A mixed mimetic spectral element model of the rotating shallow water equations on the cubed sphere

\u3cp\u3eIn a previous article [J. Comp. Phys. 357 (2018) 282–304] [4], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly varying, non-affine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in H(rot), H(div) and L\u3csub\u3e2\u3c/sub\u3e. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the generalized Piola transformation used to construct the metric terms for the physical field quantities.\u3c/p\u3

Spectral mimetic least-squares method for div-curl systems

\u3cp\u3eIn this paper the spectral mimetic least-squares method is applied to a two-dimensional div-curl system. A test problem is solved on orthogonal and curvilinear meshes and both h- and p-convergence results are presented. The resulting solutions will be pointwise divergence-free for these test problems. For N> 1 optimal convergence rates on an orthogonal and a curvilinear mesh are observed. For N= 1 the method does not converge.\u3c/p\u3

Spectral mimetic least-squares method for curl-curl systems

\u3cp\u3eOne of the most cited disadvantages of least-squares formulations is its lack of conservation. By a suitable choice of least-squares functional and the use of appropriate conforming finite dimensional function spaces, this drawback can be completely removed. Such a mimetic least-squares method is applied to a curl-curl system. Conservation properties will be proved and demonstrated by test results on two-dimensional curvilinear grids.\u3c/p\u3

A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations

\u3cp\u3eIn this work we present a mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured triangular grids. The essential ingredients to achieve this are: (i) a velocity–vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular triangular grids.\u3c/p\u3

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

\u3cp\u3eA mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.\u3c/p\u3

A mimetic spectral element solver for the Grad-Shafranov equation

\u3cp\u3eIn this work we present a robust and accurate arbitrary order solver for the fixed-boundary plasma equilibria in toroidally axisymmetric geometries. To achieve this we apply the mimetic spectral element formulation presented in [56] to the solution of the Grad-Shafranov equation. This approach combines a finite volume discretization with the mixed finite element method. In this way the discrete differential operators (∇, ∇×, ∇.) can be represented exactly and metric and all approximation errors are present in the constitutive relations. The result of this formulation is an arbitrary order method even on highly curved meshes. Additionally, the integral of the toroidal current J\u3csub\u3eφ\u3c/sub\u3e is exactly equal to the boundary integral of the poloidal field over the plasma boundary. This property can play an important role in the coupling between equilibrium and transport solvers. The proposed solver is tested on a varied set of plasma cross sections (smooth and with an X-point) and also for a wide range of pressure and toroidal magnetic flux profiles. Equilibria accurate up to machine precision are obtained. Optimal algebraic convergence rates of order p+1 and geometric convergence rates are shown for Soloviev solutions (including high Shafranov shifts), field-reversed configuration (FRC) solutions and spheromak analytical solutions. The robustness of the method is demonstrated for non-linear test cases, in particular on an equilibrium solution with a pressure pedestal.\u3c/p\u3

The impact of wave energy farms in the shoreline wave climate:Portuguese pilot zone case study using Pelamis energy wave devices

This paper describes the study of the impact of energy absorption by wave farms on the nearshore wave climate and, in special, the influence of the incident wave conditions and the number and position of the wave farms, on the nearshore wave characteristics is studied and discussed. The study was applied to the maritime zone at the West coast off Portugal, namely in front of São Pedro de Moel, where it is foreseen the deployment of offshore wave energy prototypes and farms between the 30 m and 90 m bathymetric lines, with an area of 320 Km2. In this study the REFDIF model was adapted in order to model the energy extraction by wave farms. Three different sinusoidal incident wave conditions were considered. Five different wave farm configurations, varying the position of the wave farm, its number and the width of the navigation channels at each wave farm were analysed. The results for each configuration in terms of the change of the wave characteristics (wave height and wave direction) at the nearshore are presented, compared and discussed for three representative wave conditions

Dependence on plasma shape and plasma fueling for small edge-localized mode regimes in TCV and ASDEX Upgrade

\u3cp\u3eWithin the EUROfusion MST1 work package, a series of experiments has been conducted on AUG and TCV devices to disentangle the role of plasma fueling and plasma shape for the onset of small ELM regimes. On both devices, small ELM regimes with high confinement are achieved if and only if two conditions are fulfilled at the same time. Firstly, the plasma density at the separatrix must be large enough (n\u3csub\u3ee,sep\u3c/sub\u3e/n\u3csub\u3eG\u3c/sub\u3e ∼ 0.3), leading to a pressure profile flattening at the separatrix, which stabilizes type-I ELMs. Secondly, the magnetic configuration has to be close to a double null (DN), leading to a reduction of the magnetic shear in the extreme vicinity of the separatrix. As a consequence, its stabilizing effect on ballooning modes is weakened.\u3c/p\u3