Elastostatic Adjustment Solution

Physical basis

Any redistribution of mass at the Earth’s surface, such as snow, water, or atmosphere, loads and deforms the underlying solid Earth. At timescales that are comparable to those of the main tidal constituents, such as the near-annual periods, solid Earth deformation is excellently approximated as an elastic response. This module employs the classical Green’s function approach to solving for interior Earth responses at the surface, following the so-called load Love number formalism for a radially stratified, seismologically constrained, elastically compressible Earth.

3-D crustal motions

Let $Equation 8$ (for $Equation 7$ ) be the components of the 3-D crustal displacement vector, $Equation 6$ , evaluated at geographic coordinates $Equation 5$ at time $Equation 4$ , where $Equation 3$ is the vertical displacement (up positive), $Equation 2$ is the north-south component of horizontal displacement (north positive), and $Equation 1$ is the east-west component of horizontal displacement (east positive).

For a given surface load, $Equation 9$ , with dimensions of ice equivalent height, these displacement components may be computed theoretically as follows:

$Equation 10$

where $Equation 15$ is the 3-D Green’s function vector that models the influence of a specified point load evaluated at an arc distance $Equation 14$ and direction $Equation 13$ , from load coordinate position ( $Equation 12$ ). The integral in the above equation is applied over the surface of a unit sphere $Equation 11$ .

The components of $Equation 16$ are given by:

$Equation 17$

where:

$Equation 18$ is the ice density
$Equation 19$ is the Earth’s global mean density
$Equation 21$ are the Legendre polynomials of degree $Equation 20$
$Equation 23$ and $Equation 22$ are the load Love numbers

Numerical implementation

We use Love numbers — provided by the International Association of Geodesy (available at http://www.srosat.com/iag-jsg/loveNb.php) — which are the solutions of the zero frequency momentum equations with self-gravitation for a spherically symmetric and seismologically constrained Earth structure model [see, e.g., Alterman et al., 1959]. Since $Equation 26$ converges slowly toward a constant as $Equation 25$ , the requirement for generating an accurate solution for crustal deformation is stringent, demanding truncation of the series at high degree $Equation 24$ . See [Adhikari2017] for more details.

Model parameters

The parameters relevant to the elastostatic adjustment (ESA) solution can be displayed by running:

>> md.esa

md.esa.deltathickness: thickness change: ice height equivalent [m]
md.solidearth.lovenumbers: loads required Love numbers for solid Earth deformation
md.esa.hemisphere: North-south, East-west components of 2-D horiz displacement vector: -1 south, 1 north
md.esa.degacc: accuracy (default .01 deg) for numerical discretization of the Green’s functions

Running a simulation

To run a simulation, use the following command:

>> md = solve(md, 'Esa');

The first argument is the model, the second is the nature of the simulation one wants to run.