A new model for shallow water in the low-Rossby-number limit

2002 ◽  
Vol 450 ◽  
pp. 287-296
Author(s):  
RUPERT FORD ◽  
SIMON J. A. MALHAM ◽  
MARCEL OLIVER
Keyword(s):  
Asymptotic Expansion ◽  
Shallow Water ◽  
Energy Functional ◽  
Positive Definite ◽  
Rossby Number ◽  
Computational Domain ◽  
Dirac Bracket ◽  
New Model ◽  
Projection Approach

We revisit Salmon's ‘Dirac bracket projection’ approach to constructing generalized semi-geostrophic equations. One of the obstacles to the method's applicability is that it leads to a sign-indefinite energy functional in the computational domain. In some instances this can cause severe failure of the model. We demonstrate in the simple context of shallow-water semi-geostrophy that the Hamiltonian remains positive definite when the asymptotic expansion at the heart of this method is carried to the next order. The resulting new model can be interpreted in the framework of regularization by Lagrangian averaging, which is currently receiving much attention.

Stability of jets in a shallow water layer

2015 ◽  
Vol 25 (2) ◽  
pp. 358-374 ◽  
Author(s):  
Zhanhong Wan ◽  
Saihua Huang ◽  
Zhilin Sun ◽  
Zhenjiang You
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Shallow Water ◽  
Large Scale ◽  
Bottom Topography ◽  
Bottom Friction ◽  
Rossby Number ◽  
Content Type ◽  
The Stability ◽  
Topographic Parameters

Purpose – The present work is devoted to the numerical study of the stability of shallow jet. The effects of important parameters on the stability behavior for large scale shallow jets are considered and investigated. Connections between the stability theory and observed features reported in the literature are emphasized. The paper aims to discuss these issues. Design/methodology/approach – A linear stability analysis of shallow jet incorporating the effects of bottom topography, bed friction and viscosity has been carried out by using the shallow water stability equation derived from the depth averaged shallow water equations in conjunction with both Chézy and Manning resistance formulae. Effects of the following main factors on the stability of shallow water jets are examined: Rossby number, bottom friction number, Reynolds number, topographic parameters, base velocity profile and resistance model. Special attention has been paid to the Coriolis effects on the jet stability by limiting the rotation number in the range of Ro∈[0, 1.0]. Findings – It is found that the Rossby number may either amplify or attenuate the growth of the flow instability depending on the values of the topographic parameters. There is a regime where the near cancellation of Coriolis effects due to other relevant parameters influences is responsible for enhancement of stability. The instability can be suppressed by the bottom friction when the bottom friction number is large enough. The amplification rate may become sensitive to the relatively small Reynolds number. The stability region using the Manning formula is larger than that using the Chézy formula. The combination of these effects may stabilize or destabilize the shallow jet flow. These results of the stability analysis are compared with those from the literature. Originality/value – Results of linear stability analysis on shallow jets along roughness bottom bed are presented. Different from the previous studies, this paper includes the effects of bottom topography, Rossby number, Reynolds number, resistance formula and bed friction. It is found that the influence of Reynolds number on the stability of the jet is notable for relative small value. Therefore, it is important to experimental investigators that the viscosity should be considered with comparison to the results from inviscid assumption. In contrast with the classical analysis, the use of multi-parameters of the base velocity and topographic profile gives an extension to the jet stability analysis. To characterize the large scale motion, besides the bottom friction as proposed in the related literature, the Reynolds number Re, Rossby number Ro, the topographic parameters and parameters controlling base velocity profile may also be important to the stability analysis of shallow jet flows.

scholarly journals Towards a macroscopically consistent discrete method for granular materials: Delaunay strain-based formulation

2021 ◽  
Author(s):  
Göran Frenning
Keyword(s):  
Granular Materials ◽  
Granular Media ◽  
Deformation Gradient ◽  
Energy Functional ◽  
Computational Domain ◽  
Discrete Method ◽  
Angular Momenta ◽  
Affine Deformation ◽  
External Forces ◽  
Central Forces

AbstractWe demonstrate that the Delaunay-based strain definition proposed by Bagi (Mech Mater 22:165–177, 1996) for granular media can be straightforwardly translated into a particle-based numerical method for continua. This method has a number of attractive features, including linear completeness and satisfaction of the patch test, exact conservation of linear and angular momenta in the absence of external forces and torques, and anti-symmetry of the gradient vectors for any two points not both on the boundary of the computational domain. The formulation in effect relies on nodal (particle) interpolation of the deformation gradient and is therefore inherently unstable. Drawing on the analogy with granular media, a pairwise interaction between particles is included to alleviate this issue. The underlying idea is to define a local, non-affine deformation of each bond or contact, and to introduce pairwise forces via a stored-energy functional expressed in terms of the corresponding local displacements. In this manner, a generalisation of the Ganzenmüller (Comput Methods Appl Mech Eng 286:87–106, 2015) hourglass stabilisation procedure to non-central forces is obtained. The performance of the method is demonstrated in a range of problems. This work can be considered a first step towards the development of a macroscopically consistent discrete method for granular materials.

A Flux Vector Splitting Technique for Shallow Water Equations

2003 ◽  
Author(s):  
Arturo Pacheco-Vega ◽  
J. Rafael Pacheco ◽  
Tamara Rodic´
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Hydraulic Jump ◽  
Computational Domain ◽  
Flux Vector ◽  
Governing Equations ◽  
Splitting Technique ◽  
Flux Vector Splitting ◽  
Film Flows ◽  
Upwind Finite Difference

We present a flux vector splitting (FVS) for the solution of the shallow water equations with emphasis in their application to film flows for which a hydraulic jump may exist. The governing equations and boundary conditions are transformed from the physical to the computational domain and solved in a rectangular grid. A first-order upwind finite difference scheme is used at the point of the shock while a second-order upwind differentiation is applied elsewhere. Higher-order spatial accuracy is achieved by using a MUSCL approach. Two problems, (a) one-dimensional dam break problem and (b) radial flow with jump, are investigated to show the usefulness and accuracy of the method. Results demonstrate that the method is able to predict accuratelly the hydraulic jump using shallow water theory.

Modelling the Azov Sea circulation and extreme surges in 2013-2014 using the regularized shallow water equations

2018 ◽  
Vol 33 (3) ◽  
pp. 173-185 ◽  
Author(s):  
Dmitry S. Saburin ◽  
Tatiana G. Elizarova
Keyword(s):  
Shallow Water ◽  
Numerical Method ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Numerical Algorithms ◽  
Water Area ◽  
Observation Data ◽  
Wind Effects ◽  
New Model ◽  
Azov Sea

Abstract A new model for calculation of circulation in shallow water basins is created based on the shallow water equations taking into account the Coriolis force and quadratic friction on the bottom. Wind effects are taken into account as forcing. The main feature of the model is a new numerical method based on regularized shallow water equations allowing one to construct the simple and sufficiently accurate numerical algorithms possessing a number of advantages over existing methods. The paper provides a detailed description of all construction steps of the model. The developed model was implemented for the water area of the Azov Sea. The paper presents the modelling of extreme surges in March 2013 and September 2014, the results of calculations are compared with observation data of hydrometeorological stations in Taganrog and Yeysk.

Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model

2001 ◽  
Vol 445 ◽  
pp. 93-120 ◽  
Author(s):  
G. M. REZNIK ◽  
V. ZEITLIN ◽  
M. BEN JELLOUL
Keyword(s):  
Shallow Water ◽  
Slow Component ◽  
Fast Component ◽  
Water Dynamics ◽  
Water Model ◽  
Rossby Number ◽  
Geostrophic Adjustment ◽  
Perturbation Expansions ◽  
Modulation Equation ◽  
Rotating Shallow Water

We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.

scholarly journals Water waves and integrability

2007 ◽  
Vol 365 (1858) ◽  
pp. 2267-2280 ◽  
Author(s):  
Rossen I Ivanov
Keyword(s):  
Asymptotic Expansion ◽  
Shallow Water ◽  
Water Waves ◽  
Integrable Equation ◽  
Inviscid Fluid ◽  
Shallow Water Waves ◽  
Integrable Equations ◽  
Scale Parameters ◽  
Euler’S Equations ◽  
Euler's Equations

Euler's equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymptotic expansion of Euler's equations is considered (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modelling the motion of shallow water waves are reviewed in this contribution.

The quasi-geostrophic theory of the thermal shallow water equations

2013 ◽  
Vol 723 ◽  
pp. 374-403 ◽  
Author(s):  
Emma S. Warneford ◽  
Paul J. Dellar
Keyword(s):  
Energy Conservation ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Fluid Layer ◽  
Conservation Equation ◽  
Rossby Number ◽  
Layer System ◽  
Energy Conservation Equation ◽  
Balanced Model ◽  
Horizontal Variations

AbstractThe thermal shallow water equations provide a depth-averaged description of motions in a fluid layer that permits horizontal variations in material properties. They typically arise through an equivalent barotropic approximation of a two-layer system, with a spatially varying density contrast due to an evolving temperature field in the active layer. We formalize a previous derivation of the quasi-geostrophic (QG) theory of these equations, by performing a direct asymptotic expansion for small Rossby number. We then present a second derivation as the small Rossby number limit of a balanced model that projects out high-frequency dynamics due to inertia-gravity waves. This latter derivation has wider validity, not being restricted to mid-latitude $\beta $-planes. We also derive their local energy conservation equation from the QG limit of a thermal shallow water pseudo-energy conservation equation. This derivation involves the ageostrophic correction to the leading-order geostrophic velocity that is eliminated in the usual derivation of a closed evolution equation for the QG potential vorticity. Finally, we derive the non-canonical Hamiltonian structure of the thermal QG equations from a decomposition in Rossby number of a pseudo-energy and Poisson bracket for the thermal shallow water equations.

2D multi-layer hydrodynamic and sediment transport modelling in a tidal estuary

2020 ◽  
Author(s):  
Kai-Yi Bai ◽  
Jiing-Yun You
Keyword(s):  
Sediment Transport ◽  
Shallow Water ◽  
Finite Volume ◽  
Stokes Equations ◽  
Transport Model ◽  
Three Dimensional ◽  
Sediment Concentration ◽  
Computational Domain ◽  
Operation Rules ◽  
Sediment Transport Model

<p>This study developed a multi-layer hydrodynamic and sediment transport model for simulating tides and the estuarine flows. The flow circulation in an estuary shows complicated mixing and stratification patterns due to the combined effects from currents and tides. This kind of issues becomes more important in Taiwan in line with the more and more frequent sediment flushing operation which led to high sediment concentration flow at the estuary. In some applications,  three-dimensional (3D) models solving full Navier-Stokes equations were used. However, the extremely high computational cost, especially for the large-scale environmental problems, is always a serious concern. In the past years, continuous efforts have been devoted to the development of efficient quasi-three-dimensional models under hydrostatic and Boussinesq assumptions. Following the same state-of-the-art modelling strategy, this study develops a multi-layer shallow-water and sediment transport model with finite volume method. In this model, a terrain following coordinate with high local resolution is used to vertically divide the computational domain into multiple layers to better addressing bottom topography and velocity profile. Our model is rigorously validated against several benchmark cases including winddriven circulation, subcritical flow over a hump, tidal wave propagation, and sediment transport. The grid convergence test and accuracy both are in good agreement with analytical solutions. Subsequently, the model is applied to investigate the estuary dynamics and sediment transport under different conditions, e.g., flow discharges, bottom slopes, wind shears and tidal variations. Overall, the results show a relationship between flow conditions and sediment transport. Later, some scenarios for various upstream inflow and sediment concentration will be examined to assess the reservoir operation rules. </p><p><strong>Keywords: shallow water, sediment transport, multi-layer, hydrostatic, Boussinesq Assumption, a finite volume characteristics (FVC) method </strong><br> </p><p><br> <br> <br><br> </p>

Sign in / Sign up

Export Citation Format

Share Document