Tides

1. Tides • The tide generating force and the equilibrium tide. • The origin of tidal constituents. • Tidal wave amplification across a shelf edge. • Tides in bays and semi-enclosed seas (co-oscillation). • Harmonic analysis. Tides (1) Tides (2) Tides (3) Useful books: Pugh, D.. Tides, Surges, and Mean Sea Level (Short loan) Bowden, K.. Physical Oceanography of Coastal Waters. (Short loan).

2. Space and time scales of processes which affect sea-level. The maximum space and time scales are the dimensions and the age of the earth

3. One of the most important dynamical processes in shelf seas is the tide. Ancient civilizations were aware that the rise and fall of sea level was connected somehow to the celestial motions. The earliest evidence of this knowledge probably dates back to 1500 BC. Although quite advanced scientifically, the Greeks and Romans wrote very little about the tide, presumably because the Mediterranean basin responds only slight to tidal forcing. The English Monk Bede, was aware of tidal behaviour around the Northumbrian coast in 730AD. The earliest tide table for London Bridge dates from 1250AD. Further steps towards a unified understanding of tides were made by European scientists in the 16th and 17th centuries. The fundamental physical law is the Law of Universal Gravitation which states that the force of gravitational attraction of two bodies of masses M1 and M2 separated by a distance D is given by: G is the universal gravitational constant = 6.67 x 10-11 m2/kg2 (Newton, 1687 - the original copy of Newton’s Principia is held in the Trinity College Library at Cambridge)

4. The Tide Generating Force (TGF). Remember: 1. Newton’s law of gravitation: 2. The gravitational force exerted by a body on the outside world can be treated as if all of its mass were at its centre of gravity. Earth centre of gravity P Moon centre of gravity r a C M C of G d Earth-moon system rotates about common system centre of gravity (like an asymmetric dumbbell)

5. All points on the surface of the earth (and inside the earth) rotate about the earth-moon centre of mass with the same radius. Check this movie to convince yourself. • The centripetal (or centrifugal) force required to keep all points rotating is: [N kg-1] • Only at the centre of the earth (C) does the moon’s attractive force exactly balance the required centrifugal force (think what would happen if it did not). Elsewhere there is an imbalance because the distance to the moon changes. i.e. on the previous diagram the distance PM = r, and r < d, which means the moons attractive force at point P is greater than the centrifugal force: [N kg-1]

6. r centrifugal force  P  TGF a attractive force  C M d ( = lunar zenith angle) Calculation of the magnitude of the TGF: Resolve into horizontal and vertical components at P 

7. Apply cosine rule to triangle CPM:  Also note: and …..to eventually get:

8. Knowing that d=60.26a, and Me=81.53Mm we can say: Fv is small compared to gravity, so we ignore it. Fh is small, but acts perpendicular to gravity and is therefore important…this is the TGF. i.e. the tidal variation in sea level is not caused by direct attraction under the moon, but the dragging of water from 45 away from the sub-lunar point to pile up under the position of the moon.

9. The Equilibrium Tide. (After Newton, 1687). Assumes: the ocean has a uniform depth over the globe, and there is a constant equilibrium between the TGF and the sea surface slope. dsinfinitesimal area  d a  Fh = Fh() so we expect = () Slope force = TGF 

10. Assume hydrostatic pressure: K Relate to mean sea level by integrating over the entire surface to find C :

12. Therefore: This describes an ellipsoid of revolution. HW occurs for  = 0,   =35 cm LW occurs for  = /2  =-18 cm Thus, predicted range of equilibrium tide  53 cm. This is similar to the observed range in the open ocean, but is considerably smaller than the range observed in many coastal seas.

13. The equilibrium tide analysis can be applied to the earth-sun system in exactly the same way: i.e. the sun also produces a tidal ellipsoid. Given: Mass of earth Me = 6 x 1024 kg; Mass of moon Mm = 7 x 1022 kg; Mass of Sun Msun= 2 x 1030 kg; Earth - moon distance = 4 x 108 m; Earth - Sun distance = 1.5 x 1011 m. Calculate the strength of the solar tide relative to that caused of the moon.

14. Tidal Constituents (1) - the spring-neap cycle. Imagine the earth rotating about its axis beneath either the lunar or solar tidal ellipsoid. You would see two high tides (HW) and two low tides (LW) per day. These ellipses generate the main two tidal constituents: M2 : principal lunar semidiurnal period = 12.42 hours S2 : principal solar semidiurnal period = 12.00 hours Now imagine what happens when you have both lunar and solar tides interacting.

15. Spring Tide Sun’s tidal ellipsoid Moon’s tidal ellipsoid New moon Earth Sun Full moon Total tidal ellipsoid

16. Neap Tide Moon 1st quarter Earth Sun Moon last quarter

17. Beat period TSN NB: you have to observe a signal for at least the beat period to be able to resolve the 2 contributing frequencies. Investigate the effects of the spring-neap cycle using the tsp6 software (set the port to Liverpool).

18. Tidal constituents (2) - the other 388 constituents. • If: • the moon’s orbit was exactly circular, • the moon’s orbital plane was aligned with the earth’s rotational plane, • the earth’s rotational plane was aligned with the earth’s orbital plane about the sun, • the earth’s orbit about the sun was exactly circular, • then we might only have to deal with M2 and S2. • But, that’s not the case…….

19. For instance, consider the fact that the moon’s orbital plane is at an angle of about 18.5 to the earth’s rotational plane: Daily rotation To moon P X At point P you would only see 1 HW per day, or above or below P you might see unequal HWs during the day (the diurnal inequality)  the addition of diurnal tidal constituents. Also, point X precesses around the earth with a period of 18.6 years  longer period constituents.

20. Consider also that the moon’s orbit is an ellipse, not a circle, so there is a difference between successive spring HW. Moon Earth This manifests as another semi-diurnal constituent (N2)

21. The full expansion of the equilibrium tide was started by Laplace, with further work by Kelvin, Darwin, and eventually completed by Doodson. This resulted in an equilibrium tide containing 390 tidal constituents. Symbol Name Period (hrs) Strength (M2=1.0) M2 Principal lunar 12.42 1.0000 S2 Principal solar 12.00 0.4652 N2 Larger lunar elliptic 12.66 0.1915 K2 Luni-solar declinational 11.97 0.0402 K1 Luni-solar declinational 23.93 0.1852 O1 Larger lunar declinational 25.82 0.4151 P1 Larger solar declinational 24.07 0.1932 Mf Lunar fortnightly 13.65 days 0.1723 Mm Lunar monthly 27.55 days 0.0909 Investigate the effects of different tidal constituent using the tsp6 software.

22. Tidal constituents (3) - shallow water constituents (see Pugh page 110-112). There is a group of tidal constituents, not predicted by equilibrium theory, that we need to take into account. These are the shallow water constituents, caused by the interaction of the tidal wave with a shoaling seabed. hcrest htrough hcrest hmean hmean htrough Remember: Shallow water, hcrest> hmean > htrough, so c is greater at crest than at trough. Crest catches up with trough, and wave steepens. Deep water, hcrest hmean htrough, so c = constant over wave height.

23. Steepen a sinusoidal wave  add higher harmonics of the basic wave period. The M2 tide generates M4 (period 6.21 hours), M6 (period 4.14 hours), M8 (period 3.11 hours)…..etc. Example: Southampton Water. Ultimately the tidal wave can reach a situation where the crest overtakes the trough and the wave breaks to form a tidal bore. Investigate the effects of shallow water constituents using the tsp6 software (set the port to Southampton). Qiantang River, China.

24. Summary of the equilibrium theory: 1. A simple qualitative explanation of the main features of the tide. 2. Defines the fundamental frequencies. But: 1. It is a gross simplification - based on a simplified equation of motion, without rotation, friction, or acceleration (i.e. misses most of the dynamics). 2. Very simple geometry (no continents, no shelves, constant depth). The big strength is the determination of the tidal constituents. Laplace: for any mechanical system, regardless of damping, the output frequencies are the same as the input frequencies. This is critical to the prediction of tides through harmonic analysis.