Hamiltonian particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation

We develop a particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation. The method is based on the recently proposed Hamiltonian particle-mesh (HPM) method and the interpretation of the rigid-lid approximation as a set of holonomic constraints. The suggested spatial discretization leads to a constrained Hamiltonian system of ODEs which is integrated in time using a variant of the symplectic SHAKE/RATTLE algorithm. It is demonstrated that the elimination of external gravity waves by the rigid-lid approximation can be achieved in a computationally stable and efficient way

On the rate of convergence of the Hamiltonian particle-mesh method

The Hamiltonian Particle-Mesh (HPM) method is a particle-in-cell method for compressible fluid flow with Hamiltonian structure. We present a numer- ical short-time study of the rate of convergence of HPM in terms of its three main governing parameters. We find that the rate of convergence is much better than the best available theoretical estimates. Our results indicate that HPM performs best when the number of particles is on the order of the number of grid cells, the HPM global smoothing kernel has fast decay in Fourier space, and the HPM local interpolation kernel is a cubic spline

Modelling of nonlinear wave-buoy dynamics using constrained variational methods

We consider a comprehensive mathematical and numerical strategy to couple water-wave motion with rigid ship dynamics using variational principles. We present a methodology that applies to three-dimensional potential flow water waves and ship dynamics. For simplicity, in this paper we demonstrate the method for shallow-water waves coupled to buoy motion in two dimensions, the latter being the symmetric motion of a crosssection of a ship. The novelty in the presented model is that it employs a Lagrange multiplier to impose a physical restriction on the water height under the buoy in the form of an inequality constraint. A system of evolution equations can be obtained from the model and consists of the classical shallow-water equations for shallow, incompressible and irrotational waves, and relevant equations for the dynamics of the wave-energy buoy. One of the advantages of the variational approach followed is that, when combined with symplectic integrators, it eliminates any numerical damping and preserves the discrete energy; this is confirmed in our numerical results

Compatible finite element methods for geophysical fluid dynamics

This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz decomposition, which has been used to exclude the possibility of several types of spurious oscillations from linear equations of geophysical flow. This means that compatible finite element spaces provide a useful framework for building dynamical cores. In this article we introduce the main concepts of compatible finite element spaces, and discuss their wave propagation properties. We survey some methods for discretising the transport terms that arise in dynamical core equation systems, and provide some example discretisations, briefly discussing their iterative solution. Then we focus on the recent use of compatible finite element spaces in designing structure preserving methods, surveying variational discretisations, Poisson bracket discretisations, and consistent vorticity transport.Comment: correction of some typo

Mathematical Theory and Modelling in Atmosphere-Ocean Science

Mathematical theory and modelling in atmosphere-ocean science combines a broad range of advanced mathematical and numerical techniques and research directions. This includes the asymptotic analysis of multiscale systems, the deterministic and stochastic modelling of sub-grid-scale processes, and the numerical analysis of nonlinear PDEs over a broad range of spatial and temporal scales. This workshop brought together applied mathematicians and experts in the disciplinary fields of meteorology and oceanography for a wide-ranging exchange of ideas and results in this area with the aim of fostering fundamental interdisciplinary work in this important science area

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time

Theory, Experiments, and Simulations of Internal Waves in Deep Water

This dissertation concerns internal waves occurring in density-stratified fluids wherein one layer is much deeper than the other. Such stratifications are characteristic of internal waves in the ocean, but despite the practical importance of such phenomena, the best-known models fail to capture some of the features of large-amplitude waves. In this thesis we derive a model suitable for large-amplitude waves in the deep regime. We are specifically interested in solitary internal waves, which arise as a balance between nonlinear and dispersive terms. The nonlinearity is intrinsic to the fluid dynamics system and, in contrast with well-studied models, we make no assumptions on the maximum amplitude of the waves. We will, however, exploit the fact that the waves are typically long with respect to the upper layer fluid, and average physical quantities over this layer. The dispersive part of the model comes from pressure contribution of the deep lower layer, and arises as an integral operator. On account of the nonlinearity and the nonlocality of the model, the initial value problem is analytically intractable. Thus we produce a numerical method for solving the model equations which is fast, accurate and flexible. The method exploits the physical properties of the model, specifically by using the natural variables which arise through the Hamiltonian formalism. The numerical method is shown to be pseudospectrally accurate in space and fourth-order accurate in time. Finally we compare the model results to laboratory experiments and show that the model does a reasonably good job at capturing features of large-amplitude solitary waves. These experiments are performed on a much larger scale than results available in the literature, as well as the being the first experiments using miscible fluids (fresh and salt water) in the deep regime. Some mathematical results pertaining to variational principles for stratified fluids are contained in the appendices. New results contained herein apply to two-layer models with the fluid interface in contact with the boundary of the fluid domain, and the Hamiltonian principle for the incompressible, variable-density fluid motion.Doctor of Philosoph