The Finite-Element Sea Ice Model (FESIM), used as a component of the Finite-Element
Sea ice Ocean Model, is presented. Version 2 includes the
elastic-viscous-plastic (EVP) and viscous-plastic (VP) solvers and employs a
flux corrected transport algorithm to advect the ice and snow mean
thicknesses and concentration. The EVP part also includes a modified approach
proposed recently by

The Finite-Element Sea Ice Model (FESIM) was developed as a component of the
Finite-Element Sea Ice Ocean circulation Model (FESOM) (for a recent
description see

Version 2 of the model is augmented by a new viscous-plastic (VP) solver,
while the Galerkin least-squares stabilized advection scheme inherited from
early versions of FESOM is replaced by the FE (finite-element) flux corrected transport (FCT)
scheme by

The intention of this paper is to present the description of the dynamical
part of the model (momentum balance and tracer advection), and illustrate the
performance of the solver algorithms implemented in the model. The
thermodynamical part will not be described here, as its implementation is
standard (pointwise) and is not affected by the unstructured character of the
surface mesh. It follows

Several approaches to sea ice modeling on unstructured meshes have been
proposed recently.

Sections

The sea ice motion equation is

Here

The internal ice stresses are computed assuming the VP rheology
(

In our case we deal with three tracers, the concentration

The well known difficulty in solving the ice momentum equation is related to
the internal stress term, which makes this equation very stiff and would
require time steps of fractions of a second if stepped explicitly. There are
two common ways of handling this difficulty. The first one treats a part of
stress divergence in an implicit way, with linearization for the moduli, as
suggested by

The EVP approach, as proposed by

In this notation, the EVP approach is

If not observed, the CFL limitation may lead to noisy fields of velocity
divergence and viscosities in practical applications in the areas where

One expects that if this scheme is stable and converged, it would produce
solutions identical to those of a converging VP solver, while the standard
EVP scheme may slightly deviate. We will return to this in
Sects.

FESIM implements the three approaches mentioned above, which will be referred to further as VP, EVP and mEVP. The reason for keeping all of them is twofold. First, it facilitates the comparison of results with other models which may use one of these approaches. Second, their numerical efficiency and performance depend on applications, and one may wish to select the most appropriate one for a particular application.

We first describe spatial discretization, and then the discretization in time. The easiest way of introducing the FE method is by considering transport equations. For this reason we begin with advection and then continue with the motion equation.

This section explains the FE spatial discretization, which is based on linear
continuous functions defined on triangles. The original motivation for this
choice was the ability to share the infrastructure with the ocean model,
which is based on the same discretization. The transport
Eq. (

The tracer equations are solved with the FE Taylor–Galerkin (TG) method
(see, e.g.,

In the last case the velocity is considered steady during the tracer time
step. This still provides the second order in time if velocity and tracers
are considered to be shifted by a half time step (asynchronous time
stepping). The resulting equation

To solve the tracer Eq. (

The procedure outlined above gives the equation in a so-called weak form. The
discretization is obtained by expanding scalar fields and velocities into
series:

Note that summation is implied over

The presence of a consistent mass matrix in the TG method effectively removes
a significant portion of dispersion related to the Lax–Wendroff method.
However, remaining dispersive errors may still be damaging. For this reason,
the approach is augmented to the FE-FCT method as proposed by

The rhs of the last expression is split into contributions from separate
elements. They are limited as detailed in Löhner et al. (1987) and
assembled back to recover a monotonic solution

By construction, the solution method is conserving. Indeed, because

Despite the fact that the FCT limiting doubles the computational cost of advection (compared to using solely the TG method), the burden remains small compared to the cost of solving for ice velocities.

Similar to the thicknesses and concentration, ice velocities are considered
to be linear functions on elements:

The strain rates are therefore elementwise constant. At this point we need to
take into account sphericity and peculiarities coming from the derivatives of
metric terms. We use the spherical coordinate system with poles at land to avoid
the pole singularity. In spherical coordinates

Here

Here summation is over vertices

Rigorous finite-element implementation of the momentum equation would involve
mass matrices and would be too time consuming in the case of EVP and mEVP
solvers. For that reason some simplifications are required. Luckily, mass
matrices are not important here, as no compensation of discrete errors can be
achieved with their help. We therefore use nodal quadratures in all terms
that do not involve spatial derivatives. Multiplying Eq. (

Here

Summation over

The second term on the rhs of Eq. (

Here

In the third term on the rhs of Eq. (

The last term in Eq. (

As mentioned above, large values for viscosities in the VP case would lead to
severe CFL limitations in the case of explicit time stepping. This suggests
to account for the stress term in the ice motion equation implicitly:

However, since the viscosities in

This approach is suboptimal because of the need to solve a problem
for a matrix of dimension

The now traditional way of handling this problem was proposed by

Because of the need to keep the same matrix in

All derivatives and

It is easy to see that all “metric differentiation terms” lead to the
additional contributions

The operator matrix is assembled in the standard sparse format on each time step. In order to reduce the computational load in the course of iterative solution, the matrix entries in the rows corresponding to nodes where the ice concentration is less than a small critical value are set to one at the diagonal, and zero otherwise. The rhs vector is corrected accordingly, and set to zero (default) or to the ocean velocity or to the velocity of the previous time step. The PETSc solver with ILU (incomplete lower–upper) preconditioning is used to solve the resulting matrix problem.

In theory, the tolerance does not necessarily need to be very small as the
solution procedure is repeated on every time step, and the solution cannot
diverge very much from the previous solution. However, on unstructured meshes
a small tolerance can sometimes be required to achieve an acceptable accuracy
on elements of differing size. Also, higher solver accuracy can be needed in
quasistationary regimes, to properly handle areas where

There is always some sensitivity to the mesh, domain geometry and preconditioning; users are advised to experiment with the available options of the solver.

In the EVP case Eqs. (

Here

Pseudo-time stepping of the stress equations of mEVP is given by
Eqs. (

Time stepping of momentum equations is implicit for the Coriolis term and the
part of ice–ocean stress. In the case of EVP the equations at each vertex

The expressions for the two last terms have been given above
(Eqs.

Now, when all equations are written, we can discuss the differences between
the methods. The differences between the EVP and mEVP are subtle (apart from
the difference in variables used to organize subcycling). First, (i) as can
be seen comparing Eqs. (

The model described above is routinely used with FESOM both in an
ice/ocean-only version or in a version coupled to an atmosphere model, so
that its practical performance can be judged by the results of respective
papers (see, e.g.,

The setup follows that used by

Triangular mesh used in simulations. The resolution varies from
approximately 40 to 10 km. Stability of EVP and mEVP on the fine mesh
requires that

Ice is driven by the wind stress

We start by comparing VP and mEVP solutions. In case A advection is
switched off, and we compare the convergence of solutions obtained with
different methods. In cases B and C the advection is switched on, they differ
by the value of

Figures

The difference between the two VP solutions and mEVP500 is much smaller and
is largely concentrated at the front between the moving and nearly stopped
ice. However, one sees that there is a basin-scale pattern in the velocity
difference in the bottom left panel of Fig.

Ice zonal velocity (m s

Reaching full agreement between mEVP and VP solutions is more difficult if
the ice advection is on, because errors may accumulate in this case with
time. Smaller values of

Same as in Fig.

Ice thickness

The results of case B are given in Figs.

Finally, case C (Figs.

Ice thickness (m) after 1 month of simulations in case C
(

Since the intention of

The next pair of figures (Figs.

Ice mean thickness

In summary, given the sensitivity of the field of

The ice mean thickness and concentration, in contrast, show a much more
robust behavior, and are much more consistent, even in the presence of noise
in

The finite-element discretization of sea ice dynamics employed by FESIM works in a robust way on unstructured triangular meshes. We now discuss how it relates to other unstructured-mesh discretizations proposed in the literature.

We first note that the FE

Same as Fig.

The vertex placement of variables we used is an analogue of the A grid in the
traditional (Arakawa) terminology. A different A-grid implementation with the
cell (triangle centroid) placement of variables was proposed by

The implementation adopted by FVCOM (

Finally, the discretization proposed by

Thus, despite its simplicity the discretization in FESIM deserves attention as a balanced choice. Work is planned on augmenting it with a multi-category ice functionality.

There is ongoing discussion on the convergence of traditional implementations
of VP and EVP, with indications that convergence is lacking (see, e.g.,

The stability (and convergence as a result) of (m)EVP solvers is sensitive to
the mesh size, and will generally deteriorate if the mesh is refined. Larger

Computations of stresses and their contributions to the rhs of momentum
equation are rather expensive in models formulated on unstructured meshes
(compared to their structured-mesh counterparts) mainly because of the lack
of directional splitting and, in the case presented, also because the number
of triangles is twice as large as the number of scalar degrees of freedom.
For this reason, one computation of the rhs (done

FESIM, the sea ice component of FESOM v.1.4, is described here. We focus on
the dynamical part of the model in this documentation. The new EVP solver
(mEVP) proposed by

The finite-element implementation described above can be recast in a finite-volume form, as briefly described below.

In a FV implementation one deals with median-dual cells formed around
vertices. They are formed by connecting mid-edges with centroids of mesh
cells. The area of the median-dual cell associated with vertex

Note that the gradient computed by the last formula will be slightly
different from its true FV counterpart in the spherical geometry. The latter
can be recovered by using

The modifications of the transport scheme are as well straightforward, but it is recommended to keep the consistent mass matrix of the FE case, which will augment the FV Lax–Wendroff scheme to the FE Taylor–Galerkin one. The FCT scheme in that case should follow the FE logics, because the mass matrix will mix the fluxes associated with boundaries. Other positivity preserving schemes are possible too, but have to be tested.

The code of the model can be obtained on request from the first author (sergey.danilov@awi). It has also been uploaded as a supplement to this paper.