Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes

Space-time discontinuous Galerkin discretization of rotating shallow water equations on moving grids

A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova's discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge-Kutta method. A thorough verification of the space-time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore-vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space-time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid

A water wave model with horizontal circulation and accurate dispersion

We describe a new water wave model which is variational, and combines a depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite element profile with a small number of elements (say), leading to a framework for efficient modelling of the interaction of steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical vorticity (or circulation). We show that the potential flow water wave equations and the shallow-water equations are recovered in the relevant limits, and provide approximate shock relations for the model which can be used in numerical schemes to model breaking waves

Parcel Eulerian-Lagrangian fluid dynamics for rotating geophysical flows

Parcel Eulerian-Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations and in alternative explanations of fluid parcel (in) stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcel's position. In this paper, we derive the geometric link between the parcel Eulerian-Lagrangian formulation and well-known variational and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity-stream function dynamics, the rotating two-dimensional shallow-water equations and the rotating three-dimensional compressible Euler equations

Space-time discontinuous Galerkin finite element method for shallow water flows

A space-time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in non-linear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.\u

Constrained isentropic models of tropospheric dynamics

A two-layer isentropic model consisting of a tropospheric and a stratospheric layer is simplified using perturbation analysis while preserving the Hamiltonian structure. The first approximation applies when the thickness of the stratospheric layer is much larger than the tropospheric layer, such that the Froude number of the stratospheric layer is a small number. Using leading-order perturbation theory in the Hamiltonian formulation yields a conservative one-and-a-half isentropic layer model. Furthermore, when the Rossby number in this active lower layer is small, Hamiltonian theory either directly leads to (Salmon's) L1-dynamics using a geostrophic constraint, following a more concise derivation than shown before, or yields quasigeostrophic dynamics. The extension to multilayer isentropic balanced models for use in idealized climate forecasting is discussed

Inertial waves in a rectangular parallelepiped

A study of inertial gyroscopic waves in a rotating homogeneous fluid is undertaken both theoretically and numerically. A novel approach is presented to construct a semi-analytical solution of a linear three-dimensional fluid flow in a rotating rectangular parallelepiped bounded by solid walls. The three-dimensional solution is expanded in vertical modes to reduce the dynamics to the horizontal plane. On this horizontal plane the two dimensional solution is constructed via superposition of 'inertial' analogs of surface Poincar\'{e} and Kelvin waves reflecting from the walls. The infinite sum of inertial Poincar\'{e} waves has to cancel the normal flow of two inertial Kelvin waves near the boundaries. The wave system corresponding to every vertical mode results in an eigenvalue problem. Corresponding computations for rotationally modified surface gravity waves are in agreement with numerical values obtained by Taylor (1921), Rao (1966) and also, for inertial waves, by Maas (2003) upon truncation of an infinite matrix. The present approach enhances the currently available, structurally concise modal solution introduced by Maas (2003). In contrast to Maas' approach, our solution does not have any convergence issues in the interior and does not suffer from Gibbs phenomenon at the boundaries. Additionally, an alternative finite element method is used to contrast these two semi-analytical solutions with a purely numerical one. The main differences are discussed for a particular example and one eigenfrequency

Two-dimensional magma-repository interactions

Two-dimensional simulations of magma-repository interactions reveal that the three phases --a shock tube, shock reflection and amplification, and shock attenuation and decay phase-- in a one-dimensional flow tube model have a precursor. This newly identified phase ``zero'' consists of the impact of magma against the drift roof, after breakthrough into the drift, which results into a large pressure pulse. This initial pressure pulse against the tunnel roof can be estimated using one-dimensional shock reflection models in the vertical direction. After phase zero there is a large-pressure region near the breakthrough and the subsequent phases: a shock tube phase, a shock reflection phase at the closed end of the drift and an attenuation phase are similar as in the one-dimensional flow tube model. The flow along the drift then becomes nearly one dimensional due to the large length to diameter ratio of the drift. Most notably, the pressure reflection pulse at the closed tunnel end is (much) larger than the initial pressure pulse at the tunnel roof. The presented simulations are preliminary in that they are purely two dimensional. Improved values of pressure pulses and volume fluxes of magma will be calculated in the future by incorporating the proper three-dimensional effects into averaged two-dimensional flow area models. The model considered herein is a necessary precursor of these future models