A classical Green's function approach for computing gravitationally consistent sea-level variations associated with mass redistribution on the earth's surface employed in contemporary sea-level models naturally suits the spectral methods for numerical evaluation. The capability of these methods to resolve high wave number features such as small glaciers is limited by the need for large numbers of pixels and high-degree (associated Legendre) series truncation. Incorporating a spectral model into (components of) earth system models that generally operate on a mesh system also requires repetitive forward and inverse transforms. In order to overcome these limitations, we present a method that functions efficiently on an unstructured mesh, thus capturing the physics operating at kilometer scale yet capable of simulating geophysical observables that are inherently of global scale with minimal computational cost. The goal of the current version of this model is to provide high-resolution solid-earth, gravitational, sea-level and rotational responses for earth system models operating in the domain of the earth's outer fluid envelope on timescales less than about 1 century when viscous effects can largely be ignored over most of the globe. The model has numerous important geophysical applications. For example, we compute time-varying computations of global geodetic and sea-level signatures associated with recent ice-sheet changes that are derived from space gravimetry observations. We also demonstrate the capability of our model to simultaneously resolve kilometer-scale sources of the earth's time-varying surface mass transport, derived from high-resolution modeling of polar ice sheets, and predict the corresponding local and global geodetic signatures.

Earth system modeling of climate warming scenarios and their impact on
society requires ever greater capacity to incorporate appropriate coupling of
models that traditionally have operated in isolation from one another. One
example is the necessity to couple the redistribution of earth surface mass
and energy during secular and non-secular changes. The coupling of the major
ice sheets to the earth's time-varying geoid was a main subject of Erich von
Drygalski's PhD thesis

The importance of gravitational loading and self-attraction on earth system
modeling is now demonstrated, for example, via coupling to ocean circulation

A major obstacle to efficiently coupling existing models has been their
fundamentally different computational frameworks: 3-D ice-sheet models often
operate on an unstructured mesh

In Sect. 2, we briefly review the standard Green's function approach for solving the perturbation theory of relative sea level applied to an elastically compressible and density layered self-gravitating, rotating earth. In Sect. 3, we provide our approach to evaluating key components of this theory on an anisotropic mesh and demonstrate its superiority (in terms of high-resolution capability, numerical accuracy, and computational efficiency) over contemporary pseudo-spectral methods. As example applications, in Sect. 4, we produce computations of global geodetic and sea-level signatures associated with the recent evolution of polar ice sheets. The polar ice-sheet mass budget data are derived from space gravimetry observations, and hence are of relatively low resolution (on the order of 300 km). In order to demonstrate the high-resolution capability of our model, we provide in Sect. 5 sea-level fingerprints induced by high-resolution mass change of both polar ice sheets, as modeled by ISSM's core ice-flow capability. Finally, in Sect. 6, we summarize key conclusions of this research and briefly outline its scope and limitations.

Redistribution of mass on the earth's surface caused by cryosphere and other
climate driven phenomena, such as wind stress, ocean currents, and land water
storage, perturbs the gravitational and rotational (centrifugal) potential of
the planet. Due to the fundamental properties of self-gravitation,
perturbation in these potentials induces sea-level change, solid-earth
deformation, and polar motion. If magnitudes (or trends) of mass
redistribution are known (e.g., from satellite observations), such important
geodetic signatures can be computed using a simple model of relative
sea-level variation. Following the seminal work of

For a viscoelastic earth, relative sea level at a given space on the
earth's surface and time may be defined as the difference between the
absolute sea level (i.e., sea surface without any dynamic effect of tides
and ocean currents) and the solid-earth surface, assuming that these are
measured relative to a common datum

In what follows, we assume that the redistribution of surface mass is
induced by transport of material into and out of the cryosphere and
that there is an associated viscoelastic gravitational response of the
solid earth. For the situation where it is mass transport between
continental ice and oceans, it is most convenient to define a loading
function,

The mathematical description of the gravity and loading associated
with mass transport requires perturbations in gravitational potential,

Similarly, the viscoelastic gravitational response of solid earth
following redistribution of surface mass (Eq.

In the following, we briefly present the fundamental concepts and
mathematical descriptions of gravitational and rotational potentials, as well
as the associated deformation of the solid-earth surface, required to fully
define

The general model description presented above may be applied to any earth
model, ranging from a simple rigid earth

In order to define

The terms

The surface mass redistribution and associated deformation of solid earth
also induce changes in the earth's rotational vector

In analogy with the description of

In order to define the perturbation

When the rotational perturbations are small,

From the rotational theory presented above, it is clear that

There are certain elements in the relative sea-level theory presented in
Sect. 2 that would naturally favor the spectral methods for their numerical
evaluation; expansion of non-dimensional Green's functions in the form of an
infinite sum of Legendre polynomials (Eq.

Despite the widespread application, one obvious disadvantage of
pseudo-spectral methods is that these require large numbers of terms in the
series expansion in order to accurately parameterize a slowly converging
function such as

Here we present a simple mesh-based computation of SLE on a
self-gravitating, elastically compressible, rotating earth that exploits
Green's representation of perturbation in gravitational potential and
solid-earth deformation evaluated at the surface of the earth
(Eq.

Constants and parameters used in this study. Solid-earth parameters are taken from

Crucial to evaluating

In contemporary models,

Typically,

A much better approximation than Eq. (

For

Example of unstructured mesh at earth surface. Both the

Since

Once

It is essential to compute

The system of non-homogeneous ordinary differential equations appearing in Eq. (

We assume that time-dependent variables may be expressed as the sum of their
incremental step changes. For instance,

If incremental step changes in parameters are known a priori (or computed) up
to and including the

Similarly, the following can be derived from Eq. (

If

Once

As noted earlier, we define

Similarly, the eustatic terms appearing in Eq. (

Numerical discretization of all components of SLE is now complete, and these
can be easily assembled to evaluate radial displacement of the solid-earth
surface (Eq.

Parameterization of (elastic) solid-earth deformation
caused by surface loading.

The computational algorithm used in our mesh-based model is similar to that
of pseudo-spectral models

Once gravitationally consistent solutions for change in relative sea level,

Most of our computations are done at the vertices of the mesh. Therefore, we
have to mainly deal with vectors of size ^{®} code takes about

To our knowledge, there are no standard benchmark or model intercomparison
experiments available in order to test and validate new relative sea-level
models that operate on an elastic rotating earth. However, for a suitable
set of experiments, we validate key components of our model by reproducing
relevant published results as summarized in Appendix B. We now provide
a brief comparison of our mesh model with contemporary pseudo-spectral models
in terms of computational efficiency. In the latter models, as noted earlier,
the SLE is discretized in the SH domain and individual SH coefficients are
evaluated. For a chosen spatial resolution, ^{®} version of SELEN ^{®} version of HEALPix
(^{®} version of the model, coded by the
authors (unpublished), is tested and validated against the original SELEN
model for suitable experiments. In Table 2, we compare mesh-based and
pseudo-spectral models in terms of numerical architecture and computational
cost. The latter model already demands a large computer resource to capture
even a moderate

Comparison of pseudo-spectral and mesh-based computations of the SLE.
We denote element (or pixel) size by

^{®} and
simulated in a MacBook Pro (OS X 10.9.5). We employ the Parallel Computing Toolbox^{™}
of Matlab^{®} with four local

Of several climate driven phenomena of mass redistribution on the earth's
surface, those related to the cryosphere may be of particular interest.
Space-based observations have shown that ice sheets and glaciers expel
a large volume of meltwater in an ongoing climate warming

The twin Gravity Recovery and Climate Experiment (GRACE) satellites are now
a way of monitoring and assessing earth's time-varying gravity field caused
by the climate driven surface mass redistribution and transportation of
materials within the earth's interior

The GRACE data are distributed in the form of Stokes coefficients

We compute Stokes coefficient anomalies for further processing by subtracting
the corresponding mean values (over the GRACE period) from individual Stokes
coefficients. There may be several techniques of varying complexities to
process these data

Summary of the GRACE data used in this study.

The temporal and spatial trends in the final products of the AIS and GrIS
mass balance are shown in Fig. 3. The amplitudes of temporal variability are
higher for the AIS, implying the large seasonal mass turnover there, but it
could also be due to large signal amplitudes of the GIA model used

The monthly time series of

Some important geodetic signatures of ice sheets
during the GRACE period. Rates of change in

Location of selected tide gauge stations, with Global Sea Level Observing System (GLOSS) ID.

From an ice-sheet modeling point of view,

It may be useful to evaluate the corresponding changes in absolute
gravity (gravity anomaly or disturbance), because this geodetic
variable may be measured directly using absolute gravimeters

An interesting exercise is to compare the relative contribution of individual
ice sheets to the total sea-level change. (Total sea-level change, in the
context here, is due to the combined mass evolution of both AIS and GrIS.)
For such analysis, we select 14 representative tide gauge stations, half of
which are located in the Northern Hemisphere. The descriptions of these sites
are given in Table 3 and their coordinates on the global map are shown in
Fig. 4a. For two representative sites (one for each hemisphere), Fig. 5a
shows the explicit evolution of sea-level change and the relative
contribution of AIS and GrIS. In Honolulu, the total sea level rises faster
(black line in the figure) than the GMR (red line) throughout the GRACE
period. The GrIS contribution at this site is higher (light blue fill) than
its contribution to the global mean value (blue line). The AIS influence at
this site is similar. This is summarized in the figure inset, in which we
compare the average trends in sea-level variation: the local contributions of
GrIS (

Magnitudes and trends of sea-level change at 14
selected locations. (See Table 3 for their description and
Fig. 4a to locate their position on the global map.)

Polar motion during the GRACE period. Monthly pole
positions (with respect to April 2002),

Figure 5b summarizes a similar comparison for

What we have highlighted thus far are the global predictive features that can be efficiently extracted from our flexible FE mesh system. In the section that follows we apply those model predictive features to two important geodynamical observables associated with space geodesy: polar motion and geocentric motion of the earth.

The redistribution of mass on the earth's surface (Eq.

Figure 6a and b shows 3-D plots for the monthly position of the North Pole. Complex interactions between the near-annual forcing and Chandler (433-day period) wobble results in a net polar wobble with varying amplitude. This can be seen in the figures for both ice sheets. While wobbling around the mean rotational axis, the pole also drifts away from its initial position, as indicated by trend lines in the figures. A kink in the drift direction is apparent for the AIS in about 2007, which may be linked to a similar feature observed in mass evolution of the ice sheet (see Fig. 3a).

Geocentric motion during the GRACE period. The CM-CF
shift, with respect to the April 2002 position, along the

In order to predict polar drift from our mesh-based computational framework,
we evaluate classic mass excitation functions,

Sea-level fingerprints of high-resolution ice-sheet forcing. The upper frame shows
relative sea level computed by ISSM-SEASAW at the end of a 200-year control
run of both polar ice sheets. The corresponding
ice forcings

These predictions are generally consistent with the report by

From gravitationally consistent surface mass redistribution, the mesh model
may also estimate the geocentric motion of the earth. While observationally
more elusive than polar motion, this is a fundamental parameter important to
global reference frames. The geocentric motion is caused by the shift in
relative position between the CM of the earth system and the center of figure
(CF) of the solid-earth surface, and this information is essential to
reconcile the geodetic data that are tracked from the ground stations using
absolute gravimeters and also from the passive geodetic satellites using SLR.
Let the CM-CF shift be denoted by the position vector

The ice-sheet induced components of

In Sect. 4 we have demonstrated the wide range of applications of
ISSM-SESAW. Our loading functions (Eq.

We model the dynamics of polar ice sheets over high-resolution mass
conserving beds

These solutions are obtained by running ISSM-SESAW on NASA's Pleaides
supercomputer. Due to the intensive nature of the convolution operation
(i.e., a load defined at every elemental centroid contributes to sea level
evaluated at every vertex of the mesh), a low-latency bandwidth network is
required to distribute solutions across the network in a dense pattern.
Pleaides has an InfiniBand^{®} interconnect,
with all nodes connected in a partial hypercube topology, which makes this
operation possible. Scaling is highly dependent upon the convolution loop,
which needs to be carried out in a fragmented way, to space out the
distribution of values across the cluster. This operation is CPU dependent,
but is also highly limited by the interconnect speed. Without the
discretization of SLE that operates on a flexible unstructured mesh system
(presented in this paper) and use of a massively parallelized and fully
scalable capability of ISSM-SESAW, such a high-resolution computation of
relative sea level and associated global geodetic observables would have been
virtually impossible.

While this experiment demonstrates the overall capability of our model
development, we acknowledge that the earth has non-negligible creep
deformation over the timescales of 200 years

The motivation for this study is in concert with the rapid developments in
ice-sheet modeling that have occurred over the past 5–10 years, wherein
processes that occur at a kilometer scale must be captured along with local
sea-level variability. Toward developing a coherent set of ice-sheet and
solid-earth/sea-level models that operates on a common computational
architecture provided by JPL's ISSM (

In order to explain the global model, we compute the evolution of sea-level
fingerprints and other observables, such as sea surface height, gravity
anomaly and solid-earth deformation, associated with GRACE inferred monthly
mass balance of ice sheets for a period from April 2002 to March 2015 in
a manner that is broadly familiar to the space geodesy and altimetry
communities

Global geodetic and sea-level signatures that can be computed using the mesh model may have important implications for earth system modeling. Coupling a global sea-level model to a local mesh of a 3-D ice-sheet model, for example, enhances the realistic simulation of outlet glaciers, such as Pine Island Glacier, as it provides a direct constraint to two of the important boundary conditions, namely the bedrock elevation and the sea surface height, that would be consistent with global-scale climate driven mass redistribution. There may yet be several other applications that involve continental-scale gravitational and loading interaction. However, the current model development is strictly applied to an elastically compressible and density layered self-gravitating earth and, hence, suitable for short-timescale (monthly to decadal) evaluation of variables. For relatively long-timescale (centennial or longer) computations, the model should also account for viscoelastic response of the solid earth. It may be achieved through appropriate parameterization of long-term GIA response via time-dependent viscoelastic Love numbers.

In the Supplement, we provide the source code and the necessary data set to
run the model within JPL's ISSM (

Any square-integrable function,

For the chosen

We employ this property while evaluating, for example,

We validate our model against the existing results/models through the
following three experiments: (1) sea-level change neglecting ice loads on an
elastic non-rotating earth; (2) sea-level change following instantaneous
collapse of a synthetic ice sheet on an elastic non-rotating earth; and
(3) rotational feedback to sea-level distribution. These experiments should
be sufficient to demonstrate the ability of our model to provide accurate
solutions for relative sea level on a self-gravitating, elastically
compressible, rotating earth. All results are expressed (and compared) in
terms of normalized relative sea level as follows:

Change in relative sea level neglecting ice loads on an elastic non-rotating earth.
Normalized sea level with respect to global mean value
(see Eq. B1),

Change in relative sea level induced by deglaciation on an elastic non-rotating earth.
Normalized sea level with respect to global mean value
(see Eq. B1), computed by

Rotational feedback to relative sea level.

The first experiment considers the change in sea level caused by a change in
total mass of the ocean, assuming that the mass is added from outside the
earth rather than having an ice melt origin. This experiment is motivated by
the computation documented in

In the second experiment, we compute change in relative sea level caused by
instantaneous melting of a synthetic ice sheet and compare it with the
corresponding solution obtained from
a Matlab^{®} version of SELEN

This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA) and funded through both the Cryosphere Program and the Earth Surface and Interior Focus Area as part of the GRACE Science Team and NASA Sea-level Change Team efforts. Support for S. Adhikari is through a fellowship from the NASA Post-Doctoral Program. Conversations with Jianli Chen, Richard Gross, Mathieu Morlighem, and Mike Watkins are acknowledged.Edited by: N. Kirchner