Nodal Corrections: Amplitude and Phase Modulations#
The standard harmonic prediction equation treats each tidal constituent as a pure sinusoid of fixed amplitude and phase. In reality, the amplitude and phase of each constituent are slowly modulated by the 18.61-year precession of the lunar node. Rather than include all the closely spaced “satellite” frequencies that produce this modulation, the classical approach absorbs it into time-dependent correction factors \(f\) (amplitude) and \(u\) (phase), collectively known as nodal corrections.
This section documents all the nodal correction formulae implemented in PyFES, following Schureman’s notation.
The Prediction Equation with Nodal Corrections#
The full harmonic prediction equation at universal time \(t\) is:
where:
\(H_0\) is the mean water level above the chart datum,
\(H_k\) is the amplitude of constituent \(k\) from the tidal atlas,
\(G_k\) is the phase lag of constituent \(k\),
\(V_k(t)\) is the equilibrium astronomical argument at Greenwich at time \(t\); it advances at the angular speed \(\omega_k\) of the constituent, \(V_k(t) = V_k(t_0) + \omega_k (t - t_0)\) for a reference epoch \(t_0\),
\(f_k\) is the nodal amplitude factor (dimensionless, close to 1),
\(u_k\) is the nodal phase correction (in degrees or radians).
The factors \(f_k\) and \(u_k\) vary slowly over the 18.61-year nodal cycle and are recomputed at each time step (or, in some older implementations, once per year).
Physical Origin#
The dominant source of nodal modulation is the regression of the Moon’s ascending node \(N\) with a period of 18.61 years. As the node regresses, the inclination \(I\) of the lunar orbit to the celestial equator varies between approximately 18.3° and 28.6°, modulating the tidal potential coefficients that depend on \(I\).
In the harmonic development, each major constituent is accompanied by nearby “satellite” frequencies separated by the nodal rate (\(\dot{N}' \approx 0.0022°/\text{h}\)). Instead of resolving these satellites individually (which would require very long observational records), the classical method treats each group as a single constituent whose amplitude and phase vary slowly with \(N\).
Nodal Amplitude Factor \(f\)#
The factor \(f_k\) is the ratio of the obliquity factor \(g(I)\) (which depends on \(I\) and hence on \(N\)) to a mean value \(k\) of this factor. Schureman’s constants \(k_{65}\) to \(k_{70}\) (see Mean Obliquity Constants) are not the mean of \(g(I)\) itself, but the mean over the nodal cycle of the complex coefficient \(g(I) \, e^{iu}\), i.e. the amplitude of the central line of the constituent. It is \(f_k e^{iu_k}\), not \(f_k\) alone, that averages to 1 over a nodal cycle. The coefficients \(k_{227}\) and \(k_{235}\) of the \(K_1\) and \(K_2\) formulas below are of a different kind: they are coefficients under a square root.
Long-Period Constituents#
Diurnal Constituents#
where \(k_{227,1} = 0.8965\), \(k_{227,2} = 0.6001\), \(k_{227,3} = 0.1006\). The \(K_1\) formula is more complex because this constituent combines lunar and solar terms (see Astronomical Angle Computation).
Semidiurnal Constituents#
where \(k_{235,1} = 19.0444\), \(k_{235,2} = 2.7702\), \(k_{235,3} = 0.0981\).
where \(1/R_a\) is the x1ra factor documented in
Astronomical Angle Computation. Unlike the other factors, it also depends
on the longitude of the lunar perigee, through \(p - \xi\)
(8.85-year period); so does the phase factor \(R\) of \(u(L_2)\).
Compound Constituents#
For compound waves formed by the interaction of parent constituents, the nodal factor is the product of the parents’ factors:
Solar constituents (\(S_2\), \(S_1\), etc.) have \(f = 1\) since the Sun has no nodal modulation.
Nodal Phase Correction \(u\)#
The phase correction \(u_k\) adjusts the phase of each constituent to account for the slowly varying position of the lunar node. The corrections follow systematic patterns depending on the constituent class.
Long-Period Constituents#
Diurnal Constituents#
where \(\xi\) and \(\nu\) are the longitude and right ascension of the lunar intersection, and \(\nu'\) is the combined lunar-solar correction angle (see Astronomical Angle Computation).
Semidiurnal Constituents#
where \(R\) is the phase factor for \(L_2\) and \(\nu''\) is the combined correction for \(K_2\).
Compound Constituents#
For compound waves, the phase correction is the algebraic sum of the parents’ corrections. For example:
Mean Obliquity Constants#
The denominators \(k\) in the \(f\)-factor formulae are the means, over the 18.61-year nodal cycle, of the complex coefficients \(g(I) \, e^{iu}\):
They differ from the mean of \(g(I)\) alone: for \(M_f\), the mean
of \(\sin^2 I\) is 0.1645, while \(k_{66} = 0.1578\). The values
below are those of the default build; when PyFES is built with
FES_USE_IERS_CONSTANTS, they are recomputed from the IERS 2010
constants:
Constant |
\(g(I)\), averaged with \(e^{iu}\) |
Schureman |
PyFES |
|---|---|---|---|
\(k_{65}\) |
\(\frac{2}{3} - \sin^2 I\) |
0.5021 |
0.5021 |
\(k_{66}\) |
\(\sin^2 I\) |
0.1578 |
0.1578 |
\(k_{67}\) |
\(\sin I \cos^2(I/2)\) |
0.3800 |
0.3800 |
\(k_{68}\) |
\(\sin 2I\) |
0.7214 |
0.7214 |
\(k_{69}\) |
\(\sin I \sin^2(I/2)\) |
0.0164 |
0.0164 |
\(k_{70}\) |
\(\cos^4(I/2)\) |
0.9154 |
0.9154 |
PERTH Group Modulations#
The PERTH/Doodson engine offers an alternative approach to nodal corrections. Instead of applying individual \(f\) and \(u\) factors to each constituent, related constituents within the same tidal group (sharing the same \(n_1\) and \(n_2\) coefficients) are modulated together. The group modulation sums over the sideline frequencies within each group, providing a more physically consistent treatment of the nodal cycle.
Group modulations are enabled via:
settings = pyfes.PerthSettings().with_group_modulations(True)
When group modulations are disabled, the PERTH engine applies R. Ray’s individual nodal corrections, not the Schureman expressions documented above. Each one is written as a sum of satellite lines:
where the satellite ratios \(\alpha_j\) come from the Cartwright–Tayler–Edden development of the potential and the phases \(\phi_j\) are combinations of \(N\) and \(p\). For example, \(f e^{iu}(O_1) = 1 + 0.1886 e^{iN} - 0.0058 e^{2iN} - 0.0065 e^{2ip}\). For the main constituents, the two engines agree closely: over 1990–2030, \(O_1\) differs by less than 0.9 % in \(f\) and 0.5° in \(u\), and \(M_2\) by less than 0.01 %.
References#
Schureman, P. (1940). Manual of Harmonic Analysis and Prediction of Tides, SP 98, pp. 24–26 (formulas 65–80), pp. 44–46 (formulas 213–235).
Simon, B. (2013). Marées Océaniques et Côtières (943-MOC), Ch. VI, pp. 159–163.