Tide Energy Resource Assessment San Jose State University FX Rongère April 2009

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)

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

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.

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

Balance of forces Balance of gravitation and centrifugal forces (Newton law) R r M m L L’ D G=6.67 10-11 M.m2.kg-2

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

Forces at the surface of the earth At the first and third quadrants Only the centrifugal force has a contribution to the vertical axis The tide amplitude will be proportional to the difference between Fclose and FQ Using the balance between gravitational and centrifugal forces Moon Earth K is a constant

Earth and Moon relative movement From the balance between gravitational and centrifugal forces: Period of moon rotation: Moon Orbital Period A moon day is longer than a solar day because of the rotation of the moon around the earth

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

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

M2 and S2 Neap and Spring tides simulated with Moon and Sun contributions 14.5 days Synodic period of the moon: 29.53 days

Forcing Potential The attraction potential from an 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

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

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.

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

37 Tide Harmonics M2- Principal lunar semidiurnal constituent S2- Principal solar semidiurnal constituent N2- Larger lunar elliptic semidiurnal constituent K1- Lunar diurnal constituent M4- Shallow water overtides of principal lunar constituent O1- Lunar diurnal constituent M6- Shallow water overtides of principal lunar constituent MK3- Shallow water terdiurnal S4- Shallow water overtides of principal solar constituent MN4- Shallow water quarter diurnal constituent NU2- Larger lunar evectional constituent S6- Shallow water overtides of principal solar constituent MU2- Variational constituent 2N2- Lunar elliptical semidiurnal second-order constituent OO1- Lunar diurnal LAM2- Smaller lunar evectional constituent S1- Solar diurnal constituent M1- Smaller lunar elliptic diurnal constituent J1- Smaller lunar elliptic diurnal constituent MM- Lunar monthly constituent SSA- Solar semiannual constituent SA- Solar annual constituent MSF- Lunisolar synodic fortnightly constituent MF- Lunisolar fortnightly constituent RHO- Larger lunar evectional diurnal constituent Q1- Larger lunar elliptic diurnal constituent T2- Larger solar elliptic constituent R2- Smaller solar elliptic constituent 2Q1- Larger elliptic diurnal P1- Solar diurnal constituent 2SM2- Shallow water semidiurnal constituent M3- Lunar terdiurnal constituent L2- Smaller lunar elliptic semidiurnal constituent 2MK3- Shallow water terdiurnal constituent K2- Lunisolar semidiurnal constituent M8- Shallow water eighth diurnal constituent MS4- Shallow water quarter diurnal constituent

Tide calculator Calculator developed by Kelvin in 1873

Type of Tides

Type of Tides

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

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

San Francisco Tide is Mixed San Francisco – November 2007

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

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

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

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

Gulf of California Tide amplitude may reach 9 m. in San Felipe In the bay of Fundy: Twave = 12h25 = 45,000s h = 2,000 m L = 1,500 km λ = 6,300 km λ/4 = 1,567 km c = 140 m.s-1 http://oceanografia.cicese.mx/predmar/anim/tidegc.htm

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 Brittany 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) Amphidromic point

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

Available Energy Energy balance zh zl R is the range or amplitude of the tide

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 τ is 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:

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

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 = .3 to .5 Then

Examples

Tidal Streams There are generated by the difference of water levels

Golden Gate Currents Source: http://wolfweb.unr.edu/homepage/edc/tides.html

Model Ocean Basin z2 z1 R C z1 has the same period as z2 and a phase-shift Current has the same period as z2 and is in quadrature with z1 φ=π/4 z1 has the same period as z2 and a phase-shift Current has the same period as z2 and is in quadrature with z1

Current Power The power of the current is similar to the power of the wind Available energy is proportional to the cube of the current velocity

Shear Effect

The Golden Gate Site Golden Gate maximum depth = 377 feet

Tidal Stream Resources at EPRI Study Sites Source: George Hagerman Energy from Tidal, River, and Ocean Currents and from Ocean Waves, 08 June 2007

Resource Analysis Currents: Kinetic energy Tide: Potential Energy Source: Tidal Power in the UK

Resource Analysis Source: SEI “Tidal & Current Energy Resources in Ireland”, 2006

Strangford Lough project

Minas Passage Project Source: George Hagerman Tidal Stream Energy in the Bay of Fundy, Energy Research & Development Forum 2006 Antigonish, Nova Scotia 25 May 2006

Minas Passage Project Source: George Hagerman Tidal Stream Energy in the Bay of Fundy, Energy Research & Development Forum 2006 Antigonish, Nova Scotia 25 May 2006

Minas Passage Project Current prediction is difficult Source: George Hagerman Tidal Stream Energy in the Bay of Fundy, Energy Research & Development Forum 2006 Antigonish, Nova Scotia 25 May 2006

Passamaquoddy Bay Project Source: George Hagerman Energy from Tidal, River, and Ocean Currents and from Ocean Waves, 08 June 2007

Pudget Sounds Projects Source: Brian Polagye Tidal In-Stream Energy Overview, March 6, 2007 Agate Passage Spieden Channel Guemes Channel San Juan Channel Deception Pass Rich Passage Point Wilson Marrowstone Point Tacoma Narrows Bush Point Large resource Strong currents Small resource Weaker currents 700+ MW of tidal resources identified

Preliminary Array Performance Deception Pass High Power Region 1 km Preliminary Array Performance 2 km 20 turbines (10 m diameter) Average installation depth ~30m Exceptionally strong currents may complicate installation and surveys 3 MW average electric power 11 MW rated electric power Power for 2000 homes Source: Brian Polagye Tidal In-Stream Energy Overview, March 6, 2007