# FESOM1.4

# Overview

FESOM 1.4 is the first mature global sea-ice ocean model that employs unstructured-mesh methods. It was developed mainly for usage in large scale ocean circulation and climate research. Different numerical and parametrization schemes are available in FESOM 1.4. The model has been successfully used in different ocean and sea ice studies. Until 2017/2018, FESOM applications have mainly been undertaken with this model version. It is the ocean sea ice component of the coupled climate model AWI-CM, which contributes to CMIP6.

FESOM 1.4 uses the Finite Element method to solve the primitive equations. The variational formulation with the FE method involves two basic steps. First, the partial differential equations are multiplied by a test function and integrated over the model domain. Second, the unknown variables are approximated with a sum over a finite set of basis functions. FESOM 1.4 uses the combination of continuous, piecewise linear basis functions in two dimensions for surface elevation and in three dimensions for velocity and tracers.

In 2D FESOM 1.4 uses triangular surface meshes. The 3D mesh is generated by dropping vertical lines starting from the surface 2D nodes, forming prisms which are then cut into tetrahedral elements. Except for layers adjacent to sloping ocean bottom each prism is cut into three tetrahedra; over a sloping bottom not all three tetrahedra are used in order to employ shaved cells. The model in global configurations is mainly used with z-level grids, although hybrid grids (z+sigma) are also used to better represent ice shelf cavities.

For a finite element discretization the basis functions for velocity and pressure (surface elevation in the hydrostatic case) should meet the so-called LBB condition, otherwise spurious pressure modes can be excited. The basis functions used in FESOM 1.4 for velocity and pressure do not satisfy the LBB condition, so a pressure projection method is used to stabilize the code against spurious pressure modes.

For discretization in time, the advection term in the momentum equation is solved with the so-called characteristic Galerkin method, which is effectively the explicit second-order finite-element Taylor–Galerkin method. The method is based on taking temporal discretization using Taylor expansion before applying spatial discretization. Using this method with the linear spatial discretization as mentioned above, the leading-order error of the advection equation is still second order and generates numerical dispersion, thus requiring friction for numerical stability. The horizontal viscosity is solved with the explicit Euler forward method. The vertical viscosity is solved with the Euler backward method because the forward time stepping for vertical viscosity is unstable with a typical vertical resolution and time step. To ensure solution efficiency, we solve the implicit vertical mixing operators separately from other parts of the momentum and tracer equations.

The surface elevation is solved implicitly to damp fast gravity waves, and needs iterative solvers. The Coriolis force term uses the semi-implicit method to well represent inertial oscillations. The default tracer advection scheme is an explicit flux-corrected-transport (FCT) scheme, while the vertical diffusivity uses the Euler backward method for the same reason as for vertical viscosity. The final momentum and tracer equations have only matrices of time derivative terms on the left-hand side of the equations, which can be relatively efficiently solved. Overall the dynamics and thermodynamics in the model are staggered in time with a half time step. That is, the new velocity is used to advect tracers, and the updated temperature and salinity are then used to calculate density.

# Code availability

The source code of the model available from the AWI based svn repository. One has to register at AWIForge system and send request to join AWI-CM project. At present FESOM 1.4 is developed within AWI-CM, but can be run in stand alone mode.

# Key publications

Wang, Q., Danilov, S., Sidorenko, D., Timmermann, R., Wekerle, C., Wang, X., ... & Schröter, J. (2014). The Finite Element Sea Ice-Ocean Model (FESOM) v. 1.4: formulation of an ocean general circulation model. Geoscientific Model Development, 7(2), 663-693.

Danilov, S., Kivman, G., & Schröter, J. (2004). A finite-element ocean model: principles and evaluation. Ocean Modelling, 6(2), 125-150.