallow Layer
of a Boussinesq Fluid ........................ 28
11.4.3 The Two-Layer Model .......................... 39
11.5 Concluding Discussion ................................ 46
References ................................................. 48
12 The Role of the Earth's Rotation: Oscillations in Semi-
bounded and Bounded Basins of Constant Depth ............... 49
12.1 Motivation ........................................... 49
12.2 Kelvin Waves ......................................... 50
12.2.1 Pseudo-Standing Kelvin Waves ................. 55
12.2.2 Baroclinic Kelvin Waves ...................... 58
12.3 Inertial Waves ....................................... 60
12.4 Poincare Waves ....................................... 62
12.5 Reflection from the End of a Channel Wall ............ 70
12.6 Shallow Water Waves in a Rectangle of Constant
Depth ................................................ 73
12.6.1 Frequency Relation ........................... 77
12.6.2 Modal Structure .............................. 81
12.6.3 Additional Results ........................... 85
12.7 A 'Second-Class' of Inertial Waves: 'Inertial Waves
Proper' .............................................. 87
12.7.1 Governing Equations .......................... 88
12.7.2 Plane Inertial Sverdrup (Poincare) Waves ..... 91
12.7.3 Inertial Kelvin Waves ........................ 92
12.7.4 Inertial Poincare Waves in a Channel ......... 94
12.7.5 Inertial Poincare Channel Waves Reflecting
from a Vertical Wall ......................... 96
12.7.6 Inertial Waves in Rectangular Basin of
Constant Depth ............................... 98
12.7.7 Discussion .................................. 101
12.8 Concluding Discussion ............................... 103
12.9 Appendix: Solution Scheme of Proudman-Rao to Solve
(12.1) .............................................. 104
References ................................................ 111
13 Basin-Scale Gravity Waves in Circular and Elliptical
Containers on the Rotating Earth .......................... 115
13.1 Motivation .......................................... 115
13.2 Conceptual Prerequisites ............................ 117
13.3 Circular Cylindrical Geometry ....................... 118
13.4 Three-Layer Stratification - Lake Kinneret Treated
as a Circular Cylinder of Constant Depth ............ 123
13.5 Elliptical Cylindrical Container .................... 129
13.6 Mathieu Functions ................................... 131
13.7 Elliptical Basin: Normal Mode Analysis .............. 135
13.8 Experimental Verification ........................... 147
13.9 Discussion .......................................... 151
References ................................................ 153
14 Barotropic and Baroclinic Basin-Scale Wave Dynamics
Affected by the Rotation of the Earth ..................... 155
14.1 Introduction ........................................ 155
14.2 Barotropic Basin-Wide Oscillations of Lake
Michigan ............................................ 157
14.3 Internal Seiche Dynamics in Lake Geneva ............. 164
14.3.1 Introduction ................................ 164
14.3.2 Lake Morphology and Data Handling ........... 165
14.3.3 Model Equations ............................. 166
14.3.4 Modal Analysis for the TEDM ................. 168
14.3.5 Modal Analysis for the TCDM ................. 172
14.3.6 Internal Wave Dynamics Revealed by Surface
Level Data .................................. 173
14.4 Transverse Internal Wave Motion in Lake
Ьberlingen .......................................... 179
14.4.1 Statement of the Problem .................... 179
14.4.2 Observations During the Bodensee-
Experiment 1972 ............................. 179
14.4.3 Numerical Solution for the TVD Model with
Realistic Bottom Topography ................. 182
14.5 Lake Biwa ........................................... 186
14.6 Concluding Discussion ............................... 190
References ................................................ 192
15 Higher-Order Baroclinicity (I): Two Fluid Layers with
Diffuse Interface - Three Fluid Layers with Sharp
Interfaces ................................................ 197
15.1 Motivation and Review ............................... 197
A. Laboratory Experiments on Baroclinic Solitary Waves
in a Two-Layer Fluid System with Diffusive Interface ... 199
15.2 Experimental Set-Up and Wave Generation ............. 199
15.2.1 The Wave Channel ............................ 199
15.2.2 Solitary Wave Generation and Measuring
Technique ................................... 199
15.2.3 Error Estimation ............................ 202
15.3 The Experiments ..................................... 202
15.3.1 Typical Experimental Data ................... 203
15.3.2 Results ..................................... 208
15.4 Analytical Models for the Evolution of Baroclinic
Waves ............................................... 212
15.4.1 Equations ................................... 212
15.4.2 Baroclinic modes for a two-layer system
with diffuse interface ...................... 216
15.4.3 Results of the Numerical Modeling ........... 219
B. Three-Layer Model of the North Basin of the Lake of
Lugano ................................................. 224
15.5 The Thermo-Chemical Density Structure of the North
Basin of Lake of Lugano ............................. 225
15.6 Linear Wave Dynamics of the Three-Layer Model ....... 229
15.7 Computational Results and Their Comparison with
Field Data .......................................... 233
15.7.1 Mode 1 ...................................... 234
15.7.2 Mode 2 ...................................... 240
15.7.3 Mode 3 ...................................... 242
15.7.4 Modes 4 and 5 ............................... 244
15.8 Model Sensitivity ................................... 245
15.9 Inferences .......................................... 246
15.10 Summary ............................................. 247
References ................................................ 248
16 Higher-Order Baroclinicity (II): Interpretation of Lake
Data with Rotating and Non-rotating Models ................ 251
16.1 V2-Vertical Mode of the Internal Seiche in Lake
Alpnach ............................................. 252
16.1.1 Observations ................................ 253
16.1.2 Seiche Analysis ............................. 254
16.2 Internal Seiche Climate in Lake Banyoles,
Catalonia (Spain) ................................... 258
16.2.1 Description of the Site ..................... 259
16.2.2 Methods of Computation and Data Analysis .... 260
16.2.3 Results ..................................... 261
16.3 Internal Wave Weather in Stratified Lake Biwa ....... 267
16.3.1 Methodology and Overview of Field
Results ..................................... 267
16.4 Basin-Scale Wave Motion in Lake Constance ........... 274
16.4.1 Morphology and Methodology .................. 274
16.4.2 Interpretation of the Observations .......... 278
16.5 Closing Remarks ..................................... 281
References ................................................ 284
17 Barotropic Oscillations in Lake Onega: A Lake of Complex
Geometry .................................................. 287
17.1 Lake Morphology and First Interpretations
of Water Level Measurements ......................... 287
17.2 Measured Water-Level Fluctuations and Water
Currents at Isolated Points ......................... 291
17.3 The Barotropic Eigenvalue Problem ................... 296
17.4 Numerical Results and Their Comparison with
Observations ........................................ 298
17.5 Concluding Remarks .................................. 308
17.6 Appendix: The Lanczos' Procedure in Solving
Symmetric Eigenvalue Problems ....................... 309
References ................................................ 313
18 Observation and Analysis of Internal Seiches in the
Southern Basin of Lake of Lugano .......................... 315
18.1 Introductory Remarks, Lake Morphology ............... 315
18.2 State of Stratification and Wind Forces: 15
August-15 October 1984 .............................. 318
18.3 Internal Seiche Response: Variation in Isotherm
Depth and Wind Stress ............................... 324
18.3.1 Internal Oscillations 25 August-5
September ................................... 324
18.3.2 Internal Oscillations 7-30 September ........ 326
18.3.3 Internal Seiche After 3 October ............. 329
18.3.4 Harmonic Analysis ........................... 329
18.4 Model Predictions: The Two-Layered Variable-Depth
Model ............................................... 333
18.5 Current Structure of the Internal Seiches ........... 344
18.6 Closing Remarks ..................................... 348
18.6.1 Observed Features Not Reproduced by the
TVD-Model ................................... 348
18.6.2 A Remark on the Generation of Topographic
Waves ....................................... 349
18.6.3 Barotropic-Baroclinic Coupling of the
North- and South Basin ...................... 351
References ................................................ 352
19 Topographie Waves in Enclosed Basins: Fundamentals
and Observations .......................................... 355
19.1 Review of Early Work ................................ 355
19.2 Some Observations and Proposed Interpretations ...... 358
19.2.1 Lake Michigan ............................... 358
19.2.2 Lake of Lugano (North Basin) ................ 362
19.2.3 Other Lakes and Ocean Basins ................ 365
19.3 Baroclinic Coupling: The Two-Layer Model ............ 368
19.3.1 Two-Layer-Equations ......................... 368
19.3.2 Approximations .............................. 369
19.3.3 Scale Analysis .............................. 372
19.3.4 Boundary Conditions ......................... 376
19.4 Continuous Stratification ........................... 377
19.4.1 Modal Equations ............................. 377
19.4.2 Spectral Decomposition of the Baroclinic
Fields ...................................... 382
19.4.3 Scale Analysis .............................. 389
19.5 Discussion .......................................... 391
References ................................................ 396
20 Topographic Rossby Waves in Basins of Simple Geometry ..... 399
20.1 Motivation .......................................... 399
20.2 Topographic Wave Equation in Curvilinear
Orthogonal Coordinate Systems ............................. 400
20.2.1 Preparation ................................. 400
20.2.2 Cylindrical Coordinates ..................... 403
20.2.3 Elliptical Coordinates ...................... 403
20.2.4 Natural Coordinates ......................... 405
20.2.5 Cartesian-Coordinate Correspondence
Principle ................................... 407
20.3 An Almanac of Analytical Solutions .................. 407
20.3.1 Circular Basin with Parabolic Bottom ........ 407
20.3.2 Circular Basin with a Power-Law Bottom
Profile ..................................... 411
20.3.3 Elliptic Basin with Parabolic Bottom ........ 413
20.3.4 Elliptic Basin with Exponential Bottom ...... 417
20.3.5 Topographic Vorticity Waves in Infinite
Domains ..................................... 425
20.3.6 Elliptic Island in Infinite Space ........... 432
20.4 Application of Transformation Principles ............ 434
20.4.1 Hyperbolically Curved Channels .............. 434
20.4.2 Semi-Infinite Gulf and Patched-Up
Elongated Basins ............................ 437
20.5 Discussion .......................................... 443
References ................................................ 444
21 Topographic Waves in Basins with Complex Shapes and
Complex Bathymetries ...................................... 447
21.1 Conceptual Review ................................... 447
21.2 The Method of Weighted Residuals .................... 448
21.2.1 The Method of Weighted Residuals Applied
to Topographic Waves ........................ 448
21.2.2 Symmetrisation .............................. 453
21.3 Topographic Waves in Infinite Channels .............. 454
21.3.1 Basic Concept ............................... 455
21.3.2 Dispersion Relation ......................... 458
21.3.3 Channel Solutions ........................... 463
21.4 Topographic Waves in Rectangular Basins ............. 469
21.4.1 Crude Lake Models ........................... 469
21.4.2 The Role of the Aspect Ratio ................ 472
21.4.3 Lake Model with Non-constant Depth Along
Its Thalweg ................................. 474
21.4.4 Current Patterns ............................ 485
21.5 Curved Channels ..................................... 490
21.5.1 The Method of Weighted Residuals for Lakes
with Curved Thalwegs ........................ 490
21.5.2 Dispersion Relation ......................... 493
21.5.3 TW-Wave Modes in Wedges of Annuli with
Smooth Bathymetry ........................... 495
21.6 Reflection of Topographic Waves ..................... 498
21.6.1 Reflection at a Vertical Wall ............... 499
21.6.2 Reflection at a Gulf End with Continuous
Depth Lines But Discontinuous Slope
Parameter ................................... 502
21.6.3 Reflection at a Channel End with
Continuous Depth Lines and Continuous
Slope Parameter ............................. 504
21.7 Bay Modes and Resonances ............................ 512
21.7.1 The Boundary Value Problem for TWs in
a Semi-Infinite Gulf with Exponential
Bathymetry .................................. 513
21.7.2 The Flat Channel ............................ 518
21.7.3 Channel with Shelf Topography ............... 521
21.8 Concluding Discussion ............................... 530
21.9 Appendix ............................................ 533
References ................................................ 536
22 A Class of Chrystal-Type Equations ........................ 539
22.1 Motivation .......................................... 539
22.2 Traditional Chrystal-Type Equations ................. 541
22.2.1 Homogeneous Lakes ........................... 541
22.2.2 Two-Layer Channel Model ..................... 544
22.3 Extended Channel Models: Governing Equations ........ 555
22.4 Method of Weighted Residuals ........................ 559
22.5 Derivation of a Hierarchy of Channel Equations
for Barotropic Motions in Lakes ..................... 563
22.5.1 Mass Balance ................................ 564
22.5.2 Momentum Balance ............................ 566
22.5.3 Summary ..................................... 567
22.6 Low-Order Channel Models for Curved Rotating
Elongated Lakes ..................................... 569
22.6.1 Non-rotating Basins ......................... 570
22.6.2 A First-Order Model Accounting for the
Rotation of the Earth ....................... 572
22.7 Gravity Waves in Channels and Lakes of Rectangular
Cross Section on the Rotating Earth ................. 576
22.7.1 Free Oscillations in a Non-rotating
Rectangle ................................... 576
22.7.2 Kelvin-Type Waves in an Infinitely Long
Rectangular Straight Canal .................. 579
22.7.3 Wave Solutions of the Full First-Order
System: Poincare-Like Waves ................. 581
22.7.4 Reflection of Kelvin-Type Waves at
a Barrier of a Half-Open Rectangular Canal
and Free Oscillations in Rectangles ......... 589
22.8 Ring-Shaped Basins with Constant Depth .............. 600
22.8.1 Solutions of the Two-Dimensional Tidal
Equation .................................... 600
22.8.2 First-Order Channel Model ................... 603
22.9 Higher Order Chrystal-Type Models Applied to Free
Oscillations in Natural Basins ...................... 608
22.9.1 The Nth Order Two-Point Boundary-Value
Problem for Barotropic Forced or Free
Oscillations ................................ 608
22.9.2 Integration Procedure ....................... 612
22.9.3 Barotropic Seiches of the Northern Basin
of Lake of Lugano ........................... 614
Appendix 22.A ............................................. 620
Appendix 22.В ............................................. 623
References ................................................ 626
Name Index .................................................... 629
Lake Index .................................................... 633
Subject Index ................................................. 635
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