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Topographic Waves in Channels and Lakes on the f-Plane

Topographic Waves in Channels and Lakes on the f-Plane

Thomas Stacker, Kolumban Hutter

ISBN: 978-1-118-66918-1 July 2013 American Geophysical Union 176 Pages

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Published by the American Geophysical Union as part of the Lecture Notes on Coastal and Estuarine Studies Series, Volume 21.

The last one or two decades have witnessed an increased interest in topographic Rossby waves, both from a theoretical computational as well as an observational point of view. However, even though long periodic processes were observed in lakes and ocean basins with considerable detail, it appears that interpretation in terms of physical models is not sufficiently conclusive. The reasons for this lack in understanding may be sought both, in the insufficient spatial resolution or the brevity of the time series of the available data and the inadequacy of the theoretical understanding of long periodic oscillating processes in lakes and ocean bays. Advancement will emerge from intensified studies of both aspects, but it is equally our believe that the understanding of long periodic oscillations in lakes is presently likely to profit most from a theoretical-computational study of topographic Rossby waves in enclosed basins.

1. Introduction 1

1.1 Preamble 1

1.2 Waves in waters 1

1.3 Observations - Their interpretations 5
a) Lake Michigan 5
b) Lake of Lugano (North basin) 9
c) Lake of Zurich 15
d) Lake Ontario 19
e) Other lakes and ocean basins 21

1.4 Aim of this work 24

2. Governing equations 26

2.1 Equations of adiabatic fluid flow 26

2.2 Vorticity, potential vorticity, topographic Rossby waves 28

2.3 Baroclinic coupling - the two-layer model 35
a) Prerequisites 35
b) Two-layer equations 36
c) Approximations 37
d) Scale analysis 40
e) Boundary conditions 44

2.4 Continuous stratification 45
a) Modal equations 45
b) Scale analysis 55

2.5 TW-equation in orthogonal coordinate systems 57
a) Preparation 57
b) Cylindrical coordinates 58
c) Elliptical coordinates 58
d) Natural coordinates 60
e) Cartesian-coordinate correspondence principle 61

3. Some known solutions of the TW-equation in various domains 62

3.1 Circular basin with parabolic bottom 62

3.2 Circular basin with a power-law bottom profile 65

3.3 Elliptic basin with parabolic bottom 67

3.4 Elliptic basin with exponential bottom 71
a) Basin with central island 71
b) Basin without island 75

3.5 Topographic waves in infinite domains 79
a) Straight channel 80
b) Channel with one-sided topography 80
c) Shelf 82
d) Trench 83
e) Single-step shelf 83
f) Elliptic island 85
The Method of Weighted Residuals 87

4.1 Application to the TW-equation 87

4.2 Symmetrization 93

5, Topographic waves in infinite channels 95

5. 1 Basic concept 95

5. 2 Dispersion relation 98

5. 3 Channel solutions 105

5. 4 Velocity profiles 111

5. 5 Alternative solution procedures 114

5. 6 Hyperbolically curved channels 118

6, Topographic waves in rectangular basins 122

6.1 Crude lake model 123

6.2 Lake model with non-constant thalweg 126
a) Numerical method 126
b) New types of topographic waves 130
c) Convergence and parameter dependence 134
d) The bay-type 136

7. Reflection of topographic waves 137

7.1 Reflection at a vertical wall 138

7.2 Reflection at an exponential shore 141

7.3 Reflection at a sin2-shore 143
a) Numerical method
b) Reflection patterns

8. Review and outlook: Restrictions of this study
A l i s t of unsolved problems

8. 1 A brief summary 152

8. 2 Validity of TW-equations 152

8. 3 Single or multivalued dispersion relation 153

8. 4 Bay-type modes 154

8. 5 A list of unsolved problems 155
a) On the computational side 155
b) On the physical side 155

8. 6 Measurements, observations 156

Appendix A 157

Appendix B 161

Appendix C

References 168

Author Index 17 4

Subject Index 175