Tide Energy Resource Assessment

1. Tide EnergyResource Assessment San Jose State University FX Rongère April 2008

2. Tides • Rising and falling of ocean because of the influence of the moon and the sun • Generally two times per day (semi-diurnal) tides depends a lot of the location • They may be up to 17 meters of amplitude High tide and low tide in Chausey Islands (UK)

3. Large amplitude tides High tide and low tide in the Bay of Fundy (Canada)

4. Tide amplitude around the world Solution of the Tidal Equations for the M2 and S2 Tides in the World Oceans from a Knowledge of the Tidal Potential Alone", Y. Accad, C. L. Pekeris Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 290, No. 1368 (November 28, 1978), pp. 235-266.

5. The cause of the tide • The Moon • Earth satellite 384,000 km • Diameter 3,474 km • Mass 7.35 1022 kg • The Earth • Diameter 12,740 km • Mass 598 1022 kg

6. Balance of forces • Balance of gravitation and centrifugal forces M m L L’ D G=6.67 10-11 M.m2.kg-2

7. Forces at the surface of the earth • Close to the moon • Far from the moon Moon Earth Conclusion: There are two lunar tidal ranges of equal amplitude per day Since Then

8. Forcing Potential • The attraction potential from a astronomical object is: (r,λφ) are the spherical coordinates of the observation point on the earth: Earth radius (r), Latitude (λ) and Longitude (φ) ρ is the mass of th object G gravitational constant Mo mass of the astronomical object O lo distance from the point to the object Do distance from the center of the earth to the object Ψo zenithal angle of the object

9. Sun Contribution • The sun • Diameter: 1.4 106 km • Distance: 150 106 km • Mass: 1.99 1030 kg

10. Spring tide and Neap tide • Spring tides happen at the New moon and the Full moon • Neap tides happen at the Quarters

11. Harmonic Analysis • The potential function is a very complex periodic function. It may be approximated by the sum of a series of sinusoidal functions: • Darwin and Doodson extracted these terms. Here are the dominant: • NOAA uses 37 harmonic coefficient for tide prediction

12. Earth and Moon relative movement A moon day is longer than a solar day because of the rotation of the moon around the earth

13. M2 and S2 • Neap and Spring tides simulated with M2 and S2 contributions 14 days

14. Tide calculator Calculator developed by Kelvin in 1873

15. Coriolis Force Tide as a wave • Tide is a periodic phenomena and will propagate as a wave • Laplace’s equations Motion balance Mass balance Source: Myrl C. Hendershott Long Waves and Ocean Tides

16. Tide amplitude around the world Solution of the Tidal Equations for the M2 and S2 Tides in the World Oceans from a Knowledge of the Tidal Potential Alone", Y. Accad, C. L. Pekeris Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 290, No. 1368 (November 28, 1978), pp. 235-266.

17. Type of Tides

18. Type of Tides

19. Harmonic Constituents - San Francisco Amplitude: Ai (ft) Phase: φi (Degree) Speed: ωi (360/Ti) Ti: period (hrs) Source: http://co-ops.nos.noaa.gov/data_menu.shtml?stn=9414290%20San%20Francisco,%20CA&type=Harmonic%20Constituents

20. Water level in San Francisco Moon phases 4/20 5/5 5/12 4/28

21. San Francisco Tide is Mixed • San Francisco – November 2007

22. Local Conditions • In fact, tides may change a lot from one location to another because of resonance • Example: Bay of Fundy (Canada) Bathymetry Mean tide range (m) Bay of Fundy

23. Bay of Fundy • The bay creates a resonance area for the tide wave

24. Wave propagation in a channel z Width: b Face area: A h v x x+dx Equation of motion on the x-axis Equation of mass on the x-axis

25. Wave propagation in a channel • Wave equation: • Resonance: In the bay of Fundy: Twave = 12h25 = 45,000s h = 45m L = 240km λ = 945km L/4 = 236km c = 21 m.s-1 Same phenomena in the Severn Estuary between Wales and England

26. Tidal wave in the Channel • The tidal wave enter from the South West of the Channel and propagate between the French and the British coasts: the tide is 6 hours earlier at the Pointe St Mathieu in Britany than between Dover and Calais. • The Coriolis effect amplifies the tides along the French coast (> 11m) while it decreases their range along the English coast (<3m Amphidromicpoint

27. More Complicated Cases • New Zealand Source: http://www.tecplot.com/showcase/studies/2001/niwa.htm

28. Available Energy • Energy balance zh zl

29. Annual Available Energy • The average power available during one tide is then: • To know the annual available energy, we have to average this value on all the tides τis the period of the tide (about 12h25’) R τ/2 It is assumed that the energy is captured when the tide goes up and down Parabolic function is not linear then:

30. Annual Available Energy • Because of the sun influence, the range (R=zh-zl) varies from day to day following a law close to sinusoidal • In fact the distance between the earth, the moon and the sun varies as well and the tides are never the same but we neglect this effect for that evaluation Spring tide Rn Rs Neap tide

31. Annual Available Energy • Tide range: T is the lunar month duration: 29.53 days. It covers 2 spring and 2 neap tides In general α = 2 to 3 Then

32. Examples