Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: Inertial waves

Abstract

A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions.The numerical challenges concern: (i) discretisation of a divergence-free velocity field; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved: (i) by employing Dirac's method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow -along the wall; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc's linear algebra routines with our high-level software.We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincaré waves). Additional verifications concerned semianalytical eigenmode solutions in rotating cuboids with solid walls.

A new semi-analytical solution for inertial waves in a rectangular parallelepiped

“…The solutions we have presented have been used to verify the novel numerical technique developed in Nurijanyan et al 26 for the initial-value problem of three-dimensional inertial waves in arbitrarily shaped domains. This method is geared to investigate whether wave attractors (Maas, 9 Manders and Maas, 23 and Rieutord et al 27 ) and complex eigenmodes (such as in Bokhove and Ambati 28 ) emerge in domain shapes of sufficient geometric complexity.…”

Derivation of the Hall and extended magnetohydrodynamics brackets

“…as it should, with the factor of J f being absorbed back into d 3 q when invoking (58). After some work, the q variation gives…”

“…where we have set ∆β ≡ β + − β − and expanded (1/8π)B · B ⋆ using (57), the first equation of (51), and (56). Thankfully, variations are simplified considerably by the result (58). It is also interesting that the usual delta function is replaced by a more specialized Green's function in the (1/8π)B·B ⋆ term.…”

Discrete and continuous hamiltonian systems for wave modelling

“…The solutions we have presented have been used to verify the novel numerical technique developed in [79] for the initial-value problem of threedimensional inertial waves in arbitrarily shaped domains. This method is geared to investigate whether wave attractors ( [61,69,88] ) and complex eigenmodes (such as in [17]) emerge in domain shapes of sufficient geometric complexity.…”

“…Several numerical test cases are discussed, including a simulation of inertial waves in a rotating rectangular parallelepiped and allegedly chaotic wave attractors in a tilted parallelepiped. This chapter is an extended version of the material presented in [79].…”