Tides

1. Tides Lecture 18 OEAS-604 November 30, 2011 • Outline: • Gravitational Forces on the Ocean • Equilibrium Theory of Tides and Astronomical Forcing • Semidiurnal, Mixed and Diurnal Tides • Tidal Propagation in Basins • Reflection, convergence and resonance • Tidal bores

2. Tides are the result of the imbalances in gravitational forces acting on the earth. Gravitational Force between two objects (F): G = 6.6×10-11 Newton m2/kg2 M1 = mass object #1 M2 = mass object #2 r = distance between them Considering only the moon for now: Gravitational force between earth and moon: Earth Moon r

3. The tide generating force (T) is equal to difference between the gravitational force at the center of mass of the earth and the gravitational force at other points: r x x At center of the earth : a At side of the earth facing moon: So tide generating force is inversely proportional to the cube of the distance between earth and moon. What about the Sun? Sun is 27 million times more massive than moon, but 390 times farther away. Moon over twice as important for creating tides.

4. Rotation of the Earth-moon system creates tidal bulges The moon does not rotate around the center of Earth. Earth and moon together rotate around a common center of mass about 1,650 kilometers (1,023 miles) beneath Earth’s surface. The moon’s gravity attracts the ocean toward it. The motion of Earth around the center of mass of the Earth – moon system throws up a bulge on the side of Earth opposite the moon. The combination of the two effects creates two tidal bulges.

5. Equilibrium Theory of Tides Developed based on the assumption that the Earth is completely covered in water. The action of gravity and inertia on particles at five different locations on Earth. At points (1) and (2), the gravitations attraction of the moon slightly exceeds the outward-moving tendency of inertia; the imbalance of forces causes water to move along Earth’s surface, converging at a point toward the moon. At points (3) and (4), inertia exceeds gravitational force, so water moves along Earth’s surface to converge at a point opposite the moon. Forces are balanced only at the center of Earth (point CE).

6. Rotation of the Earth Relative to the Gravitational Bulge causes the Tides (a) How Earth’s rotation beneath the tidal bulges produces high and low tides. Notice that the tidal cycle is 24 hrs 50 minutes long because the moon rises 50 minutes later each day. (b) A graph of the tides at the island in (a).

7. Lunar Day is 24 hours 50 minutes This gives two high and two low tides, each 12.42 hours apart. So tidal period is 12.42 hours (the tide is a very long wave). A lunar day is longer than a solar day. A lunar day is the time that elapses between the time the moon is highest in the sky and the next time it is highest in the sky. In a 24-hour solar day, the moon moves eastward about 12.2°. Earth must rotate another 12.2° - 50 minutes – to again place the moon at the highest position overhead. A lunar day is therefore 24 hours 50 minutes long. Because Earth must turn an additional 50 minutes for the same tidal alignment, lunar tides usually arrive 50 minutes later each day.

8. Both Sun and Moon Create Tides Remember, moon’s tide generating force is 2.2 times bigger than sun’s. Relative positions of the sun, moon, and Earth during spring and neap tides. (a) At the new and full moons, the solar and lunar tides reinforce each other, making spring tides, the highest high and lowest low tides. (b) At the first-and third-quarter moons, the sun, Earth, and moon form a right angle, creating neap tides, the lowest high and the highest low tides.

9. Just like surface waves, tidal waves can constructively and destructively interfere with each other. Wave period due to moon = 12.42 hours. This called the M2 tide. Wave period due to sun = 12.00 hours. This called the S2 tide. Tides that occur twice a day (2 highs and 2 lows) are called semi-diurnal tides. Spring Tides Neap Tides Neap Tides

10. Earth’s axis is tilted 23.5 degrees relative to its rotation about the Sun

11. Moon’s orbit around Earth is Nearly aligned (5°) with the Ecliptic (Path of Sun). 23.5 ° North of Equator moon moon 23.5 ° South of Equator declination Moon orbits Earth every 27.3 days

12. Declination of Earth’s Axis Leads to Diurnal and Mixed Tides C B A 13.6 days Observer at point A sees 2 high tides of equal size (semidiurnal). Observer at point B sees 2 high tides of different size (mixed tides). Observer at point C see only 1 high tide (diurnal tides).

13. Mixed Tides have 2 unequal highs and lows during the day. Small/No Diurnal Inequality Big Diurnal Inequality Diurnal Inequality – When two tides have different amplitudes.

14. Moon’s rotation about Earth is an ellipse. Perigee Apogee Moon’s orbit around Earth is elliptical not circular so the size of the bulge will vary with the moon’s distance from earth. This has period of 27.55 days. So there are also small variations of the tides over monthly time scales.

15. Earth’s rotation about sun is also an ellipse. January 3 July 7 Perihelion aphelion Perihelion to aphelion occurs in one anomalistic year (365.2596 days). Tide generating forces are greater in the northern hemisphere winter.

16. Dominant Diurnal and Semidiurnal Tidal Constituents Tidal elevations have been recorded using tide gauges at coastal locations around the world for many years. These time series are used to determine the amplitude and phase of all of the known tidal constituents (there are a lot more than just listed above). This information can then be used to make predictions of the tides. These are very accurate, except for ….

17. Tidal Prediction Constituent #2 Predicted tidal elevation Constituent #1 frequency And so on. amplitude phase

18. Equilibrium Theory Assumes That Earth is Completely Covered In Water. This theory provides a lot of insight into what creates the tides. But, the Earth is not totally covered in water, which makes tides a little more complicated. Tides are really long waves, so they are considered shallow water waves (L >> 20h) For very long waves like tides, orbital velocities become flat.

19. Is the Earth’s rotation important to tides? For surface gravity waves we assume acceleration is much too big for Coriolis force to be important What is the ratio of local acceleration to Coriolis acceleration for the M2 tide? From scaling this ratio is simply:

20. As a result of the Earth’s rotation, tides propagate around ocean basins. Tidal Bulge moves like a wave, that is turned by rotation of the Earth. Wave turns right in the Northern Hemisphere and left in the Southern Hemisphere. When it encounters the coastline, it becomes a coastally-trapped Kelvin Wave.

21. Solution for a Kelvin Wave Governing Equations: Investigate case where u = 0 x-momentum equation: y-momentum equation: or Depth averaged continuity: This is a wave equation, with solution of the form: After a bunch of math, the final solution is: Rossby Radius Wave speed [gh]1/2 wavenumber Along coast velocity Tidal elevation Tidal height at coast frequency depth

22. Final Solution (Kelvin Wave) Where R is the Rossby radius of deformation. The length scale over which rotational effects become important R Direction of wave propagation depth (m) x-direction (km) y-direction (km)

23. Kelvin Wave Propagation in Northern Hemisphere

24. Kelvin Wave Solution Tidal Elevation (m) Along-shore distance (m) Cross-shore distance (m) White arrows show wave orbital velocity. Kelvin wave is a progressive wave, max velocity at high tide!!!

25. Tides Propagate Around Ocean Basins with the Coastline on their Right in the Northern Hemisphere and on their Left in the Southern Hemisphere.

26. Along open coast lines, tides are usually “progressive waves”. Maximum velocity magnitude occurs at high and low water. In some embayments or estuaries, the tide is a “standing wave”. Velocity is zero at high and low water.

27. Tides in very short systems What happens in basins that are short compared to the tidal wavelength? Tidal wavelength ~ 100s km Hoffler Creek ~ 2 km Because of continuity relationship: Water surface goes up during flood. Water surface goes down during ebb.

28. There are several other mechanisms that can cause a tide to be a standing wave. In an enclosed basin, one explanation is the super-position of an incident and reflected wave:

29. Standing Waves can Resonate. resonance In physics, resonance is the tendency of a system to oscillate at maximum amplitude at certain frequencies. At these frequencies, even small periodic driving forces can produce large amplitude vibrations, because the system stores vibrational energy. Tides can produce resonance, when the basin length is ¼ the tidal wave length: Bay of Fundy has 15 meter tides (over 50 feet)

30. Interaction of tide and continental shelf Resonance? Assume depth over continental shelf is 100m. What is the wave speed? What is the wave length? What is the resonant wave length (L/4)? 50 km Tidal range at Cape Hatteras is less than 3 m. Tidal range along Georgia coast is more than 6 m. 350 km

31. Tides can also increase due to convergence: Remember the continuity equation Delaware Bay Converging channel can increase the water elevation and tides get bigger.

32. When the tidal range is big and convergence is rapid, a tidal bore can develop. tidal bore surfing