Chapter 17 - Coastal Processes and Tides

17.4 Theory of Ocean Tides

Tides have been so important for commerce and science for so many thousands of years that tides have entered our everyday language: time and tide wait for no one, the ebb and flow of events, a high-water mark, and turn the tide of battle.

1. Tides produce strong currents in many parts of the ocean. Tidal currents can have speeds of up to 5m/s in coastal waters, impeding navigation and mixing coastal waters.
2. Tidal currents generate internal waves over seamounts, continental slopes, and mid-ocean ridges. The waves dissipate tidal energy. Breaking internal waves and tidal currents are a major forces driving oceanic mixing.
3. Tidal currents can suspend bottom sediments, even in the deep ocean.
4. Earth's crust is elastic. It bends under the influence of the tidal potential. It also bends under the weight of oceanic tides. As a result, the seafloor, and the continents move up and down by about 10cm in response to the tides. The deformation of the solid Earth influence almost all precise geodetic measurements.
5. Oceanic tides lag behind the tide-generating potential. This produces forces that transfer angular momentum between Earth and the tide producing body, especially the moon. As a result of tidal forces, Earth's rotation about it's axis slows, increasing the length of day; the rotation of the moon about Earth slows, causing the moon to move slowly away from Earth; and moon's rotation about it's axis slows, causing the moon to keep the same side facing Earth as the moon rotates about Earth.
6. Tides influence the orbits of satellites. Accurate knowledge of tides is needed for computing the orbit of altimetric satellites and for correcting altimeter measurements of oceanic topography.
7. Tidal forces on other planets and stars are important for understanding many aspects of solar-system dynamics and even galactic dynamics. For example, the rotation rate of Mercury, Venus, and Io result from tidal forces.

Mariners have known for at least four thousand years that tides are related to the phase of the moon. The exact relationship, however, is hidden behind many complicating factors, and some of the greatest scientific minds of the last four centuries worked to understand, calculate, and predict tides. Galileo, Descartes, Kepler, Newton, Euler, Bernoulli, Kant, Laplace, Airy, Lord Kelvin, Jeffreys, Munk and many others contributed. Some of the first computers were developed to compute and predict tides. Ferrel built a tide-predicting machine in 1880 that was used by the U. S. Coast and Geodetic Survey to predict nineteen tidal constituents. In 1901, Harris extended the capacity to 37 constituents.

Despite all this work important questions remained: What is the amplitude and phase of the tides at any place on the ocean or along the coast? What is the speed and direction of tidal currents? What is the shape of the tides on the ocean? Where is tidal energy dissipated? Finding answers to these simple questions is difficult, and the first, accurate, global maps of deep-sea tides were only published in 1994 (LeProvost et al., 1994). The problem is hard because the tides are a self-gravitating, near-resonant, sloshing of water in a rotating, elastic, ocean basin with ridges, mountains, and submarine basins.

Predicting tides along coasts and at ports is much simpler. Data from a tide gauge plus the theory of tidal forcing gives an accurate description of tides near the tide gauge.

Tidal Potential
Tides are calculated from the hydrodynamic equations for a self-gravitating ocean on a rotating, elastic Earth. The driving force is the gradient of the gravity field of the moon and sun. The small variations in gravity arise from two separate mechanisms. To see how they work, consider the rotation of moon about Earth. If Earth were an ocean planet with no land, and if we ignore the influence of inertia and currents, the gravityb gradient produces a pair of bulges of water on Earth, one on the side facing the moon, one on the side away from the moon. A clear derivation of the forces is given by Pugh (1987) and by Dietrich, Kalle, Krauss, and Siedler (1980). Here I follow the discussion in Pugh §3.2.

Note that many oceanographic books state that the tide is produced by two processes: i) the centripetal acceleration at earth's surface as earth and moon circle around a common center of mass, and ii) the gravitational attraction of mass on earth and moon. However, the derivation of the tidal potential does not involve centripatal acceleration, and the concept is not used by the astronomical or geodetic communities.

Figure 17.10 Sketch of coordinates for determining the tide-generating potential.

To calculate the amplitude and phase of the tide on an ocean planet, we begin by calculating the tide generating potential. This is much easier than calculating the forces. Ignoring for now Earth's rotation, the rotation of moon about Earth produces a potential V_M at any point on Earth's surface

(17.5)

where the geometry is sketched in Figure 17.10, g is the gravitational constant, and M is moon's mass. From the triangle OPA in the figure,

(17.6)

Using this in (17.5) gives

(17.7)

r/R » 1/60, and (17.7) may be expanded in powers of r/ R using Legendre polynomials (Whittaker and Watson, 1963: §15.1):

(17.8)

The tidal forces are calculated from the spatial gradient of the potential, so the first term in (17.8) produces no force. The second term produces a constant force parallel to OA. This force keeps Earth in orbit about the center of mass of the Earth-moon system. The third term produces the tides, assuming the higher-order terms can be ignored. The tide-generating potential is therefore:

(17.9)

The tide-generating force can be decomposed into components perpendicular P and parallel H to the sea surface. The tides are produced by the horizontal component. "The vertical component is balanced by pressure on the sea bed, but the ratio of the horizontal force per unit mass to vertical gravity has to be balanced by an opposing slope of the sea surface, as well as by possible changes in current momentum" (Cartwright, 1999: 39, 45). The horizontal component is shown in Figure 17.11. It is:

(17.10)

where

(17.11)

The tidal potential is symmetric about the Earth-moon line, and it produces symmetric bulges.

Figure 17.11 The horizontal component of the tidal force on Earth when the tide-generating body is above the Equator at Z. From Dietrich, et al. (1980).

If we allow our ocean-covered Earth to rotate, an observer in space sees the two bulges fixed relative to the Earth-moon line as Earth rotates. To an observer on Earth, the two tidal bulges seems to rotate around Earth because moon appears to move around the sky at nearly one cycle per day. Moon produces high tides every 12 hours and 25.23 minutes on the equator if the moon is above the equator. Notice that high tides are not exactly twice per day because the moon is also rotating around Earth. Of course, the moon is above the equator only twice per lunar month, and this complicates our simple picture of the tides on an ideal ocean-covered Earth. Furthermore, moon's distance from Earth R varies because moon's orbit is elliptical and because the elliptical orbit is not fixed.

Clearly, the calculation of tides is getting more complicated than we might have thought. Before continuing on, we note that the solar tidal forces are derived in a similar way. The relative importance of the sun and moon are nearly the same. Although the sun is much more massive than moon, it is much further away.

	(17.12)
	(17.13)
	(17.14)

where R sun is the distance to the sun, S is the mass of the sun, R moon is the distance to the moon, and M is the mass of the moon.

Coordinates of Sun and Moon
Before we can proceed further we need to know the position of moon and sun relative to Earth. An accurate description of the positions in three dimensions is very difficult, and it involves learning arcane terms and concepts from celestial mechanics. Here, I paraphrase a simplified description from Pugh. See also Figure 4.1.

A natural reference system for an observer on Earth is the equatorial system described at the start of Chapter 3. In this system, declinations δ of a celestial body are measured north and south of a plane which cuts the Earth's equator.

Angular distances around the plane are measured relative to a point on this celestial equator which is fixed with respect to the stars. The point chosen for this system is the vernal equinox, also called the 'First Point of Aries'... The angle measured eastward, between Aries and the equatorial intersection of the meridian through a celestial object is called the right ascension of the object. The declination and the right ascension together define the position of the object on a celestial background...

[Another natural reference] system uses the plane of the Earth's revolution around the sun as a reference. The celestial extension of this plane, which is traced by the sun's annual apparent movement, is called the ecliptic. Conveniently, the point on this plane which is chosen for a zero reference is also the vernal equinox, at which the sun crosses the equatorial plane from south to north near 21 March each year. Celestial objects are located by their ecliptic latitude and ecliptic longitude. The angle between the two planes, of 23.45°, is called the obliquity of the ecliptic... Pugh (1987: 72).

Tidal Frequencies
Now, let's allow Earth to spin about its polar axis. The changing potential at a fixed geographic coordinate on Earth is:

(17.15)

where φ_p is latitude at which the tidal potential is calculated, d is declination of moon or sun north of the equator, and τ₁ is the hour angle of moon or sun. The hour angle is the longitude where the imaginary plane containing the sun or moon and Earth's rotation axis crosses the Equator.

The period of the solar hour angle is a solar day of 24hr 0min. The period of the lunar hour angle is a lunar day of 24hr 50.47min.

Earth's axis of rotation is inclined 23.45° with respect to the plane of Earth's orbit about the sun. This defines the ecliptic, and the suns declination varies between δ = ± 23.45° with a period of one solar year. The orientation of Earth's rotation axis precesses with respect to the stars with a period of 26,000 years. The rotation of the ecliptic plane causes δ and the vernal equinox to change slowly, and the movement called the precession of the equinoxes.

Earth's orbit about the sun is elliptical, with the sun in one focus. That point in the orbit where the distance between the sun and Earth is a minimum is called perigee. The orientation of the ellipse in the ecliptic plane changes slowly with time, causing perigee to rotate with a period of 20,900 years. Therefore R_sun varies with this period.

Moon's orbit is also elliptical, but a description of moon's orbit is much more complicated than a description of Earth's orbit. Here are the basics. The moon's orbit lies in a plane inclined at a mean angle of 5.15° relative to the plane of the ecliptic. And lunar declination varies between δ = 23.45 ± 5.15° with a period of one tropical month of 27.32 solar days. The actual inclination of moon's orbit varies between 4.97°, and 5.32°.

The shape of moon's orbit also varies. First, perigee rotates with a period of 8.85 years. The eccentricity of the orbit has a mean value of 0.0549, and it varies between 0.044 and 0.067. Second, the plane of moon's orbit rotates around Earth's axis of rotation with a period of 17.613 years. Both processes cause variations in R_moon.

Note that I am a little imprecise in defining the position of the sun and moon. Lang (1980: §5.1.2) gives much more precise definitions.

Substituting (17.15) into (17.9) gives:

(17.16)

Equation (17.16) separates the period of the lunar tidal potential into three terms with periods near 14 days, 24 hours, and 12 hours. Similarly the solar potential has periods near 180 days, 24 hours, and 12 hours. Thus there are three distinct groups of tidal frequencies: twice-daily, daily, and long period, having different latitudinal factors sin² θ, sin 2θ, and (1 - 3 cos²θ) / 2, where is the co-latitude ( 90° - φ ).

Doodson (1922) expanded (17.16) in a Fourier series using the cleverly chosen frequencies in Table 17.1. Other choices of fundamental frequencies are possible, for example the local, mean, solar time can be used instead of the local, mean, lunar time. Doodson's expansion, however, leads to an elegant decomposition of tidal constituents into groups with similar frequencies and spatial variability.

Using Doodson's expansion, each constituent of the tide has a frequency

(17.17)

where the integers n_i are the Doodson numbers. n₁ = 1, 2, 3 and n₂ - n₆ are between -5 and + 5. To avoid negative numbers, Doodson added five ton_{2 · · · 6}. Each tidal constituent, sometimes called a partial tides, has a Doodson number. For example, the principal, twice-per-day, lunar tide has the number 255.555. Because the very long-term modulation of the tides by the change in sun's perigee is so small, the last Doodson number n₆ is usually ignored.

If the tidal potential is expanded in Doodson's Fourier series, and if the ocean surface is in equilibrium with the tidal potential, the largest tidal constituents would have frequencies and amplitudes given in table 17.2. The expansion shows that tides with frequencies near one or two cycles per day are split into closely spaced lines with spacing separated by a cycle per month. Each of these lines is further split into lines with spacing separated by a cycle per year (Figure 17.12). Furthermore, each of these lines is split into lines with a spacing separated by a cycle per 8.8yr, and so on. Clearly, there are very many possible tidal constituents.

Why are the tidal lines in Figure 17.12 split into so many constituents? To answer the question, suppose moon's elliptical orbit was in the equatorial plane of Earth. Then d = 0. From (17.16), the tidal potential on the equator, where j_p = 0, is:

(17.18)

If the ellipticity of the orbit is small, R = R₀ ( 1 + e ) , and (17.18) is approximately

(17.19)

where a = ( g Mr ₂ ) / ( 4 R³ ) is a constant. e varies with a period of 27.32 days, and we can write e = b cos( 2 p f₂) where b is a small constant. With these simplifications, (17.19) can be written:

	(17.20a)
	(17.20b)

which has a spectrum with three lines at 2 f₁ and 2 f₁ ± f₂. Therefore, the slow modulation of the amplitude of the tidal potential at two cycles per lunar day causes the potential to be split into three frequencies. This is the way amplitude modulated AM radio works. If we add in the the slow changes in the shape of the orbit, we get still more terms even in this very idealized case of a moon in an equatorial orbit.

If you are very observant, you will have noticed that the spectrum of the tide waves in Figure 17.12 does not look like the spectrum of ocean waves in Figure 16.6. Ocean waves have all possible frequencies, and their spectrum is continuous. Tides have precise frequencies determined by the orbit of Earth and moon, and their spectrum is not continuous. It consists of discrete lines.

Figure 17.12 Upper: Spectrum of equilibrium tides with frequencies near twice per day. The spectrum is split into groups separated by a cycle per month (0.55deg/hr). Lower: Expanded spectrum of the S2 group, showing splitting at a cycle per year (0.04deg/hr). The finest splitting in this figure is at a cycle per 8.847 years (0.0046deg/hr).

Doodson's expansion included 399 constituents, of which 100 are long period, 160 are daily, 115 are twice per day, and 14 are thrice per day. Most have very small amplitudes, and only the largest are included in table 17.2. The largest tides were named by Sir George Darwin (1911) and the names are included in the table. Thus, for example, the principal, twice-per-day, lunar tide, which has Doodson number 255.555, is the M₂ tide, called the M-two tide.

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