Multilayer Shallow Water Equations Research Articles

A fully coupled dynamic water-mooring line system: Numerical implementation and applications

Several numerical challenges exist in the analysis of water-mooring line systems which require robust, yet practical, methods to address this type of fully coupled nonlinear dynamic problems. The present study proposes a novel class of numerical techniques for the formulation and implementation of a fully coupled dynamic system which involves water flows and catenary mooring line system. In particular, the three-dimensional water flow model is replaced by a simplified multilayer shallow water system with mass exchange terms between the layers including frictional forces at the bed topography and wind-driven forces at the water free-surface. Coupling conditions between the multilayer shallow water model and the mooring line system are also investigated in the current work. As numerical solvers we implement a fast finite volume method for the multilayer shallow water equations and a nonlinear dynamic analysis for the mooring line based on elastic catenary cable elements. Efficient calculations of the interaction forces between the shallow water flow and the submerged mooring line system and associated numerical implementations are also discussed. The accuracy and computational advantages of the proposed fully coupled system are verified using a series of well-established benchmark problems and wind-driven flows over both flat and non-flat beds. The computational results obtained show high performance the developed model and demonstrate the ability of the method to simulate fully coupled dynamic water-mooring line systems.

A semi-Lagrangian Splitting framework for the simulation of non-hydrostatic free-surface flows

Incorporation of the free-surface dynamics plays an important role in the numerical modelling of non-hydrostatic flows. In the present work a novel semi-Lagrangian splitting (SLS) scheme for the free-surface Euler system is proposed.Using the operator splitting technique, the three main components emerge: the multilayer Shallow Water Equations (mSWE) (enforcing the free-surface kinematics), the Vertical Remeshing Operator (VRO) (that blends the Eulerian and Lagrangian techniques) and the Pressure Correction Operator (PCO) (that is a classic elliptic equation used in projection-type pressure coupling schemes). The first two are solved by two separate Finite Volume schemes, while the latter is resolved in a Finite Difference manner.In order to assess the SLS’s performance, numerical simulations of periodic waves and solitons propagating over variable bathymetry are performed. The last case concerns the propagation of waves on top of a shear current, showcasing the use of the SLS on rotational flows. The results are compared with experiments and analytic relations and the method is proven to be robust and efficient.

A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography

A well-balanced Runge-Kutta discontinuous Galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms. Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkin method is a locally conservative finite element method whose approximate solutions are discontinuous across the inter-element boundaries. The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-LobattoLegendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as a high-order version of upwind finite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standard finite element methods fail. To deal with the source terms we also implement a high-order splitting operator for the time integration. The accuracy of the proposed Runge-Kutta discontinuous Galerkin method is examined for several examples of multilayer free-surface flows over both flat and non-flat beds. The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method. The proposed method is also applied to solve a recirculation flow problem in the Strait of Gibraltar.

Parallel exponential time differencing methods for geophysical flow simulations

Two ocean models are considered for geophysical flow simulations: the multi-layer shallow water equations and the multi-layer primitive equations. For the former, we investigate the parallel performance of exponential time differencing (ETD) methods, including exponential Rosenbrock–Euler, ETD2wave, and B-ETD2wave. For the latter, we take advantage of the splitting of barotropic and baroclinic modes and propose a new two-level method in which an ETD method is applied to solve the fast barotropic mode. These methods could improve the computational efficiency of numerical simulations because ETD methods allow for much larger time step sizes than traditional explicit time-stepping techniques that are commonly used in existing computational ocean models. Several standard benchmark tests for ocean modeling are performed and comparison of the numerical results demonstrates a great potential of applying the parallel ETD methods for simulating real-world geophysical flows.

A surrogate model for efficient quantification of uncertainties in multilayer shallow water flows

In this study, we investigate the implementation of a Proper Orthogonal Decomposition (POD) Polynomial Chaos Expansion (PCE) POD-PCE surrogate model for the propagation and quantification of the uncertainty in hydraulic modelling. The considered model consists of a system of multilayer shallow water equations with a mass exchange between the layers and over stochastic beds. As a numerical solver, we propose a finite volume characteristics method that does not require eigenstructure of the system in its implementation. The method is fast, accurate and can be used for both slowly and rapidly hydraulic simulations. The propagation and influence of several uncertainty parameters are quantified in the considered numerical methods for multilayer shallow water flows. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial Chaos expansions for efficient uncertainty quantification of complex hydraulic problems with a large number of random variables. Numerical results are shown for several test examples including a dam-break problem over a flat bed, and a wind-driven recirculation flow on flat and non-flat bottoms. Results are also presented for the case study of a recirculation flow problem in the Strait of Gibraltar. The results demonstrate the robustness of the uncertainty quantification method compared to the standard Monte-Carlo simulations. The results presented in this study suggest that the use of surrogate modelling may save a considerable amount of the necessary computational cost for all the considered cases.

Simulation of Three-Dimensional Free-SurfaceFlows Using Two-Dimensional Multilayer Shallow Water Equations

Simulation of Three-Dimensional Free-SurfaceFlows Using Two-Dimensional Multilayer Shallow Water Equations

New Numerical Algorithm for the Multi-Layer Shallow Water Equations Based on the Hyperbolic Decomposition and the CABARET Scheme

New Numerical Algorithm for the Multi-Layer Shallow Water Equations Based on the Hyperbolic Decomposition and the CABARET Scheme

Novyy chislennyy algoritm dlya uravneniy mnogosloynoy melkoy vody na osnove giperbolicheskoy dekompozitsii i skhemy KABARE

Novyy chislennyy algoritm dlya uravneniy mnogosloynoy melkoy vody na osnove giperbolicheskoy dekompozitsii i skhemy KABARE

Efficient multilayer shallow-water simulation system based on GPUs

The computational simulation of shallow stratified fluids is a very active research topic because these types of systems are very common in a variety of natural environments. The simulation of such systems can be modeled using multilayer shallow-water equations but do impose important computational requirements, especially when applied to large domains.General Purpose Computing on Graphics Processing Units (GPGPU) has become a vivid research field due to the arrival of massively parallel hardware platforms (based on graphics cards) and adequate programming frameworks which have allowed important speed-up factors with respect to not only sequential but also parallel CPU based simulation systems.In this work we present simulation of shallow stratified fluids with an arbitrary number of layers using GPUs. The designed system does fully adapt to the many-core architecture of modern GPUs and several experiments have been carried out to illustrate its scalability and behavior on different GPU models. We propose a new multilayer computational scheme for an underlying 2D mathematical model. This scheme is capable of handling an arbitrary number of layers. The system adds no overhead when used for two-layer scenarios, compared to an existing 2D system specifically designed for just two layers.Our proposal is aimed at creating a GPU-based computational scheme suitable for the simulation of multilayer large-scale real-world scenarios.

An energy and potential enstrophy conserving numerical scheme for the multi-layer shallow water equations with complete Coriolis force

We present an energy- and potential enstrophy-conserving scheme for the non-traditional shallow water equations that include the complete Coriolis force and topography. These integral conservation properties follow from material conservation of potential vorticity in the continuous shallow water equations. The latter property cannot be preserved by a discretisation on a fixed Eulerian grid, but exact conservation of a discrete energy and a discrete potential enstrophy seems to be an effective substitute that prevents any distortion of the forward and inverse cascades in quasi-two dimensional turbulence through spurious sources and sinks of energy and potential enstrophy, and also increases the robustness of the scheme against nonlinear instabilities. We exploit the existing Arakawa-Lamb scheme for the traditional shallow water equations, reformulated by Salmon as a discretisation of the Hamiltonian and Poisson bracket for this system. The non-rotating, traditional, and our non-traditional shallow water equations all share the same continuous Hamiltonian structure and Poisson bracket, provided one distinguishes between the particle velocity and the canonical momentum per unit mass. We have determined a suitable discretisation of the non-traditional canonical momentum, which includes additional coupling between the layer thickness and velocity fields, and modified the discrete kinetic energy to suppress an internal symmetric computational instability that otherwise arises for multiple layers. The resulting scheme exhibits the expected second-order convergence under spatial grid refinement. We also show that the drifts in the discrete total energy and potential enstrophy due to temporal truncation error may be reduced to machine precision under suitable refinement of the timestep using the third-order Adams-Bashforth or fourth-order Runge-Kutta integration schemes. Hamiltonian-derived scheme exactly conserves energy and potential enstrophy.Energy/enstrophy conserved at machine accuracy for sufficiently small time step.Can be used to incorporate full Coriolis force into existing isopycnal models.

Approximate Osher–Solomon schemes for hyperbolic systems

This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher–Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) [19,20]. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems for ideal gas dynamics, magnetohydrodynamics and multilayer shallow water equations. The numerical tests indicate that the proposed schemes are robust, stable and accurate with a satisfactory time step restriction, and provide an efficient alternative for approximating time-dependent solutions in which the spectral decomposition is computationally expensive.

A Class of Incomplete Riemann Solvers Based on Uniform Rational Approximations to the Absolute Value Function

In this paper we propose a new class of incomplete Riemann solvers, based on approximations in the $$L^\infty $$ L ? -norm to the absolute value function in $$[-1,1]$$ [ - 1 , 1 ] by means of rational functions, for the numerical approximation of the solution of hyperbolic systems of conservation laws. The main idea relies on the construction of a numerical approximation to the viscosity matrix by using an appropriate rational real function $$R(x)$$ R ( x ) , that approximates the function $$|x|$$ | x | uniformly in $$[-1,1]$$ [ - 1 , 1 ] , evaluated at the Jacobian of the fluxes of the hyperbolic system computed at some average value (for example, Roe averages). In addition to the Jacobians of the fluxes we shall use either the maximum in absolute value of the characteristic speeds in each cell or an upper bound of them. Thus, the resulting approximate Riemann solver is incomplete in the sense that we do not use the complete spectral decomposition of the Jacobian. Moreover, the new class of Riemann solvers consists of a hierarchy of schemes running from the more dissipative to the less dissipative ones, and having as limiting case a Roe-like scheme. According to the order of the approximation of the generating rational function used, the degree of dissipation can be dosed for particular applications. We study different rational approximations: Newman-type functions, iterative generated Halley functions, and also Chebyshev polynomial approximants. We test our basic algorithms for different initial value Riemann problems for ideal gas dynamics (HD) and magnetohydrodynamics (MHD) to observe their behavior with respect to challenging scenarios in numerical simulations, including some standard numerical pathologies (e. g., heat conduction, postshock oscillations and overheating) and the formation of compound waves in ideal MHD. We also examine our proposed schemes, by computing the numerical approximation of different initial value problems for nonconservative multilayer shallow water equations, where it has been observed that intermediate waves can be properly captured for an appropriate degree of approximation of the generating rational function used. Our numerical tests indicate that the proposed schemes are robust, running stable and accurate with a satisfactory time step restriction (CFL constant), and the computational cost is more advantageous with respect to schemes that use a complete spectral decomposition of the Jacobians.

A simple multi-layer finite volume solver for density-driven shallow water flows

A simple solver is proposed for the numerical solution of density-driven multi-layer shallow water flows. The governing equations consist on coupling the multi-layer shallow water equations for the hydraulic variables with suspended sediment transport equations for the concentration variables. The layers can be formed in the shallow water model based on the variation of water density which may depend on the water temperature and salinity. At each time step, the method consists of two stages to update the numerical solution. In the first stage, the multi-layer shallow water equations are rewritten in a non-conservative form and the intermediate solutions are calculated using the modified method of characteristics. In the second stage, the numerical fluxes are reconstructed from the intermediate solutions in the first stage and used in the conservative form of the multi-layer shallow water equations. The proposed method avoids Riemann problem solvers and it is suitable for multi-layer shallow water equations on non-flat topography. Several numerical results are presented to illustrate the performance of the proposed finite volume method. The computed results confirm its capability to solve multi-layer shallow water equations for density-driven flows over flat and non-flat bottom topography.

Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear

AbstractWe analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid.

Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves

AbstractWe investigate the behaviour of linear plane waves in multilayer shallow water equations that include a complete treatment of the Coriolis force. These equations improve upon the conventional shallow water equations, based on the traditional approximation, that include only the part of the Coriolis force due to the locally vertical component of the rotation vector. Including the complete Coriolis force leads to dramatic changes in the structure of long linear plane waves. It allows subinertial waves to exist with frequencies below the inertial frequency, the minimum frequency for which waves exist under the traditional approximation. These subinertial waves are characterized by a distinguished limit in which the horizontal pressure gradient becomes comparable to the upwellings and downwellings driven by the non-traditional Coriolis term in the vertical momentum equation. The subinertial waves connect wave modes that remain separate in the conventional multilayer shallow water equations, such as the surface and internal waves in a two-layer system. Eastward-propagating surface waves in a two-layer system connect with westward-propagating internal waves, and vice versa, via the long subinertial waves. The long subinertial waves cannot be classified as either surface or internal waves, due to the phase difference between the disturbances to the interfaces in these waves.

Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane

We derive equations to describe the flow of multiple superposed layers of inviscid, incompressible fluids with constant densities over prescribed topography in a rotating frame. Motivated by geophysical applications, these equations incorporate the complete Coriolis force. We do not make the widely used ‘traditional approximation’ that omits the contribution to the Coriolis force from the locally horizontal part of the rotation vector. Our derivation is performed by averaging the governing Euler equations over each layer, and from two different forms of Hamilton's variational principle that differ in their treatment of the coupling between layers. The coupling may be included implicitly through the map from Lagrangian particle labels to particle coordinates, or explicitly by adding terms representing the work done on each layer by the pressure exerted by the layers above. The latter approach requires additional terms in the Lagrangian, but extends more easily to many layers. We show that our equations obey the expected conservation laws for energy, momentum and potential vorticity. The conserved momentum and potential vorticity are modified by non-traditional effects. The vertical component of the rotation vector that appears in the potential vorticity for each layer under the traditional approximation is replaced by the component perpendicular to the layer's midsurface. The momentum includes an additional contribution that reflects changes in angular momentum caused by changes in a fluid element's distance from the rotation axis as it is displaced vertically. Again, this effect is absent in the traditional approximation.

Multilayer shallow water flow using lattice Boltzmann method with high performance computing

A multilayer lattice Boltzmann (LB) model is introduced to solve three-dimensional wind-driven shallow water flow problems. The multilayer LB model avoids the expensive Navier–Stokes equations and obtains stratified horizontal flow velocities as vertical velocities are relatively small and the flow is still within the shallow water regime. A single relaxation time BGK method is used to solve each layer coupled by the vertical viscosity forcing term. To increase solution stability, an implicit step is suggested to obtain flow velocities. The main advantage of using the LBM is that after selecting appropriate equilibrium distribution functions, the LB algorithm is only slightly modified for each layer and retains all the simplicities of the LBM within the high performance computing (HPC) environment. The performance of the parallel LB model for the multilayer shallow water equations is investigated on CPU-based HPC environments using OpenMP. We found that the explicit loop control with cache optimization in LBM gives better performance on execution time, speedup and efficiency than the implicit loop control as the number of processors increases. Numerical examples are presented to verify the multilayer LB model against analytical solutions. We demonstrate the model’s capability of calculating lateral and vertical distributions of velocities for wind-driven circulation over non-uniform bathymetry.

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws

Linear Baroclinic Instability in Extended Regime Geostrophic Models

The linear wave and baroclinic instability properties of various geostrophic models valid when the Rossby number is small are investigated. The models are the ‘‘L1’’ dynamics, the ‘‘geostrophic potential vorticity’’ equations, and the more familiar quasigeostrophic and planetary geostrophic equations. Multilayer shallow water equations are used as a control. The goal is to determine whether these models accurately portray linear baroclinic instability properties in various geophysically relevant parameter regimes, in a highly idealized and limited set of cases. The L1 and geostrophic potential vorticity models are properly balanced (devoid of inertio-gravity waves, except possibly at solid boundaries), valid on the b plane, and contain both quasigeostrophy and planetary geostrophy as limits in different parameter regimes; hence, they are appropriate models for phenomena that span the deformation and planetary scales of motion. The L1 model also includes the ‘‘frontal geostrophic’’ equations as a third limit. In fact, the choice to investigate such relatively unfamiliar models is motivated precisely by their applicability to multiple scales of motion. The models are cast in multilayer form, and the dispersion properties and eigenfunctions of wave modes and baroclinic instabilities produced are found numerically. It is found that both the L1 and geostrophic potential vorticity models have sensible linear stability properties with no artifactual instabilities or divergences. Their growth rates are very close to those of the shallow water equations in both quasigeostrophic and planetary geostrophic parameter regimes. The growth rate of baroclinic instability in the planetary geostrophic equations is shown to be generally less than the growth rate of the other models near the deformation radius. The growth rate of the planetary geostrophic equations diverges at high wavenumbers, but it is shown how this is ameliorated by the presence of the relative vorticity term in the geostrophic potential vorticity equations.