Solid Earth elastic response
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 
(for 
) be the components of the 3-D crustal displacement vector, )
, evaluated at geographic coordinates )
at time 
, where 
is the vertical displacement (up positive), 
is the north-south component of horizontal displacement (north positive), and 
is the east-west component of horizontal displacement (east positive).
For a given surface load, )
, with dimensions of ice equivalent height, these displacement components may be computed theoretically as follows:
where )
is the 3-D Green's function vector that models the influence of a specified point load evaluated at an arc distance 
and direction 
, from load coordinate position (
). The integral in the above equation is applied over the surface of a unit sphere 
.
The components of 
are given by:
where:
is the ice density
is the Earth's global mean density
are the Legendre polynomials of degree 
and 
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 converges slowly toward a constant as 
, the requirement for generating an accurate solution for crustal deformation is stringent, demanding truncation of the series at high degree 
. 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 deformationmd.esa.hemisphere: North-south, East-west components of 2-D horiz displacement vector: -1 south, 1 northmd.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.
References
-
S. Adhikari, E. R. Ivins, and E. Larour.
Mass transport waves amplified by intense Greenland melt and
detected in solid Earth deformation.
Geophys. Res. Lett., 44(10):4965-4975.
