We solve standard primitive equations in the Boussinesq, hydrostatic, and traditional approximations (see, e.g., ). The solution is sought in the domain Q^=Q×[0,tf], where tf 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 ∂Q1 and the open boundary ∂Q2, Q={x,y,z;x,y∈Ω,-h(x,y)≤z<ζ(x,y,t)}, 0≤t≤tf. Here ζ is the surface elevation and h the bottom topography. We seek the vector of unknown q=(u,w,ζ,T,S), where u=(u,v) is the horizontal velocity, w the vertical velocity, T the potential temperature, and S the salinity, satisfying the equations ∂u∂t+∂∂xi(uui)+∂∂z(uw)+1ρ0∇p+fk×u=∂∂zϑ∂u∂z+∇⋅(K∇)u,∇⋅u+∂∂zw=0,∂p∂z=-gρ,∂Θj∂t+∂∂xi(uiΘj)+∂∂z(wΘj)=∂∂zϑΘ∂Θj∂z+∇(KΘ∇)Θj. Here i=1,2, x1=x, x2=y, u1=u, and u2=v, and summation is implied over the repeating indices i; p is the pressure; and j=1,2 with Θ1=T and Θ2=S represents the potential temperature and salinity, respectively. The seawater density is determined by the equation of state ρ=ρ(T,S,p), and ρ0 is the reference density; f is the Coriolis parameter; k is the vertical unit vector; ϑ and K are the coefficients of vertical and horizontal turbulent momentum exchange, respectively; ϑΘ and KΘ are the respective diffusion coefficients; and g is the acceleration due to gravity.

Writing ρ(x,y,z,t)=ρ0+ρ′(x,y,z,t), where ρ′ is the density fluctuation, we obtain, integrating Eq. (), p-patm=∫zζρgdz=gρ0(ζ-z)+g∫zζρ′dz, where patm is the atmospheric pressure. The horizontal pressure gradient is then expressed as the sum of barotropic, baroclinic, and atmospheric pressure gradients: ρ0-1∇p=g∇ζ+gρ0-1∇I+ρ0-1∇patm,I=∫zζρ′dz. 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. ()–() 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 ϑ=lb, where l is the scale of turbulence, ϑΘ=cρϑ, ε=cεb2/ϑ, and cε=0.046 . Prandtl's number cρ 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 ∂b∂t-ϑuz2+cρgρ0-1∂ρ∂z+cεb2/ϑ=αb∂∂zϑ∂b∂z, with the boundary conditions b|h=B1|u|2,ϑbz|ζ=γζu*ζ3, where αb=0.73, B1=16.6, and γζ=0.4×10-3; u*ζ=(ρ/ρa)1/2u* is the dynamical velocity in water near the surface, ρa the air density, and u* the dynamic velocity of water on the interface between air and water.

Dissipative term is written as b2/ϑ=2bν+1bν-(b2)ν/ϑν, where ν is the index of iterations. Equation () 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⁡bν+1-bν/bν<ϖ, where ϖ is a small value O(10-6). More details on the solution of this equation are given in .

To determine the turbulence scale l in the presence of surface and bottom boundary layers we use the Montgomery formula : l=κHZhZζ, where H=h+ζ is the full water depth, Zh=z+h+zh, Zζ=-z+ζ+zζ, κ≃0.4 is the von Kármán constant, z the layer depth, and zh and zζ 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 Z0=1-β1H-2ZhZζ, 0≤β1≤4 : l=κHZhZζZ0.

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

The model uses either a constant bottom friction coefficient Cd, or it is computed through the specified bottom roughness height zh. 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 and has the following form: Cd=ln⁡0.5hb+zh/zh/κ-2, where hb is the thickness of the bottom layer. It is also possible to prescribe Cd or zh as a function of the horizontal coordinate at the initialization step.

The boundary conditions for the dynamical Eqs. ()–() are those of no-slip on the solid boundary ∂Q1, u∂Q1=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 ∂Q2. Such tapering makes the equations quasi-hyperbolic at the open boundary so that the formulation of one condition (for example, for the elevation, ζ∂Q2=ζΓ) is possible .

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 ϑ∂u∂zζ=τζ/ρ0,ϑ∂u∂z-h=τh/ρ0=Cd|uh|uh. The first one sets the surface momentum flux to the wind stress at the surface (τζ), and the second one sets the bottom momentum flux to the frictional flux at the bottom (τh), with uh the bottom velocity.

Now we turn to the boundary conditions for the scalar quantities obeying Eq. (). 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 ∂Q1 and the bottom z=-h.

The conditions at the open boundary ∂Q2 are given for outflow and inflow; see, e.g., and : Θt+aΘx+bΘy=-1τΘ-ΘΓ, where ΘΓ 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=-ΘtΘx/G, b=-ΘtΘy/G, and G=[(∂Θ/∂x)2+(∂Θ/∂y)2]-1 , Θ propagates out of the domain, and one sets τ=τ0. If it propagates into the domain, a and b are set to zero and τ=τΓ, with τΓ≪τ0. 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: ϑΘ∂T∂zζ=Q^(x,y,t)/ρ0cp,ϑΘ∂S∂zζ=0, where Q^ is the heat flux excluding shortwave radiation, which has been included as a volume heat source in the temperature equation, with cp 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.

The divergence operator on scalar control volumes is computed as ∫v∇⋅△udS=∑e=e(v)△unℓl+△unℓr, 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. ). With vectors sl and sr connecting the midpoint of edge e with the cell centers on the left and on the right, we get (nℓ)l=k×sl and similar, but with the minus sign for the right element (k is a unit vertical vector). The mean cell values, for example layer thickness on the cell, can be defined as △c=∑v=v(c)△vwcv, where wcv=1/3 on triangles and wcv=Scv/Sc for quads (Sc cell area and Scv, the part of it in the scalar control volume around the vertex).

Gradients of scalar quantities are needed on cells and are computed as ∫c∇ζdS=∑e=e(c)nℓζe, 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: L=∑n=n(c)uc-un-αx,αyrcn2=min⁡. Here rcn=xcn,ycn 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 acnx=xcnY2-ycnXY/d and acny=ycnX2-xcnXY/d, acting on velocity differences and returning the derivatives. Here d=X2Y2-(XY)2, X2=∑n=n(c)xcn2, Y2=∑n=n(c)ycn2, and XY=∑n=n(c)xcnycn.

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. ). The second one is central and is based on scalar control volumes, with subsequent averaging to cells. In the upwind implementation we write ∫c∇⋅u△udS=∑e=e(c)unℓ△ue.

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 uce and une, and upwind will be written in the form 2u=uce1+sgn(un)+une1-sgn(un), where un=nuce+une/2.

The other form is adapted from . 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: ∫v∇⋅u△udS=∑e=e(v)unℓ△ul+unℓ△ur.

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 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 uτ3-D/2. In both cases a linear reconstruction of the tracer field for each scalar control volume is performed: ΘR(x,y)=Θ0(xv,yv)+Θx(x-xv)+Θy(y-yv), where Θ0 is the tracer value at vertex, Θx and Θy are the gradients averaged to vertex locations, and xv and yv the coordinates of vertex v. The fluxes for scalar control volume faces associated with edge e are computed as ∑e=e(v)unℓ△ΘRl+unℓ△ΘRr.

The estimate of the tracer is made at the midpoints of the left and right segments and at points displaced by uτ3-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: Θe±=Θvi±ℓe(∇Θ)e±/2, where i=1,2 are the indices of edge vertices, and gradients are computed as (∇Θ)e+=23(∇Θ)c+13(∇Θ)uand(∇Θ)e-=23(∇Θ)c+13(∇Θ)d. 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 Qe=unℓ△l+unℓ△r associated with edge e, which leaves the control volume ν1 (see Fig. ), is QeΘe=12QeΘe++Θe-+12(1-γ)QeΘe++Θe-, 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.  and ) 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 ∫c∇A∇udS=∑e=e(c)Aℓn∇ue. The estimate of velocity gradient on edge e is symmetrized following the standard practice 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: (∇u)e=((∇u)c+(∇u)n)/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 n=rcn/rcn+n-rcn/rcn, where rcn is the vector connecting the centroid of cells c and n. The derivative in the direction of rcn is just the difference between the neighboring velocities divided by the distance, which is explicitly used to correct n∇u. It is easy to show that on rectangular quads or equilateral triangles (n and rcn are collinear) the second term of the expression above will disappear. This is the harmonic discretization and a biharmonic version is obtained by applying the procedure twice.

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, Fc=-1τf∑n(c)un-uc, where τf 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 .

For modeling wetting and drying we use the method proposed by . 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 wet,ifDwd=h+ζ+hl>Dmin⁡dry,ifDwd=h+ζ+hl≤Dmin⁡, where Dmin⁡ is the critical depth and hl is the topography. Each cell is treated as wet,ifDwd=min⁡hv(c)+max⁡ζv(c)>Dmin⁡dry,ifDwd=min⁡hv(c)+max⁡ζv(c)≤Dmin⁡, where hv and ζv 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.

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.,. 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.,. 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. .

We constructed three different meshes (Fig. ) 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 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 . The experiment is forced by prescribing elevation due to an M2 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 M2 wave. The last tidal period was analyzed. Quasi-stationary behavior is already established in the second tidal period. The simulated M2 wave is essentially nonlinear during the tidal cycle judged by the difference in amplitude of two tidal half-cycles.

Figure  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 τf 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. ). Note that the velocities and layer thickness are small in these areas, so the difference between kinetic energies in Fig. c and d is small.

The average currents, sea level, and residual circulation simulated on MESH-1 are presented in Fig. . The results of this experiment show good agreement with the previously published results of .

An example spectrum of level oscillations on station List-auf-Sylt from model results is presented in Fig. . The amplitude of the M2 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 (M4) 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  presents the range of fluctuations for the whole period. As is seen, the main tidal wave M2 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. , 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.  and , 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. ). 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 . The related estuarine circulation , strong lateral salinity, and nutrient gradients and river plumes are important aspects of this area. In our simulations, the mesh consists of mainly quadrilateral cells. The mesh is constructed with Gmsh using the Blossom-Quad method . 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. ). 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 and daily resolved observed river runoff . Salinity and temperature data on the open boundary were extracted from hindcast simulations based on TRIM-NP . The sea surface elevation at the open boundary was prescribed in terms of amplitudes and phase for M2 and M4 tidal waves derived from the previous simulations of the North Sea . 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 M2 tidal wave is presented in Fig. . 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. ). 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 M2 wave are well reproduced by the model. We characterize its accuracy by the total vector error: μ=1N∑n=1NA*cos⁡φ*-Acos⁡φ2+A*sin⁡φ*-Asin⁡φ2n1/2, 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 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.  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 ), and an extended wetting and drying area around this station. The simulation is in good agreement with tidal filtered mean SSS (Fig. ). The model represents the summer flood event during June–July well.

Figure  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) . As can be seen from the plot, there is a high consistency of the simulated results with observational data.

We examine the computational efficiency by comparing the CPU time needed to simulate five tidal periods of an M2 wave on MESH-1 and MESH-2 in the Sylt–Rømø experiment, as presented in Fig. . 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 M2 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  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.