2. Solid Earth Tides

The total gravitational potential at a position on the Earth’s surface due to a celestial object is directly related to the distance between the Earth and the object, and the mass of that object [1, 77]. Similar to ocean tides, solid Earth tides (or body tides) are tidal deformations due to changes in the gravitational potential [1, 17, 45, 48]. However, while ocean tides are apparent to observers on the coast, solid Earth tides are typically more difficult to observe due to the reference frame of the observer moving. The Earth’s tidal deformation is largely considered to be elastic and to a very high degree instantaneous.

2.1. Love and Shida Numbers

The elastic response of the Earth to the tidal potential is calculated using Love/Shida numbers [Equation 2.1]. Love and Shida numbers describe the elastic response of the Earth in terms of vertical displacement (\(h\)), gravitational potential (\(k\)) and horizontal displacement (\(l\)) [36, 50].

(2.1)\[\begin{split}S_r &= \frac{1}{g}\sum_{l=2}^{\infty} h_l V_l \\ S_\varphi &= \frac{1}{g}\sum_{l=2}^{\infty} l_l \frac{\partial V_l}{\partial\varphi} \\ S_\lambda &= \frac{1}{g}\sum_{l=2}^{\infty} l_l \frac{\partial V_l}{\sin\varphi\partial\lambda}\end{split}\]

Combinations of Love/Shida numbers can be used to calculate additional quantities, such as the displacement of the Earth’s ocean surface with respect to the Earth’s tidally deformed crust [4, 50].

(2.2)\[\gamma_2 = 1 + k_2 - h_2\]

For a spherical, non-rotating Earth, the Love and Shida numbers are largely independent of tidal frequency as the tidal periods are longer than the Earth’s free oscillation periods [4, 76, 77]. However, for a rotating, ellipsoidal Earth, the Love and Shida numbers have some dependence on tidal frequency, with resonances particularly in the diurnal band [36, 60, 76, 77]. pyTMD computes these frequency-dependent corrections along with the dissipative mantle anelasticity corrections following Mathews et al. [43] and Wahr [77].

../_images/love-numbers.png

Figure 2.1: Diurnal frequency dependence of Love/Shida numbers from Wahr [76]

2.2. Methods

Within pyTMD, the tidal deformation of the Earth can be modeled using two methods:

  1. pyTMD.predict.solid_earth_tide() uses ephemerides and the formalism described in the IERS Conventions, which are based on Wahr [77] and Mathews et al. [43]

  2. pyTMD.predict.body_tide() uses tide potential catalogs [81] and the spherical harmonic formalism described in Cartwright and Tayler [12].

For the ephemerides method, analytical approximate positions for the sun and moon can be calculated, or high-resolution numerical ephemerides for the sun and moon can be downloaded from the Jet Propulsion Laboratory. These astronomical positions are used to estimate the instantaneous tide potential impacting the solid Earth [47].

Most tide potential catalogs have for each tidal constituent, the spherical harmonic degree and order, the Doodson arguments and the potential amplitude [12]. Some catalogs additionally include the potentials induced by the motions of the closest planetary bodies [see Tide Potential Catalogs] and higher degree harmonics [see Spherical Harmonics].

2.2.1. Ephemerides Method

From Mathews et al. [43], the steps for using the ephemerides method to calculate the solid Earth tide displacements at the location and times of interest are:

  1. Compute positions of the sun and moon in ECEF coordinates (\(X\), \(Y\) and \(Z\))

  2. Calculate frequency-independent deformations using “nominal” Love/Shida number values

  3. Include contributions from the Love/Shida number out-of-phase (imaginary) components

  4. Include contributions from the Love/Shida number latitudinal dependency for degree 2

  5. Include contributions from the Love/Shida numbers frequency dependence

  6. Convert the permanent tide system (if necessary)

2.2.2. Catalog Method

Similarly, from Cartwright and Tayler [12], the steps for using the catalog method are:

  1. Read the tide potential catalog [see Tide Potential Catalogs]

  2. Calculate the Arguments and potentially the planetary arguments

  3. For each constituent potential in the catalog:

    1. Calculate the angular frequency (\(\Omega\)) of the constituent

    2. Calculate the frequency and latitude dependent Love/Shida numbers

    3. Calculate the spherical harmonics for the degree and order of the constituent

    4. Calculate the equilibrium phase of the constituent [see Equilibrium Theory]

    5. Rotate the spherical harmonics by the equilibrium phase [see Spherical Harmonics]

    6. Scale the rotated spherical harmonics by the constituent amplitude and the appropriate Love/Shida numbers [see Equation 2.1] to calculate the induced elastic deformation

    7. Add contributions to the total tidal displacement

2.3. Permanent Tide

In addition to the ups and downs of tides, there is a considerable portion of tidal potential and displacement that does not vary in time, a “permanent tide” that is due to the Earth being in the presence of the Sun and Moon (and other planetary bodies). The Earth is lower in polar areas and higher in equatorial areas than it would without those gravitational effects. The IERS formalism for determining station locations is to remove all cyclical and permanent components of the tides, which is known as a “tide-free” system. This is the default “tide-system” within pyTMD. Alternatively, the permanent tide components can be added back in order to calculate the station locations in a “mean tide” state. The radial difference in terms of latitude between the “mean tide” and “tide-free” systems is:

(2.3)\[\delta r(\varphi) = -0.120582 \left(\frac{3}{2} \sin^2 \varphi - \frac{1}{2} \right)\]