scholarly journals A Mathematical Model of Stratified Geophysical Fluid Flows Over Variable Bottom Topography

2020 ◽  
Vol 3 (3b) ◽  
pp. 112-137
Author(s):  
SI Iornumbe ◽  
T Tivde ◽  
RA Chia
Keyword(s):  
Mathematical Model ◽  
Shallow Water ◽  
Wave Motion ◽  
Wave Speed ◽  
Bottom Topography ◽  
Nonlinear Partial Differential Equations ◽  
Fluid Flows ◽  
Variable Bottom ◽  
Conservation Of Momentum ◽  
Model Equations

In this paper, a mathematical model of stratified geophysical fluid flow over variable bottom topography was derived for shallow water. The equations are derived from the principles of conservation of mass and conservation of momentum. The force acting on the fluid is gravity, represented by the gravitational constant. A system of six nonlinear partial differential equations was obtained as the model equations. The solutions of these models were obtained using perturbation method. The presence of the coriolis force in the shallow water equations were shown as the causes of the deflection of fluid parcels in the direction of wave motion and causes gravity waves to disperse. As water depth decreases due to varied bottom topography, the wave amplitude were shown to increase while the wavelength and wave speed decreases resulting in overturning of the wave. The results are presented graphically.

scholarly journals Characteristics of Very Low Frequency Sound Propagation in Full Waveguides of Shallow Water

Sensors ◽  
2020 ◽  
Vol 21 (1) ◽  
pp. 192
Author(s):  
Nansong Li ◽  
Hanhao Zhu ◽  
Xiaohan Wang ◽  
Rui Xiao ◽  
Yangyang Xue ◽  
...  
Keyword(s):  
Shallow Water ◽  
Wave Speed ◽  
Sound Propagation ◽  
Sea Water ◽  
Bottom Topography ◽  
Low Frequency ◽  
Acoustic Energy ◽  
Propagation Characteristics ◽  
Sea Bottom ◽  
Proposed Model

This work is concerned with the characteristics of very low frequency sound propagation (VLF, ≤100 Hz) in the shallow marine environment. Under these conditions, the classical hypothesis of considering the sea bottom as a fluid environment is no longer appropriate, and the sound propagation characteristics at the sea bottom should be also considered. Hence, based on the finite element method (FEM), and setting the sea bottom as an elastic medium, a proposed model which unifies the sea water and sea bottom is established, and the propagation characteristics in full waveguides of shallow water can be synchronously discussed. Using this model, the effects of the sea bottom topography and the various geoacoustic parameters on VLF sound propagation and its corresponding mechanisms are investigated through numerical examples and acoustic theory. The simulation results demonstrate the adaptability of the proposed model to complex shallow water waveguides and the accuracy of the calculated acoustic field. For the sea bottom topography, the greater the inclination angle of an up-sloping sea bottom, the stronger the leak of acoustic energy to the sea bottom, and the more rapid the attenuation of the acoustic energy in sea water. The effect of a down-sloping sea bottom on acoustic energy is the opposite. Moreover, the greater the pressure wave (P-wave) speed in the sea bottom, the more acoustic energy remains in the water rather than leaking into the bottom; the influence laws of the density and the shear wave (S-wave) speed in the sea bottom are opposite.

scholarly journals Mathematical Model of Geophysical Fluid Flow over Variable Bottom Topography

2020 ◽  
Vol 3 (2) ◽  
pp. 186-199
Author(s):  
SI Iornumbe ◽  
GCE Mbah ◽  
RA Chia
Keyword(s):  
Mathematical Model ◽  
Fluid Flow ◽  
Partial Differential Equations ◽  
Perturbation Method ◽  
Coriolis Force ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Variable Bottom ◽  
Nonlinear Shallow Water Equations ◽  
Two Space Dimensions

In this paper, the bottom topography of a geophysical fluid flow is modelled in the presence of Coriolis force by the nonlinear shallow water equations. These equations, which are a system of three partial differential equations in two space dimensions, are solved using the perturbation method. The Effects of the Coriolis force and the bottom topography for particular initial flows on the velocity components and different kind of flow patterns possible in geophysical fluid flow have been studied and the results illustrated graphically.

A multi-layer model for nonlinear internal wave propagation in shallow water

2012 ◽  
Vol 695 ◽  
pp. 341-365 ◽  
Author(s):  
Philip L.-F. Liu ◽  
Xiaoming Wang
Keyword(s):  
Wave Propagation ◽  
Internal Wave ◽  
Shallow Water ◽  
Internal Waves ◽  
Bottom Topography ◽  
Laboratory Data ◽  
Constant Density ◽  
Layer Model ◽  
Fluid System ◽  
Nonlinear Internal Wave

AbstractIn this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an $N$-layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

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