Modern Theory and Practice of Tide Analysis and Tidal Power
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MODERN THEORY AND PRACTICE OF TIDE ANALYSIS AND TIDAL POWER Professor Zygmunt Kowalik College of Fisheries and Ocean Sciences, University of Alaska, Fairbanks Dr John L Luick South Australian Research and Development Institute, Adelaide Copyright © 2019 Individuals are free and welcome to copy and distribute this publication in whole or part, as long as the original authors are duly acknowledged. First edition: August 2019 Publisher: Austides Consulting 34 Chitunga Road Eden Hills South Australia, 5050 Australia www.austides.com Title page photo: Flooding from Hurricane Florence, North Carolina, September 2018. By permission. i Table of Contents Chapter I: Tidal forces .......................... 5 1. Basics of celestial mechanics . 5 2. Tide–producing forces . 8 3. Equilibrium tides . 13 4. The orbits of Earth and Moon . 20 5. The harmonic constituents of the equilibrium tide . 26 6. The origins of major tidal variations . 34 7.Nodalphase............................36 8. Longitude formulas . 38 References..............................40 Chapter II: Basic equations .......................43 1. Rectangular and spherical system of coordinates . 43 2. Total tidal potential . 45 3. Energy equation . 48 4. Free tidal waves in a channel . 50 5. Energy considerations . 53 5a.Energyflux..........................53 5b. Kinetic and potential energy . 55 6. Free tidal waves in two-dimensional geometry . 56 7.Kelvinwave............................59 8.Sverdrupwave...........................62 9.Shelfwaves............................64 10. Tidal motion enhancement around islands and seamounts . 66 10a. Trapped tidal motion around islands . 68 10b. Residual tidal circulation around islands . 70 References..............................74 Chapter III: Tidal current ........................77 1. Bottom boundary layer . 77 2. Vertical changes in the eddy viscosity . 81 3. Influence of the stratification . 84 4. St. Lawrence Island region . 88 5. Tidal data and comparison of full records against the vertically averaged model . 91 6. Structure of temporal and vertical variability . 95 7.Rotaryspectra...........................98 8. Elliptical motion along the vertical direction . 100 9. Tidal ellipses . 101 10. Nonlinear effects in the tidal currents . 106 11. Nonlinear tidal interactions in the Sea of Okhotsk . 109 ii References............................. 117 Chapter IV: Tide distribution and tidal power ............. 120 1.Introduction........................... 120 2. Short description of tides in the World Ocean . 121 3. Short description of extreme tide ranges . 126 4. Rudimentary notions related to transfer of energy . 133 5. Tidal energy balance in local water bodies . 135 6. Tidal power from bays of high tidal energy . 140 7. Tidal power from the sea level difference . 142 8. Tidal power from currents . 147 8.1 The future of tidal power . 147 8.2 Power generation from currents . 149 9.Conclusion............................ 153 10. The last page from Gibrats Lenergie des marees . 154 11. Acknowledgements . 154 References............................. 155 Chapter V: Physical characteristics of tides. Tide analysis and prediction ............................ 158 1. Simple superposition of the tidal constituents . 158 2. Types of tide as deduced from constituents . 162 3. Phase and frequency conventions for the tidal analysis/prediction . 166 4. The analysis procedure . 168 5. A sample prediction . 169 6. Nyquist frequency and Rayleigh criterion . 173 7. Inference of constituents . 174 References............................. 175 Chapter VI: Tidal terminology ..................... 177 ( page number is given for the first item in alphabetical arrangement) . absolute sea level . 177 beatfrequency........................... 181 cadastre.............................. 182 day................................ 185 earthtide............................. 187 formfactor............................. 189 geopotential............................ 189 high water full and change (HWF&C) . 190 inference of constituents . 191 kappaphase............................ 193 lunisolartide............................ 194 month............................... 195 NGWLMS............................. 195 perigeantide............................ 196 quadrature............................. 198 iii radiational tides . 198 satellite altimetry . 201 TASK-2000............................ 209 Universal Time (UT) . 215 Van de Casteele test . 215 year................................ 216 References............................. 216 iv Preface This book is the result of a collaboration between the authors that began back in 1990. We present an introduction to the oceanography of tides. This is a large subject containing results of many authors. We therefore have confined our presentation to the results we have taught and published. CHAPTER I provides a short introduction to tide-producing forces. Their origin is con- nected to the rotational motion of moon and Earth and Earth and sun. The origin of the major tidal harmonic constituents are also linked to the Earths rotation. The harmonic modela for the tides proposed by Thomson and developed by Darwin and Doodson are described step by step and connected directly to the main orbital and rotational periods. These include: the mean solar day, the sidereal month (27.321582 mean solar days), the tropical year, (365.24219879 mean solar days), the period of the lunar perigee (8.8475420 years), the period of the lunar node (18.613188 years), and the period of the solar perihe- lion (20,940 years). The origin of the major tidal variations is presented, in particular, how the 18.6 year lunar nodal cycle is accounted for in the analysis of lunar tidal constituents by use of amplitude (f) and phase (u) corrections. CHAPTER II introduces hydrodynamics on a rotating Earth and the system of coordinates used in computing tides. As tides belong to the class of the long waves a brief summary of the essentials of long wave dynamics is given. Basic notions of long wave propagation such as phase velocity, energy conservation and energy flux are introduced. Special attention is given to the energy flux, which is an effective tool in tidal investigation. To understand the main features of tide propagation, we begin by constructing the dispersive equation. Out of this equation, the simplified cases such as Sverdrup waves, Poincare waves and Kelvin waves are derived. Modification of the tide by bathymetry is shown in the context of trapped shelf waves along the shore, and around the seamounts and islands. Since trapping in shallow water often generates strong currents, we describe the nonlinear interactions that lead to residual tidal currents. CHAPTER III is concerned with the vertical and horizontal structure of the tidal currents. We start by describing the bottom boundary layer (BBL) through analytical and numerical solutions. These solutions show that the vertical scale of the BBL depends on the tidal ellipse rotation. For clockwise and counter-clockwise rotation, two different scales are derived. The scale also differs for the diurnal and semidiurnal tides, and can depend on the proximity of the tidal periods to the inertial period. To examine the primary characteristics of the tidal current distribution in the vertical, we present year-long analyses of moorings around St. Lawrence Island (Bering Sea). The analyses show very different behavior of the BBL for M2 and K1 constituents confirming analytical solutions. In very shallow waters, both semidiurnal and diurnal currents are significantly magnified, resulting in enhanced local mixing and tidal fronts. In proximity to the shelf, islands, headlands and around banks, as a consequence of nonlinear interactions, tidal motion can generate new frequencies (so called compound tides and overtides) and residual steady currents (e.g. permanent eddies). These phenomena are shown to occur in the Okhotsk Sea, where the diurnal tide predominates. CHAPTER IV describes various aspects of tidal power generation. Since modern tidal v plants tap both potential and kinetic energy, knowledge of tide distribution in the World Ocean is critical in development of tidal power on a site-specific basis. To illustrate basic tidal physics which govern the large tidal levels, six locations with very large tides were identified. For exploitation of tidal energy it is important to understand the balance of tidal energy at the specific sites. The balance involves an accounting of the energy inflow and outflow to and from the local domain, and sources and sinks of energy within the local domain. An example of the energy balance approach is discussed for the Okhotsk Sea, where one of the worlds largest tidal ranges (13.9 m) have been recorded. The use of potential energy generated by the sea level difference to produce electricity is explained and connected to the tidal cycle. Such use of tidal power has been implemented very slowly, because of the high costs of tidal power plant construction. The newer approach is to use tidal currents, similar to the way windmills are used to tap wind energy. Simple theory shows that the maximum power available from the moving fluid is proportional to fluid density times velocity cubed. Because density of sea water is approximately 900 times greater than that of air, the same amount of energy as that generated by wind can be achieved by water with much slower movement. While the development of tidal energy will proceed regardless, it is important to develop a new branch of tidal dynamics which will help to better understand