Inviscid Shallow Water Equations Research Articles

A new hypothesis on the intrusion of the Loop Current

The unsteady and inviscid shallow water equations of motion and continuity for a northward moving Loop Current (LC) on the f-plane were integrated to derive an equation for the rate of LC area increase. The dA/dt, where A is the LC surface area, is a function of kinetic energy change across the LC and absolute vorticity. This equation agrees with previous results and is proposed as a new hyphothesis to explain the LC intrusion: a reduction in kinetic energy from west to east across the LC causes mass to be stored inside the Gulf of Mexico (GOM) because the outflow is smaller than the inflow and is manifested as a positive dA/dt indicating a growing and intruding LC into the GOM. The necessary kinetic energy reduction for driving the LC intrusion is caused by relative vorticity and dissipative energy mechanisms. Observations from the intruding LC showed a horizontal reduction of kinetic energy as suggested herein. We discussed that relative vorticity, dissipative forces, and baroclinic and baratropic instabilities induce a kinetic energy reduction across the LC causing an intrusion. Because of the possibility of several kinetic and potential energy-changing mechanisms acting jointly, finding a single mechanism or “trigger” to explain the LC intrusion seems difficult. Additionally, it seems that the relative vorticity values within the LC act to keep dA/dt close to its steady state value. The proposed hypothesis sheds light on the correlation between the LC intrusion and relative vorticity found by Candela et al. (2002). This hypothesis is diagnostic in the sense that it suggests a mechanism or mechanisms that induce a kinetic energy reduction across the LC and the intrusion. Numerical models are ideally suited to evaluate such mechanism(s), to verify the hypothesis, and help design field experiments to test it in the real ocean.

On the Efficiency of Staggered C-Grid Discretization for the Inviscid Shallow Water Equations from the Perspective of Nonstandard Calculus

This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables (surface elevation and normal velocity components) is capable of dealing with flow discontinuities. After a brief introduction of hyperreals through the notion of infinitesimal increments, a nonstandard rendition of the governing equations is derived that essentially turns into a finite procedure and permits a convenient way of modeling the hydraulic jumps in open channel flow. A central result of this paper is that the discrete formulations thus obtained are distinguished by the topological structures of the solution fields and subsequently provide a natural framework for the staggered discretization of the governing equations. Another key of the present study is to demonstrate that the discretization naturally regularizes the solution of the inviscid flow passing through the hydraulic jump without the need of non-physical dissipation. The underlying justification is provided by analytically studying the distributions of the flow variables across an infinitesimal thin hydraulic jump along with the use of hyperreal Heaviside step functions. This main finding is shown to be useful to comprehend the importance of the application of staggered finite difference schemes to accurately predict rapidly varying free-surface flows. A numerical experiment is provided to confirm this result.

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation

Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. We assess the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.

Non-autoregressive time-series methods for stable parametric reduced-order models

Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. We evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover—and critical for many applications of these surrogates—inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.

Numerical Study of Spurious Inertial Modes in Shallow Water Models for a Variable Bathymetry

In this study we perform a modal analysis of the linear inviscid shallow water equations using a non constant bathymetry, continuous and discontinuous Galerkin approximations. By extracting the discrete eigenvalues of the resulting algebraic linear system written on the form of a generalized eigenvalue / eigenvector problem we first show that the regular variation of the bathymetry does not prevent the presence of spurious inertial modes when centered finite element pairs are used. Secondly, we show that such spurious modes are not present in discontinuous Galerkin discretizations when all variables are approximated in the same descrete space. Such spurious inertial modes have been found very damageable for the quality of inertia-gravity and Rossby modes in ocean modelling.

The role of the Rankine-Hugoniot relations in staggered finite difference schemes for the shallow water equations

The purpose of this work is to point out the relevance of the Rankine-Hugoniot jump relations regarding the numerical solution of the inviscid shallow water equations. To arrive at physically relevant solutions in rapidly varied flow, it is of crucial importance that continuity of mass flux and momentum flux across a steady discontinuity is fulfilled at the discrete level. By adopting this viewpoint, finite difference schemes can be studied that may be well suited to solve shallow water flow problems involving discontinuities, while they are not based on a characteristic decomposition of the governed hyperbolic equations. Three schemes on staggered grids with either the water level or the water depth at the cell centre and the flow velocity or the depth-integrated velocity at the cell interface are examined. They differ in (1) the character of the transport velocity to bias the discretization of the advective acceleration term in the upwind direction, and (2) the determination of the water depth at the cell face with which the depth-integrated velocity must be linked to the flow velocity. A detailed analysis is provided and aimed at highlighting the necessity of fulfilling the Rankine-Hugoniot jump conditions for preventing the odd-even decoupling problem. The accuracy and robustness of three selected schemes is assessed by means of convergence tests, three idealized 1D test problems with exact solutions and a 1D laboratory experiment of the breaking, runup and rundown of a solitary wave on a sloping beach. Numerical results reveal that schemes satisfying exactly the jump conditions display improved performance over schemes which do not share this property. Also, these results support strong evidence on the link between not fulfilling the jump conditions and the appearance of odd-even oscillations.

An Entropy-Stable Residual Distribution Scheme for the System of Two-Dimensional Inviscid Shallow Water Equations

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.

A purely Lagrangian method for simulating the shallow water equations on a sphere using smooth particle hydrodynamics

It has long been suggested that a purely Lagrangian solution to global-scale atmospheric/oceanic flows can potentially outperform tradition Eulerian schemes. Meanwhile, a demonstration of a scalable and practical framework remains elusive. Motivated by recent progress in particle-based methods when applied to convection dominated flows, this work presents a fully Lagrangian method for solving the inviscid shallow water equations on a rotating sphere in a smooth particle hydrodynamics framework. To avoid singularities at the poles, the governing equations are solved in Cartesian coordinates, augmented with a Lagrange multiplier to ensure that fluid particles are constrained to the surface of the sphere. An underlying grid in spherical coordinates is used to facilitate efficient neighbor detection and parallelization. The method is applied to a suite of canonical test cases, and conservation, accuracy, and parallel performance are assessed.

Solving the linear inviscid shallow water equations in one dimension, with variable depth, using a recursion formula

When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

Relative entropy for compressible Navier-Stokes equations with density dependent viscosities and various applications

This paper provides the full proof of the results announced by the authors in [C. R. Acad. Sciences (2016)]. We introduce an original relative entropy for compressible Navier-Stokes equations with density dependent viscosities and discuss some possible applications such as inviscid limit or low Mach number limit. We first consider the case µ() = µµ and λ() = 0 and a pressure law under the form p() = aa γ with γ > 1, which corresponds in particular to the formulation of the viscous shallow water equations. We present some mathematical results related to the weak-strong uniqueness, the convergence to a dissipative solution of compressible or incompressible Euler equations. Moreover, we show the convergence of the viscous shallow water equations to the inviscid shallow water equations in the vanishing viscosity limit and further prove convergence to the incompressible Euler system in the low Mach limit. This extends results with constant viscosities recently initiated by E. Feireisl, B.

The 2d Nonlinear Fully Hyperbolic Inviscid Shallow Water Equations in a Rectangle

In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.

The nonlinear 2D supercritical inviscid shallow water equations in a rectangle

In this article we consider the inviscid two-dimensional shallow water equations in a rectangle. The flow occurs near a stationary solution in the so called supercritical regime and we establish short term existence of smooth solutions for the corresponding initial and boundary value problem.

The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction

In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle $(0,1)_x \times (0,1)_y$. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the $y$-direction, we propose the boundary conditions for the $x$-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.

The Linearized 2D Inviscid Shallow Water Equations in a Rectangle: Boundary Conditions and Well-Posedness

We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference velocities $(u_0,v_0)$ and reference height $\phi_0$ (sub- or super-critical flow at each part of the boundary).

Multilevel finite volume methods and boundary conditions for geophysical flows

This review article concerns multi-level methods for finite volume discretizations. They are presented in the context of the non-viscous shallow water equations in space dimension one and two, although the ideas are of more general validity. The second, partly related topic, addressed in this article is that of boundary conditions also presented in the context of the inviscid shallow water equations. Beside algorithmic presentation, the article contains the discussion of some numerical stability issues and the results of some numerical simulations. The multi-level method is applied numerically to the equations of shallow water using a central-upwind scheme for finite volumes and we prove the stability for the linear shallow water equations in different situations. The question of the boundary conditions is illustrated for the two-dimensional equations in the reference case of the so-called Rossby soliton.

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

AbstractWe construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

A discontinuous/continuous low order finite element shallow water model on the sphere

We study the applicability of a new finite element in atmosphere and ocean modeling. The finite element under investigation combines a second order continuous representation for the scalar field with a first order discontinuous representation for the velocity field and is therefore different from continuous and discontinuous Galerkin finite element approaches. The specific choice of low order approximation spaces is attractive because it satisfies the Ladyzhenskaya–Babuska–Brezzi condition and is, at the same time, able to represent the crucially important geostrophic balance. The finite element is used to solve the viscous and inviscid shallow water equations on a rotating sphere. We introduce the spherical geometry via a stereographic projection. The projection leads to a manageable number of additional terms, the associated scaling factors can be exactly represented by second order polynomials. We perform numerical experiments considering steady and unsteady zonal flow, flow over topography, and an unstable zonal jet stream. For ocean applications, the wind driven Stommel gyre is simulated. The experiments are performed on icosahedral geodesic grids and analyzed with respect to convergence rates, conservation properties, and energy and enstrophy spectra. The results match quite well with results published in the literature and encourage further investigation of this type of element for three-dimensional atmosphere/ocean modeling.

High-order discontinuous element-based schemes for the inviscid shallow water equations: Spectral multidomain penalty and discontinuous Galerkin methods

Two commonly used types of high-order-accuracy element-based schemes, collocation-based spectral multidomain penalty methods (SMPM) and nodal discontinuous Galerkin methods (DGM), are compared in the framework of the inviscid shallow water equations. Differences and similarities in formulation are identified, with the primary difference being the dissipative term in the Rusanov form of the numerical flux for the DGM that provides additional numerical stability; however, it should be emphasized that to arrive at this equivalence between SMPM and DGM requires making specific choices in the construction of both methods; these choices are addressed. In general, both methods offer a multitude of choices in the penalty terms used to introduce boundary conditions and stabilize the numerical solution. The resulting specialized class of SMPM and DGM are then applied to a suite of six commonly considered geophysical flow test cases, three linear and three non-linear; we also include results for a classical continuous Galerkin (i.e., spectral element) method for comparison. Both the analysis and numerical experiments show that the SMPM and DGM are essentially identical; both methods can be shown to be equivalent for very special choices of quadrature rules and Riemann solvers in the DGM along with special choices in the type of penalty term in the SMPM. Although we only focus our studies on the inviscid shallow water equations the results presented should be applicable to other systems of nonlinear hyperbolic equations (such as the compressible Euler equations) and extendable to the compressible and incompressible Navier–Stokes equations, where viscous terms are included.

Upwind-biased FORCE schemes with applications to free-surface shallow flows

In this paper we develop numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple-space dimensions on structured meshes. The proposed method is an extension of the multidimensional FORCE flux developed by Toro et al. (2009) [14]. Here we introduce upwind bias by modifying the shape of the staggered mesh of the original FORCE method. The upwind bias is evaluated using an estimate of the largest eigenvalue, which in any case is needed for selecting a time step. The resulting basic flux is first-order accurate and monotone. For the linear advection equation, the proposed UFORCE method reproduces exactly the upwind Godunov method. Extension to non-linear systems has been done empirically via the two-dimensional inviscid shallow water equations. Second order of accuracy in space and time on structured meshes is obtained in the framework of finite volume methods. The proposed method improves the accuracy of the solution for small Courant numbers and intermediate waves associated with linearly degenerate fields (contact discontinuities, shear waves and material interfaces). It achieves comparable accuracy to that of upwind methods with approximate Riemann solvers, though retaining the simplicity and efficiency of centred methods. The performance of the schemes is assessed on a suite of test problems for the two-dimensional shallow water equations.