Earth Tide-Generating Potential (TGP)

• The Earth Tide-Generating Potential (TGP) is defined as the gravitational influence by external celestial bodies, such as the Moon and Sun, which modulate Earth’s tidal forces and deformation.
• Spectral decomposition of TGP reveals long-periodic constituents and advanced harmonic modeling that enhance predictions of Earth rotation, crustal deformation, and sea level variations.
• Recent advances in mathematical formulations and calibration with gravimeter data refine geodetic, seismic, and oceanographic applications through improved modeling of Love numbers and tidal dissipation.

The Earth Tide-Generating Potential (TGP) describes the gravitational potential produced on and within the Earth by external celestial bodies, primarily the Moon and the Sun, and to lesser degrees, other planets. The TGP modulates a wide variety of terrestrial phenomena, including the deformation of Earth's crust, variation in local gravity, oceanic tidal flows, and long-period changes detectable in geodetic and geophysical measurements. Its mathematical formalism forms the foundation for precise tidal modeling, reference system definition, and the analysis of tidal dissipation and orbital evolution over both short and geologic time scales.

1. Mathematical Formulation of the Tide-Generating Potential

The TGP is quantified by summing the gravitational potentials from astronomical sources, commonly focusing on the dominant $l = 2$ terms from the Moon and Sun. The general form for the potential at a site $M$ is:

$$V(t) = \sum_{n=1}^{\infty} \left(\frac{r}{R_E}\right)^n \sum_{m=0}^n \bar{P}_{nm}(\sin \phi) [C_{nm}(t)\cos m(\lambda+\theta(t)) + S_{nm}(t)\sin m(\lambda+\theta(t))]$$

where $R_E$ is Earth's equatorial radius, $\bar{P}_{nm}$ are normalized associated Legendre polynomials, and $C_{nm}(t)$, $S_{nm}(t)$ are time-dependent coefficients encapsulating the ephemerides of the perturbing bodies and Earth orientation (Kudryavtsev et al., 25 Aug 2025). In practical tidal modeling, the potential is often reduced for specific bands, e.g., long-periodic terms (with $m=0$), or semi-diurnal and diurnal constituents where the time dependence arises from planetary positions, lunar node and perigee cycles, and Earth’s precession.

For purposes of physical modeling, the instantaneous forcing potential from a companion mass $m_2$ at distance $d$ is given:

$$V = \frac{1}{4} g R \left(\frac{m_2}{m_1}\right) \left(\frac{R}{d}\right)^3 (\cos^2 \delta) P_2^2(\cos\theta) \cos(2\phi - 2\omega t)$$

where $g$ is gravity, $R$ Earth's radius, $\delta$ declination, $P_2^2$ an associated Legendre polynomial, $\phi$ longitude, $\theta$ colatitude, and $\omega$ the tidal frequency (Wei, 2020).

2. Spectral Decomposition and Long-Period Harmonic Development

Recent advances in the harmonic development of the Earth TGP have led to comprehensive catalogues of long-periodic terms, extending well beyond the classical short-term constituents. By employing modified spectral analysis on numerically integrated TGP values over >30,000 years (utilizing NASA JPL DE441 ephemerides), researchers have identified 38 distinct terms with periods exceeding 18 years and amplitudes above $10^{-8}~{\rm m}^2\,{\rm s}^{-2}$ (Kudryavtsev et al., 25 Aug 2025).

Key features include:

• Identification of new long-periodic waves, e.g., a $7.4$ kyr term with amplitude $3\times 10^{-5}~{\rm m}^2\,{\rm s}^{-2}$.
• Resolution of closely spaced terms near the lunar nodal cycle ($\sim18.61$ yr) into separate components (18.55, 18.58, 18.65, 18.68 yr).
• Inclusion of general precession (period $\sim25.7$ kyr) in harmonic arguments, affecting term phase and amplitude for millennial-scale modeling.

These findings expand the utility of TGP spectra in understanding long-term tidal modulation of Earth deformation, rotation, and its link to external astronomical cycles.

Period (yr)	Amplitude (${\rm m}^2\,{\rm s}^{-2}$)	Principal Argument Description
18.61	$>10^{-7}$	Lunar nodal cycle; $N'$ and interactions
7,400	$3\times10^{-5}$	Solar perigee harmonics + general precession $2P_S+p_A$
25,700	$10^{-8}$	General precession $p_A$

3. Geometric and Physical Interpretation of Tidal Displacement

Traditional approaches model Earth’s response to TGP via expansions in empirical Love numbers coupled with the potential equations. A recent geometric model instead treats the solid Earth as a prolate ellipsoid whose major axis is physically aligned to the instantaneous Earth–Moon (or Earth–Sun) line. The ellipsoid’s geometry (major/minor axes, flattening) and instantaneous displacement $H(t)$ are directly expressed in terms of lunar and solar angles, with deformation parameters modulated by real-time astronomical positions (Yang et al., 11 Jul 2024):

$H(t) = H_m(t) + H_s(t)$

$$H_m(t) = V[(R + M_e)^2 \cos^2\alpha + (R - M_s)^2 \sin^2\alpha] - R$$

Here $R$ is Earth's mean radius, $M_e$, $M_s$ are elongation/shortening terms (modulated by distance and latitude factors), and $\alpha$ is the lunar angle. This model, validated against 23 years of superconducting gravimeter data, achieves RMS gravity residuals of $6.47\,\mu$Gal compared to $30.77\,\mu$Gal for IERS 2010-based models, providing higher precision in tidal displacement and its conversion to gravity change.

4. Rheological and Dynamical Models of Tidal Lag

MacDonald’s (1964) tidal theory introduced a constant angular lag $\delta$ in the true anomaly, yielding a tidal potential $W = k_2 W_2(f-\delta)$ where the lag is geometric and not related to frequency. Fourier analysis exposes that each tidal constituent acquires distinct phase lags, lacking a simple rheology (e.g., “lag ∝ frequency”). Singer’s (1968) patch adjusts for non-uniform anomaly rates but does not unify lag–frequency dependence. Only with Williams and Efroimsky’s (2012) modification—delay imposed in mean anomaly—does the model recover Darwin’s rheology, where lag for each constituent obeys

$\delta_i = n_i T$

with $n_i$ the frequency and $T$ a time lag (Ferraz-Mello, 2013). This proportionality is fundamental for modern tidal dissipation theory and impacts orbital evolution models.

5. Oceanic Tides, Stratification, and Resonant Enhancement

Three-dimensional models of planetary oceanic tides explicitly include compressibility, vertical stratification (Brunt–Väisälä frequency $N$), and spherical geometry. Analytic solutions display strong dependence of the tidal response on forcing frequency $\sigma$, ocean depth $H$, and stratification $N$ (Auclair-Desrotour et al., 2018). Surface gravity waves dominate in shallow oceans, while deep and stratified oceans exhibit pronounced internal gravity wave resonances. Dissipation peaks (quantified by tidal quality factor $Q$ and the imaginary part of Love numbers) occur when $\sigma$ matches eigenfrequencies, with amplitude scaling sharply as $1/\sigma_R$ (Rayleigh drag). Implications include accelerated planetary rotational evolution and variable spin-orbit configurations as a function of ocean structure.

Ocean Regime	Resonance Dominance	Dissipation Scaling
Shallow (small $H$)	Surface gravity waves $\sqrt{gH}$	Surface/barotropic
Deep/Stratified	Internal gravity waves $N^2 H/g$	Internal/baroclinic

6. Secular Evolution and Dynamical Tide Coupling

Dynamical tides represent the resonant response of various fluid modes (surface gravity, inertial waves) to orbital forcing. Shallow ocean models reveal that tidal dissipation can increase with orbital separation $a$ due to dynamical tide–orbital coupling, in contrast to classical $a^{-6}$ scaling. Resonant phases, where orbital forcing aligns with fluid eigenfrequencies, enhance dissipation $D_0$ and tidal torque $\Gamma$ (Wei, 3 Feb 2025), thereby accelerating both Earth–Moon recession and terrestrial spin-down over secular time scales.

Basic equations of exchange:

$\dot{L}_o = \Gamma, \quad \dot{L}_1 = -\Gamma$

$D = -\Gamma(\omega_o - \omega_1)$

Secular evolution must account for non-monotonic dissipation due to resonance passages, fundamentally impacting models of tidal history and planetary system architecture.

7. Applications in Geodesy, Seismology, and Geophysics

TGP underpins numerous application domains:

• Geodesy: The permanent tide (mean-tide system) is crucial for establishing the International Height Reference Frame (IHRF), with the reference potential $W_0 = 62\,636\,853.4$ m²/s² fixed by convention (Mäkinen, 2020). Correction formulas bridge between tide-free, zero-tide, and mean-tide systems for coordinates and global gravity models.
• Seismic Forecasting: Diurnal averages and Sun–Moon “extremes” of the TGP serve as triggers for time windows of heightened seismicity. Combined with geomagnetic precursor signals, the method achieves reliability up to 91% in regional earthquake forecasting (window $\pm 2.7$ days), as verified across multiple INTERMAGNET stations (Mavrodiev et al., 2016).
• Precision Accelerometry: Modern gravimeters—MEMS-based, diamagnetic-levitated, and superconducting—use tidal signals as calibration standards. Devices achieve a correlation up to $0.979$ with theoretical models, bias stabilities $\sim8.18\,\mu$Gal at 400 s, and drift as low as $61\,\mu$Gal/day (Prasad et al., 2021, Leng et al., 23 Mar 2024), enabling cost-effective long-term gravimetry and geophysical monitoring.
• Geodynamo Studies: The potential of lunar tides to sustain the early geodynamo was tested; however, scaling laws ($B \propto \beta^{4/3}$ for weak interaction, $B \propto \beta$ for strong forcing) suggest tidal forcing alone could not explain ancient field amplitudes, implying dominant roles for thermal or compositional convection (Vidal et al., 23 Jun 2025).

8. Implications, Catalogue Development, and Future Directions

Comprehensive catalogues of long-periodic TGP terms (KC25) now include up to 38 components longer than 18 years, with explicit evaluation of the general precession and improved resolution of lunar nodal-related terms (Kudryavtsev et al., 25 Aug 2025). Accurate TGP modeling is now critical for:

• Refinement of geodetic datums and height reference frames.
• Improved oceanic and solid earth tide models, directly influencing SSH determination, vertical land motion, and climate studies.
• Deeper understanding of tidal modulation in Earth's deformation, rotation, seismicity, and historical geodynamics.
• The development of chip-scale gravimeters and mobile gravity sensing platforms capable of surveying underground density and forecasting hazards.

In conclusion, the Earth Tide-Generating Potential is a foundational construct in geodesy, geophysics, oceanography, astronomy, and planetary science. Recent advances in mathematical representation, physical interpretation, instrumentation, and catalogue development have significantly sharpened its precision and expanded its relevance across observational and theoretical domains.