A robust well-balanced scheme for multi-layer shallow water equations

International audienceThe numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves nonconservative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete entropy inequality and the well-balanced property, without any theoretical difficulty related to the loss of hyperbolicity. However, this scheme has been shown to often give wrong solutions. We introduce here a variant of the splitting scheme, that has the overall property of being conservative in the total momentum. It is based on a source-centered hydrostatic scheme for the one-layer shallow water system, a variant of the hydrostatic scheme. The final method enables to treat an arbitrary number of layers, with arbitrary densities and arbitrary topography. It has no restriction concerning complex eigenvalues, it is well-balanced and it is able to treat vacuum, it satisfies a semi-discrete entropy inequality. The scheme is fast to execute, as is the one-layer hydrostatic method

A Well-Balanced Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations

We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients---the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. Motivated by future applications to large-scale motions in the ocean and atmosphere, the model is considered on the tangent plane to a rotating planet both in mid-latitudes and at the Equator. The numerical scheme is shown to be capable of quite accurately maintaining the equilibrium states in the presence of nontrivial topography and rotation. Prior to numerical simulations, an analysis of the TRSW model based on the use of Lagrangian variables is presented, allowing one to obtain criteria of existence and uniqueness of the equilibrium state, of the wave-breaking and shock formation, and of instability development out of given initial conditions. The established criteria are confirmed in the conducted numerical experiments

Generation of inertia-gravity waves by pulsating lens-like axisymmetric vortices

We consider interactions between the two most important components of the atmosphere and ocean dynamics: slowly evolving vortical motion and inertia-gravity waves in rotating stratified axisymmetric flows. Any steady axisymmetric solution for a finite volume anticyclonic vortex with outcropping isopycnals is known to correspond to a set of self-similar analytical time-periodic pulson solutions assuming flows in surrounding fluid is negligible. Here we analyze the flow patterns generated in homogeneous fluid below stratified pulsating lens-like vortex and its feedback on the upper layer vortex

Geostrophic instabilities of a front in a stratified rotating fluid

Les instabilités d'une région de front dans un fluide stratifié en rotation sont étudiées. On s'intéresse plus particulièrement à l'instabilité dite de Rossby-Kelvin, qui existe grâce au couplage d'une onde de Rossby et d'une onde Kelvin. La stabilité linéaire d'un front dans un fluide à deux couches est étudiée numériquement par la méthode de collocation. Les résultats de Sakai (1989) pour le cas de deux couches d'égale hauteur sont confirmés et ceci valide notre approche. Celle-ci permet d'étendre ces résultats aux cas non-symétriques et au cas où le front intersecte le fond ou la surface. Ensuite, la stabilité d'un front dans un fluide continuement stratifié est analysée par des simulations numériques à l'aide du modèle méso-échelle WRF (Weather Research and Forecast). L'existence de l'instabilité de Rossby-Kelvin dans un fluide stratifié est ainsi confirmée, avec des taux de croissance comparables au cas du fluide à deux couches

Geostrophic adjustement of density fronts: what do we learn from recent laboratory experiments ?

We present here a rapid review on recent laboratory investigations on geostrophic adjustment of density fronts. Several configurations were studied: warm core lens, cyclonic-anticyclonic PV patches and uniform PV front. The geostrophic adjustment is the first dynamical process which converts a significant fraction of the potential energy input of the atmosphere and the ocean into kinetic energy. According to the cases we studied we have shown that during this rapid adjustment toward a quasi-equilibrium state, an important part of the initial energy could be transferred to wave motion or dissipated by small-scale non-hydrostatic instabilities. A mean adjusted state is always reached after one or two inertial period. Even if a strong wave activity is present in the initial region of unbalance, the time-averaged mean flow could nevertheless be adjusted. We have shown that the wave modes frequency is concentrated around the inertial frequency. Besides, some anticyclonic structures may also exhibit sub-inertial o

Variational discretization for rotating stratified fluids

In this paper we develop and test a structure-preserving discretization scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25,9] and is derived via a discrete version of the Euler-Poincaré variational formulation of rotating stratified fluids. The resulting variational integrator allows for a discrete version of Kelvin circulation theorem, is applicable to irregular meshes and, being symplectic, exhibits excellent long term energy behavior. We then report a series of preliminary tests for rotating stratified flows in configurations that are symmetric with respect to translation along one of the spatial directions. In the benchmark processes of hydrostatic and/or geostrophic adjustments, these tests show that the slow and fast component of the flow are correctly reproduced. The harder test of inertial instability is in full agreement with the common knowledge of the process of development and saturation of this instability, while preserving energy nearly perfectly and respecting conservation laws

Nonlinear dynamics of rotating shallow water: methods and advances

The rotating shallow water (RSW) model is of wide use as a conceptual tool in geophysical fluid dynamics (GFD), because, in spite of its simplicity, it contains all essential ingredients of atmosphere and ocean dynamics at the synoptic scale, especially in its two- (or multi-) layer version. The book describes recent advances in understanding (in the framework of RSW and related models) of some fundamental GFD problems, such as existence of the slow manifold, dynamical splitting of fast (inertia-gravity waves) and slow (vortices, Rossby waves) motions, nonlinear geostrophic adjustment and w

Geophysical fluid dynamics: understanding (almost) everything with rotating shallow water models

The book explains the key notions and fundamental processes in the dynamics of the fluid envelopes of the Earth (transposable to other planets), and methods of their analysis, from the unifying viewpoint of rotating shallow-water model (RSW). The model, in its one- or two-layer versions, plays a distinguished role in geophysical fluid dynamics, having been used for around a century for conceptual understanding of various phenomena, for elaboration of approaches and methods, to be applied later in more complete models, for development and testing of numerical codes and schemes of data assimilations, and many other purposes. Principles of modelling of large-scale atmospheric and oceanic flows, and corresponding approximations, are explained and it is shown how single- and multi-layer versions of RSW arise from the primitive equations by vertical averaging, and how further time-averaging produces celebrated quasi-geostrophic reductions of the model. Key concepts of geophysical fluid dynamics are exposed and interpreted in RSW terms, and fundamentals of vortex and wave dynamics are explained in Part 1 of the book, which is supplied with exercises and can be used as a textbook. Solutions of the problems are available at Editorial Office by request. In-depth treatment of dynamical processes, with special accent on the primordial process of geostrophic adjustment, on instabilities in geophysical flows, vortex and wave turbulence and on nonlinear wave interactions follows in Part 2. Recently arisen new approaches in, and applications of RSW, including moist-convective processes constitute Part 3