Glacial Isostatic Adjustment (GIA)

Physical basis

The ISSM/GIA model assumes that the ice sheet rests on top of the solid Earth, which is considered to be a simple two-layered incompressible continuum with upper elastic lithosphere floating on the viscoelastic (Maxwell material) mantle half-space. Coordinate transformations allow simple axisymmetric solutions for the deformation of pre-stressed solid Earth (subject to a normal surface traction of ice/ocean) to retrieve semi-analytical solutions of vertical displacement at the lithosphere surface.

Vertical surface displacement

Vertical displacement at the lithosphere surface (i.e., ice/ocean-bedrock interface), LaTeX equation, is the most relevant field variable for GIA assessment. For brevity, hereinafter, this is referred to as the GIA solution. Semi-analytical GIA solution is given by [Ivins1999]:


LaTeX equation

where:

  • LaTeX equation is the radial distance from the centre of the cylindrical disc load
  • LaTeX equation is the evaluation time
  • LaTeX equation is the Hankel transform variable of LaTeX equation (or wavenumber)
  • LaTeX equation is the radius of the cylindrical disc load
  • LaTeX equation is the shear modulus of elasticity of lithosphere
  • LaTeX equation is the lithosphere density
  • LaTeX equation is the vertical component of the gravity vector
  • LaTeX equation is the LaTeX equation-th order Bessel function of the first kind
  • LaTeX equation accounts for the integrated influence of ice loading history (cf. Figure 1) at the evaluation time LaTeX equation. (Note that LaTeX equation is the LaTeX equation-th order Hankel transform of function LaTeX equation.)
Figure2_IJ99.jpg
Schematic of evolution of piecewise continuous load height, LaTeX equation, with LaTeX equation linear segments (from [Ivins1999]). For LaTeX equation-th segment, we can compute LaTeX equation and LaTeX equation (cf. Eqs. 3--4) based on the ice load at time LaTeX equation and LaTeX equation. At LaTeX equation, for example, ice load at the lithosphere surface is given by LaTeX equation, where LaTeX equation is the ice density.

Assuming LaTeX equation, the term LaTeX equation can be written as follows


LaTeX equation

For LaTeX equation,


LaTeX equation

and for LaTeX equation (i.e. the last load segment),


LaTeX equation

where:

  • LaTeX equation is the slope of the linear LaTeX equation-th load segment
  • LaTeX equation is the LaTeX equation-intercept of the linear LaTeX equation-th load segment
  • LaTeX equation is the inverse decay time
  • LaTeX equation is the amplitude factor

For LaTeX equation, the inverse decay times are given by


LaTeX equation

and the amplitude factors by


LaTeX equation

Parameters appearing in Eqs. (5) and (6) are defined as follows


LaTeX equation

where:

  • LaTeX equation is the lithosphere thickness
  • LaTeX equation is the Maxwell relaxation time
  • LaTeX equation is the effective viscosity of mantle
  • LaTeX equation is the shear modulus of elasticity of mantle
  • parameters with primes, e.g. LaTeX equation, are dimensionless (listed in Table 1)

with the following dimensionless parameters:

  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation

where:

  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation
  • LaTeX equation

The following set of non-dimensionlized parameters are defined, as needed to express dimensionless terms listed in Table 2


LaTeX equation

where:

  • LaTeX equation is the mantle density

Numerical implementation

In the Cartesian frame of ISSM, we treat the size of ice load as the property of mesh element and compute the GIA solution at each node of the element [Adhikari2014]. Individual 2-D (LaTeX equation-plane) mesh elements are defined as the equivalence of footprint (i.e., projection onto the LaTeX equation-plane) of cylindrical disc loads, ensuring that the corresponding element and disc both share the same origin and plan-form area. The height of ice load is then assigned to each element such that the average normal tractional force on the corresponding area of bedrock is conserved. At each node within the domain, the final GIA solutions are computed by integrating the solutions due to individual disc loads, defined as the property of mesh elements.

Model parameters

The parameters relevant to the GIA solution can be displayed by typing:

>> md.gia
  • md.gia.mantle_viscosity: mantle viscosity (in Pa s)
  • md.gia.lithosphere_thickness: lithosphere thickness (in km)
  • md.gia.cross_section_shape: shape of the cylindrical disc load; 1: square-edged (default) 2: elliptical

The solution will also use the following model fields:

  • md.materials.lithosphere_shear_modulus: shear modulus of lithosphere (in Pa)
  • md.materials.lithosphere_density: lithosphere density (in g/cmLaTeX equation)
  • md.materials.mantle_shear_modulus: shear modulus of mantle (in Pa)
  • md.materials.mantle_density: mantle density (in g/cmLaTeX equation)
  • md.timestepping.start_time: GIA evaluation time LaTeX equation (in yr)
  • md.timestepping.final_time: LaTeX equation in Figure 1 (in yr).
  • md.geometry.thickness: ice loading history in the LaTeX equation matrix form; the LaTeX equation-th row, for example, should be defined as LaTeX equation (cf. Figure 1).

ISSM Configuration

To activate the GIA model, add the following in the configuration script and compile ISSM:

--with-math77-dir="$ISSM_DIR/externalpackages/math77/install"

Running a simulation

To run a simulation, use the following command:

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

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

References

  • S. Adhikari, E. Ivins, E. Larour, H. Seroussi, M. Morlighem, and S. Nowicki. Future antarctic bed topography and its implications for ice sheet dynamic. Solid Earth, 5(1):569--584, 2014.
  • Erik R. Ivins and Thomas S. James. Simple models for late Holocene and present-day Patagonian glacier fluctuations and predictions of a geodetically detectable isostatic response. Geophys. J. Int., 138(3):601--624, 1999.