The Tide-Generating Potential

The Tide-Generating Potential#

The physical origin of ocean tides lies in the differential gravitational attraction exerted by the Moon and Sun on the Earth. At the Earth’s centre, the gravitational pull of a celestial body is exactly balanced by the orbital acceleration of the Earth–Moon or Earth–Sun system. At any other point on the Earth’s surface, a residual force remains: the tide-generating force. This section derives the mathematical form of the corresponding potential, which underpins the entire harmonic prediction method implemented in PyFES.

Tide-Generating Force#

Consider a celestial body \(A\) of mass \(M_A\) at distance \(R\) from the Earth’s centre \(T\). At a point \(P\) on the Earth’s surface, at geocentric distance \(a\) (the Earth’s mean radius) and geocentric zenith distance \(z\) from the sub-body point, the tide-generating force per unit mass is the difference between the gravitational attraction at \(P\) and the uniform attraction at \(T\).

The distance from \(A\) to \(P\) is:

\[r = \sqrt{R^2 + a^2 - 2 a R \cos z}\]

The tide-generating force is directed along the line \(AP\) (the tractive component) and can be decomposed into a radial (vertical) and two horizontal components. For tidal prediction, it is more convenient to work with the potential of this force rather than the force itself.

Potential Expansion in Legendre Polynomials#

The gravitational potential of body \(A\) at point \(P\), minus the uniform potential at the Earth’s centre, can be expanded in a series of Legendre polynomials:

\[W = \frac{G M_A}{R} \sum_{n=2}^{\infty} \left(\frac{a}{R}\right)^n P_n(\cos z)\]

where:

  • \(G\) is the gravitational constant,

  • \(a/R\) is the ratio of the Earth’s radius to the distance of the celestial body (the lunar parallax \(a/R \approx 1/60\) for the Moon),

  • \(P_n(\cos z)\) is the Legendre polynomial of degree \(n\),

  • the \(n=1\) term vanishes because it represents a uniform translation of the entire Earth.

Since \((a/R)^n\) decreases rapidly with \(n\), only the first few terms are significant. For the Moon, the \(n=2\) term dominates, and the \(n=3\) term contributes at the level of a few percent. For the Sun, the much larger distance makes only the \(n=2\) term non-negligible. In Doodson’s original development, terms with amplitudes below \(10^{-5}\) of the dominant \(M_2\) constituent were discarded.

The Basic Tidal Potential#

Retaining only the dominant \(n=2\) term, the tidal potential at colatitude \(\varphi\) and geocentric zenith distance \(z\) is:

\[W_2 = \frac{3}{2} \frac{G M_A}{R} \left(\frac{a}{R}\right)^2 \left(\cos^2 z - \frac{1}{3}\right)\]

This can be rewritten using the basic factor \(U\):

\[U = \frac{3}{2} \frac{M_A}{M_T} \left(\frac{a}{R_0}\right)^3 g a\]

where \(M_T\) is the Earth’s mass, \(R_0\) is the mean distance of the celestial body, and \(g\) is the gravitational acceleration at the Earth’s surface. The ratio \(U_{\text{Moon}} / U_{\text{Sun}} \approx 2.2\) quantifies the Moon’s dominant role in tide generation.

Decomposition into Laplace Species#

The key step linking the potential to observable tidal frequencies is the decomposition of \(\cos^2 z\) into functions of the observer’s latitude \(\varphi\) and the celestial body’s coordinates. Using the spherical harmonic addition theorem, the second-degree potential separates into three terms, known as the Laplace species:

\[W_2 = W_{2,0} + W_{2,1} + W_{2,2}\]

Each species corresponds to a different dependence on the hour angle \(\tau\) (the angular position of the body relative to the observer’s meridian):

Long-period species (\(m = 0\)):

\[W_{2,0} \propto \left(\frac{1}{2} - \frac{3}{2} \sin^2 \varphi\right) \left(\frac{2}{3} - \cos^2 \delta \right)\]

This term depends only on the body’s declination \(\delta\), not on the hour angle. It produces oscillations with periods from a fortnight to years (e.g., \(M_f\), \(M_m\), \(S_{sa}\)). The latitude factor vanishes at \(\varphi = 35.26°\) and is maximal at the poles.

Diurnal species (\(m = 1\)):

\[W_{2,1} \propto \sin 2\varphi \cdot \sin 2\delta \cdot \cos \tau\]

This term oscillates once per day (\(\cos \tau\)) and produces the diurnal constituents (\(O_1\), \(K_1\), \(P_1\), \(Q_1\)). The latitude factor \(\sin 2\varphi\) is maximal at \(\varphi = 45°\) and vanishes at the equator and poles.

Semidiurnal species (\(m = 2\)):

\[W_{2,2} \propto \cos^2 \varphi \cdot \cos^2 \delta \cdot \cos 2\tau\]

This term oscillates twice per day (\(\cos 2\tau\)) and produces the dominant semidiurnal constituents (\(M_2\), \(S_2\), \(N_2\), \(K_2\)). The latitude factor \(\cos^2 \varphi\) is maximal at the equator and vanishes at the poles.

Connection to PyFES#

The tidal constituents stored in pyfes.darwin.WaveTable and pyfes.perth.WaveTable are the individual spectral lines obtained by further expanding each species into its harmonic components (see Harmonic Development and the Doodson Classification). Each constituent’s frequency is determined by the rates of change of the astronomical angles appearing in the expansion, and its theoretical amplitude derives from the coefficients of the potential development.

The WaveInterface property type distinguishes between SHORT_PERIOD (diurnal and semidiurnal species) and LONG_PERIOD (long-period species), directly reflecting the Laplace species classification.

References#

  • Simon, B. (2013). Marées Océaniques et Côtières (943-MOC), Ch. III.

  • Schureman, P. (1940). Manual of Harmonic Analysis and Prediction of Tides, SP 98, pp. 10–28.

  • Doodson, A. T. (1921). The Harmonic Development of the Tide-Generating Potential. Proc. Roy. Soc. London A, 100(704), 305–329.

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