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 5 m/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 force driving oceanic mixing.
3. Tidal mixing helps drive the deep circulation, and it influences climate and abrupt climate change.
4. Tidal currents can suspend bottom sediments, even in the deep ocean.
5. 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 sea floor and the continents move up and down by about 10 cm in response to the tides. The deformation of the solid Earth influences almost all precise geodetic measurements.
6. 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 its 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 the Moon’s rotation about its axis slows, causing the Moon to keep the same side facing Earth as it rotates about Earth.
7. 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.
8. 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 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 easier. 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. If the Earth were an ocean planet with no land, and if we ignore the influence of inertia and currents, the gravity gradient produces a pair of bulges of water on Earth, one on the side facing the Moon or Sun, one on the side away from that body. 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 (1987: §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 the Earth and Moon circle around a common center of mass, and ii) the gravitational attraction of mass on Earth and the Moon. However, the derivation of the tidal potential does not involve centripetal acceleration, and the concept is not used by the astronomical or geodetic communities.

Figure \(\PageIndex{1}\): 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 the Moon about Earth produces a potential \(V_{M}\) at any point on Earth’s surface \[V_{M} = -\frac{\gamma M}{r_{1}} \nonumber \]

where the geometry is sketched in figure \(\PageIndex{1}\), \(\gamma\) is the gravitational constant, and \(M\) is the Moon’s mass. From the triangle \(OPA\) in the figure, \[r_{1}^{2} = r^{2} + R^{2} - 2rR \cos \varphi \nonumber \]

Using this in \((\PageIndex{1})\) gives \[V_{M} = -\frac{\gamma M}{R} \left[1 - 2 \left(\frac{r}{R}\right) \cos \varphi + \left(\frac{r}{R}\right)^{2}\right]^{-1/2} \nonumber \]

\(r/R \approx 1/60\), and \((\PageIndex{3})\) may be expanded in powers of \(r/R\) using Legendre polynomials (Whittaker and Watson, 1963: §15.1): \[V_{M} = -\frac{\gamma M}{R} \left[1 + \left(\frac{r}{R}\right) \cos \varphi + \left(\frac{r}{R}\right)^{2} \left(\frac{1}{2}\right) \left(3 \cos^{2} \varphi - 1\right) + \cdots \right] \nonumber \]

The tidal forces are calculated from the spatial gradient of the potential. The first term in \((\PageIndex{4})\) produces no force. The second term, when differentiated with respect to \((r \cos \varphi)\), produces a constant force \(\gamma M/R^{2}\) parallel to \(OA\) that keeps Earth in orbit around 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: \[V = -\frac{\gamma M r^{2}}{2R^{3}} \left(e \cos^{2} \varphi - 1\right) \nonumber \]

Figure \(\PageIndex{2}\): The horizontal component of the tidal force on Earth when the tide-generating body is above the equator at \(Z\). After Dietrich et al. (1980: 413).

The tide-generating force can be decomposed into components perpendicular, \(P\), and parallel, \(H\), to the sea surface. 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, shown in figure \(\PageIndex{2}\), is: \[H = -\frac{1}{r} \frac{\partial V}{\partial \varphi} = \frac{2G}{r} \sin 2 \varphi \nonumber \]

where \[G = \frac{3}{4} \gamma M \left(\frac{r^{2}}{R^{3}}\right) \nonumber \]

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

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 the Moon appears to move around the sky at nearly one cycle per day. The 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, the Moon’s distance from Earth, \(R\), varies because the 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 the Moon are nearly the same. Although the Sun is much more massive than the Moon, it is much further away. \[\begin{align} G_{sun} = G_{S} &= \frac{3}{4} \gamma S \left(\frac{r^{2}}{R_{sun}^{3}}\right) \\ G_{moon} = G_{M} &= \frac{3}{4} \gamma M \left(\frac{r^{2}}{R_{moon}^{3}}\right) \\ \frac{G_{S}}{G_{M}} &= 0.46051 \end{align} \nonumber \]

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 the Moon and the 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 (1987). See also figure \(4.1.1\).

A natural reference system for an observer on Earth is the equatorial system described at the start of Chapter 3. In this system, declinations \(\delta\) 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 around 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: \[\cos \varphi = \sin \varphi_{p} \sin \delta + \cos \varphi_{p} \cos \delta \cos \left(\tau_{1} - 180^{\circ}\right) \nonumber \]

where \(\varphi_{p}\) is the latitude at which the tidal potential is calculated, \(\delta\) is the declination of the Moon or Sun north of the equator, and \(\tau_{1}\) is the hour angle of the Moon or Sun. The hour angle is the longitude where the imaginary plane containing both Earth’s rotation axis and the Sun or Moon crosses the Equator.

The period of the solar hour angle is a solar day of 24 hr 0 m. The period of the lunar hour angle is a lunar day of 24 hr 50.47 m. Earth’s axis of rotation is inclined \(23.45^{\circ}\) with respect to the plane of Earth’s orbit about the sun. This defines the ecliptic, and the Sun’s declination varies between \(\delta = \pm 23.45^{\circ}\) 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 \(\delta\) and the vernal equinox to change slowly, and the movement is 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 20,942 years. Therefore \(R_{sun}\) varies with this period.

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

The shape of the 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 the Moon’s orbit rotates around Earth’s axis of rotation with a period of 18.613 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 \((\PageIndex{11})\) into \((\PageIndex{5})\) gives: \[V = \frac{\gamma Mr^{2}}{R^{3}} \frac{1}{4} \left[\left(3 \sin^{2} \varphi_{p} - 1\right) \left(3 \sin^{2} \delta - 1\right) + 3 \sin 2\varphi_{p} \ \sin 2\delta \ \cos \tau_{1} + 3 \cos^{2} \varphi_{p} \ \cos^{2} \delta \ \cos 2\tau_{1}\right] \nonumber \]

Equation \((\PageIndex{12})\) 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 of \(\sin^{2} \theta\), \(\sin 2\theta\), and \(\left(1 - 3 \cos^{2} \theta\right)/2\), where \(\theta\) is the co-latitude \(\left(90^{\circ} - \varphi\right)\).

Doodson (1922) expanded \((\PageIndex{12})\) in a Fourier series using the cleverly chosen frequencies in Table \(\PageIndex{1}\). 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 \[f = n_{1}f_{1} + n_{2}f_{2} + n_{3}f_{3} + n_{4}f_{4} + n_{5}f_{5} + n_{6}f_{6} \nonumber \]

Table \(\PageIndex{1}\). Fundamental Tidal Frequencies

	Frequency \((^{\circ}/\text{hr})\)		Period	Source
\(f_{1}\)	\(14.49205211\)	\(1\)	lunar day	Local mean lunar time
\(f_{2}\)	\(0.54901653\)	\(1\)	month	Moon’s mean longitude
\(f_{3}\)	\(0.04106864\)	\(1\)	year	Sun’s mean longitude
\(f_{4}\)	\(0.00464184\)	\(8.847\)	years	Longitude of Moon’s perigee
\(f_{5}\)	\(-0.00220641\)	\(18.613\)	years	Longitude of Moon’s ascending node
\(f_{6}\)	\(0.00000196\)	\(20,940\)	years	Longitude of Sun’s perigee

where the integers \(n_{i}\) are the Doodson numbers. \(n_{1} = 1, 2, 3\) and \(n_{2}\text{-}n_{6}\) are between \(-5\) and \(+5\). To avoid negative numbers, Doodson added five to \(n_{2 \cdots 6}\). Each tidal wave having a particular frequency given by its Doodson number is called a tidal constituent, sometimes called a partial tide. 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 \(n_{6}\) is usually ignored.

If the ocean surface is in equilibrium with the tidal potential, which means we ignore inertia and currents and assume no land (Cartwright 1999: 274), the largest tidal constituents would have amplitudes given in Table \(\PageIndex{2}\). Notice 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 separated by a cycle per year (figure \(\PageIndex{3}\)). Furthermore, each of these lines is split into lines separated by a cycle per 8.8 yr, and so on.

Table \(\PageIndex{2}\). Principal Tidal Constituents.

Tidal Species	Name	\(n_{1}\)	\(n_{2}\)	\(n_{3}\)	\(n_{4}\)	\(n_{5}\)	Equilibrium Amplitude^† \((m)\)	Period \((hr)\)
Semidiurnal	\(n_{1} = 2\)
Principal lunar	\(M_{2}\)	\(2\)	\(0\)	\(0\)	\(0\)	\(0\)	0.242334	12.4206
Principal solar	\(S_{2}\)	\(2\)	\(2\)	\(-2\)	\(0\)	\(0\)	0.112841	12.0000
Lunar elliptic	\(N_{2}\)	\(2\)	\(-1\)	\(0\)	\(0\)	\(0\)	0.046398	12.6584
Lunisolar	\(K_{2}\)	\(2\)	\(2\)	\(0\)	\(0\)	\(0\)	0.030704	11.9673
Diurnal	\(n_{1} = 1\)
Lunisolar	\(K_{1}\)	\(1\)	\(-1\)	\(0\)	\(0\)	\(0\)	0.141565	23.9344
Principal lunar	\(O_{1}\)	\(1\)	\(-1\)	\(0\)	\(0\)	\(0\)	0.100514	25.8194
Principal solar	\(P_{1}\)	\(1\)	\(1\)	\(-2\)	\(0\)	\(0\)	0.046843	24.0659
Elliptic lunar	\(Q_{1}\)	\(1\)	\(-2\)	\(0\)	\(1\)	\(0\)	0.019256	26.8684
Long Period	\(n_{1} = 0\)
Fortnightly	\(Mf\)	\(0\)	\(2\)	\(0\)	\(0\)	\(0\)	0.041742	327.85
Monthly	\(Mm\)	\(0\)	\(1\)	\(0\)	\(1\)	\(0\)	0.022026	661.31
Semiannual	\(Ssa\)	\(0\)	\(0\)	\(2\)	\(0\)	\(0\)	0.019446	4383.05
^†Amplitudes from Apel (1987)

Clearly, there are very many possible tidal constituents.

=

Figure \(\PageIndex{3}\): Upper: Spectrum of equilibrium tides with frequencies near twice per day. The spectrum is split into groups separated by a cycle per month (0.55 deg/hr). Lower: Expanded spectrum of the \(S_{2}\) group, showing splitting at a cycle per year (0.04 deg/hr). The finest splitting in this figure is at a cycle per 8.847 years (0.0046 deg/hr). From Richard Eanes, Center for Space Research, University of Texas.

Why is the tide split into the many constituents shown in figure \(\PageIndex{3}\)? To answer the question, suppose the Moon’s elliptical orbit was in the equatorial plane of Earth. Then \(\delta = 0\). From \((\PageIndex{12})\), the tidal potential on the equator, where \(\varphi_{p} = 0\), is: \[V = \frac{\gamma Mr^{2}}{R^{3}} \frac{1}{4} \cos (4 \pi f_{1}) \nonumber \]

If the ellipticity of the orbit is small, \(R = R_{0}(1+\epsilon)\), and \((\PageIndex{14})\) is approximately \[V = a(1 - 3\epsilon) \cos (4 \pi f_{1}) \nonumber \]

where \(a = (\gamma Mr^{2}) / (4 R^{3})\) is a constant. \(\epsilon\) varies with a period of 27.32 days, and we can write \(\epsilon = b \cos (2 \pi f_{2})\) ) where b is a small constant. With these simplifications, \((\PageIndex{15})\) can be written: \[\begin{array}{l} V = a \cos (4 \pi f_{1}) - 3ab \cos (2 \pi f_{2}) \cos (4 \pi f_{1}) \\ V = a \cos (4 \pi f_{2}) - 3ab \left[\cos 2 \pi (2 f_{1} - f_{2}) + \cos 2\pi (2 f_{1} + f_{2})\right] \end{array} \nonumber \]

which has a spectrum with three lines at \(2f_{1}\) and \(2f_{1} \pm f_{2}\). 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 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 tidal spectrum in figure \(\PageIndex{3}\) does not look like the ocean-wave spectrum of ocean waves in figure \(16.3.4\). 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.

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 \(\PageIndex{2}\). 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 \(M_{2}\) tide, called the M-two tide.