Following , we split the horizontal velocity field into depth-averaged u‾ and deviation u′=u-u‾ components. The depth-averaged momentum equation is then defined as ∂u‾∂t+fez∧u‾+g∇hη=G, where G is a forcing term used to couple the 2-D and 3-D modes. This equation is complemented with the depth-averaged continuity (free surface) equation: ∂η∂t+∇h⋅(Hu‾)=0. The 2-D system (Eqs. –) contains the fast-propagating, rotational surface gravity waves. The corresponding equation for u′ is obtained by subtracting Eq. () from Eq. () : ∂u′∂t+∇h⋅(uu)+∂(wu)∂z+fez∧u′+Fpg=∇h⋅νh∇hu+∂∂zν∂u∂z-G. Note that the advection and viscosity terms are included in Eq. () without splitting, based on the assumption that these processes are slow enough to be captured with long time steps. The Coriolis term, on the other hand, only contains the slow modes. The vertical velocity w only appears in the advection term, which is not split, and thus there is no need to split w.

The 2-D and 3-D modes are coupled using the additional term G . First, the 3-D momentum equation (Eq. ) is solved with G=0, resulting in a velocity field u′ that has a non-zero depth average, generated by the advection and viscosity terms (that depend on u‾). We then compute the depth-averaged u′‾ and apply a correction: G=u′‾/Δt,u′←u′-GΔt to enforce zero depth average. By definition, the field G is a constant over the vertical, and it will be used as a forcing term in the 2-D momentum equation (Eq. ) in the subsequent solve. This procedure ensures that Eqs. () and () sum up to Eq. () and ∫u′dz0.

In this paper, a linear equation of state is used: ρ(T,S)=ρ0-αTT-T0+βSS-S0, where αT and βS are the thermal expansion and saline contraction coefficients, respectively, and T0 and S0 are reference temperature and salinity. In all the test cases presented herein, salinity does not contribute to water density (βS=0). Thetis also implements a full non-linear equation of state .

Baroclinic flows require some form of viscosity to filter out grid-scale noise. In this paper, we only consider Laplacian horizontal viscosity, set to a constant νh=UΔx/Reh corresponding to the velocity scale U, horizontal mesh resolution Δx, and the desired grid Reynolds number Reh. Here, the velocity scale U is taken as a global constant specific to each test case. Unless otherwise specified, the horizontal diffusivity of tracers is zero.

In most test cases, vertical viscosity is set to a constant. In certain cases, we use the gradient Richardson number dependent parameterization by : ν=ν0(1+αRi)n+νb,μ=ν1+αRi+μb, where Ri=N2/M2 is the gradient Richardson number, N is the buoyancy frequency, and M is the vertical shear frequency. The background values are set to νb=μb=2×10-5 m2 s-1, while maximum viscosity is set to ν0=2×10-2 m2 s-1; the dimensionless parameters are α=10 and n=2 . More sophisticated turbulence closures will be addressed in future work.

The prognostic variables of the coupled 2-D–3-D system (Eqs. , , , ) are η, u‾, u′, T, and S. Diagnostic variables include the vertical velocity w, water density ρ′, baroclinic head r, and internal pressure gradient Fpg. The choice of function spaces where these variables reside is crucial for numerical stability and accuracy.

Our discretization is based on the linear discontinuous Galerkin function space, P1DG. The 2-D system is discretized with a P1DG-P1DG velocity–pressure finite element pair: water elevation and both components of the depth-averaged velocity are approximated in P1DG space, i.e., η∈H2-D=P1DG, u∈U2-D=[P1DG]2. When embedded with appropriate Riemann fluxes at element interfaces, the P1DG-P1DG element pair is well suited for rotational shallow water problems .

Achieving an accurate and monotone 3-D tracer advection scheme is one of our main design criteria. The tracers, therefore, are also considered within a discontinuous function space, T, S∈H=P1DG×P1DG (here, the × operator stands for the Cartesian product of function spaces in the extruded mesh: horizontal × vertical function space). Tracer consistency (sometimes called local tracer conservation) is a necessary condition for monotonicity; it ensures that a constant tracer field does not exhibit spurious local extrema. In practice, it implies that the discrete tracer equation must reduce to the discrete continuity equation for a constant tracer. In this work, we satisfy this property by requiring the vertical velocity to belong to the tracer space H . In addition, compatibility between the 2-D and 3-D momentum equations requires that the 3-D horizontal velocity must be P1DG in the horizontal direction. We therefore set u′∈U=[P1DG×P1DG]2 as well. The used function spaces are summarized in Table .

Note that this choice of function spaces is not mimetic : the discrete system does not preserve all the properties of the continuous equations; for example, enstrophy is not conserved exactly. As the coastal ocean is generally very dissipative, maintaining mimetic properties is, however, not crucial. It is possible to define a mimetic discretization as well, for example, using Raviart–Thomas elements for the velocity, i.e., element pair RT1-P1DG . Our preliminary experiments, however, indicate that this choice significantly increases the computational cost of the system, without a corresponding improvement in accuracy. Formal assessment of the performance of mimetic discretizations in coastal ocean applications will be investigated in the future.

In the weak forms, we use the following notation for volume and interface integrals: 〈•〉Ω=∫Ω•dx,〈〈•〉〉∂Ω=∫∂Ω•ds. In interface terms, we additionally use the average {{⋅}} and jump [[⋅]] operators for scalar a and vector u fields: {{a}}=12a++a-,{{u}}=12u++u-,[[an]]=a+n++a-n-,[[u⋅n]]=u+⋅n++u-⋅n-,[[un]]=u+n++u-n-, where the superscripts “+” and “-” arbitrarily label the values on either side of the interface, and n is the outward unit normal vector of each element on the interface.

Let T stand for the triangulation of the 2-D domain Γ0. The set of element interfaces is denoted by I={k∩k′|k,k′∈T}, and n=(nx, ny) the outward unit normal vector of an interface e∈I. For brevity, boundary conditions are omitted from the weak forms.

Let ϕ2-D∈H2-D and ψ2-D∈U2-D be test functions in the 2-D function spaces. The weak formulation of the 2-D system then reads: find η∈H2-D, u‾∈U2-D such that 〈∂η∂tϕ2-D〉Γ0+〈〈H*u‾*⋅ϕ2-Dn〉〉I-〈(Hu‾)⋅∇hϕ2-D〉Γ0=0, 〈∂u‾∂t⋅ψ2-D〉Γ0+〈fez∧u‾⋅ψ2-D〉Γ0+〈〈gη*ψ2-D⋅n〉〉I-〈gη∇h⋅ψ2-D〉Γ0=〈G⋅ψ2-D〉Γ0,∀ϕ2-D∈H2-D,ψ2-D∈U2-D. Here, the divergence ∇h⋅(Hu‾) and external gradient g∇hη terms have been integrated by parts. The resulting interface terms are defined on the element edges where the state variables η, u‾ are not uniquely defined. The values η*, u‾* are obtained from an approximate Riemann solver; here, we use the linear Roe solution η*{{η}}+H/g[[u‾⋅n]] and u‾*={{u‾}}+g/H[[ηn]] .

Let P denote the set of prisms of the 3-D domain Ω, obtained from a vertical extrusion of Γ0. The set of horizontal and vertical interfaces is denoted by Ih and Iv, respectively. Let ψ∈U be a test function. The weak formulation of the 3-D momentum equation then reads: find u∈U such that 〈∂u′∂t⋅ψ〉Ω-〈∇hψ:(uu)〉Ω+〈〈uup⋅ψnh⋅{{u}}〉〉Ih∪Iv-〈(wu)⋅∂ψ∂z〉Ω+〈〈uup⋅ψnz{{w}}〉〉Ih+〈fez∧u′⋅ψ〉Ω+〈Fpg⋅ψ〉Ω+〈〈γlf[[u]]⋅[[ψ]]〉〉Ih∪Iv=Dh(u,ψ)+Dv(u,ψ),∀ψ∈U. Here, the advection and viscosity terms have been integrated by parts see; the colon operator is the Frobenius inner product, A:B=∑i,jAi,jBi,j, and uup stands for the upwind value at the interface. The internal pressure gradient term has been augmented with the Lax–Friedrichs flux with parameter γlf={{|u|}}. Adding such a flux is required to stabilize the internal pressure gradient: it reduces noise in the velocity field and decreases spurious numerical mixing in baroclinic applications. The Dh, Dv terms denote the diffusion operators introduced later.

The weak formulation of the tracer equations is derived analogously: find T∈H such that 〈∂T∂tϕ〉Ω-〈Tu⋅∇hϕ〉Ω+〈〈Tupϕnh⋅{{u}}〉〉Ih∪Iv-〈(Tw)∂ϕ∂z〉Ω+〈〈Tupϕnz{{w}}〉〉Iv=Dh(T,ϕ)+Dv(T,ϕ),∀ϕ∈H. Note that we do not employ the Lax–Friedrichs flux in the tracer equation.

The presented discretization is unstable for elliptic operators, and the diffusion operators require additional stabilization. Here, we use the symmetric interior penalty Galerkin (SIPG) method . The SIPG formulation of the tracer diffusion operators read Dh(T,ϕ)=-〈μh∇hϕ⋅∇hT〉Ω+〈〈μh∇hT⋅ϕnh〉〉Ih∪Iv+〈〈μh∇hϕ⋅Tnh〉〉Ih∪Iv-〈〈{{σ}}μhTnh⋅ϕnh〉〉Ih∪Iv, Dv(T,ϕ)=-〈μ∂T∂z∂ϕ∂z〉Ω+〈〈μ∂T∂zϕnz〉〉Ih+〈〈μ∂ϕ∂zTnz〉〉Ih-〈〈{{σ}}{{μ}}Tnzϕnz〉〉Ih. For the viscosity terms, we get Dh(u,ψ)=-〈νh∇hψ:∇huT〉Ω+〈〈ψnh⋅νh∇hu〉〉Ih∪Iv+〈〈unh⋅νh∇hψ〉〉Ih∪Iv-〈〈{{σ}}νhunhψnh〉〉Ih∪Iv, Dv(u,ψ)=-〈ν∂ψ∂z⋅∂u∂z〉Ω+〈〈ψnz⋅ν∂u∂z〉〉Ih+〈〈unz⋅ν∂ψ∂z〉〉Ih-〈〈{{σ}}{{ν}}unz⋅ψnz〉〉Ih. The penalty factor σ is defined as σ=γp(p+1)L , where p is the degree of the basis functions, γ is a factor depending on mesh quality, and L is the local element length scale in the normal direction of the interface. Let hh and hv denote the horizontal and vertical element sizes, and Δ=diag(hh, hh, hv). We then define L=n⋅Δ⋅n=hh(nx2+ny2)+hvnz2 . In this paper, we use γ=5.

The vertical velocity w is computed diagnostically from the continuity equation (Eq. ) by solving the weak form: find w∈H such that 〈〈wnzφ〉〉Γs+〈〈{{w}}φnz〉〉Ih-〈w∂φ∂z〉Ω=〈u⋅∇hφ〉Ω-〈〈{{u}}⋅φnh〉〉Ih∪Iv-〈〈u⋅φnh〉〉Γs,∀φ∈H, where both the left- and right-hand sides have been integrated by parts. Note that the terms on the bottom surface Γb vanish due to the impermeability constraint u⋅nh+wnz=0.

The water density is computed diagnostically using the equation of state. We use the same P1DG×P1DG function space for tracers and water density. In this work, we use a linear equation of state (Eq. ), and consequently density can be computed locally (at each node of the tracer field). In general, however, the equation of state is non-linear, and the density is projected on the ρ field.

The baroclinic head is computed from Eq. () by integrating ρ′ over the vertical. In practice, we solve equation ∂r∂z=ρ′/ρ0 weakly with the appropriate boundary conditions: 〈〈rnzφ〉〉Γs+〈〈rupφnz〉〉Ih-〈r∂φ∂z〉Ω=〈1ρ0ρ′φ〉Ω. Here, the left-hand side has been integrated by parts, and rup denotes the value on the prism above the interface. Note that the free surface terms vanish because r=0 on Γs by definition. We use function space P1DG×P2 for r to alleviate internal pressure gradient errors .

Finally, taking a test function ψ∈U, we compute the internal pressure gradient with the weak form 〈Fpg⋅ψ〉Ω=-〈gr∇h⋅ψ〉Ω+〈〈g{{r}}ψ⋅nh〉〉Ih∪Iv+〈〈grψ⋅nh〉〉Γs∪Γb,∀ψ∈U, where the right-hand side has been integrated by parts. Usually, Fpg belongs to the same space as the horizontal velocity, i.e., [P1DG×P1DG]2. However, to reduce bathymetry induced internal pressure gradient errors, it is possible to use a quadratic horizontal space, i.e., r∈P2DG×P2 and Fpg∈[P2DG×P1DG]2. In this paper, we use a linear Fpg field unless otherwise specified.

We use vertex-based P1DG slope limiters for three-dimensional variables to ensure positivity. The limiter is applied to both tracer and horizontal velocity fields after each update of the advection operator as discussed in the next section.

To accurately account for the free surface movement, one must move the mesh in the vertical direction. In this work, we adopt the ALE method . Here, we describe a mesh update procedure that stretches (or compresses) the mesh uniformly over the vertical direction. The ALE formulation, however, allows more complex mesh-moving methods as well, such as the (approximate) tracking of isopycnals .

In three dimensions, an ALE update consists of solving an advection–diffusion equation between two domains, Ωn and Ωn+1. Here, the domain is uniquely defined by the surface elevation field, such that for any time level n the surface Γsn matches ηn. Due to the chosen discretization, the elevation field η is discontinuous, yet we wish to maintain a conforming mesh, i.e., a continuous coordinate field z. This is achieved by projecting the elevation field ηn to a continuous space and updating the geometry with the continuous field ηcgn. The projection induces a small discrepancy between the elevation field and the 3-D domain, but its effect remains negligible in practical applications because jumps in the elevation field are typically small.

Let Ωref be the reference domain corresponding to unperturbed elevation field ηcg=0, and zref∈[-h, 0] its vertical coordinate. Applying a uniform mesh stretching, the time-dependent mesh coordinates can then be written as zn=zref+ηcgnzref+hh∈-h,ηcgn. The mesh velocity is obtained as wm=∂z∂t. In practice, the consecutive fields ηcgn+1 and ηcgn are known so we can evaluate wmn+1=ηcgn+1-ηcgnΔtzref+hh. Given the mesh velocity, a conservative ALE update can be written as ddt〈Tϕ〉Ω=〈FTT,u,w-wmϕ〉Ω, for a generic tracer equation ∂T∂t=FT(T, u, w).

The coupled 2-D–3-D system is advanced in time with a two-stage ALE time integration scheme. For convenience, we rewrite the 3-D momentum and tracer equations as ∂T∂t=FT(T,u,w)+GT(T),∂u∂t=FuFpg,u,w+Gu(u), where FT and Fu denote all the terms that are treated explicitly, while GT and Gu contain all the implicit terms. In this work, only vertical diffusion (Eq. ), vertical viscosity (Eq. ), and bottom friction terms are treated implicitly.

The explicit 3-D equations are advanced in time with a second-order SSP Runge–Kutta scheme, SSPRK(2,2) . For a generic problem (∂c∂t=F(c)), the scheme reads c(1)=cn+ΔtFcn,cn+1=cn+12ΔtFcn+12ΔtFc(1).

When applied to the explicit 3-D momentum and tracer equations, Eqs. () and (), both of these stages are ALE updates where the mesh is updated from geometry Ωn to Ω(1) and then Ωn+1. The ALE formulation of the explicit 3-D tracer equation can then be written as 〈T(1)ϕ〉Ω(1)=〈Tnϕ〉Ωn+Δt〈FTTn,un,wn-wm(1)ϕ〉Ωn, 〈T̃n+1ϕ〉Ωn+1=〈Tnϕ〉Ωn+12Δt〈FTTn,un,wn-wm(1)ϕ〉Ωn+12Δt〈FTT(1),u(1),w(1)-wmn+1ϕ〉Ω(1), where the vertical velocity is adjusted by the mesh velocity wm.

After the SSPRK update, the implicit terms are advanced with the backward Euler method. This step is computed in domain Ωn+1: 〈Tn+1ϕ〉Ωn+1=〈T̃n+1ϕ〉Ωn+1+Δt〈GTTn+1ϕ〉Ωn+1. The 3-D momentum equation is treated analogously.

The 2-D equations are advanced in time with an implicit scheme to circumvent the strict time step constraint imposed by surface gravity waves. To ensure consistency between the movement of the 3-D mesh and the 2-D mode, the 2-D time integration scheme must be compatible with the aforementioned SSPRK(2,2) method. Here, we use a combination of a forward Euler and trapezoidal steps: c(1)=cn+ΔtFcn,cn+1=cn+12ΔtFcn+Fcn+1. Denoting the tendencies of the 2-D system (Eqs. –) by Fη and Fu‾, respectively, we can write the 2-D solver as 〈η(1)ϕ2-D〉Γ0=〈ηnϕ2-D〉Γ0+Δt〈Fηηn,u‾nϕ2-D〉Γ0,〈u‾(1)⋅ψ2-D〉Γ0=〈u‾n⋅ψ2-D〉Γ0+Δt〈Fu‾ηn,u‾n⋅ψ2-D〉Γ0, 〈ηn+1ϕ2-D〉Γ0=〈ηnϕ2-D〉Γ0+Δt2〈Fηηn,u‾n+FηHn,u‾n+1ϕ2-D〉Γ0, 〈u‾n+1⋅ψ2-D〉Γ0=〈u‾n⋅ψ2-D〉Γ0+Δt2〈Fu‾ηn,u‾n+Fu‾ηn+1,u‾n+1⋅ψ2-D〉Γ0. The second implicit stage is linearized by treating the total depth H explicitly in Eq. ().

The 2-D system is solved first, resulting in an updated elevation field (η(1) and ηn+1 for the two stages, respectively) and consequently mesh geometry (Ω(1) and Ωn+1). Once the mesh geometry is known, it is straightforward to compute the corresponding mesh velocity wm and perform a 3-D ALE update.

The time integration method is second order for all the terms. The whole algorithm is summarized in Algorithm 1.

The maximal admissible time step is constrained by the stability of the coupled time integrator. The presented SSPRK(2,2) scheme has a CFL (Courant–Friedrichs–Lewy) factor 1. The 2-D scheme (Eq. ) and the implicit vertical solver (Eq. ), on the other hand, are unconditionally stable. This implies that the coupled system is stable under the same conditions as the explicit SSP scheme on its own.

The horizontal advection term imposes a constraint: Δtadv=σhLhU, where Lh is the horizontal element size, U is the maximal horizontal velocity scale, and σh is a length scaling factor. For the presented P1DG discretization, we take Lh as the square root of the triangle area. For rectangular P1DG elements and second-order RK schemes, the scaling factor is approximately σh=0.33 . In this work, we use σh=0.05 for all the diagnostic test cases. In strongly stratified flows, internal waves may impose a stricter constraint: Δtiw=σhLhCiw+U, where Ciw is the speed of the internal waves.

Analogously, the time step constraint for vertical advection is Δtw=σvLzW, where Lz is the element height, W is the vertical velocity scale, and σv=0.125 is the scaling factor.

Given a horizontal viscosity scale Nh, the explicit viscosity operator imposes a constraint: Δtvisc=σviscσhLh2Nh, which may become stringent for small elements and large viscosity values. The scaling factor σvisc depends on the used stabilization scheme; here, a value of σvisc=2 is used. The constraint for horizontal diffusivity is analogous.

In the simulations presented herein, the minimal admissible time step is evaluated on the mesh based on constant a-priori velocity and viscosity scales. The time step is kept constant throughout the simulation.

We first evaluate the performance of the solver in a barotropic standing wave test case. The domain is a Lx=60 km long rectangular channel, 625 m wide, and 100 m deep. All lateral boundaries are closed. Initially, the water is at rest. A 10 m tall sinusoidal elevation perturbation is prescribed along the channel (ηa=-η0cos⁡(2πx/Lx), η0=10 m), leading to a non-linear wave as the simulation progresses.

The simulation is run for two full wave periods, approximately 3831.31 s. To investigate tracer conservation and consistency properties, two passive tracers are included: salinity is set to a constant 4.5 psu, while temperature varies between 5.0 and 15.0 ∘C along the channel (T=5sin⁡(2πx/Lx)+10 ∘C).

The domain was discretized with a split-quad mesh using 40 elements along the channel (1500 m edge length) and four vertical layers. The time step is Δt=95.78 s, chosen to meet the horizontal advection condition.

During the simulation, the volume of the 3-D domain was conserved to accuracy O(10-15). The “2-D volume”, i.e., the integral of the elevation field, was conserved to accuracy O(10-16). Salinity remained at constant 4.5 psu with a small O(10-9) deviation. The total mass of salinity and temperature were both conserved to accuracy O(10-12). Over- and undershoots in the temperature field were negligible due to the slope limiters. Without the limiter, temperature overshoots were O(10-2). These results show that the model indeed fully conserves volume and tracers and does not exhibit overshoots. Moreover, the tracer consistency property is satisfied, verifying the integrity of the ALE formulation.

To investigate the order of convergence of the solver, we used a smaller initial elevation perturbation (η0=1 cm). In this case, the resulting standing wave is close to linear. At the end of the simulation, the solution was compared to the analytical solution of the linear wave equation (which coincides with the initial condition) by computing the L2 error, E(η)=(∫Ω(η-ηa)2dx)1/2.

We ran the simulation, varying the horizontal mesh resolution between 3 km and 300 m; the number of vertical levels varied between 2 and 20. In each case, the channel was made one element wide, and the time step was chosen to meet the CFL criterion for horizontal advection. At the end of the simulation, the L2 error was computed for water elevation and velocity (see Fig. ). The velocity field shows the expected second-order convergence, whereas elevation converges at a rate of 3.2. It is known that P1DG shallow water equations models may exhibit superconvergence properties, especially for the elevation field . Here, our results verify that the solver behaves as expected and yields second-order accuracy under barotropic forcing.

Verifying model accuracy under baroclinic forcing is more challenging as no analytical solutions are available. Here, we use the method of manufactured solutions MMS; to construct a steady-state test case that allows us to verify the correctness of the discrete baroclinic operators. The domain is a Lx=15 by Ly=10 km large and h=40 m deep rectangular box. All lateral boundaries are closed. We prescribe initial velocity and temperature fields: ua=12sin⁡2πLxxcos⁡3zh,va=13cos⁡πyLysin⁡z2h,Ta=15sin⁡πxLxsin⁡πyLycos⁡zh+15. These functions were chosen to be analytic (infinitely differentiable) and fully three-dimensional to better quantify the spatial discretization error.

Salinity is set to a constant 35 psu. We use the linear equation of state (Eq. ) with ρ0=1000 kg m-3, αT=0.2 kg m-3 ∘C-1, and T0=5 ∘C-1. For the sake of simplicity, bathymetry is constant and elevation is set to zero initially. Coriolis frequency was set to a constant f=10-4 s-1. Bottom friction, viscosity, and diffusivity are omitted.

Without any additional forcing, the initial conditions lead to a time-dependent solution. Following the MMS strategy, we add analytical source terms in the dynamic equations that cancel all the active terms in the equations, leading to a steady-state solution. The remaining error is purely the discretization error of the advection, pressure gradient, and Coriolis operators. The source terms are derived analytically and projected to the corresponding function space. The analytical formulae are given in Appendix .

The coarsest mesh contains four elements both in x and y directions and two vertical levels. We refine the mesh up to 10 times (40 elements and 20 vertical levels) and compute the L2 error of the prognostic fields against the exact solutions. In each case, the model is run for 50 iterations with a time step chosen to meet the CFL condition.

The variation of the L2 errors with resolution is shown in Fig. . All the prognostic variables exhibit the correct second-order convergence rate. The diagnostic vertical velocity field (which depends on the divergence of u) converges linearly as expected. Therefore, we conclude that advection, pressure gradient, and Coriolis terms are discretized correctly. We have also developed similar MMS tests for the diffusivity/viscosity operators and the bottom friction term, all of which show second-order convergence as well (not shown).

The validity of the baroclinic solver and its level of spurious mixing is investigated with the standard lock exchange test case . Here, we follow the setup of and : The domain is a 64 km long and 1 km wide rectangular channel. Water depth is 20 m. Initially, the left-hand side of the domain is filled with dense water mass (T=5.0 ∘C) compared to the water on the right (T=30.0 ∘C). Salinity is kept at constant 35 psu. We use the same linear equation of state as in Sect. , resulting in a density difference of Δρ=5.0 kg m-3. The domain is discretized with a regular split-quad mesh. The triangle edge length is 500 m and 20 equidistant σ levels are used in the vertical direction.

Stabilizing the internal pressure gradient requires some form of friction. To this end, we apply a constant Laplacian horizontal viscosity, using values ν=1.0, 10.0, 100.0, and 200.0 m2 s-1. These values correspond to grid Reynolds numbers Reh=UΔx/ν=250.0, 25.0, 2.5, and 1.25, respectively, where the characteristic velocity scale of the exchange flow is U=0.5 m s-1. Vertical viscosity is set to a constant 10-4 m2 s-1. Bottom friction is omitted.

Figure  shows the initial density field and solution after 17 h of simulation for the three cases. Higher background viscosity leads to a less noisy velocity field and therefore sharper density front. The sharpness and shape of the fronts are similar to results presented in the literature e.g., Fig. 5 in. The low viscosity cases (Reh=25250) exhibit an internal wave at the front which significantly increases the overall mixing.

Assuming that, in the absence of bottom friction, all available potential energy is transformed into kinetic energy, the maximum front propagation speed can be estimated as c=1/2gHΔρ/ρ0 . Figure a shows the propagation of the front location at the bottom of the domain (the front at the surface behaves comparably). All three simulations are in good agreement with the theoretical propagation speed. The simulated front propagation speed is underestimated by roughly 5 %, indicating loss of energy due to mixing. These results are similar to results reported in the literature; e.g.,  show similar performance for ROMS, MITgcm, and MOM.

Figure b shows the maximum over- and undershoots in the temperature field during the simulation. Even in the low viscosity case (Reh=250), the overshoots are of order 10-5 ∘C, indicating that the tracer advection scheme is indeed close to monotone, due to the SSP time integration method and slope limiters. Note that, if the slope limiter is omitted, the overshoots can reach 30 ∘C.

To diagnose the role of spurious mixing, we use the reference potential energy (RPE; ). RPE is computed as the vertical center of mass of a sorted density field ρ*: RPE = g∫ρ*(z+h)dx. The ρ* field is defined as the unique, stratified density field where the densest water parcels are distributed over the bottom, and density increases monotonically over the water column. As such, ρ* is the steady-state density distribution, and RPE represents the portion of potential energy that cannot be transformed into kinetic energy. Mixing the two water masses increases RPE (the center of mass), and thus the amount of unavailable potential energy increases. Figure c shows the evolution of normalized RPE, RPE‾(t)=(RPE(t)-RPE(0))/RPE(0), during the simulation. At t=17 h, the values are 0.612, 1.13, 2.35, 3.11×10-5 for the four simulations. These results are in good agreement with those reported with MPAS-Ocean model : with the same mesh resolution, MPAS-Ocean shows slightly larger normalized RPE, for example, at t=17 h RPE‾≈3.5×10-5 in the case of Reh=25. The difference is likely due to the different spatial discretization (P1DG instead of finite volumes) or differences in the numerical viscosity operator. Applying slope limiters to the velocity field is not necessary for numerical stability, but it reduces high-frequency noise in the velocity field and hence results in lower RPE values.

In order to investigate the role of the Lax–Friedrichs flux on numerical mixing, we ran the lock exchange test case with zero viscosity. After 17 h of simulation, the RPE value was approximately 3.2×10-5. When the Lax–Friedrichs flux was omitted, a similar RPE value was obtained with viscosity ν=3.125 m2 s-1. Therefore, in this particular test case, the Lax–Friedrichs flux introduces mixing that is roughly equivalent to 3 m2 s-1 viscosity, corresponding to Reh=80. When viscosity was non-zero, it was evident from the numerical simulations that the Lax–Friedrichs flux has a negligible impact on numerical mixing if Re < 10 (not shown).

We investigate the model's ability to generate baroclinic eddies with the eddying channel test case of . This test case is an idealization of the Antarctic Circumpolar Current, the domain spanning 500 and 160 km in the meridional and zonal directions, respectively. The domain is 1000 m deep. At the zonal boundaries, periodic boundary conditions are applied; northern and southern boundaries are closed. The Coriolis parameter is taken as a constant 1.2×10-4 s-1.

Initially, the domain is linearly stratified with warmer water at the surface. In addition, the northern half of the domain is warmer, with a narrow sinusoidal transition band separating the warm (northern) and cold (southern) water masses (Fig. ; see for the definition of the initial temperature field). Water temperature ranges between 10 and 20 ∘C. A linear equation of state is used with ρ0=1000 kg m-3, αT=0.2 kg m-3 ∘C-1 and T0=5 ∘C. Salinity is kept at constant 35 psu and it does not affect density (βS=0). Bottom friction is parameterized by a constant drag coefficient of CD=0.01.

The baroclinic Rossby radius of deformation is 20 km . Horizontal mesh resolution is constant in space. We use a regular split-quad mesh with two different mesh resolutions: eddy-permitting 10 km and a finer 4 km resolution. In the vertical direction, 26 and 40 equidistant sigma levels are used in the two cases, respectively. Simulations are carried out with different values of horizontal viscosity, with the grid Reynolds number ranging from 2 to 100. The different setups are summarized in Table . Vertical viscosity is set to a constant 10-4 m2 s-1.

As the simulation progresses, baroclinic eddies develop at the center of the domain, quickly propagating elsewhere. This is a spin-down experiment, i.e., the domain is a closed system with no forcing at the boundaries. Therefore, all the energy in the system originates from the initial potential energy, which is being dissipated during the simulation; again, the RPE is used as a metric for the energy transfer or the loss of energy due to mixing.

Figure  shows the surface temperature fields at various time intervals up to 200 days after the initialization for different values of horizontal viscosity. As expected, the model captures more mesoscale features as viscosity is decreased. Qualitatively, the results are in agreement with ROMS and MITgcm results , as well as MPAS-Ocean , all of which use a comparable Laplacian scheme for horizontal viscosity.

The evolution of the normalized RPE during the simulation is shown in Fig. a for the 4 km mesh resolution. The amount of mixing clearly depends on the grid Reynolds number, with RPE being roughly twice as high for Reh=20 compared to Reh=2. The average rate of change of RPE, averaged over days 3 to 319, is shown in Fig. b for all the simulations. As expected, the rate of change increases with larger grid Reynolds number and with a coarser mesh. These RPE metrics are in good agreement with results in the literature. At Reh=20 Thetis dRPE/dt, values are 4.3×10-4 and 2.2×10-4 W m-2, for the 10 and 4 km resolutions. The corresponding values for MITgcm, Modular Ocean Model (MOM), and Parallel Ocean Program (POP) (averaged over days 3 to 319) are larger: at least 8×10-4 and 3×10-4 W m-2, respectively Fig. 12. reported similar values for MITgcm and MOM. On a hexagonal mesh, MPAS-Ocean yields smaller dRPE/dt values: approximately 2×10-4 and 7×10-5 W m-2 for the two resolutions, respectively values averaged over days 1–320; see Fig. 12 in. With a quad mesh, however, MPAS-Ocean values are approximately 2×10-4 W m-2 for both resolutions and therefore close to Thetis performance.

The test cases were run on a Linux cluster with 16-core Intel Xeon E5620 processors and Mellanox Infiniband interconnect. The 320-day simulation took roughly 42 h to run on 96 cores with the 4 km resolution mesh and 140.26 s time step. It should be noted, however, that the time step employed here is smaller than the maximal allowed time step. We also carried out a strong scaling test with the 4 km mesh. In the scaling test, the simulation was run for 40 time steps, recording the total elapsed wall-clock time and time spent in different parts of the solver. Figure a shows the overall speed-up up to 96 processors. The scaling efficiency drops to roughly 50 % at 96 cores, when the local degree of freedom count for the tracer field is 25 000 (see black line in Fig. b). This scaling efficiency is close to typical Firedrake performance .

The scaling efficiency of the separate solvers is plotted with colored lines in Fig. b. The implicit vertical diffusion/viscosity solvers perform best due to the fact that the problem is purely local without any horizontal dependencies. The explicit momentum solver scales almost as well, whereas the explicit tracer solver scales worse. The implicit 2-D solver (assembly and linear solve) scales the poorest because the problem is relatively small; at 96 cores, there are only around 940 degrees of freedom in the (u‾, η) system per core. We have also experimented with explicit 2-D solvers, but they do not scale significantly better compared to the two-stage implicit scheme used herein.

To further assess the CPU cost, we compared Thetis timing against the SLIM 3-D model which uses a similar DG formulation but is implemented in C/C++. The wall-clock time, and parallel efficiency used by both Thetis and SLIM 3-D are presented in Appendix . The setup, mesh, and time step were identical for the two models. On a single core, Thetis runs 3.3 times faster. On 24 cores, the ratio is 4.0, and on 144 cores Thetis is still 2.2 times faster than SLIM 3-D. This highlights the fact that Firedrake can deliver good parallel performance compared to models written in lower level languages.

It should be noted, however, that Thetis performance is currently not fully optimized. We expect that the performance can be significantly improved both in terms of serial and strong scaling performance. These will be addressed in future work.

Next, we investigate the model's ability to simulate density-driven overflows with the DOME benchmark . The domain is a 1100 by 600 km large basin, whose depth varies linearly from 600 m at the northern boundary to 3600 m in the middle of the domain (see Fig. ). To avoid boundary condition issues, we have extended the domain to the west by 120 km. At the northern boundary, there is a 100 km wide and 200 km long inlet. Initially, the basin is stably stratified with a linear temperature variation from 10 ∘C in the deepest part of the basin to 20 ∘C at the surface. We use the linear equation of state with ρ0=1000 kg m-3, αT=0.2 kg m-3 ∘C-1, and T0=10 ∘C, resulting in a Δρ=2.0 kg m-3 density difference.

At the inlet, a dense inflow (temperature 10 ∘C) is prescribed in the bottom layer, with the surface layer being at 20 ∘C. The inflow is in geostrophic balance, the thickness of the bottom layer being roughly 300 m on the eastern end of the boundary diminishing exponentially westward . The total inflow in the bottom layer is 5 Sv (5×106 m3 s-1), the surface layer being static. During the simulation, the fate of the inflowing waters is tracked with a passive tracer that is initially zero in the basin and unity at the inlet. Initially, the tracer field is set to the inflow conditions in the northern part of the basin (y>650 km). Velocity is set to zero everywhere. The eastern and southern boundaries of the basin are closed. The western boundary is open with radiation boundary conditions and a 100 km wide band where the temperature is relaxed to the initial condition.

The domain is discretized with an unstructured grid (Fig. ). Horizontal mesh resolution is 6 km near the northern boundary, increasing southward. Overall, 24 vertical sigma levels are used. Over the slope, the mesh resolution was designed to result in a hydrostatic consistency metric (r<1.5) . Horizontal viscosity is set to a constant 50 m2 s-1, which corresponds to Reh≈200 at the inlet. Horizontal diffusivity is constant at 10 m2 s-1. Vertical viscosity and diffusivity are parameterized by the Pacanowski–Philander scheme as described in Sect. . Bottom friction is parameterized with a quadratic drag coefficient Cd=2×10-3 . A quadratic function space is used for the baroclinic head and internal pressure gradient as discussed in Sect. .

As the inflowing current reaches the basin, it turns to the west and forms a coastal plume that is approximately 150 km wide (Fig. ). The plume detaches from the lateral boundary as it flows westward and along the bottom slope. As the dense water mass meets the stratified ocean, the plume becomes unstable and starts to generate eddies and internal waves. The most vigorous eddies are found in the first 300 km after the inlet (x=500–800 km), after which the plume is more mixed and quiescent. Overall, the plume is shallow; most of the passive tracer is concentrated within 200 m of the bottom. Qualitatively, the extent and propagation of the plume, and its eddy structure are in good agreement with the literature e.g.,. The results show that Thetis is able to represent eddying flows over sloping bathymetry, generating and maintaining strong gradients between water masses. The sharpest fronts in the simulation encompass only one or two elements.

Figure  shows the distribution of the inflowing tracer concentration as a function of water density and the x axis. The inflowing waters are initially very dense but get mixed to lower density as the plume advances along the coast. The histogram shows that the plume volume is low in the first 150 km after the inlet (x=650–800 km) where the plume accelerates. After x=650 km, the plume slows down and starts to accumulate in volume. The density of the main plume occupies ranges from 0.8 to 1.5 kg m-3, the peak being around 1.28 kg m-3. The rate of entrainment can be used as a metric for mixing. Results herein are similar to those presented in literature: present a mean density anomaly of 1.5 kg m-3 for their terrain-following FESOM model configuration.

The 47-day simulation took roughly 42 h to run on 90 cores with a 39.65 s time step on the same Linux cluster.