Mathematical and Numerical Analysis of a Tsunami Problem

2003 ◽  
Vol 13 (10) ◽  
pp. 1489-1514 ◽  
Author(s):  
B. di Martino ◽  
F. Flori ◽  
C. Giacomoni ◽  
P. Orenga
Keyword(s):  
Numerical Analysis ◽  
Shallow Water ◽  
Existence And Uniqueness ◽  
Large Scale ◽  
Uniqueness Result ◽  
Water Model ◽  
Shallow Water Model ◽  
Mean Velocity ◽  
Plate Equations ◽  
Tsunami Model

In this paper, we present a tsunami model based on the displacement of the lithosphere and the mathematical and numerical analysis of this model. More precisely, we give an existence and uniqueness result for a problem which models the flow and formation of waves at the time of a submarine earthquake in the vicinity of the coast. We propose a model which describes the behavior of the fluid using a bi-dimensional shallow-water model by means of a depth-mean velocity formulation. The ocean is coupled to the Earth's crust whose movement is assumed to be controlled on a large scale by plate equations. Finally, we give some numerical results showing the formation of a tsunami.

scholarly journals Improved moist-convective rotating shallow water model and its application to instabilities of hurricane-like vortices

2018 ◽  
Author(s):  
LMD
Keyword(s):  
Tropical Cyclone ◽  
Shallow Water ◽  
Water Vapour ◽  
Large Scale ◽  
Precipitable Water ◽  
Water Model ◽  
Moist Convection ◽  
Shallow Water Model ◽  
Atmospheric Motions ◽  
Rotating Shallow Water

We show how the two-layer moist-convective rotating shallow water model (mcRSW), which proved to be a simple and robust tool for studying effects of moist convection on large-scale atmospheric motions, can be improved by including, in addition to the water vapour, precipitable water, and the effects of vaporisation, entrainment, and precipitation. Thus improved mcRSW becomes cloud-resolving. It is applied, as an illustration, to model the development of instabilities of tropical cyclone-like vortices.

scholarly journals A Shallow Water Model for Convective Self-Aggregation

2020 ◽  
Author(s):  
Da Yang
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Large Scale ◽  
Wave Pattern ◽  
Water Model ◽  
Shallow Water Model ◽  
Modeling Framework ◽  
Convective Storms ◽  
Wide Range ◽  
Self Aggregation

AbstractRandomly distributed convective storms can self-aggregate in the absence of large-scale forcings. Here we present a 1D shallow water model to study the convective self-aggregation. This model simulates the dynamics of the planetary boundary layer and represents convection as a triggered process. Once triggered, convection lasts for finite time and occupies finite length. We show that the model can successfully simulate self-aggregation, and that the results are robust to a wide range of parameter values. In the simulations, convection excites gravity waves. The gravity waves then form a standing wave pattern, separating the domain into convectively active and inactive regions. We analyze the available potential energy (APE) budget and show that convection generates APE, providing energy for self-aggregation. By performing dimensional analysis, we develop a scaling theory for the size of convective aggregation, which is set by the gravity wave speed, damping timescale, and number density of convective storms. This paper provides a simple modeling framework to further study convective self-aggregation.

scholarly journals Triggered Convection, Gravity Waves, and the MJO: A Shallow-Water Model

2013 ◽  
Vol 70 (8) ◽  
pp. 2476-2486 ◽  
Author(s):  
Da Yang ◽  
Andrew P. Ingersoll
Keyword(s):  
Shallow Water ◽  
Large Scale ◽  
Low Frequency ◽  
Propagation Speed ◽  
Water Model ◽  
Shallow Water Model ◽  
Horizontal Scale ◽  
Frequency Wave ◽  
Large Scale Structures ◽  
Scale Structures

Abstract The Madden–Julian oscillation (MJO) is the dominant mode of intraseasonal variability in the tropics. Despite its primary importance, a generally accepted theory that accounts for fundamental features of the MJO, including its propagation speed, planetary horizontal scale, multiscale features, and quadrupole structures, remains elusive. In this study, the authors use a shallow-water model to simulate the MJO. In this model, convection is parameterized as a short-duration localized mass source and is triggered when the layer thickness falls below a critical value. Radiation is parameterized as a steady uniform mass sink. The following MJO-like signals are observed in the simulations: 1) slow eastward-propagating large-scale disturbances, which show up as low-frequency, low-wavenumber features with eastward propagation in the spectral domain, 2) multiscale structures in the time–longitude (Hovmöller) domain, and 3) quadrupole vortex structures in the longitude–latitude (map view) domain. The authors propose that the simulated MJO signal is an interference pattern of westward and eastward inertia–gravity (WIG and EIG) waves. Its propagation speed is half of the speed difference between the WIG and EIG waves. The horizontal scale of its large-scale envelope is determined by the bandwidth of the excited waves, and the bandwidth is controlled by the number density of convection events. In this model, convection events trigger other convection events, thereby aggregating into large-scale structures, but there is no feedback of the large-scale structures onto the convection events. The results suggest that the MJO is not so much a low-frequency wave, in which convection acts as a quasi-equilibrium adjustment, but is more a pattern of high-frequency waves that interact directly with the convection.

scholarly journals Eastward-moving convection-enhanced modons in shallow water in the equatorial tangent plane

2019 ◽  
Author(s):  
M. Rostami
Keyword(s):  
Shallow Water ◽  
Numerical Scheme ◽  
Large Scale ◽  
Tangent Plane ◽  
Water Model ◽  
Moist Convection ◽  
Shallow Water Model ◽  
Beta Plane ◽  
Vapour Condensation ◽  
Velocity Approximation

We report a discovery of steady long-living slowly eastward moving large-scale coherent twin cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing analytical asymptotic modon solutions in the non-divergent velocity approximation and then show by simulations with a high-resolution numerical scheme that such configurations evolve into steady dipolar solutions of the full model. In the atmospheric context, the modons persist in the presence of moist convection, being accompanied and enhanced by specific patterns of water-vapour condensation.

ANALYSIS OF SOME SHALLOW WATER PROBLEMS WITH RIGID-LID HYPOTHESIS

2001 ◽  
Vol 11 (06) ◽  
pp. 979-999 ◽  
Author(s):  
B. DI MARTINO ◽  
C. GIACOMONI ◽  
P. ORENGA
Keyword(s):  
Shallow Water ◽  
Water Model ◽  
Shallow Water Model ◽  
Viscous Effects ◽  
Weak Formulation ◽  
Mean Velocity ◽  
Numerical Resolution ◽  
Comparative Results ◽  
Geophysical Flow ◽  
Water Problems

We present in this paper the analysis and the comparison of two two-dimensional geophysical flow problems using a rigid-lid approximation (i.e. we do not take into account the variation of surface elevation ζ). A first rigid-lid shallow water model (noted SWRL) is obtained by neglecting the variation of the surface in a weak formulation of a usual viscous shallow water model in depth-mean velocity formulation (noted SWFS for shallow water with free surface). We establish some existence results for this model and we propose a numerical resolution method. The second model we consider is an adaptation of the lake equations proposed by Levermore, Oliver and Titi,4 in which we take into account the viscous effects, in order to compare the two approaches. For the numerical resolution, we apply the curl operator on these equations and we propose a numerical algorithm to solve this problem, that we note SWLV (shallow water for the lake with viscosity). We present finally some comparative results in an idealized configuration between the SWRL, SWLV and SWFS models (in the case where the rigid-lid approximation seems to be reasonable).

scholarly journals Successive bifurcations in a shallow-water model applied to the wind-driven ocean circulation

1995 ◽  
Vol 2 (3/4) ◽  
pp. 241-268 ◽  
Author(s):  
S. Speich ◽  
H. Dijkstra ◽  
M. Ghil
Keyword(s):  
Shallow Water ◽  
Ocean Circulation ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Steady State Solution ◽  
Stationary Solutions ◽  
Water Model ◽  
Shallow Water Model ◽  
Current Systems ◽  
Double Gyre

Abstract. Climate - the "coarse-gridded" state of the coupled ocean - atmosphere system - varies on many time and space scales. The challenge is to relate such variation to specific mechanisms and to produce verifiable quantitative explanations. In this paper, we study the oceanic component of the climate system and, in particular, the different circulation regimes of the mid-latitude win driven ocean on the interannual time scale. These circulations are dominated by two counterrotating, basis scale gyres: subtropical and subpolar. Numerical techniques of bifurcation theory are used to stud the multiplicity and stability of the steady-state solution of a wind-driven, double-gyre, reduced-gravity, shallow water model. Branches of stationary solutions and their linear stability are calculated systematically as parameter are varied. This is one of the first geophysical studies i which such techniques are applied to a dynamical system with tens of thousands of degrees of freedom. Multiple stationary solutions obtain as a result of nonlinear interactions between the two main recirculating cell (cyclonic and anticyclonic) of the large- scale double-gyre flow. These equilibria appear for realistic values of the forcing and dissipation parameters. They undergo Hop bifurcation and transition to aperiodic solutions eventually occurs. The periodic and chaotic behaviour is probably related to an increased number of vorticity cells interaction with each other. A preliminary comparison with observations of the Gulf Stream and Kuroshio Extensions suggests that the intern variability of our simulated mid-latitude ocean is a important factor in the observed interannual variability o these two current systems.

scholarly journals On a 1D-Shallow Water Model: Existence of solution and numerical simulations

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Olivier Besson ◽  
Soulèye Kane ◽  
Mamadou Sy
Keyword(s):  
Numerical Simulations ◽  
Shallow Water ◽  
Existence And Uniqueness ◽  
Existence Of Solution ◽  
Water Model ◽  
Shallow Water Model ◽  
Flood Prediction ◽  
International Audience ◽  
Linearized System ◽  
Model Existence

International audience The study of a 1D-shallow water model, obtained in a height-flow formulation, is presented. It takes viscosity into account and can be used for the flood prediction in rivers. For a linearized system, the existence and uniqueness of a global solution is proved. Finally, various numerical results are presented regarding the linear and non linear case. Nous dérivons les équations de Saint-Venant complètes avec la formulation hauteur-débit. La viscosité est prise en compte dans le modèle. Pour le système linéarisé, l’existence et l’unicité de solution globale est montrée. Des resultats numériques sont présentés aussi bien dans le cas linéaire que non linéaire.

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