Darwin Engine

Darwin Engine#

The Darwin engine implements the classical FES/Darwin harmonic prediction approach. It uses Schureman/Darwin nodal corrections and admittance calculations following established oceanographic conventions.

Overview#

The FES/Darwin engine is the historical prediction engine for FES (Finite Element Solution) tidal atlases. It employs Darwin’s notation system for tidal constituents, where each constituent is expressed through combinations of fundamental astronomical variables.

This engine is specifically designed for use with FES2014 and FES2022 tidal atlases and follows the traditional oceanographic conventions established by Darwin and Schureman.

Darwin Harmonic Notation#

The Darwin notation system represents tidal constituents through their astronomical arguments expressed as combinations of five fundamental astronomical variables:

Fundamental Astronomical Arguments:

  • s: Mean longitude of the moon

  • h: Mean longitude of the sun

  • p: Longitude of the moon’s perigee

  • N: Negative longitude of the moon’s ascending node

  • p₁: Longitude of the sun’s perihelion (solar perigee)

Each constituent is defined by its angular speed (degrees per hour) and its astronomical argument expressed as a linear combination of these variables.

Example - M₂ Constituent:

The principal lunar semidiurnal tide M₂ has an astronomical argument:

\[V(M_2) = 2(\tau + h - s + p)\]

where \(\tau\) is the Greenwich hour angle, representing Earth’s rotation.

Nodal Corrections#

The Darwin engine applies Schureman’s nodal modulation to account for the 18.6-year lunar nodal cycle. This cycle arises from the precession of the moon’s orbital plane.

For each constituent, two correction factors are applied:

  • f (nodal factor): Amplitude modulation factor

  • u (nodal angle): Phase correction angle

These corrections vary slowly over the 18.6-year cycle and are computed using Schureman’s formulations. The predicted height for a constituent is:

\[h(t) = f \cdot H \cos(V(t) + \kappa - u)\]

where:

  • \(H\) is the constituent amplitude from the tidal atlas

  • \(\kappa\) is the phase lag from the tidal atlas

  • \(V(t)\) is the astronomical argument at time t

Admittance and Inference#

The Darwin engine handles minor constituents not explicitly provided in the tidal atlas through traditional admittance relationships. The admittance is the complex ratio between a minor and major constituent:

\[Z = \frac{H_{\text{minor}}}{H_{\text{major}}} e^{i(\kappa_{\text{minor}} - \kappa_{\text{major}})}\]

This allows PyFES to estimate amplitudes and phases for many additional constituents beyond those in the atlas, improving prediction accuracy.

Supported Constituents#

The Darwin engine supports 142 tidal constituents, including:

  • Long-period tides: Sa, Ssa, Mm, Mf, etc.

  • Diurnal tides: K₁, O₁, P₁, Q₁, etc.

  • Semidiurnal tides: M₂, S₂, N₂, K₂, etc.

  • Terdiurnal tides: M₃, MK₃, etc.

  • Compound tides: M₄, M₆, M₈, etc.

For the complete list with speeds and astronomical arguments, see FES Constituents.

Compatible Tidal Models#

The Darwin engine is designed for use with FES tidal atlases:

  • FES2014: 34 constituents provided in the atlas

  • FES2022: 34 constituents provided in the atlas

These models provide amplitude and phase for major constituents. PyFES uses admittance to infer the remaining constituents for complete tidal predictions.

Configuration#

To use the Darwin engine, set engine: fes in your YAML configuration file:

tide:
  lgp:
    engine: fes
    path: ${FES_DATA}/fes2022b/ocean_tide.nc
    codes: codes
    constituents: [2N2, K1, K2, M2, N2, O1, P1, Q1, S1, S2]

radial:
  cartesian:
    engine: fes
    paths:
      M2: ${FES_DATA}/fes2022b_load/m2.nc
      S2: ${FES_DATA}/fes2022b_load/s2.nc

Runtime settings for the Darwin engine:

import pyfes

settings = (
    pyfes.FesRuntimeSettings()
    .with_astronomic_formulae(pyfes.Formulae.SCHUREMAN_ORDER_1)
    .with_time_tolerance(3600.0)
    .with_num_threads(0)
)

Astronomic Formulae Options#

The Darwin engine supports multiple formulations for computing astronomical angles:

  • SCHUREMAN_ORDER_1 (default): Schureman’s first-order formulations

  • SCHUREMAN_ORDER_3: Schureman’s third-order formulations (higher accuracy)

  • IERS: IERS conventions (for consistency with modern ephemerides)

The default SCHUREMAN_ORDER_1 provides a good balance of accuracy and compatibility with historical FES predictions.

References#

The Darwin engine implements formulations documented in:

  • Schureman, P. (1940). Manual of Harmonic Analysis and Prediction of Tides. U.S. Coast and Geodetic Survey, Special Publication No. 98.

  • Darwin, G. H. (1907). The Harmonic Analysis of Tidal Observations. Scientific Papers, Vol. 1.

For detailed constituent formulations following Schureman’s reference, see Constituents Reference.

See Also#

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