Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

Self Cite

We present an energy conserving space discretisation of the rotating shallow water equations using compatible finite elements. It is based on an energy and enstrophy conserving Hamiltonian formulation as described in McRae and Cotter (2014), and extends it to include upwinding in the velocity and depth advection to increase stability. Upwinding for velocity in an energy conserving context was introduced for the incompressible Euler equations in Natale and Cotter (2017), while upwinding in the depth field in a Hamiltonian finite element context is newly described here. The energy conserving property is validated by coupling the spatial discretisation to an energy conserving time discretisation. Further, the discretisation is demonstrated to lead to an improved field development with respect to stability when upwinding in the depth field is included.

Section: 9)mentioning

confidence: 99%

Energy conserving upwinded compatible finite element schemes for the rotating shallow water equations

Wimmer

Cotter

Bauer

2020

Self Cite

“…which is precisely the correct time derivative term. Applying this approach and using functional derivatives (65) - (67) Due to the choice of spaces, (77) holds pointwise, not just in an integral sense. Additionally, (65) can be directly substituted into (78), leaving only the auxiliary equations (66) and (67) to be solved.…”

Section: Discrete Equations Of Motionmentioning

confidence: 99%

A quasi-Hamiltonian discretization of the thermal shallow water equations

Eldred

Dubos

Kritsikis

2019

The rotating shallow water (RSW) equations are the usual testbed for the development of numerical methods for three-dimensional atmospheric and oceanic models. However, an arguably more useful set of equations are the thermal shallow water equations (TSW), which introduce an additional thermodynamic scalar but retain the single layer, two-dimensional structure of the RSW. As a stepping stone towards a three-dimensional atmospheric dynamical core, this work presents a quasi-Hamiltonian discretization of the thermal shallow water equations using compatible Galerkin methods, building on previous work done for the shallow water equations. Structure-preserving or quasi-Hamiltonian discretizations methods, that discretize the Hamiltonian structure of the equations of motion rather than the equations of motion themselves, have proven to be a powerful tool for the development of models with discrete conservation properties. By combining these ideas with an energy-conserving Poisson time integrator and a careful choice of Galerkin spaces, a large set of desirable properties can be achieved. In particular, for the first time total mass, buoyancy and energy are conserved to machine precision in the fully discrete model.

“…Since this test function is applied to the entire equation, this does not alter the residual formulation, only the nature of the test functions, and hence the scheme is expected to remain consistent at the appropriate order. SUPG was first applied to the Euler equations in streamfunction-vorticity formulation by Tezduyar et al (1988); Tezduyar (1989), and the multiscale behaviour of the resulting scheme was examined by Natale and Cotter (2017). The SUPG modification of the energy-enstrophy conserved shallow water scheme of this paper is obtained by replacing q δ in Equation (51) by q * given by…”

Section: Energy-conserving Enstrophy-dissipating Schemementioning

confidence: 99%

“…Eldred et al (2016) constructed compatible spaces from splines that allow higher-order approximations constructed around the low-order C-grid data structure, and Lee et al (2017) used mimetic spectral elements. In the context of imcompressible two-dimensional turbulence, Natale and Cotter (2017) considered consistent energy-conserving/enstrophy-dissipating finite element schemes, including a formulation that extends to a consistent energy-conserving enstrophy-dissipating version of the McRae and Cotter (2014) scheme, and showed that these schemes have favourable turbulent backscatter properties.…”

Section: Introductionmentioning

confidence: 99%

Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

Bauer¹,

Cotter²

2018

We describe an energy-enstrophy conserving discretisation for the rotating shallow water equations with slip boundary conditions. This relaxes the assumption of boundary-free domains (periodic solutions or the surface of a sphere, for example) in the energy-enstrophy conserving formulation of McRae and Cotter (2014). This discretisation requires extra prognostic vorticity variables on the boundary in addition to the prognostic velocity and layer depth variables. The energy-enstrophy conservation properties hold for any appropriate set of compatible finite element spaces defined on arbitrary meshes with arbitrary boundaries. We demonstrate the conservation properties of the scheme with numerical solutions on a rotating hemisphere. arXiv:1801.00691v3 [math.NA]