We describe FESOM-C, the coastal branch of the Finite-volumE Sea ice – Ocean Model (FESOM2), which shares with FESOM2 many numerical aspects, in particular its finite-volume cell-vertex discretization. Its dynamical core differs in the implementation of time stepping, the use of a terrain-following vertical coordinate, and the formulation for hybrid meshes composed of triangles and quads. The first two distinctions were critical for coding FESOM-C as an independent branch. The hybrid mesh capability improves numerical efficiency, since quadrilateral cells have fewer edges than triangular cells. They do not suffer from spurious inertial modes of the triangular cell-vertex discretization and need less dissipation. The hybrid mesh capability allows one to use quasi-quadrilateral unstructured meshes, with triangular cells included only to join quadrilateral patches of different resolution or instead of strongly deformed quadrilateral cells. The description of the model numerical part is complemented by test cases illustrating the model performance.

Many practical problems in oceanography require regional focus on coastal dynamics. Although global ocean circulation models formulated on unstructured meshes may in principle provide local refinement, such models are as a rule based on assumptions that are not necessarily valid in coastal areas. The limitations on dynamics coming from the need to resolve thin layers, maintain stability for sea surface elevations comparable to water layer thickness, or simulate the processes of wetting and drying make the numerical approaches traditionally used in coastal models different from those used in large-scale models. For this reason, combining coastal and large-scale functionality in a single unstructured-mesh model, although possible, would still imply a combination of different algorithms and physical parameterizations. Furthermore, on unstructured meshes, the numerical stability of open boundaries, needed in regional configurations, sometimes requires masking certain terms in motion equations close to open boundaries. This would be an unnecessary complication for a large-scale unstructured-mesh model that is as a rule global.

The main goal of the development described in this paper was to design a
tool, dubbed FESOM-C, that is close to FESOM2

Our special focus is on using hybrid meshes. In essence, the capability of
hybrid meshes is built on the finite-volume method. Indeed, computations of
fluxes are commonly implemented as cycles over edges, and the edge-based
infrastructure is immune to the polygonal type of mesh cells. However,
because of staggering, it is still convenient to keep some computations on
cells, which then depend on the cell type. Furthermore, high-order transport
algorithms might also be sensitive to the cell geometry. We limit the allowed
polygons to triangles and quads. Although there is no principal limitation on
the polygon type, triangles and quads are versatile enough in practice for
the cell-vertex discretization. Our motivation for using quads is twofold

Many unstructured-mesh coastal ocean models were proposed recently

We formulate the main equations and their discretization in the three
following sections. Section

We solve standard primitive equations in the Boussinesq, hydrostatic, and
traditional approximations (see, e.g.,

Writing

The default scheme to compute the vertical viscosity and diffusivity in the
system of Eqs. (

Dissipative term is written as

To determine the turbulence scale

In addition to the default scheme, one may select a scheme provided by the
General Ocean Turbulence Model (GOTM)

The model uses either a constant bottom friction coefficient

The boundary conditions for the dynamical Eqs. (

The other approach is to adapt the external information. It is applied to scalar fields and will be explained further.

Note that despite simplifications, barotropic and baroclinic perturbations may still disagree at the open boundary, leading to instabilities in its vicinity. In this case an additional buffer zone is introduced with locally increased horizontal diffusion and bottom friction.

Dynamic boundary conditions on the top and bottom specify the momentum fluxes
entering the ocean. Neglecting the contributions from horizontal viscosities,
we write

Now we turn to the boundary conditions for the scalar quantities obeying
Eq. (

The conditions at the open boundary

At the surface the fluxes are due to the interaction with the atmosphere:

As is common in coastal models, we split the fast and slow motions into,
respectively, barotropic and baroclinic subsystems

The numerical algorithm passes through several stages. In the first stage,
based on the current temperature and salinity fields (time step

At the second stage, the predictor values of the three-dimensional horizontal
velocity are determined as

To carry out mode splitting, we write the horizontal velocity as the sum of
the vertically averaged one

The bottom friction is taken as

The system of vertically averaged equations is stepped explicitly (except
for the bottom friction) through

At the “corrector” step, the 3-D velocities are corrected to the surface
elevation at

The final step in the dynamical part calculates the transformed vertical
velocity

New horizontal velocities, the so-called “filtered” ones, are used for
advection of a tracer. They are given by the sum of the filtered depth mean
and the baroclinic part of the “predicted” velocities

The equation for temperature is taken in the conservation form

In simulations of coastal dynamics it is often necessary to simulate flooding
and drying events. Explicit time stepping methods of solving the external
mode are well suited for this

In the finite-volume method, the governing equations are integrated over
control volumes, and the divergence terms, by virtue of the Gauss theorem, are
expressed as the sums of respective fluxes through the boundaries of control
volumes. For the cell-vertex discretization the scalar control volumes are
formed by connecting cell centroids with the centers of edges, which gives
the so-called median-dual control volumes around mesh vertices. The vector
control volumes are the mesh cells (triangles or quads) themselves, as
schematically shown in Fig.

Schematic of mesh structure. Velocities are located at centroids
(red circles) and elevation at vertices (blue circles). A scalar control
volume associated with vertex

The basic structure to describe the mesh is the array of edges given by their
vertices

In the vertical direction we introduce a

The lower and upper horizontal faces correspond to the planes

The vertical grid spacing is recalculated on each baroclinic time step for
the vertices where

The

The

We implemented two options for horizontal momentum advection in the flux
form. The first one is linear reconstruction upwind based on cell
control volumes (see Fig.

For edge

The other form is adapted from

The notation here follows that for the divergence. No velocity reconstruction is involved. These estimates are then averaged to the centers of cells. In both variants of advection form the fluid thickness is estimated at cell centers.

Horizontal advection and diffusion terms are discretized explicitly in time.
Three advection schemes have been implemented. The first two are based on
linear reconstruction for control volume and are therefore second order. The
linear reconstruction upwind scheme and the Miura scheme

The estimate of the tracer is made at the midpoints of the left and right
segments and at points displaced by

The third approach used in the model is based on the gradient reconstruction.
The idea of this approach is to estimate the tracer at mid-edge locations
with
a linear reconstruction using the combination of centered and upwind
gradients:

The advective flux of scalar quantity

A quadratic upwind reconstruction is used in the vertical with the flux
boundary conditions on surface (Eqs.

The advection schemes are coded so that their order can be reduced toward the first-order upwind for a very thin water layer to increase stability in the presence of wetting and drying.

Potential and kinetic energy. Panels

Consider the operator

A simpler algorithm is implemented to control grid-scale noise in the
horizontal velocity. It consists of adding to the right hand for the momentum
equation (2-D and 3-D flow) a term coupling the nearest velocities,

For modeling wetting and drying we use the method proposed by

In this section we present the results of two model experiments. The first considers tidal circulation in the Sylt–Rømø Bight. This area has a complex morphometry with big zones of wetting–drying and large incoming tidal waves. In this case our intention is to test the functioning of meshes of various kinds. The second experiment simulates the southeast part of the North Sea. For this area, an annual simulation of barotropic–baroclinic dynamics with realistic boundary conditions on open and surface boundaries is carried out and compared to observations. We note that a large number of simpler experiments, including those in which analytical solutions are known, were carried out in the course of model development to test and tune the model accuracy and stability. Lessons learned from these were taken into account. We omit their discussion in favor of realistic simulations.

To test the code sensitivity to the type of grid and grid quality, we computed barotropic tidally driven circulation in the Sylt–Rømø Bight in the Wadden Sea.

It is a popular area for experiments and test cases

We constructed three different meshes (Fig.

Simulations on each mesh were continued until reaching the steady state in
the tidal cycle of the

Figure

The average currents, sea level, and residual circulation simulated on MESH-1
are presented in Fig.

An example spectrum of level oscillations on station List-auf-Sylt from model
results is presented in Fig.

Of particular interest is the convergence of the solution on different meshes.
For comparison the solutions simulated on MESH-2 and MESH-3 were interpolated to
MESH-1. The comparison was performed for the full tidal cycle and is
shown in Fig.

Histograms of the difference between solutions for the tidal cycle
of the

Spatial distribution of the elevation differences for a full tidal
period for MESH-1 and MESH-2

For the solutions on MESH-1 and MESH-3 values at more than 80 % of points
agree within the range of

Spatial patterns of the differences for elevations and velocities simulated
on different meshes are presented in Figs.

Spatial distribution of the difference between the horizontal
velocities for the full tidal period of an

The area of the southeast North Sea experiment with mesh (black lines). The red dot indicates the position of the Cuxhaven station. This mesh includes 31 406 quads and 32 triangles.

The simulated

Modeled (blue line) and observed (gray dots and dashed black lines)
sea surface salinity (SSS) at the Cuxhaven station. The station is positioned
at the mouth of the Elbe River between stations 9 and 13 in
Fig.

Here we present the results of realistic simulations of circulation in the
southeastern part of the North Sea. The area of simulations is limited by the
Dogger Bank and Horns Rev (Denmark) on the north and the border between Belgium
and the Netherlands on the west. It is characterized by complex bathymetry
with strong tidal dynamics

Sea surface salinity on 26 June 2013. Filled contours are model
results, and colored lines are observational data from FerryBox (FunnyGirl)

Bathymetry from the EMODnet Bathymetry Consortium (2016) has been used.
Model runs were forced by 6-hourly atmospheric data from NCEP/NCAR Reanalysis

The validation of simulated amplitudes and phases of the

To validate the simulated temperature and salinity we used data from the
COSYNA database

The observations are from the station located in the mouth of the Elbe River
near the coast. They are characterized by a tidal amplitude in excess of
1.5 m, a horizontal salinity gradient of 0.35 PSU km

Figure

We examine the computational efficiency by comparing the CPU time needed to
simulate five tidal periods of an

CPU time on two meshes,
MESH-1 (black line) and MESH-2 (red line), for the Sylt–Rømø
experiment. The CPU time for 3-D velocity

Temperature section along the channel as simulated on the
quadrilateral mesh

The 3-D velocity part takes approximately the same CPU time as the
computation of vertically averaged velocity and elevation (external mode).
Operations on elements, which include the Coriolis and bottom friction terms
as well as computations of the gradients of velocity and scalars, are
approximately twice as cheap on quadrilateral meshes as expected.
Computations of viscosity and momentum transport are carried out in a cycle
over edges, which is 1.5 times shorter for meshes made of quadrilateral
elements and warrants a similar gain of

The presence of open boundaries is a distinctive feature of regional models. The implementation of robust algorithms for the open boundary is more complicated on unstructured triangular meshes than on structured quadrilateral meshes. For example, it is more difficult to cleanly assess the propagation of perturbations toward the boundary in this case. In addition, spurious inertial modes can be excited on triangular meshes in the case of the cell-vertex discretization used by us, which in practice leads to additional instabilities in the vicinity of the open boundary. The ability to use hybrid meshes is very helpful in this case. Indeed, even if the mesh is predominantly triangular, the vicinity of the open boundary can be constructed of quadrilateral elements.

We illustrate improvements of the dynamics in the vicinity of the open
boundary by simulating baroclinic tidal dynamics in an idealized channel with
an underwater sill. The channel is 12 km in length and 3 km in width, with
a maximum depth of 200 m near the open boundary. The sill, with a height
of 150 m, is located in the central part of the channel. The flow is forced
at the open boundaries by a tide with the period of an

Three meshes were used for these simulations. The first one is a quadrilateral mesh with a horizontal resolution of 200 m refined to 20 m in the vicinity of the underwater sill. The second one is a purely triangular mesh obtained from the quadrilateral mesh by splitting quads into triangles. The third mesh is predominantly triangular, but for the zones close to the open boundary it is also quadrilateral.

Figure

We described the numerical implementation of the three-dimensional unstructured-mesh model FESOM-C, relying on FESOM2 and intended for coastal simulations. The model is based on a finite-volume cell-vertex discretization and works on hybrid unstructured meshes composed of triangles and quads.

We illustrated the model performance with two test simulations.

Sylt–Rømø Bight is a closed Wadden Sea basin characterized by a complex morphometry and high tidal activity. A sensitivity study was carried out to elucidate the dependence of simulated surface elevation and horizontal velocity on mesh type and quality. The elevation simulated in zones of wetting and drying may depend on the mesh structure, which may lead to distinctions in the simulated energy on different meshes. The total energy comparison shows that on the triangular MESH-2, having approximately the same number of vertices as MESH-1, the solution is more dissipative; higher dissipation is generally needed to stabilize it against spurious inertial modes.

The second experiment deals with the southeastern part of the North Sea. Computation relied on the boundary information from hindcast simulations by the TRIM-NP and realistic atmospheric forcing from NCEP/NCAR. Modeling results agree both qualitatively and quantitatively with observations for the full period of simulation.

Future development of the FESOM-C will include coupling with the global
FESOM2

The version of FESOM-C v.2 used to carry out
simulations reported here can be accessed from

AA is the developer of the FESOM-C model with support from VF, IK, and SD. VF, IK, and AA carried out the experiment. AA wrote the paper with support from SD, VF, and IK. SH and NR carried out the code optimization and parallelization. KHW and HB contributed with discussions of many preliminary results. KHW helped supervise the project. All authors discussed the results and commented on the paper at all stages.

The authors declare that they have no conflict of interest.

We are grateful to Jens Schröter, Wolfgang Hiller, Peter Lemke, and Thomas Jung for supporting our work. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Robert Marsh Reviewed by: two anonymous referee