Notes to GEOF346 Tidal Dynamics and Sea Level Variations

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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