User:Egm6322.s09.bit.la/edp

= Introduction =

Overview about estuaries
Estuaries are classified as a semi-enclosed coastal body of water that have a free connection with the open ocean and where the salt water is gradually diluted by the fresh water coming from the river discharge. Estuaries have a great importance for animal species and also they are used as habitats, since that is in this environment that some aquatic animals, mainly fishes and crustacean (shrimp), use as nesting. Besides the environmental importance, estuaries are fundamental to the humanity. The influence of men in this environment considerably increase in the last decades, the consequence growth of the population in the cities next to the estuaries (nowadays approximately 60% of the world population live around these areas), is putting in risk the sustainable development of these environments. The spread use of the estuaries around the world is because they are good places to build harbors, an important access to interior areas, fertile areas to agriculture and their waters are periodically renewed by the influence of tides.

Their environmental conditions are very special, which make them highly vulnerable and they can be easily affected with abrupt changes in some of the environmental parameters. The estuary dynamics is particularly complex, due the influence of flood tides, ebb tides and river discharge.

Project goals
The aim of this report is to solve a partial differential equation (PDE) that describe the circulation in a cross section of an estuary using the balance between the friction and the barotropic term with different types of bathymetry.

=Modeling with PDE`s=

Due the complexity of the natural phenomenon and the difficult of make repetitions in laboratories or in the field, the application of numerical models is being used more often with the intent to represent the physics in the real world.

The partial differential equations represent a quantitative description of natural phenomenos. The researchers have to determine a set of valid equations and apply the correct initial and boundary conditions. This is called the mathematical model and, in practical applications, it is sufficient to find a solution to a specific set of data.

Equation that Govern the Estuarine Circulation
The cross estuary circulation can be described by the equation of momentum or equation of motion (1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}-fv=-g\frac{\partial \eta}{\partial x}-\frac{g}{\rho }\int_{z}^{\eta }\frac{\partial \rho}{\partial y}dz+\frac{\partial }{\partial x}\left [ A_x\frac{\partial u}{\partial x} \right ]+\frac{\partial }{\partial y}\left [ A_y\frac{\partial u}{\partial y} \right ]+\frac{\partial }{\partial z}\left [ A_z\frac{\partial u}{\partial z} \right ] (1)$$

Where the terms in the equation are:

$$\frac{\partial u}{\partial t}$$ - local acceleration

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}$$ - advective acceleration

$$fv$$ - Coriolis term

$$g\frac{\partial \eta}{\partial x}$$ - Barotropic component of the pressure gradient

$$\frac{g}{\rho }\int_{z}^{\eta }\frac{\partial \rho}{\partial y}dz$$ - Baroclinic component of the pressure gradient

$$\frac{\partial }{\partial x}\left [ A_x\frac{\partial u}{\partial x} \right ]+\frac{\partial }{\partial y}\left [ A_y\frac{\partial u}{\partial y} \right ]+\frac{\partial }{\partial z}\left [ A_z\frac{\partial u}{\partial z} \right ]$$ - Friction terms

The equation (1) will be the starting point for the following analysis

Consideration
In order to achieve the objective of the study and calculate the balance between the pressure gradient (barotropic) and the friction term we can make some assumption. These assumptions are also based in an analysis of scale that measure the important parameters that will contribute to the estuarine circulation.

Using the continuity equation (2) and the Boussinesq approximation, the accelerations terms are disconsidered

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0 (2)$$

The estuary will be considered with homogeneous waters, meaning that the density in the cross section of the estuary is constant or that the density doesn't change in the cross section direction (the baroclinic term of the pressure gradient in not important).

The estuary that will be used as an example is consider in shallow water and the Coriolis term in this case doesn't have an important component that will generate circulation.

The first two elements of the friction term will not be taking into account since we are analysing the cross section component of the velocit

Applying all assumptions described above

The equilibrium of the forces between the barotropic pressure and friction that will dominate the estuary is:

$$\frac{\partial^2 u }{\partial z^2}=\frac{\partial \eta }{\partial x} (3)$$

Boundary Conditions
It is important to determine the boundary conditions to the model carefully The coordinate system used is the most common used when you are describing a estuary

At z=-h

u=0

At z=0

$$\frac{\partial u}{\partial z}$$=1

Applying the boundary conditions and integrating twice the equation (3)

$$\int \frac{\partial^2 u }{\partial z^2}\partial z=\int \frac{\partial \eta }{\partial x}\partial z$$

we get: $$ \frac{\partial u }{\partial z}=z\frac{\partial \eta }{\partial x}+C_1$$

where: $$C_1$$=constant

Applying the superior boundary condition:

$$\frac{\partial u }{\partial z}=1=C_1$$

And the inferior boundary condition

u=0 at z=-h

$$\frac{\partial u }{\partial z}=-h\frac{\partial \eta }{\partial x}+1$$

$$\left (\frac{\partial \eta }{\partial x} \right )_{-h}=\frac{1}{h}$$

Getting:

$$\frac{\partial u }{\partial z}=z\frac{\partial \eta }{\partial x}+1$$

Integrating again

$$\int \frac{\partial u }{\partial z}=\int z\frac{\partial \eta }{\partial x}+1$$

Finally get the equation that will be used in the Matlab program

$$u=\frac{\partial \eta }{\partial x}\frac{(z^2-h(y)^2)}{2}+(z+h(y))$$

And the volume transport:

$$U=\int_{-h}^{0}udz=\frac{h^2}{2}-\frac{\partial \eta }{\partial x}\frac{h^3}{3}$$

Without net volume transport:

$$\int_{-1}^{1}Udy=\int_{-1}^{1}\frac{h^2}{2}dy-\frac{\partial \eta }{\partial x} \int_{-1}^{1}{\partial x}\frac{h^3}{3}dy=0$$

Creating an average in the cross-section transect

$$\frac{\partial \eta }{\partial x}=\frac{3}{2}\frac{h^2}{h^3}$$

Results of the Model
The figures 2 to 5 are the results of after the application of the program in Matlab (Annex A) to solve the problem.

The figures 2 to 5 show the patterns of velocity in the cross-section velocity using different shapes of the bathymetry. The negative values (blue colors) show inflow in the bottom layers and the positive values (red colors) show outflow in the surface.









Conclusion
The Matlab program developed to solve the PDE that describe the balance between friction and the barotropic component of the velocity showed important results. The method was very satisfactory describing the natural phenomenon. The comparison between the theoretical results obtained with different bathymetries and different estuaries showed a good agreement. Meaning that this methodology can be applied to describe the circulation in this environment.

=References=

=Annex =

Matlab code used

%% Wind Driven solution for different bathymetries

clear all close all

y=[-1:0.005:1]; z=[0:-0.0025:-1]; hy1=(1-(y.^2)); % for 1-y^2 hy2=(1-(y.^6)); % for 1-y^6 hy3=exp(-6*(y.^2)); %for exp(-6y^2) hy4=1-abs(y); % for 1-abs(y) slope1=1.5*((mean((hy1).^2))/(mean((hy1).^3))); slope2=1.5*((mean((hy2).^2))/(mean((hy2).^3))); slope3=1.5*((mean((hy3).^2))/(mean((hy3).^3))); slope4=1.5*((mean((hy4).^2))/(mean((hy4).^3)));

% for 1-y^2 for a=1:401 for b=1:401 if(-z(1,b)<=hy1(1,a)); u1(a,b)=(slope1*(z(1,b).^2-hy1(1,a).^2)/2)+(z(1,b)+hy1(1,a)); else u1(a,b)=nan; end end end

figure(1) rotu1=rot90(u1,-1); contourf(y,z,rotu1);hold on %plot(y,-hy1,'white','linewidth',10); colorbar title({'Wind Induced Flow','Bathymetry=1-y^2'}) xlabel('Nondimensional distance') ylabel('Nondimensional depth') print -djpeg fig1.jpeg

% for 1-y^6 for a=1:401 for b=1:401 if(-z(1,b)<=hy2(1,a)); u2(a,b)=(slope2*(z(1,b)^2-hy2(1,a)^2)/2)+(z(1,b)+hy2(1,a)); else u2(a,b)=nan; end end end

figure(2) rotu2=rot90(u2,-1); contourf(y,z,rotu2);hold on %plot(y,-hy2,'white','linewidth',10) colorbar title({'Wind Induced Flow','Bathymetry=1-y^6'}) xlabel('Nondimensional distance') ylabel('Nondimensional depth') print -djpeg fig2.jpeg

%for exp(-6y^2) for a=1:401 for b=1:401 if(-z(1,b)<=hy3(1,a)); u3(a,b)=(slope3*(z(1,b)^2-hy3(1,a)^2)/2)+(z(1,b)+hy3(1,a)); else u3(a,b)=nan; end end end

figure(3) rotu3=rot90(u3,-1); contourf(y,z,rotu3);hold on %plot(y,-hy3,'white','linewidth',10) colorbar title({'Wind Induced Flow','Bathymetry=exp-(6y^2)'}) xlabel('Nondimensional distance') ylabel('Nondimensional depth') print -djpeg fig3.jpeg

% for 1-abs(y) for a=1:401 for b=1:401 if(-z(1,b)<=hy4(1,a)); u4(a,b)=(slope4*(z(1,b)^2-hy4(1,a)^2)/2)+(z(1,b)+hy4(1,a)); else u4(a,b)=nan; end end end

figure(4) rotu4=rot90(u4,-1); contourf(y,z,rotu4);hold on %plot(y,-hy4,'white','linewidth',10) colorbar title({'Wind Induced Flow','Bathymetry=1-abs(y)'}) xlabel('Nondimensional distance') ylabel('Nondimensional depth') print -djpeg fig4.jpeg

=Signature=

Egm6322.s09.bit.la 16:54, 30 April 2009 (UTC)