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**Model description paper**
21 Mar 2019

**Model description paper** | 21 Mar 2019

FESOM-C: coastal dynamics on hybrid meshes

^{1}Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany^{2}Institute of Coastal Research, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany^{3}Shirshov Institute of Oceanology RAS, Moscow, Russia^{4}A. M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Russia^{5}Jacobs University, Bremen, Germany

^{1}Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany^{2}Institute of Coastal Research, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany^{3}Shirshov Institute of Oceanology RAS, Moscow, Russia^{4}A. M. Obukhov Institute of Atmospheric Physics RAS, Moscow, Russia^{5}Jacobs University, Bremen, Germany

Abstract

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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.

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Androsov, A., Fofonova, V., Kuznetsov, I., Danilov, S., Rakowsky, N., Harig, S., Brix, H., and Wiltshire, K. H.: FESOM-C v.2: coastal dynamics on hybrid unstructured meshes, Geosci. Model Dev., 12, 1009-1028, https://doi.org/10.5194/gmd-12-1009-2019, 2019.

1 Introduction

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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 (Danilov et al., 2017) in its basic principles, but can be used as a coastal model. Its routines handling the mesh infrastructure are derived from FESOM2. However, the time stepping, vertical discretization, and particular algorithms, detailed below, are different. FESOM-C relies on a terrain-following vertical coordinate (vs. the arbitrary Lagrangian–Eulerian (ALE) vertical coordinate of FESOM2), but takes a step further with respect to the mesh structure. It is designed to work on hybrid meshes composed of triangles and quads. Some decisions, such as the lack of the ALE at the present stage, are only motivated by the desire to keep the code as simple as possible through the initial phase of its development and maintenance. The code is based on the cell-vertex finite-volume discretization, the same as FESOM2 (Danilov et al., 2017) and FVCOM (Chen et al., 2003). It places scalar quantities at mesh vertices and the horizontal velocities at cell centroids.

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 (Danilov and Androsov, 2015). First, quadrilateral meshes have 1.5 times fewer edges than triangular meshes, which speeds up computations because cycles over edges become shorter. The second reason is the intrinsic problem of the triangular cell-vertex discretization – the presence of spurious inertial modes (see, e.g., Le Roux, 2012) and decoupling between the nearest horizontal velocities. Although both can be controlled by lateral viscosity, the control leads to higher viscous dissipation over the triangular portions of the mesh. The hybrid meshes can be designed so that triangular cells are included only to optimally match the resolution or are even absent altogether. For example, FESOM-C can be run on curvilinear meshes combining smooth changes in the shape of quadrilateral cells with smoothly approximated coastlines. One can also think of meshes for which triangular patches are only used to provide transitions between quadrilateral parts of different resolution, implementing an effective nesting approach.

Many unstructured-mesh coastal ocean models were proposed recently (e.g., Casulli and Walters, 2000; Chen et al., 2003; Fringer et al., 2006; Zhang and Baptista, 2008; Zhang et al., 2016). It will take some time for FESOM-C to catch up in terms of functionality. The decision on the development of FESOM-C was largely motivated by the desire to fit in the existing modeling infrastructure (mesh design, analysis tools, input–output organization), and not by any deficiency in existing models. The real workload was substantially reduced through the use or modification of the existing FESOM2 routines.

We formulate the main equations and their discretization in the three following sections. Section 5 presents results of test simulations, followed by a discussion and conclusions.

2 Model formulation

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We solve standard primitive equations in the Boussinesq, hydrostatic, and
traditional approximations (see, e.g., Marshall et al., 1997). The
solution is sought in the domain $\widehat{Q}=Q\times [\mathrm{0},{t}_{\mathrm{f}}]$,
where *t*_{f} is the time interval. The boundary ∂*Q* of
domain *Q* is formed by the free water surface, the bottom boundaries, and
lateral boundaries composed of the solid part ∂*Q*_{1} and the open
boundary ∂*Q*_{2}, $Q=\mathit{\{}x,y,z;x,y\in \mathrm{\Omega},-h(x,y)\le z<\mathit{\zeta}(x,y,t\left)\mathit{\right\}}$, $\mathrm{0}\le t\le {t}_{\mathrm{f}}$. Here *ζ* is the surface
elevation and *h* the bottom topography. We seek the vector of
unknown $\mathit{q}=(\mathit{u},w,\mathit{\zeta},T,S)$, where $\mathit{u}=(u,v)$ is the horizontal velocity, *w* the vertical
velocity, *T* the potential temperature, and *S* the salinity, satisfying the
equations

$$\begin{array}{ll}{\displaystyle}& {\displaystyle \frac{\partial \mathit{u}}{\partial t}}+{\displaystyle \frac{\partial}{\partial {x}_{i}}}\left(\mathit{u}{u}_{i}\right)+{\displaystyle \frac{\partial}{\partial z}}\left(\mathit{u}w\right)+{\displaystyle \frac{\mathrm{1}}{{\mathit{\rho}}_{\mathrm{0}}}}\mathrm{\nabla}p+f\mathit{k}\times \mathit{u}\\ \text{(1)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\displaystyle \frac{\partial}{\partial z}}\mathit{\vartheta}{\displaystyle \frac{\partial \mathit{u}}{\partial z}}+\mathrm{\nabla}\cdot \left(K\mathrm{\nabla}\right)\mathit{u},\text{(2)}& {\displaystyle}& {\displaystyle}\mathrm{\nabla}\cdot \mathit{u}+{\displaystyle \frac{\partial}{\partial z}}w=\mathrm{0},\text{(3)}& {\displaystyle}& {\displaystyle \frac{\partial p}{\partial z}}=-g\mathit{\rho},{\displaystyle}& {\displaystyle \frac{\partial {\mathrm{\Theta}}_{j}}{\partial t}}+{\displaystyle \frac{\partial}{\partial {x}_{i}}}\left({\mathit{u}}_{i}{\mathrm{\Theta}}_{j}\right)+{\displaystyle \frac{\partial}{\partial z}}\left(w{\mathrm{\Theta}}_{j}\right)={\displaystyle \frac{\partial}{\partial z}}{\mathit{\vartheta}}_{\mathrm{\Theta}}{\displaystyle \frac{\partial {\mathrm{\Theta}}_{j}}{\partial z}}\\ \text{(4)}& {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+\mathrm{\nabla}\left({K}_{\mathrm{\Theta}}\mathrm{\nabla}\right){\mathrm{\Theta}}_{j}.\end{array}$$

Here $i=\mathrm{1},\mathrm{2}$, *x*_{1}=*x*, *x*_{2}=*y*, *u*_{1}=*u*, and *u*_{2}=*v*, and summation is
implied over the repeating indices *i*; *p* is the pressure; and $j=\mathrm{1},\mathrm{2}$ with
Θ_{1}=*T* and Θ_{2}=*S* represents the potential temperature and
salinity, respectively. The seawater density is determined by the equation of
state $\mathit{\rho}=\mathit{\rho}(T,S,p)$, and *ρ*_{0} is the reference density; *f* is the
Coriolis parameter; ** k** is the vertical unit vector;

Writing

$$\begin{array}{}\text{(5)}& {\displaystyle}\mathit{\rho}(x,y,z,t)={\mathit{\rho}}_{\mathrm{0}}+{\mathit{\rho}}^{\prime}(x,y,z,t),\end{array}$$

where *ρ*^{′} is the density fluctuation, we obtain, integrating
Eq. (3),

$$}p-{p}_{\mathrm{atm}}=\underset{z}{\overset{\mathit{\zeta}}{\int}}\mathit{\rho}g\mathrm{d}z=g{\mathit{\rho}}_{\mathrm{0}}(\mathit{\zeta}-z)+g\underset{z}{\overset{\mathit{\zeta}}{\int}}{\mathit{\rho}}^{\prime}\mathrm{d}z,$$

where *p*_{atm} is the atmospheric pressure. The horizontal pressure
gradient is then expressed as the sum of barotropic, baroclinic, and
atmospheric pressure gradients:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}p=g\mathrm{\nabla}\mathit{\zeta}+g{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}I+{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}{p}_{\mathrm{atm}},\\ \text{(6)}& {\displaystyle}& {\displaystyle}I=\underset{z}{\overset{\mathit{\zeta}}{\int}}{\mathit{\rho}}^{\prime}\mathrm{d}z.\end{array}$$

Note that horizontal derivatives here are taken at fixed *z*.

The default scheme to compute the vertical viscosity and diffusivity in the
system of Eqs. (1)–(4) is based on the
Prandtl–Kolmogorov hypothesis of incomplete similarity. According to it, the
turbulent kinetic energy *b*, the coefficient of turbulent mixing
*ϑ*,
and dissipation of turbulent energy *ε* are connected as $\mathit{\vartheta}=l\sqrt{b}$, where *l* is the scale of turbulence,
*ϑ*_{Θ}=*c*_{ρ}*ϑ*, $\mathit{\epsilon}={c}_{\mathit{\epsilon}}{b}^{\mathrm{2}}/\mathit{\vartheta}$, and *c*_{ε}=0.046 (Cebeci and Smith, 1974).
Prandtl's number *c*_{ρ} is commonly chosen as 0.1 and sets the
relationship between the coefficients of turbulent diffusion and viscosity.
The equation describing the balance of turbulent kinetic energy is obtained
by parameterizing the energy production and dissipation in the equation for
turbulent kinetic energy *b* as

$$\begin{array}{}\text{(7)}& {\displaystyle \frac{\partial b}{\partial t}}-\mathit{\vartheta}\left({\left|{\mathit{u}}_{z}\right|}^{\mathrm{2}}+{c}_{\mathit{\rho}}g{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}{\displaystyle \frac{\partial \mathit{\rho}}{\partial z}}\right)+{c}_{\mathit{\epsilon}}{b}^{\mathrm{2}}/\mathit{\vartheta}={\mathit{\alpha}}_{b}{\displaystyle \frac{\partial}{\partial z}}\mathit{\vartheta}{\displaystyle \frac{\partial b}{\partial z}},\end{array}$$

with the boundary conditions

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}b{|}_{h}={B}_{\mathrm{1}}|\mathit{u}{|}^{\mathrm{2}},\\ {\displaystyle}& {\displaystyle}\mathit{\vartheta}{b}_{z}{|}_{\mathit{\zeta}}={\mathit{\gamma}}_{\mathit{\zeta}}{u}_{*\mathit{\zeta}}^{\mathrm{3}},\end{array}$$

where *α*_{b}=0.73, *B*_{1}=16.6, and ${\mathit{\gamma}}_{\mathit{\zeta}}=\mathrm{0.4}\times {\mathrm{10}}^{-\mathrm{3}}$;
${u}_{*\mathit{\zeta}}=(\mathit{\rho}/{\mathit{\rho}}_{\mathrm{a}}{)}^{\mathrm{1}/\mathrm{2}}{u}_{*}$ is the dynamical velocity in
water near the surface, *ρ*_{a} the air density, and *u*_{*} the dynamic
velocity of water on the interface between air and water.

Dissipative term is written as

$$}{b}^{\mathrm{2}}/\mathit{\vartheta}=\left(\mathrm{2}{b}^{\mathit{\nu}+\mathrm{1}}{b}^{\mathit{\nu}}-({b}^{\mathrm{2}}{)}^{\mathit{\nu}}\right)/{\mathit{\vartheta}}^{\mathit{\nu}},$$

where *ν* is the index of iterations. Equation (7) is solved by
a three-point Thomas scheme in the vertical direction with the boundary
conditions given above. Iterations are carried out until convergence
determined by the condition

$$}max\left|\left({b}^{\mathit{\nu}+\mathrm{1}}-{b}^{\mathit{\nu}}\right)/{b}^{\mathit{\nu}}\right|<\mathit{\varpi},$$

where *ϖ* is a small value *O*(10^{−6}). More details on the solution of
this equation are given in Voltzinger et al. (1989).

To determine the turbulence scale *l* in the presence of surface and bottom
boundary layers we use the Montgomery formula (Reid, 1957):

$$}l={\displaystyle \frac{\mathit{\kappa}}{H}}{Z}_{h}{Z}_{\mathit{\zeta}},$$

where $H=h+\mathit{\zeta}$ is the full water depth, ${Z}_{h}=z+h+{z}_{h}$,
${Z}_{\mathit{\zeta}}=-z+\mathit{\zeta}+{z}_{\mathit{\zeta}}$, *κ*≃0.4 is the von Kármán
constant, *z* the layer depth, and *z*_{h} and *z*_{ζ} are the roughness
parameters for the bottom and free surface, respectively. To remove turbulent
mixing in layers that are distant from interfaces we modify the Montgomery
formula by introducing the cutoff function ${Z}_{\mathrm{0}}=\mathrm{1}-{\mathit{\beta}}_{\mathrm{1}}{H}^{-\mathrm{2}}{Z}_{h}{Z}_{\mathit{\zeta}}$, $\mathrm{0}\le {\mathit{\beta}}_{\mathrm{1}}\le \mathrm{4}$ (Voltzinger, 1985):

$$}l={\displaystyle \frac{\mathit{\kappa}}{H}}{Z}_{h}{Z}_{\mathit{\zeta}}{Z}_{\mathrm{0}}.$$

In addition to the default scheme, one may select a scheme provided by the General Ocean Turbulence Model (GOTM) (Burchard et al., 1999) implemented into the FESOM-C code for computing vertical eddy viscosity and diffusion for momentum and tracer equations. GOTM includes large number of well-tested turbulence models with at least one member of every relevant model family (empirical models, energy models, two-equation models, algebraic stress models, K-profile parameterizations, etc.) and treats every single water column independently. An essential part of GOTM includes one-point second-order schemes (Umlauf and Burchard, 2005; Umlauf et al., 2005, 2007).

The model uses either a constant bottom friction coefficient *C*_{d},
or it is computed through the specified bottom roughness height *z*_{h}. The
first option is preferable if the vertical resolution everywhere in the
domain does not resolve the logarithmic layer or when the vertically averaged
equations are solved. In the second option the bottom friction coefficient is
computed according to Blumberg and Mellor (1987) and has the following form:

$$}{C}_{\mathrm{d}}={\left(\mathrm{ln}\left(\left(\mathrm{0.5}{h}_{\mathrm{b}}+{z}_{h}\right)/{z}_{h}\right)/\mathit{\kappa}\right)}^{-\mathrm{2}},$$

where *h*_{b} is the thickness of the bottom layer. It is also
possible to prescribe *C*_{d} or *z*_{h} as a function of the horizontal
coordinate at the initialization step.

The boundary conditions for the dynamical Eqs. (1)–(2)
are those of no-slip on the solid boundary ∂*Q*_{1},

$$}\mathit{u}\left|{}_{\partial {Q}_{\mathrm{1}}}\right.=\mathrm{0}.$$

As is well known, the formulation of open boundary conditions faces difficulties.
They are related to either the lack or incompleteness of information demanded
by the theory, for example on velocity components at the open boundary.
Furthermore, whatever the external information, it may contradict the
solution inside the computational domain, leading to instabilities that are
frequently expressed as small-scale vortex structures forming near the open
boundary. The procedure reconciling the external information with the
solution inside the domain becomes of paramount importance. We use two
approaches. The first one is to use a function whereby advection and
horizontal diffusion are smoothly tapered to zero in the close vicinity of
open boundary ∂*Q*_{2}. Such tapering makes the equations
quasi-hyperbolic at the open boundary so that the formulation of one
condition (for example, for the elevation, $\mathit{\zeta}\left|{}_{\partial {Q}_{\mathrm{2}}}\right.={\mathit{\zeta}}_{\mathrm{\Gamma}}$) is possible (Androsov et al., 1995).

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

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathit{\vartheta}{\displaystyle \frac{\partial \mathit{u}}{\partial z}}\left|{}_{\mathit{\zeta}}\right.={\mathit{\tau}}_{\mathit{\zeta}}/{\mathit{\rho}}_{\mathrm{0}},\\ {\displaystyle}& {\displaystyle}\mathit{\vartheta}{\displaystyle \frac{\partial \mathit{u}}{\partial z}}\left|{}_{-h}\right.={\mathit{\tau}}_{h}/{\mathit{\rho}}_{\mathrm{0}}={C}_{\mathrm{d}}\left|{\mathit{u}}_{h}\right|{\mathit{u}}_{h}.\end{array}$$

The first one sets the surface momentum flux to the wind stress at the
surface (*τ*_{ζ}), and the second one sets the bottom momentum flux
to the frictional flux at the bottom (*τ*_{h}), with *u*_{h} the
bottom velocity.

Now we turn to the boundary conditions for the scalar quantities obeying
Eq. (4). This is a three-dimensional parabolic equation and the
boundary conditions are determined by its leading (diffusive) terms. We
impose the no-flux condition on the solid boundary ∂*Q*_{1} and the
bottom $z=-h$.

The conditions at the open boundary ∂*Q*_{2} are given for outflow and
inflow; see, e.g., Barnier et al. (1995) and Marchesiello et al. (2001):

$$}{\mathrm{\Theta}}_{t}+a{\mathrm{\Theta}}_{x}+b{\mathrm{\Theta}}_{y}=-{\displaystyle \frac{\mathrm{1}}{\mathit{\tau}}}\left(\mathrm{\Theta}-{\mathrm{\Theta}}_{\mathrm{\Gamma}}\right),$$

where Θ_{Γ} is the given field value, usually a climatological
one or relying on data from a global numerical model or observations. If the
phase velocity components and $a=-{\mathrm{\Theta}}_{t}{\mathrm{\Theta}}_{x}/G$, $b=-{\mathrm{\Theta}}_{t}{\mathrm{\Theta}}_{y}/G$, and $G=\left[\right(\partial \mathrm{\Theta}/\partial x{)}^{\mathrm{2}}+(\partial \mathrm{\Theta}/\partial y{)}^{\mathrm{2}}{]}^{-\mathrm{1}}$ (Raymond and Kuo, 1984), Θ propagates out of the domain,
and one sets *τ*=*τ*_{0}. If it propagates into the domain, *a* and *b*
are set to zero and *τ*=*τ*_{Γ}, with *τ*_{Γ}≪*τ*_{0}.
The parameter *τ* is determined experimentally and commonly is from hours
to days. In the FESOM-C such an adaptive boundary condition is routinely
applied for temperature and salinity, yet it can also be used for any
components of a solution.

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

$$\begin{array}{}\text{(8)}& {\displaystyle}& {\displaystyle}{\mathit{\vartheta}}_{\mathrm{\Theta}}{\displaystyle \frac{\partial T}{\partial z}}\left|{}_{\mathit{\zeta}}\right.=\widehat{Q}(x,y,t)/{\mathit{\rho}}_{\mathrm{0}}{c}_{p},\text{(9)}& {\displaystyle}& {\displaystyle}{\mathit{\vartheta}}_{\mathrm{\Theta}}{\displaystyle \frac{\partial S}{\partial z}}\left|{}_{\mathit{\zeta}}\right.=\mathrm{0},\end{array}$$

where $\widehat{Q}$ is the heat flux excluding shortwave radiation, which has
been included as a volume heat source in the temperature equation, with *c*_{p}
the specific heat of seawater. The impact of the precipitation–evaporation
has been included as a volume source in the continuity equation. In the
presence of rivers, their discharge is added either as a prescribed inflow at
the open boundary in the river mouth or as volume sources of mass, heat, and
momentum distributed in the vicinity of the open boundary. In the first case
it might create an initial shock in elevation, so the second method is safer.

3 Temporal discretization

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As is common in coastal models, we split the fast and slow motions into,
respectively, barotropic and baroclinic subsystems
(Lazure and Dumas, 2008; Higdon, 2008; Gadd, 1978; Blumberg and Mellor, 1987; Deleersnijder and Roland, 1993).
The reason for this splitting is that surface gravity waves (external mode)
are fast and impose severe limitations on the time step, whereas the internal
dynamics can be computed with a much larger time step. The time step for the
external mode *τ*_{2-D} is limited by the speed of surface gravity waves, and that for the internal
mode, *τ*_{3-D}, by the speed of internal waves or advection. The
ratio ${M}_{t}={\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}/{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}$ depends on applications,
but is commonly between 10 and 30. In practice, additional limitations are
due to vertical advection or wetting and drying processes. We will further
use the indices *k* and *n* to enumerate the internal and external time
steps, respectively.

The numerical algorithm passes through several stages. In the first stage,
based on the current temperature and salinity fields (time step *k*), the
pressure is computed from hydrostatic equilibrium Eq. (6) and then
used to compute the baroclinic pressure gradient ${\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}p=g\mathrm{\nabla}{\mathit{\zeta}}^{k}+g{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}{I}^{k}+{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}{p}_{\mathrm{atm}}$. We use an asynchronous time stepping, assuming that
integration of temperature and salinity is half-step shifted with respect to
momentum. The index *k* on *I* implies that it is centered between *k* and
*k*+1 of momentum integration. The elevation in the expression above is taken
at time step *k*, which makes the entire estimate for ∇*p* only
first-order accurate with respect to time.

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

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\stackrel{\mathrm{\u0303}}{\mathit{u}}}^{k+\mathrm{1}}-{\mathit{u}}^{k}={\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}{\left(-f\mathit{k}\times \mathit{u}-\mathrm{\nabla}\cdot \mathit{u}\mathit{u}+\mathrm{\nabla}\cdot \left(K\mathrm{\nabla}\mathit{u}\right)\right)}^{\mathrm{AB}\mathrm{3}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}p+{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}{\partial}_{z}\mathit{\vartheta}{\partial}_{z}{\stackrel{\mathrm{\u0303}}{\mathit{u}}}^{k+\mathrm{1}}-{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}{\partial}_{z}{\left(w\mathit{u}\right)}^{\mathrm{AB}\mathrm{3}}.\end{array}$$

Here *K* is the coefficient of horizontal viscosity, and AB3 implies the
third-order Adams–Bashforth estimate. The horizontal viscosity operator can
be made biharmonic or replaced with filtering as discussed in the next
chapter.

To carry out mode splitting, we write the horizontal velocity as the sum of
the vertically averaged one $\stackrel{\mathrm{\u203e}}{\mathit{u}}$ and the deviation thereof
(pulsation) *u*^{′}:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathit{u}=\stackrel{\mathrm{\u203e}}{\mathit{u}}+{\mathit{u}}^{\prime},\\ {\displaystyle}& {\displaystyle}\stackrel{\mathrm{\u203e}}{\mathit{u}}={\displaystyle \frac{\mathrm{1}}{H}}\underset{-h}{\overset{\mathit{\zeta}}{\int}}\mathit{u}\mathrm{d}z,\\ {\displaystyle}& {\displaystyle}\underset{-h}{\overset{\mathit{\zeta}}{\int}}{\mathit{u}}^{\prime}=\mathrm{0}.\end{array}$$

By integrating the system in Eqs. (1)–(3) vertically between the bottom and surface, with regard for the kinematic boundary conditions ${\partial}_{t}\mathit{\zeta}+\mathit{u}\mathrm{\nabla}\mathit{\zeta}=w$ on the surface, $-\mathit{u}\mathrm{\nabla}h=w$ at the bottom, and time discretization, we get

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\left({\mathit{\zeta}}^{n+\mathrm{1}}-{\mathit{\zeta}}^{n}\right)+{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}\mathrm{\nabla}{\left(H\stackrel{\mathrm{\u203e}}{\mathit{u}}\right)}^{\mathrm{AB}\mathrm{3}}=\mathrm{0},\\ {\displaystyle}& {\displaystyle}{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{n+\mathrm{1}}-{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{n}={\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}{\left(-f\mathit{k}\times \stackrel{\mathrm{\u203e}}{\mathit{u}}-\mathrm{\nabla}\cdot \stackrel{\mathrm{\u203e}}{\mathit{u}}\phantom{\rule{0.125em}{0ex}}\stackrel{\mathrm{\u203e}}{\mathit{u}}+\mathrm{\nabla}\cdot \left(K\mathrm{\nabla}\stackrel{\mathrm{\u203e}}{\mathit{u}}\right)\right)}^{\mathrm{AB}\mathrm{3}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}(g\mathrm{\nabla}\mathit{\zeta}{)}^{\mathrm{AM}\mathrm{4}}+{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}\left({\mathit{\tau}}_{\mathit{\zeta}}/{\mathit{\rho}}_{\mathrm{0}}-{\mathit{\tau}}_{h}\right)-{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}{R}_{\mathrm{3}\text{-}\mathrm{D}}^{\prime}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}-{\mathit{\tau}}_{\mathrm{2}\text{-}\mathrm{D}}g{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\nabla}{\stackrel{\mathrm{\u203e}}{I}}^{k}.\end{array}$$

Here a specific version of AB3 is used: ${\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{\mathrm{AB}\mathrm{3}}=(\mathrm{3}/\mathrm{2}+\mathit{\beta}){\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{k}-(\mathrm{1}/\mathrm{2}+\mathrm{2}\mathit{\beta}){\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{k-\mathrm{1}}+\mathit{\beta}{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{k-\mathrm{2}}$, with *β*=0.281105 for stability reasons
(Shchepetkin and McWilliams, 2005); AM4 implies the Adams–Multon estimate
${\mathit{\zeta}}^{\mathrm{AM}\mathrm{4}}=\mathit{\delta}{\mathit{\zeta}}^{n+\mathrm{1}}+\left(\mathrm{1}-\mathit{\delta}-\mathit{\gamma}-\mathit{\u03f5}\right){\mathit{\zeta}}^{n}+\mathit{\gamma}{\mathit{\zeta}}^{n-\mathrm{1}}+\mathit{\u03f5}{\mathit{\zeta}}^{n-\mathrm{2}}$, taken with
*δ*=0.614, *γ*=0.088, and *ϵ*=0.013
(Shchepetkin and McWilliams, 2005). In the equations above *τ*_{ζ} and
*τ*_{h} are the surface (wind) and bottom stresses, respectively, and
$\mathrm{\nabla}\stackrel{\mathrm{\u203e}}{I}$ is the vertically integrated gradient of baroclinic
pressure. The term ${R}_{\mathrm{3}\text{-}\mathrm{D}}^{\prime}$ contains momentum advection and
horizontal dissipation of the pulsation velocity integrated vertically:

$$}{R}_{\mathrm{3}\text{-}\mathrm{D}}^{\prime}={\displaystyle \frac{\mathrm{1}}{{H}^{k}}}{\left[\underset{-h}{\overset{\mathit{\zeta}}{\int}}\mathrm{\nabla}\cdot {\mathit{u}}^{\mathbf{\prime}}{\mathit{u}}^{\mathbf{\prime}}-\underset{-h}{\overset{\mathit{\zeta}}{\int}}\mathrm{\nabla}\cdot \left(K\mathrm{\nabla}{\mathit{u}}^{\mathbf{\prime}}\right)\right]}^{k}.$$

In this expression *H*^{k} is the total fluid depth at time step *k*,
${H}^{k}=h+{\mathit{\zeta}}^{k}$. This term is computed only on the baroclinic time step and
kept constant through the integration of the internal mode.

The bottom friction is taken as

$$}{\mathit{\tau}}_{h}={C}_{\mathrm{d}}{\left|\stackrel{\mathrm{\u203e}}{\mathit{u}}\right|}^{n}\left({\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{n+\mathrm{1}}-{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{n}\right)/{H}^{n+\mathrm{1}}+{C}_{\mathrm{d}}|{\stackrel{\mathrm{\u0303}}{\mathit{u}}}_{h}{|}^{k+\mathrm{1}}{\stackrel{\mathrm{\u0303}}{\mathit{u}}}_{h}^{k+\mathrm{1}}/{H}^{n+\mathrm{1}}.$$

The first part of bottom friction is needed to increase stability, while the second part estimates the correct friction, with ${\stackrel{\mathrm{\u0303}}{\mathit{u}}}_{h}^{k+\mathrm{1}}$ the horizontal velocity vector in the bottom cell on the predictor time step.

The system of vertically averaged equations is stepped explicitly (except
for the bottom friction) through *M*_{t} time steps of duration
*τ*_{2-D} (index *n*) to “catch up” the *k*+1 baroclinic time
step. The update of elevation is made first, followed by the update of
vertically integrated momentum equations.

At the “corrector” step, the 3-D velocities are corrected to the surface
elevation at *k*+1:

$${\mathit{u}}^{k+\mathrm{1}}={\displaystyle \frac{{\mathrm{\u25b3}}_{i}^{k}}{{\mathrm{\u25b3}}_{i}^{k+\mathrm{1}}}}{\stackrel{\mathrm{\u0303}}{\mathit{u}}}^{k+\mathrm{1}}+\left({\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{k+\mathrm{1}}-{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{P}\right),$$

with ${\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{P}=\frac{\mathrm{1}}{{H}^{k+\mathrm{1}}}{\sum}_{-h}^{\mathit{\zeta}}\left({\stackrel{\mathrm{\u0303}}{\mathit{u}}}^{k+\mathrm{1}}{\mathrm{\u25b3}}_{i}^{k}\right)$, where *i* is the vertical
index. Here ${\mathrm{\u25b3}}_{i}^{k}$ and ${\mathrm{\u25b3}}_{i}^{k+\mathrm{1}}$ are the thicknesses of
the *i*th layer calculated on respective baroclinic time steps. The layer
thickness is ${\mathrm{\u25b3}}_{i}^{k}={\mathrm{\u25b3}}_{i}{H}^{k}$, where △_{i} is the
unperturbed vertical grid spacing. This correction removes the barotropic
component of the predicted velocity and combines the result with the computed
barotropic velocity. We will suppress the layer index *i* where it is
unambiguous.

The final step in the dynamical part calculates the transformed vertical
velocity *w*^{k+1} from 3-D continuity Eq. (2). It is used in the
next predictor step. Note that in the predictor step the computations of
vertical viscosity are implicit.

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 (Deleersnijder, 1993),

$$}{\mathit{u}}_{F}^{k+\mathrm{1}}={\displaystyle \frac{{\mathrm{\u25b3}}_{i}^{k}}{{\mathrm{\u25b3}}_{i}^{k+\mathrm{1}}}}{\stackrel{\mathrm{\u0303}}{\mathit{u}}}^{k+\mathrm{1}}+\left({\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{F}-{\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{P}\right),$$

with ${\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{F}=\frac{\mathrm{1}}{{M}_{t}{H}^{k+\mathrm{1}}}{\sum}_{n=\mathrm{1}}^{n={M}_{t}}\left({\stackrel{\mathrm{\u203e}}{\mathit{u}}}^{n}{H}^{n}\right)$. The procedure of “filtering” removes possible high-frequency components in the barotropic velocity. It also improves accuracy because it in essence works toward centering the contribution of the elevation gradient. Once the filtered velocity is computed, the vertical velocity is updated to match it.

The equation for temperature is taken in the conservation form

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\u25b3}}^{k+\mathrm{1}}{T}^{k+\mathrm{1}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}={\mathrm{\u25b3}}^{k}{T}^{k}-{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}\left[\mathrm{\nabla}\cdot \left({\mathit{u}}_{F}^{k+\mathrm{1}}{\mathrm{\u25b3}}^{k}{T}^{*}\right)+{w}_{\mathrm{Ft}}{T}_{\mathrm{t}}^{*}-{w}_{\mathrm{Fb}}{T}_{\mathrm{b}}^{*}\right]\\ {\displaystyle}& {\displaystyle}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}+D+{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}R+{\mathit{\tau}}_{\mathrm{3}\text{-}\mathrm{D}}C,\end{array}$$

where *D* combines the terms related to diffusion, *w*_{Ft},*w*_{Fb}, *T*_{t}, and *T*_{b} are the vertical transport
velocity and temperature on top and bottom of the layer, and *T*^{*} is computed
through the second-order Adams–Bashforth estimate. *R* is the boundary
thermal flux (either from the surface or due to river discharge). The last term
in the equation above is

$$}C=T{\displaystyle \frac{\partial {\mathrm{\u25b3}}_{i}}{\partial t}}=-T\left(\mathrm{\nabla}{\left({\mathrm{\u25b3}}_{i}{\mathit{u}}_{F}\right)}^{k+\mathrm{1}}+{\partial}_{z}{\left({\mathrm{\u25b3}}_{i}{w}_{F}\right)}^{k+\mathrm{1}}\right).$$

Its two constituents combine to zero because of continuity. Keeping this term makes sense if computations of advection are split into horizontal and vertical substeps. The salinity is treated similarly.

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 (Luyten et al., 1999; Blumberg and Mellor, 1987; Shchepetkin and McWilliams, 2005). The algorithm to account for wetting and drying will be presented in the next section. We only note that computations are performed on each time step of the external mode.

4 Spatial discretization

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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. 1.

The basic structure to describe the mesh is the array of edges given by their
vertices *v*_{1} and *v*_{2}, and the array of two pointers *c*_{1} and *c*_{2} to
the cells on the left and on the right of the edge. There is no difference
between triangles, quads, or hybrid meshes in the cycles that assemble
fluxes. Quads and triangles are described through four indices to the vertices
forming them; in the case of triangles the fourth index equals the first one.
The treatment of triangles and quads differs slightly in computations of
gradients as detailed below. We will use symbolic notation *e*(*c*) for the
list of edges forming cell *c*, *e*(*v*) for the list of edges connected to
vertex *v*, and *v*(*c*) for the list of vertices defining cell of element *c*.

In the vertical direction we introduce a *σ* coordinate
(Phillips, 1957):

$$}\mathit{\sigma}={\displaystyle \frac{z+h}{h+\mathit{\zeta}}},\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\mathrm{1}\le \mathit{\sigma}\le \mathrm{0}.$$

The lower and upper horizontal faces correspond to the planes *σ*=0 and
*σ*=1, respectively. The vertical grid spacing is defined by the
selected set of *σ*_{i}. The spacing of *σ*_{i} is horizontally uniform
in the present implementation (but it can be varying) and can be selected as
equidistant or based on a parabolic function with high vertical resolution
near the surface and bottom in the vertical,

$$}{\mathit{\sigma}}_{i}=-{\left({\displaystyle \frac{i-\mathrm{1}}{N-\mathrm{1}}}\right)}^{\mathit{\varrho}}+\mathrm{1},$$

where *N* is the number of vertical layers. Here *ϱ*=1(2) gives the
uniform (parabolic) distribution of vertical layers. One more possibility to
use refined resolution near the bottom or surface is implemented through the
formula by Burchard and Bolding (2002):

$$}{\mathit{\sigma}}_{i}={\displaystyle \frac{\mathrm{tanh}\left[\left({L}_{h}+{L}_{\mathit{\zeta}}\right)\frac{(N-i)}{(N-\mathrm{1})}-{L}_{h}\right]+\mathrm{tanh}{L}_{h}}{\mathrm{tanh}{L}_{h}+\mathrm{tanh}{L}_{\mathit{\zeta}}}}-\mathrm{1},$$

where *L*_{h} and *L*_{ζ} are the number of layers near the bottom and
surface, respectively.

The vertical grid spacing is recalculated on each baroclinic time step for
the vertices where *ζ* is defined. It is interpolated from vertices to
cells and to edges. The vectors of horizontal velocity and tracers are located
in the middle of vertical layers (index $i+\mathrm{1}/\mathrm{2}$), but the vertical velocity
is at full layers.

The *divergence operator* on scalar control volumes is computed as

$$\underset{v}{\int}\mathrm{\nabla}\cdot \left(\mathrm{\u25b3}\mathit{u}\right)\mathrm{d}S=\sum _{e=e\left(v\right)}\left[{\left(\mathrm{\u25b3}\mathit{u}\mathit{n}\mathrm{\ell}\right)}_{\mathrm{l}}+{\left(\mathrm{\u25b3}\mathit{u}\mathit{n}\mathrm{\ell}\right)}_{\mathrm{r}}\right],$$

where the cycle is over edges containing vertex *v*, the indices l and r
imply that the estimates are made on the left and right segments of the
control volume boundary attached to the center of edge *e*, ** n** is
the outer normal, and ℓ the length of the segment (see
Fig. 1). With vectors

*Gradients of scalar* quantities are needed on cells and are computed
as

$$}\underset{c}{\int}\mathrm{\nabla}\mathit{\zeta}\mathrm{d}S=\sum _{e=e\left(c\right)}{\left(\mathit{n}\mathrm{\ell}\mathit{\zeta}\right)}_{e},$$

where summation is over the edges of cell *c*, the normal and length are
related to the edges, and *ζ* is estimated as the mean over edge
vertices.

The *gradients of velocities* on cells can be needed for the
computation of viscosity and the momentum advection term. They are computed
through the least squares fit based on the velocities on neighboring cells:

$$\mathfrak{L}=\sum _{n=n\left(c\right)}{\left({\mathit{u}}_{c}-{\mathit{u}}_{n}-\left({\mathit{\alpha}}_{x},{\mathit{\alpha}}_{y}\right){\mathit{r}}_{\mathrm{cn}}\right)}^{\mathrm{2}}=min.$$

Here ${\mathit{r}}_{\mathrm{cn}}=\left({x}_{\mathrm{cn}},{y}_{\mathrm{cn}}\right)$ is
the vector connecting the center of *c* to that of its neighbor *n*. Their
solution can be reformulated in terms of two matrices (computed once and
stored) with coefficients ${a}_{\mathrm{cn}}^{x}=\left({x}_{\mathrm{cn}}{Y}^{\mathrm{2}}-{y}_{\mathrm{cn}}XY\right)/d$ and ${a}_{\mathrm{cn}}^{y}=\left({y}_{\mathrm{cn}}{X}^{\mathrm{2}}-{x}_{\mathrm{cn}}XY\right)/d$, acting on velocity differences and returning
the derivatives. Here $d={X}^{\mathrm{2}}{Y}^{\mathrm{2}}-(XY{)}^{\mathrm{2}}$, ${X}^{\mathrm{2}}={\sum}_{n=n\left(c\right)}{x}_{\mathrm{cn}}^{\mathrm{2}}$,
${Y}^{\mathrm{2}}={\sum}_{n=n\left(c\right)}{y}_{\mathrm{cn}}^{\mathrm{2}}$, and
$XY={\sum}_{n=n\left(c\right)}{x}_{\mathrm{cn}}{y}_{\mathrm{cn}}$.

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. 1). The second one is central and is based on scalar control volumes, with subsequent averaging to cells. In the upwind implementation we write

$$}\underset{c}{\int}\mathrm{\nabla}\cdot \left(\mathit{u}\mathrm{\u25b3}\mathit{u}\right)\mathrm{d}S=\sum _{e=e\left(c\right)}{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}\mathit{u}\right)}_{e}.$$

For edge *e*, linear velocity reconstructions on the elements on its two
sides are estimated at the edge center. One of the cells is *c*, and let *n*
be its neighbor across *e*. The respective velocity estimates will be denoted
as *u*_{ce} and *u*_{ne}, and upwind will be
written in the form $\mathrm{2}\mathit{u}={\mathit{u}}_{\mathrm{ce}}\left(\mathrm{1}+\text{sgn}\left(\mathit{u}\mathit{n}\right)\right)+{\mathit{u}}_{\mathrm{ne}}\left(\mathrm{1}-\text{sgn}\left(\mathit{u}\mathit{n}\right)\right)$, where $\mathit{u}\mathit{n}=\mathit{n}\left(\left({\mathit{u}}_{\mathrm{ce}}+{\mathit{u}}_{\mathrm{ne}}\right)\right)/\mathrm{2}$.

The other form is adapted from Danilov (2012). It provides additional smoothing for momentum advection by computing flux divergence for larger control volumes. In this case we first estimate the momentum flux term on scalar control volumes:

$$}\underset{v}{\int}\mathrm{\nabla}\cdot \left(\mathit{u}\mathrm{\u25b3}\mathit{u}\right)\mathrm{d}S=\sum _{e=e\left(v\right)}\left[{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}\mathit{u}\right)}_{\mathrm{l}}+{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}\mathit{u}\right)}_{\mathrm{r}}\right].$$

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 (Miura, 2007) differ
in the implementation of time stepping. The first needs the
Adams–Bashforth method to be second order with respect to time. The
scheme by Miura reaches this by estimating the tracer at a point displaced by
*u**τ*_{3-D}∕2. In both cases a linear reconstruction of
the tracer field for each scalar control volume is performed:

$$}{\mathrm{\Theta}}_{R}(x,y)={\mathrm{\Theta}}_{\mathrm{0}}({x}_{v},{y}_{v})+{\mathrm{\Theta}}_{x}(x-{x}_{v})+{\mathrm{\Theta}}_{y}(y-{y}_{v}),$$

where Θ_{0} is the tracer value at vertex, Θ_{x} and Θ_{y}
are the gradients averaged to vertex locations, and *x*_{v} and *y*_{v} the
coordinates of vertex *v*. The fluxes for scalar control volume faces
associated with edge *e* are computed as

$$}\sum _{e=e\left(v\right)}\left(\left[{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}{\mathrm{\Theta}}_{R}\right)}_{\mathrm{l}}+{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}{\mathrm{\Theta}}_{R}\right)}_{\mathrm{r}}\right]\right).$$

The estimate of the tracer is made at the midpoints of the left and right
segments and at points displaced by *u**τ*_{3-D}∕2 from
them,
respectively.

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: ${\mathrm{\Theta}}_{e}^{\pm}={\mathrm{\Theta}}_{{v}_{i}}\pm {\mathrm{\ell}}_{e}(\mathrm{\nabla}\mathrm{\Theta}{)}_{e}^{\pm}/\mathrm{2}$, where $i=\mathrm{1},\mathrm{2}$ are the indices of edge vertices, and gradients are computed as

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}(\mathrm{\nabla}\mathrm{\Theta}{)}_{e}^{+}={\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}(\mathrm{\nabla}\mathrm{\Theta}{)}^{\mathrm{c}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}(\mathrm{\nabla}\mathrm{\Theta}{)}^{\mathrm{u}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{and}\\ {\displaystyle}& {\displaystyle}(\mathrm{\nabla}\mathrm{\Theta}{)}_{e}^{-}={\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}(\mathrm{\nabla}\mathrm{\Theta}{)}^{\mathrm{c}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}(\mathrm{\nabla}\mathrm{\Theta}{)}^{\mathrm{d}}.\end{array}$$

Here, the upper index c means centered estimates, and u and d imply the estimates on the up- and down-edge cells.

The advective flux of scalar quantity Θ through the face of the scalar
volume $\left({Q}_{e}=\left[{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}\right)}_{\mathrm{l}}+{\left(\mathit{u}\mathit{n}\mathrm{\ell}\mathrm{\u25b3}\right)}_{\mathrm{r}}\right]\right)$ associated
with edge *e*, which leaves the control volume *ν*_{1} (see
Fig. 1), is

$$}{Q}_{e}{\mathrm{\Theta}}_{e}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{Q}_{e}\left({\mathrm{\Theta}}_{e}^{+}+{\mathrm{\Theta}}_{e}^{-}\right)+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(\mathrm{1}-\mathit{\gamma})\left|{Q}_{e}\right|\left({\mathrm{\Theta}}_{e}^{+}+{\mathrm{\Theta}}_{e}^{-}\right),$$

where *γ* is the parameter controlling the upwind dissipation. Taking
*γ*=0 gives the third-order upwind method, whereas *γ*=1 gives the
centered fourth-order estimate.

A quadratic upwind reconstruction is used in the vertical with the flux boundary conditions on surface (Eqs. 8 and 9) and zero flux at the bottom. Other options for horizontal and vertical advection, including limiters, will be introduced in the future.

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.

Consider the operator ∇*A*∇** u**. Its computation follows
the rule

$$\underset{c}{\int}\mathrm{\nabla}A\mathrm{\nabla}\mathit{u}\mathrm{d}S=\sum _{e=e\left(c\right)}A\mathrm{\ell}{\left(\mathit{n}\mathrm{\nabla}\mathit{u}\right)}_{e}.$$

The estimate of velocity gradient on edge *e* is symmetrized following the
standard practice (Danilov, 2012) over the values on neighboring cells.
“Symmetrized” means that the estimate on edge *e* is the mean of horizontal
velocity gradients computed on elements *c* and *n* (notation from article)
with the common edge *e*: $(\mathrm{\nabla}\mathit{u}{)}_{e}=((\mathrm{\nabla}\mathit{u}{)}_{c}+(\mathrm{\nabla}\mathit{u}{)}_{n})/\mathrm{2}$. The consequence of this symmetrization is that on regular
meshes (formed of equilateral triangles or rectangular quads) the information
from the nearest neighbors is lost. Any irregularity in velocity on the
nearest cells will not be penalized. Although unfavorable for both quads and
triangles, it has far-reaching implications for the latter: it cannot
efficiently remove the decoupling between the nearest velocities, which may
occur for triangular cells. This fact is well known, and the modification of
the scheme above that improves coupling between the nearest neighbors
consists of using the identity

$$\mathit{n}={\mathit{r}}_{\mathrm{cn}}/\left|{\mathit{r}}_{\mathrm{cn}}\right|+\left(\mathit{n}-{\mathit{r}}_{\mathrm{cn}}/\left|{\mathit{r}}_{\mathrm{cn}}\right|\right),$$

where *r*_{cn} is the vector connecting the centroid of cells
*c* and *n*. The derivative in the direction of *r*_{cn} is just
the difference between the neighboring velocities divided by the distance,
which is explicitly used to correct ** n**∇

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,

$${F}_{c}=-\left({\displaystyle \frac{\mathrm{1}}{{\mathit{\tau}}_{\mathrm{f}}}}\right)\sum _{n\left(c\right)}\left({\mathit{u}}_{n}-{\mathit{u}}_{c}\right),$$

where *τ*_{f} is a timescale selected experimentally. On regular meshes this
term is equivalent to the Laplacian operator. On general meshes it deviates
from the Laplacian, yet after some trivial adjustments it warrants momentum
conservation and energy dissipation (Danilov and Androsov, 2015).

For modeling wetting and drying we use the method proposed by Stelling and Duinmeijer (2003). The idea of this method is to accurately track the moving shoreline by employing the upwind water depth in the flux computations. The criterion for a vertex to be wet or dry is taken as

$$}\left\{\begin{array}{lcl}\text{wet},& \text{if}& {D}_{\mathrm{wd}}=h+\mathit{\zeta}+{h}_{l}>{D}_{min}\\ \text{dry},& \text{if}& {D}_{\mathrm{wd}}=h+\mathit{\zeta}+{h}_{l}\le {D}_{min},\end{array}\right.$$

where *D*_{min} is the critical depth and *h*_{l} is the topography. Each cell
is treated as

$$}\left\{\begin{array}{lcl}\text{wet},& \text{if}& {D}_{\mathrm{wd}}=min\phantom{\rule{0.125em}{0ex}}{h}_{v\left(c\right)}+max{\mathit{\zeta}}_{v\left(c\right)}>{D}_{min}\\ \text{dry},& \text{if}& {D}_{\mathrm{wd}}=min\phantom{\rule{0.125em}{0ex}}{h}_{v\left(c\right)}+max{\mathit{\zeta}}_{v\left(c\right)}\le {D}_{min},\end{array}\right.$$

where *h*_{v} and *ζ*_{v} are the depth and sea surface height at the
vertices *v*(*c*) of the cell *c*. When a cell is treated as dry, the velocity
at its center is set to zero and no volume flux passes through the boundaries
of scalar control volumes inside this cell.

5 Numerical simulations

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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 (e.g., Lumborg and Pejrup, 2005; Ruiz-Villarreal et al., 2005; Burchard et al., 2008; Purkiani et al., 2014). The Sylt and Rømø islands are connected to the mainland by artificial dams, creating a relatively small semi-enclosed bight with a circulation pattern well known from observations and modeling (e.g., Becherer et al., 2011; Purkiani et al., 2014). It is a tidally energetic region with a water depth down to 30 m, characterized by wide intertidal flats and a rugged coastline. Water is exchanged with the open sea through a relatively narrow (up to 1.5 km wide) and deep (up to 30 m) tidal inlet, Lister Dyb. The bathymetry data for the area were provided by Hans Burchard (personal communication, 2015) and are presented in Fig. 2.

We constructed three different meshes (Fig. 2) for our
experiments. The first one is a nearly regular quadrilateral mesh,
complemented by triangles that straighten the coastline (MESH-1). Its spatial
resolution is 200 m. The second mesh is purely triangular (MESH-2) with
resolution varying from ∼820 to ∼90 m. The third mesh was
generated by the Gmsh mesh generator (Geuzaine and Remacle, 2009) and includes 34 820 quads
and 31 triangles with a minimum cell size of 30 m and maximum size of ∼260 m (MESH-3). All meshes have 21 nonuniform sigma layers in the vertical
direction (refined near the surface and bottom). The wetting–drying option
is turned on. We apply the *k*−*ϵ* turbulence closure model with
transport equations for the turbulent kinetic energy and the turbulence
dissipation rate using the GOTM library. The second-moment closure is
represented by algebraic relations suggested by Cheng et al. (2002). The
experiment is forced by prescribing elevation due to an *M*_{2} tidal wave at
the open boundary (western and northern boundaries of the domain) provided by
Hans Burchard (personal communication, 2015).

Simulations on each mesh were continued until reaching the steady state in
the tidal cycle of the *M*_{2} wave. The last tidal period was analyzed.
Quasi-stationary behavior is already established in the second tidal period.
The simulated *M*_{2} wave is essentially nonlinear during the tidal cycle
judged by the difference in amplitude of two tidal half-cycles.

Figure 3 shows the behavior of potential and kinetic energies
in the entire domain, and the right panels show the energies computed
over the areas deeper than 1 m. The results are sensitive to the meshes,
which is explained further. The smallest tidal energy is simulated on the
triangular mesh (MESH-2). The reason is that with the same value of the
timescale *τ*_{f} in the filter used by us in these simulations, the
effective viscous dissipation is much higher on a triangular mesh than on
quadrilateral meshes of similar resolution. However, the solutions on
quadrilateral meshes are also different, and this time the reason is the
difference in the details of representing very shallow areas on meshes of
various resolution (MESH-3 is finer than MESH-1). The difference between the
simulations on two quadrilateral meshes is related to the potential energy
and comes from the difference in the elevation simulated in the areas subject
to wetting and drying (see Fig. 8). Note that the
velocities and layer thickness are small in these areas, so the difference
between kinetic energies in
Fig. 3c and d is small.

The average currents, sea level, and residual circulation simulated on MESH-1 are presented in Fig. 4. The results of this experiment show good agreement with the previously published results of Ruiz-Villarreal et al. (2005).

An example spectrum of level oscillations on station List-auf-Sylt from model
results is presented in Fig. 5. The amplitude of the
*M*_{2} wave on quad meshes (MESH-1 and MESH-3) slightly exceeds 80 cm and is
a bit smaller on MESH-2. Similar behavior is seen for the second harmonics
(*M*_{4}) expressing nonlinear effects in this region. We tried to compare
model simulation with the observations
(https://www.pegelonline.wsv.de/gast/start, last access: 28 February
2019). For comparison, the observations were taken for the first half of
January 2018. Figure 6 presents the range of
fluctuations for the whole period. As is seen, the main tidal wave *M*_{2} has
a smaller amplitude (about 70 cm) than in simulations. However, the
high-frequency part of the spectrum is very noisy because of atmospheric
loading and winds. If the analysis is performed for separate tidal cycles in
cases of strong wind and no wind, the correspondence with observations is
recovered in the second case.

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. 7, which presents histograms of the differences.

For the solutions on MESH-1 and MESH-3 values at more than 80 % of points
agree within the range of ±1 cm for the elevation (the maximum tidal
wave exceeds 1 m) and within the range of ±1 cm s^{−1} for the
velocity (the maximum horizontal velocity is about 120 cm s^{−1}). Thus,
the agreement between simulations on quadrilateral MESH-1 and MESH-3 is also
maintained on a local level. The agreement becomes worse when comparing
solutions on triangular MESH-2 and quadrilateral MESH-1. Here the share of
points with larger deviations is noticeably larger.

Spatial patterns of the differences for elevations and velocities simulated on different meshes are presented in Figs. 8 and 9, respectively. Substantial differences for the elevation are located in wetting and drying zones. This is related to the sensitivity of the wetting and drying algorithm to the cell geometry. For the horizontal velocity the difference between the solutions is defined by the resolution of bottom topography in the most energetically active zone on the quadrilateral meshes (see residual circulation in Fig. 4). The difference between the triangular grid and quadrilateral grid has a noisy character and is seen in the regions of strongest depth gradients.

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 (Maßmann et al., 2010; Idier et al., 2017). The related estuarine circulation (Burchard et al., 2008; Flöser et al., 2011), strong lateral salinity, and nutrient gradients and river plumes (Voynova et al., 2017; Kerimoglu et al., 2017) are important aspects of this area. In our simulations, the mesh consists of mainly quadrilateral cells. The mesh is constructed with Gmsh (Geuzaine and Remacle, 2009) using the Blossom-Quad method (Remacle et al., 2012). It includes 31 406 quads and only 32 triangles. The mesh resolution (defined as the distance between vertices) varies between 0.5 and 1 km in the area close to the coast and Elbe estuary, coarsening to and 4–5 km at the open boundary (Fig. 10). The mesh contains five sigma layers in the vertical.

Bathymetry from the EMODnet Bathymetry Consortium (2016) has been used.
Model runs were forced by 6-hourly atmospheric data from NCEP/NCAR Reanalysis
(Kalnay et al., 1996) and daily resolved observed river runoff
(Radach and Pätsch, 2007; Pätsch and Lenhart, 2011). Salinity and temperature data on
the open boundary were extracted from hindcast simulations based on TRIM-NP
(Weisse et al., 2015). The sea surface elevation at the open boundary was
prescribed in terms of amplitudes and phase for *M*_{2} and *M*_{4} tidal waves
derived from the previous simulations of the North Sea
(Maßmann et al., 2010; Danilov and Androsov, 2015). Data for temperature and salinity
from the TRIM-NP model were used to initialize model runs for 1 year. The
results of these runs were used as initial conditions for a 10-month final
simulation.

The validation of simulated amplitudes and phases of the *M*_{2} tidal wave is
presented in Fig. 11. This wave is the main tidal constituent
in this region. It enters the domain at the western boundary and propagates
along the coast as a Kelvin wave. The phase field is characterized by two
amphidromic points. We used the observed values from Ole Baltazar Andersen
(personal communication, 2008). for the comparison. The simulated amplitudes
are generally slightly smaller than the observed ones
(Fig. 11). The deviations in amplitudes can be explained by
uncertainty in model bathymetry and the use of a constant bottom friction
coefficient. The phases of the *M*_{2} wave are well reproduced by the model.
We characterize its accuracy by the total vector error:

$$}\mathit{\mu}={\displaystyle \frac{\mathrm{1}}{N}}\sum _{n=\mathrm{1}}^{N}{\left({\left({A}_{*}\mathrm{cos}{\mathit{\phi}}_{*}-A\mathrm{cos}\mathit{\phi}\right)}^{\mathrm{2}}+{\left({A}_{*}\mathrm{sin}{\mathit{\phi}}_{*}-A\mathrm{sin}\mathit{\phi}\right)}^{\mathrm{2}}\right)}_{n}^{\mathrm{1}/\mathrm{2}},$$

where *A*_{*}, *φ*_{*} and *A*, *φ* are the observed and
computed amplitudes and phases, respectively, at *N* stations. The total
vector error is 0.24 m for 53 stations in the entire simulated domain, which
presents a reasonably good result for this region given the domain size. From
the results of comparison it is seen that observations at some stations, such
as station 7 in the open sea, differ considerably from the amplitude and
phase at the close stations. The comparison will improve if such outlier
stations are excluded.

To validate the simulated temperature and salinity we used data from the COSYNA database (Baschek et al., 2017) and ICES database (http://www.ices.dk, last access: 28 February 2019). Comparisons of modeled surface temperature and salinity show good Pearson correlation coefficients of 0.98 and 0.9 with RMSD values of 1.24 and 0.98, respectively. The model can represent both seasonal changes in sea surface temperature (SST) and salinity (SSS), as well as lateral gradients (not shown) reasonably well. The modeled and observed SSS for Cuxhaven station is presented in Fig. 12 for simulations with the Miura advection scheme.

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^{−1} (during spring
tide up to 0.45 PSU km^{−1}) (https://www.portal-tideelbe.de, last
access: 28 February 2019, and Kappenberg et al., 2018), and an extended
wetting and drying area around this station. The simulation is in good
agreement with tidal filtered mean SSS (Fig. 12). The model
represents the summer flood event during June–July well.

Figure 13 shows the calculated surface salinity field in part of the simulated domain on 26 June 2013 in comparison with the observational data from FerryBox (FunnyGirl) (Petersen, 2014). As can be seen from the plot, there is a high consistency of the simulated results with observational data.

6 Discussion

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We examine the computational efficiency by comparing the CPU time needed to
simulate five tidal periods of an *M*_{2} wave on MESH-1 and MESH-2 in the
Sylt–Rømø experiment, as presented in Fig. 14. The
number of vertices of the quadrilateral MESH-1 is approximately ∼1.13
that of triangular MESH-2, but the numbers of elements relate as ∼0.57. We have found that the total CPU times are in an approximate ratio of 1.62
(triangles ∕ quads). The simulations were performed with the same time steps.

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 ∼1.5 in performance on quadrilateral meshes. In our simulation, the net gain was ∼1.62 times on MESH-1 compared to MESH-2, despite the fact that the number of vertices is 13 % larger than on MESH-2. The model is stable on the quadrilateral meshes with smaller horizontal viscosity, which is also an advantage.

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 *M*_{2} wave and amplitude
of 1 cm, applied in antiphase. The left part of the channel contains denser
waters than the right one.

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 15 illustrates that at times close to the maximum inflow (8 h 20 m), a strong computational instability due to the interaction between baroclinic and barotropic flow components evolves on the right open boundary on the triangular mesh, eventually leading to the blowup of the solution (see the left insert). However, by replacing triangles in a small domain adjacent to the open boundary with quadrilateral cells we stabilize the numerical solution (see the right insert), which allows us to cleanly handle the directions normal and tangent to the boundary.

7 Conclusions

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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 (Danilov et al., 2017), the addition of monotonic high-order schemes and sea ice from FESOM2, and various modules that would increase the functionality of FESOM-C.

Code and data availability

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Code and data availability.

The version of FESOM-C v.2 used to carry out simulations reported here can be accessed from https://doi.org/10.5281/zenodo.2085177. The General Ocean Turbulence Model (GOTM) (Burchard et al., 1999) implemented into the FESOM-C code is published under the GNU Public License and can be freely used. The meshes are constructed with the Gmsh software (Geuzaine and Remacle, 2009). Bathymetry used in the model simulation (Southeast North Sea experiment) is received from the EMODnet Bathymetry Consortium (2016) database (https://doi.org/10.12770/c7b53704-999d-4721-b1a3-04ec60c87238) and is freely available online. The TRIM-NP model is used to initialize runs for 1 year (Weisse et al., 2015). NCEP/NCAR reanalysis atmospheric forcing data (Kalnay et al., 1996) used in the model are freely available online.

Author contributions

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Author contributions.

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.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

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

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Short summary

We present a description of a coastal ocean circulation model designed to work on variable-resolution meshes made of triangular and quadrilateral cells. This hybrid mesh functionality allows for higher numerical performance and less dissipative solutions.

We present a description of a coastal ocean circulation model designed to work on...

Geoscientific Model Development

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