Long-time shallow-water equations with a varying bottom

1997 ◽  
Vol 349 ◽  
pp. 173-189 ◽  
Author(s):  
ROBERTO CAMASSA ◽  
DARRYL D. HOLM ◽  
C. DAVID LEVERMORE
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Euler Equations ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Three Dimensional ◽  
Local Conservation ◽  
First Order ◽  
Long Time ◽  
Conservation Laws Of Energy

We present and discuss new shallow-water equations that model the long-time effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid possessing a free surface and moving under the force of gravity. We consider the regime where the Froude number ε is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained from the ε→0 limit of the Euler equations (the rigid-lid approximation) at the first order of an asymptotic expansion in δ2. These equations possess local conservation laws of energy and vorticity which reflect exact layer mean conservation laws of the three-dimensional Euler equations. In addition, they convect potential vorticity and have a Hamilton's principle formulation. We contrast them with the Green–Naghdi equations.

A domain decomposition method for the three-dimensional shallow water equations

1995 ◽  
Vol 3 (4-5) ◽  
pp. 307-325 ◽  
Author(s):  
E.D. de Goede ◽  
J. Groeneweg ◽  
K.H. Tan ◽  
M.J.A. Borsboom ◽  
G.S. Stelling
Keyword(s):  
Domain Decomposition ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Three Dimensional

Effect of Dissipative Terms on the Quality of Two and Three-Dimensional Euler Flow Solutions

2008 ◽  
Author(s):  
Seyed Saied Bahrainian
Keyword(s):  
Euler Equations ◽  
Hyperbolic Conservation Laws ◽  
Three Dimensional ◽  
Strong Shock ◽  
Computational Domain ◽  
Third Order ◽  
First Order ◽  
Proper Values ◽  
Dissipative Terms

The Euler equations are a set of non-dissipative hyperbolic conservation laws that can become unstable near regions of severe pressure variation. To prevent oscillations near shockwaves, these equations require artificial dissipation terms to be added to the discretized equations. A combination of first-order and third-order dissipative terms control the stability of the flow solutions. The assigned magnitude of these dissipative terms can have a direct effect on the quality of the flow solution. To examine these effects, subsonic and transonic solutions of the Euler equations for a flow passed a circular cylinder has been investigated. Triangular and tetrahedral unstructured grids were employed to discretize the computational domain. Unsteady Euler equations are then marched through time to reach a steady solution using a modified Runge-Kutta scheme. Optimal values of the dissipative terms were investigated for several flow conditions. For example, at a free stream Mach number of 0.45 strong shock waves were captured on the cylinder by using values of 0.25 and 0.0039 for the first-order and third-order dissipative terms. In addition to the shock capturing effect, it has been shown that smooth pressure coefficients can be obtained with the proper values for the dissipative terms.

scholarly journals Thermomechanically coupled modelling for land-terminating glaciers: a comparison of two-dimensional, first-order and three-dimensional, full-Stokes approaches

2015 ◽  
Vol 61 (228) ◽  
pp. 702-712 ◽  
Author(s):  
Tong Zhang ◽  
Lili Ju ◽  
Wei Leng ◽  
Stephen Price ◽  
Max Gunzburger
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Integration Time ◽  
Two Dimensional ◽  
Shape Factors ◽  
Coupled Modelling ◽  
First Order ◽  
Glacier Changes ◽  
Basal Sliding ◽  
Long Time

AbstractFor many regions, glacier inaccessibility results in sparse geometric datasets for use as model initial conditions (e.g. along the central flowline only). In these cases, two-dimensional (2-D) flowline models are often used to study glacier dynamics. Here we systematically investigate the applicability of a 2-D, first-order Stokes approximation flowline model (FLM), modified by shape factors, for the simulation of land-terminating glaciers by comparing it with a 3-D, ‘full’-Stokes ice-flow model (FSM). Based on steady-state and transient, thermomechanically uncoupled and coupled computational experiments, we explore the sensitivities of the FLM and FSM to ice geometry, temperature and forward model integration time. We find that, compared to the FSM, the FLM generally produces slower horizontal velocities, due to simplifications inherent to the FLM and to the underestimation of the shape factor. For polythermal glaciers, those with temperate ice zones, or when basal sliding is important, we find significant differences between simulation results when using the FLM versus the FSM. Over time, initially small differences between the FLM and FSM become much larger, particularly near cold/temperate ice transition surfaces. Long time integrations further increase small initial differences between the two models. We conclude that the FLM should be applied with caution when modelling glacier changes under a warming climate or over long periods of time.

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