scholarly journals On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

2019 ◽  
Author(s):  
Roberto Camassa ◽  
◽  
Gregorio Falqui ◽  
Giovanni Ortenzi ◽  
Marco Pedroni ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Extended Self ◽  
Self Similar ◽  
Similar Solutions

scholarly journals Brief Communication: A nonlinear self-similar solution to barotropic flow over varying topography

2017 ◽  
Author(s):  
Ruy Ibanez ◽  
Joseph Kuehl ◽  
Kalyan Shrestha ◽  
William Anderson
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Analytic Solution ◽  
Error Function ◽  
Lambert W Function ◽  
Linear Solution ◽  
Barotropic Flow ◽  
Self Similar Solution ◽  
Nonlinear Advection ◽  
Self Similar

Abstract. Beginning from the Shallow Water Equations (SWE), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways: 1) The solution is valid for intensifying jets. 2) The influence of nonlinear advection is included. The SWE are scaled to the case of a topographically controlled jet, then solved by introducing a similarity variable, η = cxnxyny. The nonlinear solution, valid for topographies h = h0 − αxy3, takes the form of the Lambert W Function for pseudo velocity. The linear solution, valid for topographies h = h0 − αxy−γ, takes the form of the Error Function for transport. Kuehl's results considered the case −1 ≤ γ 

On axisymmetric intrusive gravity currents: the approach to self-similarity solutions of the shallow-water equations in a linearly stratified ambient

2007 ◽  
Vol 463 (2085) ◽  
pp. 2165-2183 ◽  
Author(s):  
Tamar Zemach ◽  
Marius Ungarish
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Conditions ◽  
Fixed Ratio ◽  
Analytical Prediction ◽  
Ambient Fluid ◽  
Time Motion ◽  
Ring Shape ◽  
Long Time ◽  
Self Similar

The axisymmetric intrusion of a fixed volume of fluid, which is released from rest and then propagates radially at the neutral buoyancy level in a deep linearly stratified ambient fluid, is investigated. Attention is focused on the development of self-similar propagation. The shallow-water equations representing the high-Reynolds-number motion are used. For the long-time motion, an analytical similarity solution indicates propagation with t 1/3 , but the shape is peculiar: the intrusion propagates like a ring with a fixed ratio of inner to outer radii; the inner domain contains clear ambient fluid. To verify the similarity analytical prediction, a long-time finite-difference solution with realistic initial conditions was performed. To avoid accumulation of numerical errors, the problem was reformulated in terms of new variables. It is shown that the numerical solution has a ‘tail-ring’ shape. The ‘tail’ decays like t −2 and the ‘ring’ tends to the analytical similarity prediction. The initial geometry of the lock does not influence this result. Comparison with the non-stratified case is also presented. It was found that for the non-stratified case, there is a stage of propagation in which the intrusion has a similar ‘tail-ring’ form; however, this stage is only a transient to a self-similar shape which is different from that obtained for the stratified ambient.

MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

2017 ◽  
Author(s):  
Xiao-Hua Zhu ◽  
Xiao-Hua Zhu ◽  
Ze-Nan Zhu ◽  
Ze-Nan Zhu ◽  
Xinyu Guo ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Mean Amplitude ◽  
Momentum Equation ◽  
Tidal Currents ◽  
Acoustic Tomography ◽  
Mean Values ◽  
Residual Currents ◽  
Water Depths ◽  
Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

Sign in / Sign up

Export Citation Format

Share Document