A Mathematical Model of Stratified Geophysical Fluid Flows over Variable Bottom Topography

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 g . 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.


Introduction
Stratified fluids are very present in nature, present in almost any heterogeneous fluid body. Examples include thermal stratification of reservoirs and oceans, salinity stratification in estuaries, rivers, groundwater reservoirs, and oceans, heterogeneous mixtures in industrial, food, and manufacturing processing, density stratification of the atmosphere and many others. In the presence of gravity, these density differences have a dramatic impact on the dynamics and mixing of heterogeneous fluids. For instance, thermal stratification in reservoirs can reduce the vertical mixing of oxygen to the point that bottom water becomes anoxic through the action of biological processes. Preventing, predicting, and solving such reservoir problem requires an understanding of the dynamics of stratified fluids. Therefore a stratified fluid is one in which the fluid density  is a function of space   z y x , , and time t . The term stratification refers to the variation in the density field. Density differences arise through many sources. These can include differences in temperature, dissolved phases (solids, liquids, and gases), suspended solids, and pressure differences. The development of layers within the profile of a water body is a common phenomenon of stratification, (Pedlosky, 1987).
There are vast numbers of publications covering the different fields of classical geophysical fluid dynamics. General reviews and background information may be found in several textbooks; (Gill 1982;Mellor 1996;Pedlosky 1987 andPerks 1988). In the present context, with emphasis on more local scale geophysical flows, we may consider the most relevant literature in waterly flows as follows: for local waterly flows, analysis and laboratory experiments may be found in (Boyer andDavies 2000, Riley andLelong 2000), while modeling and applications ranging from local tidal flow to circulations in various restricted waters are given in (Moe et al., 2002). Dellar and Salmon (2005) derived an extended set of shallow water equations that describe a thin inviscid fluid layer above fixed topography in a frame rotating about an arbitrary axis and also derived models that treat the ocean or atmosphere as a stack of layers with variable thickness. Within each layer, the density is either assumed to be uniform or may vary horizontally due to temperature gradients.
According to Karelsky et al. (2000), the generalization of classical shallow water theory to the case of flows over an irregular bottom. They showed that the simple self-similar solutions that are characteristic for the classical problem exist only if the underlying surface has a uniform slope.
A mathematical simulation to determine the water velocity in the Lake Mariut, taking into consideration its concentration and the distribution of the temperature along it, by applying the fractional steps method for the numerical solution of the shallow water equations was developed in (Abd-el-Malek et al., 2007). The 2D shallow water equations was used to describe water flows with a free surface under the influence of gravity (with gravitational acceleration ) and the Coriolis force due to the earth's rotation with angular velocity, (Vallis, 2006). The shortcoming of the model is that it does not take into account the density stratification which is present in the geophysical flow. Iornumbe et al.,(2020) considered the model of the bottom topography of a geophysical fluid flow in the presence of Coriolis force, without stratification and a system of three nonlinear partial differential equations in two dimensions was modelled and solved using the perturbation method. Since stratified fluids are very present in nature, present in almost any heterogeneous fluid body, therefore, we considered it in this paper.
In this paper, a mathematical model for the stratified geophysical fluid flow over variable bottom topography is derived taken into account the density stratification which is present in the geophysical flow and is an extension of the work of Iornumbe et al. (2020).
and the velocity in the ydirection as can be chosen arbitrarily, it is usually positioned at mean water level. 6. Measuring down from this plane, the bottom of the harbour is at depth is the equation for the bottom surface, also known as the variable bottom topography or bathymetry, the depth of which is usually assumed to vary with x and y . 7. The stratified fluid is one in which the fluid density  is a function of space   z y x , , and time t

Model Equations
According to Iornumbe et al. (2020); the momentum and Continuity equation for the two-dimensional shallow water flow model taking into account the effects of topography and the Earth's rotation was described as below: The momentum equations in the two directions are; And the Continuity Equation as; We now consider the dynamics of multiple layers of fluid stacked on top of each other. This is a crude representation of continuous stratification, but it turns out to be a powerful model of many geophysical interesting phenomena. The pressure is continuous across the interface, but the density jumps discontinuously and this allows the horizontal velocity to have a corresponding discontinuity. In each layer pressure is given by the hydrostatic approximation, and so anywhere in the interior. We can find the pressure by integrating the hydrostatic approximation down from the top. Thus, hydrostatic approximation is,  g dz dp   4 x Figure 1: The multilayer stratified shallow water system. The layers are numbered from the top down. The coordinates of the interfaces are denoted by , and the layer thicknesses i h .
Integrating equation    For three layer model, pressure can be given as;

Method of Solution
We will consider the model equations can be written as can be written as can be written as: can be written as: , we obtain 1 1 can be written as: can be written as: can be expressed in series form as: and compare the powers of f .              2 Integrating this with respect to t , we have;    Similarly,

Results and Discussion
Using Maple software, the following results were obtained graphically to help interpret the result of the modeled equations. Thus; . Furthermore, the result shows that in in all cases, the higher the force due to rotation, the faster the rate of flow.

128| A Mathematical Model of Stratified Geophysical Fluid Flows over Variable Bottom Topography
The graphs of figure  For the top layer, we may note that sunlight, wind and other environmental factors may increase the temperature of the upper layer thereby changing the density of it and thus making the flow faster than that of the lower layer.These correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. A surface wave travels at the free surface of a fluid. The maximum velocity of the wave and the maximum displacement of fluid particles occur at the free surface of the fluid. To be noted is the fact that there is more congestion in the second layer as there are more obstructions in the dowmward part than in the upper part and the density is denser there than in the upper part . If there is any barrier at the water bed, the effect is felt in the second layer more than in the upper layer. Since the second layer is closer to the bed the rate of randomness and fluctuation is higher there than that at the surface layer. The flow in this layer is known as internal wave which travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface) as internal waves depend on the density-stratification of the fluid.  were also experienced. This shows that, the flow velocity is high on the upper layer than in the down layer . Figure 7&8 shows the flow velocity of the first layer   t y x u , , 1 and   t y x u , , 2 in the horizontal  x direction and at different times t . The flow is irregular and random at different points in time and space. The figures described an overturned (that is inverted) character. The nature of the overturning regions is strongly dependent on the strength of stratification which caused vertically layered flow structures in stratified turbulent fluid thereby producing local shear instabilities due to sharp vertical gradients and subsequently overturning.  in the first layer. The obstructed waves inturn become barriers to other incoming waves which then causes more congestion in the second layer than in the first layer thereby decreasing the rate of flow velocity across the water bed and thus resulting in the disorderliness and turbulence in flow as shown.

Conclusion
We were able to derive the mathematical models for the stratified geophysical fluid flow over variable bottom topography. We obtained a system of six nonlinear partial differential equations in twodimensions with Coriolis force and bottom topography included and these were solved using perturbation method. The presence of the Coriolis force in the shallow water equations were shown to cause the deflection of fluid parcels in the direction of wave motion and causes gravity waves to disperse. As water depth decreases due to bottom topography, the wave amplitude increases, the wavelength and wave speed decreases resulting in overturning of the wave. The effect of the Coriolis force is seen in the oscillatory motion in the direction of wave motion which causes gravity waves to disperse.
Fluids in the geophysical flow are generally in motion. The surface resistance in the planetary boundary layer or the wind driven force of the rivers and oceanic surface layer means that the motion varies with amplitude. Despite the stable density stratification, the velocity varies adequately fast with height, destabilizing the flow and resulting in violent overturning of the density surfaces. Geophysical fluid flow with varied bottom topography strongly influences stratified flows because fluid parcels are typically deflected in both the horizontal and vertical direction by the obstacle. This behavior is parameterized by the Froude number. The Froude number is a measure of the likelihood of the flow to be deflected vertically against the density stratification. However, the response of a stratified flow to topography is also dependent upon the shape of the obstacle, but for simple two dimensional obstacles, our study has found regimes in which violent overturning of density surfaces can occur.
In addition, the vertical deflections of fluid parcels generate internal waves which can propagate vertically in the stratified flow. However, owing to the variation of velocity and stratification with height in the shallow water, it is likely that these waves will break and form regions of turbulent overturning at some level.