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Notes to GEOF346 Tidal Dynamics and Sea Level Variations Notes to GEOF346 Tidal Dynamics and Sea Level Variations Fall 2021 Helge Drange Geophysical Institute, University of Bergen helge.drange@gfi.uib.no Mainly off-hour typing { so give a word regarding errors, inconsistencies, etc... Also give a word in case of specific requests (derivations, unclear issues), and I will try to update the notes accordingly. The note will be regularly updated { so keep printing at a minimum... The notation (closely) follows that used in the text book: Sea-Level Science by D. Pugh and P. Woodworth, Cambridge, 2014 (https://doi.org/10.1017/CBO9781139235778) Parts of the derivations follow Forelesninger over tidevann og tidevannsstrømmer. Teori, opptreden, analyse og prediksjon, by Herman G. Gade, Geofysisk institutt, Universitetet i Bergen Note: • Symbols/expressions in red are, to the best of the author's knowledge, cor- rections to typos in the text book. • For simplicity, the notation (614) sin 2 a = 2 sin a cos a implies that, in this case, equation (614) has been adopted in the displayed expression. • Norwegian word/phrases are given in quotation marks in footnotes. • A (far from) complete Norwegian dictionary is provided in appendix I. • ...and the index section is only slowly being updated. • Last, but not least: Comments to the compendium are always appreciated! 1 October 4, 2021 2 Contents I Tides, basic astronomical periods and frequencies 8 1 Supporting literature 8 1.1 General text . .8 1.2 Tidal energy . .8 2 Definition of the tides 8 3 Solar periods and frequencies 9 4 Lunar periods and frequencies 10 II Equilibrium theory and derivations thereof 12 5 Equilibrium theory, direct method 12 5.1 Geometry of the Earth-Moon system . 12 5.2 Centre of mass of the Earth-Moon system . 12 5.3 Newton's law of universal gravitation . 14 5.4 Gravitational forces and accelerations in the Earth-Moon system . 14 5.5 Earth's movement around the Earth-Moon centre of mass . 15 5.6 A (somewhat complicating) side note . 15 5.7 Net gravity on the Earth caused by the Moon . 17 5.8 Introducing the zenith angle and simplifying . 17 5.9 Radial and tangential components of the tidal acceleration . 18 5.9.1 Geometric approach . 18 5.9.2 Using the definition of the dot and cross products . 19 5.9.3 The radial component can be ignored; the tangential component is important 20 5.10 Surface elevation caused by the Moon . 20 5.10.1 Determining the integration constant for the Equilibrium Tide surface el- evation . 21 6 The combined surface elevation caused by the Moon and the Sun 23 6.1 Amplitudes . 23 6.2 Relative contribution from the Sun and the Solar planets . 23 6.3 Tidal potential . 24 6.4 Gravitational potential . 25 7 Introducing latitude, hour and declination angles 26 7.1 The three leading lunar tidal components . 26 7.2 The Equilibrium Tide caused by the Moon . 28 7.3 Properties of the three leading lunar components . 28 7.3.1 Declination or nodal component, ζ0 ..................... 29 7.3.2 Diurnal component, ζ1 ............................. 29 7.3.3 Semi-diurnal component, ζ2 .......................... 29 8 Sidereal and synodic periods 30 8.1 The Moon's sidereal and synodic monthly periods . 30 3 8.2 The synodic lunar day . 32 8.3 The Earth's daily sidereal period . 33 8.4 The synodic lunar day as seen from a fixed star . 34 8.5 Key periods and frequencies for the Moon-Earth-Sun system . 35 8.6 The lunar and solar hour angles expressed by the sidereal time . 36 9 Decomposing the solar and lunar tides into a series of simple harmonic con- stituents 38 9.1 Geometry of the Earth-Sun system . 38 9.1.1 Angular speed of the Earth and the Sun . 38 9.1.2 Expressing sin 2ds as sin ds ......................... 40 9.1.3 Expressing ζ1 in terms of simple harmonics . 41 9.1.4 Expressing ζ2 in terms of simple harmonics . 42 9.2 Sun's leading tidal constituents . 43 9.3 Moon's leading tidal constituents . 43 9.4 Doodson's system for labelling tidal constituents . 44 10 The spring/neap tide 46 10.1 Derivation . 47 10.1.1 Amplitude and range . 48 10.1.2 Timing of maximum amplitude . 48 10.1.3 Age of the tide . 48 10.1.4 A note . 50 11 Tidal forcing in the momentum equation 51 12 Laplace's Tidal Equations 52 13 The tidal force expressed in terms of the tidal potential 53 13.1 The potential of a conservative force . 53 13.2 Defining the tidal potential . 53 13.2.1 Gradient of the factor q=q3 .......................... 53 13.2.2 Gradient of the factor R=R3 ......................... 54 13.2.3 Resulting tidal potential . 55 13.3 Tidal potential expressed in terms of the zenith angle . 55 13.3.1 Radial and tangential tidal forces per unit mass . 56 14 Introducing the tidal potential in the primitive momentum equations 57 15 Including the effect of elliptic orbits 58 15.1 The Sun-Earth system1 ................................ 58 15.2 Kepler's second law . 60 15.3 Kepler's first law . 61 15.4 Kepler's third law . 62 15.5 Mean anomaly E0 ................................... 63 15.6 Real vs mean length of the radius vector, and Kepler's equation . 63 15.6.1 Rs vs Rs .................................... 63 15.6.2 Kepler's equation . 64 1Mainly based on R. Fitzpatrick (): An Introduction to Celestial Mechanics, M. Hendershott (): Lecture 1: Introduction to ocean tides, and Murray & Dermott (1999), Solar Syatem Dynamics, Chap. 2. 4 15.6.3 Solving Kepler's equation . 66 15.6.4 Rs=Rs expressed by means of E0 ...................... 66 15.6.5 Introducing the ecliptic longitude and the longitude of perihelion . 67 15.7 Right Ascension . 68 15.8 Equation of time . 70 15.8.1 Equation of center . 70 15.8.2 Longitudinal differences . 71 15.8.3 Equation of time, final expression . 71 15.8.4 Equation of time, graphical representation . 71 III Tidal harmonic analysis 74 16 Background 74 17 Implementation, sea surface height variations 77 17.1 Extraction of the M2 tide . 78 17.2 Note 1 . 78 17.3 Note 2 . 78 17.4 Extraction of the S2 tide . 79 17.5 Extraction of the N2 tide . 79 17.6 Extraction of the K2 tide . 79 17.7 Extraction of the O1 tide . 79 17.8 Extraction of the K1 tide . 79 17.9 Note 3 . 80 17.10Note 4 . 81 18 Tidal current analysis 81 18.1 Direction of flow θ and current speed q ....................... 81 18.2 Semi-major and semi-minor axes . 84 18.3 Time of maximum current speed . 85 18.4 Direction of maximum current speed . 85 18.5 Direction of rotation . 85 IV Tidal related wave dynamics 87 19 The shallow water equations 87 19.1 Starting point and configuration . 87 19.2 The continuity equation . 87 19.3 The momentum equations . 89 19.4 The final set of equations . 89 20 Gravity waves 91 20.1 One-dimensional gravity waves . 91 20.1.1 Typical wavelengths for tidal waves . 91 20.2 One-dimensional gravity waves in a closed channel . 92 20.3 One-dimensional gravity waves in a semi-closed channel . 93 20.3.1 Resonance . 94 5 21 Sverdrup waves and other related waves 95 21.1 Derivation . 95 21.2 Case ! =0 ....................................... 96 2 2 2 2 21.3 Case ! = f + c0 kh ................................. 96 21.3.1 The resulting wave motion . 97 21.3.2 Approximate length of the tidal axes . 100 21.4 Case f =0 ....................................... 101 21.5 Case k2 f 2=(gD) (short waves) . 101 21.6 Case k2 f 2=(gD) (long waves) . 102 21.7 Interaction between tidal ellipses and internal oscillations . 102 V Appendix 104 A Some key parameters of the Earth, Moon, Sun system 104 A.1 Mass . 104 A.2 Length and distance . 104 B Spherical coordinates 105 B.1 Two commonly used spherical coordinate systems . 105 B.2 Volume and surface elements in spherical coordinates . ..
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