1. TIDES Tide - generic term to define alternating rise and fall in sea level with respect to land and is produced by the balance between the gravitational acceleration (of the moon and sun mainly) and the centrifugal acceleration. Tide also occurs in large lakes, in the atmosphere, and within the solid crust Gravitational Force (Newton’s Law of Gravitation): F = GmM/R 2 G = 6.67×10 -11 N m 2 /kg 2

2. Centrifugal Force Center of mass of Earth-Moon system ~1,700 km from Earth’s surface (because Earth is 81 times heavier than Moon) EQUILIBRIUM TIDE Moon’s Gravitational Force (changes from one side of the earth to the other) F = GmM/R 2 Tide Generating Force (Difference between centrifugal and gravitational)

3. How strong is the Tide-Generating Force? P A B S Tide-generating Force at A : Gravitational Force at A : Centrifugal Force at A : Imbalance (Tide-generating force at A ): Tide-generating Force at B :

4. The mass of the sun is 2x10 27 metric tons while that of the moon is only 7.3x10 19 metric tons. The sun is 390 times farther away from the earth than is the moon. The relative Tide Generating Force on Earth = [(2x10 27 /7.3x10 19 )]/(390 3 ) or = 2.7x10 7 /5.9x10 7 = 0.46 or 46% How strong is the Tide-Generating Force? P A B S Tide-generating Force at A :Tide-generating Force at B :

5. EQUILIBRIUM TIDE

7. What alters the range and phase of tides produced by Equilibrium Theory? Non-astronomical factors: coastline configuration bathymetry atmospheric forcing (wind velocity and barometric pressure) hydrography may alter speed, produce resonance effects and seiching, storm surges In the open ocean, tidally induced variations of sea level are a few cm. When the tidal wave moves to the continental shelf and into confining channels, the variations may become greater.

8. Keep in mind that tidal waves travel as shallow (long) waves How so? Typical wavelengths = 4500 km (semidiurnal wave traveling over 1000 m of water) Ratio of depth / wavelength = 1 / 4500 Then, their phase speed is: C = [ gH ] 0.5 The tide observed at any location is the superposition of several constituents that arise from diverse tidal forcing mechanisms. Main constituents:

9. The Form factor F = [ K 1 + O 1 ] / [ M 2 + S 2 ] is customarily used to characterize the tide. When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal When 1.25 < F < 3.00 the tide is mixed - mainly diurnal

10. F > 3 the tide is diurnal F < 0.25 the tide is semidiurnal When 1.25 < F < 3.00 the tide is mixed - mainly diurnal When 0.25 < F < 1.25 the tide is mixed - mainly semidiurnal Superposition of constituents generates modulation - e.g. fortnightly, monthly This applies for both sea level and velocity

12. Subtidal modulation by two tidal constituents

13. In Ponce de León Inlet: M 2 = 0.41 m; N 2 = 0.09 m; O 1 : 0.06 m; S 2 : 0.06 m; K 1 = 0.08 m F = [K 1 + 0 1 ] / [S 2 + M 2 ] = 0.30 GNV

14. PanamaCity In Panama City, FL: M 2 = 0.085 m; N 2 = 0.017 m; O 1 : 0.442 m; S 2 : 0.035 m; K 1 = 0.461 m F = [K 1 + 0 1 ] / [S 2 + M 2 ] = 7.52

15. From Pinet (1998)

16. Co-oscillation Independent tide - caused by gravitational and centrifugal forces directly on the waters of a basin -- usually negligible effect for typical dimensions of semienclosed basins Co-oscillating tide - caused by the ocean tide at the entrance to a basin as driving force The wave propagates into the basin and may be subject to RESONANCE and RECTIFICATION -- alters tidal flows and produces subtidal motions

17. Resonance of Tidal Wave At the mouth x = L, Substituting into at x = L L For resonance to exist, the denominator should tend to zero, i.e., and The natural period of oscillation is then:

18. L u For an estuary with length < λ /4, u is zero at the head and maximum at the mouth For longer estuaries u is zero at x = 0, λ / 2, 3 λ / 2,… or where sin κx = 0 and maximum at x = λ /4, 3 λ / 4, 5 λ / 4, …, i.e., where sin κx is max

19. Merion’s Formula Mode 1 (n =1) H (m)L (km)C (m/s)T N (h) Long Island Sound 2018014 Chesapeake Bay 102501028 Bay of Fundy 702502610.7

20. Effects of Rotation on a Progressive Tidal Wave in a Semi-enclosed basin Solution: R = C / f KELVIN WAVE

21. Effects of Rotation on a Standing Tidal Wave in a Semi-enclosed Basin Two Kelvin waves of equal amplitude progressing in opposite directions. Instead of having lines of no motion, we are now reduced to a central region - -- amphidromic region-- of no motion at the origin. The interference of two geostrophically controlled simple harmonic waves produces a change from a linear standing wave to a rotary wave.

22. Two Kelvin Waves in Opposite Directions Distance (m)

23. Two Kelvin Waves in Opposite Directions km

28. Effects of Bottom Friction on an amphydromic system (Parker, 1990)

29. Virtual Amphidromes (Parker, 1990)

30. Virtual amphidromes in Chesapeake Bay