Two-layer Shallow Water System Research Articles

Discussion on different numerical treatments on the loss of hyperbolicity for the two-layer shallow water system

This paper focus on the numerical approximation of two-layer shallow water system. First, a new approximation of the eigenvalues of the system is proposed, which satisfies some interesting properties. From this approximation, we give an accurate estimation of the non-hyperbolic region, which improves significantly the one computed with the classic eigenvalues approximation. In particular, we estimate both the lower and upper boundaries of the non-hyperbolic region. We also give a simple algorithm that allows us to compute bounds for the external eigenvalues, even when complex eigenvalues arise. We design an efficient FV solver depending on the hyperbolic nature of the system, that combines the PVM-2U-FL scheme and a new solver introduced in this work, named IFCP-FL, which degenerates to the Lax–Wendroff method in smooth areas. Different strategies for the numerical treatment for the loss of hyperbolicity are considered and discussed. Two of them are based on adding a friction term, depending on the classic or the new eigenvalues approximations, and the third one considers a simplified hyperbolic model in areas with complex eigenvalues. Some numerical tests are performed, including the case of two-layer fluids with different ratio of densities. The application to a two-layer model for submarine landslides is also considered. In the latter case, we show how the treatments based on adding friction are not appropriate for this kind of applications, whereas the treatment based on changing to a hyperbolic model produces much better results.

On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations

This works concerns the study of well-balanced Lagrange-projection schemes applied to the two-layer shallow water system. In particular, a formulation of the mathematical model in Lagrangian coordinates is proposed. The HLL method is then applied to a simplified version of the resulting Lagrangian system. Furthermore, based on the acoustic-transport splitting interpretation, another approximate Riemann solver for the acoustic-Lagrangian step is described. Both an explicit and an implicit-explicit method are proposed, where the latter can allow very fast simulations in sub-critical regimes. Finally, we show numerical simulations in which the outputs are compared with the IFCP method’s results.

Path-conservative central-upwind schemes for nonconservative hyperbolic systems

We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box’’ solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution.

Finite-volume schemes for shallow-water equations

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

Simulation of tsunamis generated by landslides using adaptive mesh refinement on GPU

Adaptive mesh refinement (AMR) is a widely used technique to accelerate computationally intensive simulations, which consists of dynamically increasing the spatial resolution of the areas of interest of the domain as the simulation advances. During the last years there have appeared many publications that tackle the implementation of AMR-based applications in GPUs in order to take advantage of their massively parallel architecture. In this paper we present the first AMR-based application implemented on GPU for the simulation of tsunamis generated by landslides by using a two-layer shallow water system. We also propose a new strategy for the interpolation and projection of the values of the fine cells in the AMR algorithm based on the fluctuations of the state values instead of the usual approach of considering the current state values. Numerical experiments on artificial and realistic problems show the validity and efficiency of the solver.

Dynamic process simulation with a Savage-Hutter type model for the intrusion of landslide into river

Natural damming of rivers by mass movements is a very common and potentially dangerous phenomena which has been documented all over the world. In this paper, a two-layer model of Savage-Hutter type is presented to simulate the dynamic procedure for the intrusion of landslide into rivers. The two-layer shallow water system is derived by depth averaging the incompressible Navier-Stokes equations with the hydrostatic assumption. A high order accuracy scheme based on the finite volume method is proposed to solve the presented model equations. Several numerical tests are performed to verify the realiability and feasibility of the proposed model. The numerical results indicate that the proposed method can be competent for simulating the dynamic process of landslide intrusion into the river. The interaction effect between both layers has a significant impact on the landslide movement, water fluctuation and wave propagation.

A two-layer model for simulating landslide dam over mobile river beds

Landslides can block mountainous streams and form landslide dams to threaten downstream residents. It is necessary for reliable methods to predict landslide dam dynamic for risk assessment. In this paper, we present a two-layer model of Savage-Hutter type to simulate the dynamic evolution of landslide dam which take account of the erosion of river bed. The two-layer shallow water system is derived by depth-averaging the incompressible Navier-Stokes equations with the hydrostatic assumption integrated of the erosion model of river bed. The effect of excess pore water pressure is considered in the erosion process. A high order accuracy scheme based on Roe-type solver is used to discretize the present model. Finally, several numerical tests are performed to verify the stability of the algorithm and reliability of the model. Numerical results indicate that the erosion effect enhances the huge destructiveness of landslide and increase the possibility of river blocked by landslides. The impact of excess pore water pressure on erosion process should be considered.

THREE-LAYER APPROXIMATION OF TWO-LAYER SHALLOW WATER EQUATIONS

Two-layer shallow water equations describe flows that consist of two layers of inviscid fluid of different (constant) densities flowing over bottom topography. Unlike the single-layer shallow water system, the two-layer one is only conditionally hyperbolic: the system loses its hyperbolicity because of the momentum exchange terms between the layers and as a result its solutions may develop instabilities. We study a three-layer approximation of the two-layer shallow water equations by introducing an intermediate layer of a small depth. We examine the hyperbolicity range of the three-layer model and demonstrate that while it still may lose hyperbolicity, the three-layer approximation may improve stability properties of the two-layer shallow water system.

Airborne Measurements of Coastal Jet Transition around Point Conception, California

Abstract Low-level winds along the Californian coast during spring and early summer are typically strong and contained within the cool, well-mixed marine boundary layer (MBL). A temperature inversion separates the MBL from the warmer free troposphere. This setup is often represented by a two-layer shallow-water system with a lateral boundary. Near a prominent point such as Point Conception, California, the fast-moving MBL flow is supercritical and can exhibit distinct features including a compression bulge and an expansion fan. Measurements from the University of Wyoming King Air research aircraft on 19 May 2012 during the Precision Atmospheric MBL Experiment (PreAMBLE) captured wind in excess of 14 m s−1 off of Point Conception under clear skies and wind ~2 m s−1 east of San Miguel in the California Bight. A compression bulge was identified upwind of Point Conception. When the flow rounds the point, the MBL undergoes a near collapse and there is a spike in MBL height embedded in the general decrease of MBL height with greater turbulence just downwind that is associated with greater mixing through the inversion layer. Lidar and in situ measurements reveal that transport of continental aerosol is present near the pronounced MBL height change and that there is a complex vertical structure within the Santa Barbara Channel. Horizontal pressure gradients are obtained by measuring the slope of an isobaric surface. Observations of wind and pressure perturbations are able to be linked through a simple Bernoulli relationship. Variation of MBL depth explains most, but not all of the variation of the isobaric surface.

Simulation of the Lock-Exchange Hydraulics using the Discontinuous Galerkin Method

simulations of the lock-exchange hydraulics have been carried out using a discontinuous Galerkin finite element method. The basic water circulation in the lock-exchange hydraulics consists in an upper layer of cold, fresh surface water and an opposite deep current of warmer, salty outflowing water. The governing equations are the well- established two-layer shallow water system including bathymetric forces. The considered discontinuous Galerkin method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across interelement boundaries; this property renders the method ideally suited for the hp-adaptivity. The proposed method can handle complex topography using unstructured grids and it satisfies the conservation property. Several numerical results are presented to demonstrate the high resolution of the proposed method and to confirm its capability to provide accurate and efficient simulations for the lock-exchange hydraulics. KeywordsGalerkin method; Two-layer shallow water equations; Finite element; Lock-exchange hydraulics.

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

The goal of this article is to design a new approximate Riemann solver for the two-layer shallow water system which is fast compared to Roe schemes and accurate compared to Lax-Friedrichs, FORCE, or GFORCE schemes (see Castro et al. in Math. Comput. 79:1427–1472, 2010). This Riemann solver is based on a suitable decomposition of a Roe matrix (see Toumi in J. Comput. Phys. 102(2):360–373, 1992) by means of a parabolic viscosity matrix (see Degond et al. in C. R. Acad. Sci. Paris 1 328:479–483, 1999) that captures some information concerning the intermediate characteristic fields. The corresponding first order numerical scheme, which is called IFCP (Intermediate Field Capturing Parabola) is linearly L ∞-stable, well-balanced, and it doesn’t require an entropy-fix technique. Some numerical experiments are presented to compare the behavior of this new scheme with Roe and GFORCE methods.

Numerical Treatment of the Loss of Hyperbolicity of the Two-Layer Shallow-Water System

This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an `optimal' amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.

Two-Layer Shallow Water System: A Relaxation Approach

The two-layer shallow water system is an averaged flow model. It forms a nonconservative system which is only conditionally hyperbolic. The coupling between the layers, due to the hydrostatic pressure assumption, does not provide explicit access to the system eigenstructure, which is inconvenient for Riemann solution based numerical schemes. We consider a relaxation approach which offers greater decoupling and accessible eigenstructure. The stability of the model is discussed. Numerical results are shown for unsteady flows as well as for smooth and nonsmooth steady flows.

An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.

WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.

Rossby Wave Instability and Apparent Phase Speeds in Large Ocean Basins

Abstract The stability of baroclinic Rossby waves in large ocean basins is examined, and the quasigeostrophic (QG) results of LaCasce and Pedlosky are generalized. First, stability equations are derived for perturbations on large-scale waves, using the two-layer shallow-water system. These equations resemble the QG stability equations, except that they retain the variation of the internal deformation radius with latitude. The equations are solved numerically for different initial conditions through eigenmode calculations and time stepping. The fastest-growing eigenmodes are intensified at high latitudes, and the slower-growing modes are intensified at lower latitudes. All of the modes have meridional scales and growth times that are comparable to the deformation radius in the latitude range where the eigenmode is intensified. This is what one would expect if one had applied QG theory in latitude bands. The evolution of large-scale waves was then simulated using the Regional Ocean Modeling System primitive equation model. The results are consistent with the theoretical predictions, with deformation-scale perturbations growing at rates inversely proportional to the local deformation radius. The waves succumb to the perturbations at the mid- to high latitudes, but are able to cross the basin at low latitudes before doing so. Also, the barotropic waves produced by the instability propagate faster than the baroclinic long-wave speed, which may explain the discrepancy in speeds noted by Chelton and Schlax.

AQ-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system

The goal of this paper is to construct a rst-order upwind scheme for solving the system of partial dierential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water fluids. This is done by generalizing a numerical scheme presented by Berm udez and V azquez-Cend on (3, 26, 27) for solving one-layer shallow water equations, consisting in a Q-scheme with a suitable treatment of the source terms. The diculty in the two layer system comes from the coupling terms involving some derivatives of the unknowns. Due to these terms, a numerical scheme obtained by performing the upwinding of each layer, independently from the other one, can be unconditionally unstable. In order to dene a suitable numerical scheme with global upwinding, we rst consider an abstract system that generalizes the problem under study. This system is not a system of conservation laws but, nevertheless, Roe's method can be applied to obtain an upwind scheme based on Approximate Riemann State Solvers. Following this, we present some numerical tests to validate the resulting schemes and to highlight the fact that, in general, numerical schemes obtained by applying a Q-scheme to each separate conservation law of the system do not yield good results. First, a simple system of coupled Burgers' equations is considered. Then, the Q-scheme obtained is applied to the two-layer shallow water system.