An idealized 1½-layer isentropic model with convection and precipitation for satellite data assimilation research. Part II: model derivation

In this part II paper we present a fully consistent analytical derivation of the ‘dry’ isentropic 1½-layer shallow water model described and used in part I of this study, with no convection and precipitation. The mathematical derivation presented here is based on a combined asymptotic and slaved Hamiltonian analysis which is used to resolve an apparent inconsistency arising from the application of a rigid-lid approximation to an isentropic two-layer shallow water model. Real observations based on radiosonde data are used to justify the scaling assumptions used throughout the paper, as well as in part I. Eventually, a fully consistent isentropic 1½-layer model emerges from imposing fluid at rest (v1 = 0) and zero Montgomery potential (M1 = 0) in the upper layer of an isentropic two-layer model.

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

Parallel Algorithms of Well-Balanced and Weighted Average Flux for Shallow Water Model Using CUDA

Compute Unified Device Architecture (CUDA) implementations are presented of a well-balanced finite volume method for solving a shallow water model. The CUDA platform allows programs to run parallel on GPU. Four versions of the CUDA algorithm are presented in addition to a CPU implementation. Each version is improved from the previous one. We present the following techniques for optimizing a CUDA program: limiting register usage, changing the global memory access pattern, and using loop unroll. The accuracy of all programs is investigated in 3 test cases: a circular dam break on a dry bed, a circular dam break on a wet bed, and a dam break flow over three humps. The last parallel version shows 3.84x speedup over the first CUDA implementation. We use our program to simulate a real-world problem based on an assumed partial breakage of the Srinakarin Dam located in Kanchanaburi province, Thailand. The simulation shows that the strong interaction between massive water flows and bottom elevations under wet and dry conditions is well captured by the well-balanced scheme, while the optimized parallel program produces a 57.32x speedup over the serial version.

Shallow water model for pollutant distribution in the Ypacarai Lake

In this paper, we analyze the distribution of a non-reactive contaminant in Ypacarai Lake. We propose a shallow-water model that considers wind-induced currents, inflow and outflow conditions in the tributaries, and bottom effects due to the lakebed. The hydrodynamic is based on the depth-averaged Navier-Stokes equations considering wind stresses as force terms which are functions of the wind velocity. Bed (bottom) stress is based on Manning's equation, the lakebed characteristics, and wind velocities. The contaminant transportation is modeled by a 2D convection-diffusion equation taking into consideration water level. Comparisons between the simulation of the model, analytical solutions, and laboratory results confirm that the model captures the complex dynamic phenomenology of the lake. In the simulations, one can see the regions with the highest risk of accumulation of contaminants. It is observed the effect of each term and how it can be used them to mitigate the impact of the pollutants.

Dynamical Splitting of Spot-producing Magnetic Rings in a Nonlinear Shallow-water Model

We explore the fundamental physics of narrow toroidal rings during their nonlinear magnetohydrodynamic evolution at tachocline depths. Using a shallow-water model, we simulate the nonlinear evolution of spot-producing toroidal rings of 6° latitudinal width and a peak field of 15 kG. We find that the rings split; the split time depends on the latitude of each ring. Ring splitting occurs fastest, within a few weeks, at latitudes 20°–25°. Rossby waves work as perturbations to drive the instability of spot-producing toroidal rings; the ring split is caused by the “mixed stress” or cross-correlations of perturbation velocities and magnetic fields, which carry magnetic energy and flux from the ring peak to its shoulders, leading to the ring split. The two split rings migrate away from each other, the high-latitude counterpart slipping poleward faster due to migrating mixed stress and magnetic curvature stress. Broader toroidal bands do not split. Much stronger rings, despite being narrow, do not split due to rigidity from stronger magnetic fields within the ring. Magnetogram analysis indicates the emergence of active regions sometimes at the same longitudes but separated in latitude by 20° or more, which could be evidence of active regions emerging from split rings, which consistently contribute to observed high-latitude excursions of butterfly wings during the ascending, peak, and descending phases of a solar cycle. Observational studies in the future can determine how often new spots are found at higher latitudes than their lower-latitude counterparts and how the combinations influence solar eruptions and space weather events.

An Invariant-Preserving Scheme for the Viscous Burgers- Poisson System

We formulate and analyze a new finite difference scheme for a shallow water model in the form of viscous Burgers-Poisson system with periodic boundary conditions. The proposed scheme belongs to a family of three-level linearized finite difference methods. It is proved to preserve both momentum and energy in the discrete sense. In addition, we proved that the method converges uniformly and has second order of accuracy in space. The analysis given in this work is the first time a pointwise error estimation is done on a second-order finite difference operator applied to the Burgers-Poisson system. We validate our findings by performing various numerical simulations on both viscous and inviscous problems. These numerical examples show the efficacy of the proposed method and confirm the proven theoretical results.