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

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 (November 28, 1978), pp

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

6. Balance of forces
Balance of gravitation and centrifugal forces (Newton law) R r M m L L’ D
G= 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. 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

9. 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

10. Sun Contribution The sun Diameter: 1.4 106 km Distance: 150 106 km
Mass: kg

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

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

13. 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

14. 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

15. 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 (November 28, 1978), pp

16. 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

17. 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

18. Tide calculator
Calculator developed by Kelvin in 1873

19. Type of Tides

20. Type of Tides

21. Harmonic Constituents - San Francisco
Amplitude:
Ai (ft)
Phase:
φi (Degree)
Speed:
ωi (360/Ti)
Ti: period (hrs)
Source:

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

23. San Francisco Tide is Mixed
San Francisco – November 2007

24. 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

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

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

27. 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

28. 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

29. 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

30. More Complicated Cases
New Zealand
Source:

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

32. 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:

33. 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

34. Annual Available Energy
Tide range:
T is the lunar month duration: days. It covers 2 spring and 2 neap tides
In general α = .3 to .5
Then

35. Examples

36. Tidal Streams
There are generated by the difference of water levels

37. Golden Gate Currents
Source:

38. 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

39. 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

40. Shear Effect

41. The Golden Gate Site
Golden Gate
maximum depth = 377 feet

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

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

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

45. Strangford Lough project

46. 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

47. 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

48. 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

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

50. 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

51. 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