16 
In this chapter the gravitational potential field is developed for \esc. 
In this chapter the gravitational potential field is developed for \esc. 
17 
Gravitational fields are present in many modelling scenarios, including 
Gravitational fields are present in many modelling scenarios, including 
18 
geophysical investigations, planetary motion and attraction and microparticle 
geophysical investigations, planetary motion and attraction and microparticle 
19 
interactions. Gravitational fields also presents the opportunity to demonstrate 
interactions. Gravitational fields also present an opportunity to demonstrate 
20 
the saving and visualisation of vector data for Mayavi. 
the saving and visualisation of vector data for Mayavi. 
21 


22 
The gravitational potential $U$ at a point $P$ due to a region with a mass 
The gravitational potential $U$ at a point $P$ due to a region with a mass 
26 
\nabla^2 U(P) = 4\pi\gamma\rho(P) 
\nabla^2 U(P) = 4\pi\gamma\rho(P) 
27 
\end{equation} 
\end{equation} 
28 
where $\gamma$ is the gravitational constant. 
where $\gamma$ is the gravitational constant. 
29 
Consider now the \esc general form, which can simply be related to 
Consider now the \esc general form, which has a simple relationship to 
30 
\autoref{eqn:poisson} using two coefficients. The result is 
\autoref{eqn:poisson}. There are only two nonzero coefficients, $A$ and $Y$, 
31 

thus the relevant terms of the general form are reduced to; 
32 
\begin{equation} 
\begin{equation} 
33 
\left(A_{jl} u_{,l} \right)_{,j} = Y 
\left(A_{jl} u_{,l} \right)_{,j} = Y 
34 
\end{equation} 
\end{equation} 
35 
one recognises that the LHS side is equivalent to 
One recognises that the LHS side is equivalent to 
36 
\begin{equation} \label{eqn:ex10a} 
\begin{equation} \label{eqn:ex10a} 
37 
\nabla A \nabla u 
\nabla A \nabla u 
38 
\end{equation} 
\end{equation} 
39 
If $A=\delta_{jl}$ then \autoref{eqn:ex10a} is equivalent to 
and when a $A=\delta_{jl}$, \autoref{eqn:ex10a} is equivalent to 
40 
\begin{equation*} 
\begin{equation*} 
41 
\nabla^2 u 
\nabla^2 u 
42 
\end{equation*} 
\end{equation*} 
43 
and thus Poisson's \autoref{eqn:poisson} satisfies the general form when 
Thus Poisson's \autoref{eqn:poisson} satisfies the general form when 
44 
\begin{equation} 
\begin{equation} 
45 
A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho 
A=\delta_{jl} \text{ and } Y= 4\pi\gamma\rho 
46 
\end{equation} 
\end{equation} 
47 
At least one boundary point must be set for the problem to be solvable. For this 
The boundary condition is the last parameter that requires consideration. The 
48 
example we have set all of the boundaries to zero. The normal flux condition is 
potential $U$ is related to the mass of a sphere by 
49 
also zero by default. For a more realistic and complicated models it may be 
\begin{equation} 
50 

U(P)=\gamma \frac{m}{r^2} 
51 

\end{equation} where $m$ is the mass of the sphere and $r$ is the distance from 
52 

the center of the mass to the observation point $P$. Plainly, the magnitude 
53 

of the potential is goverened by an inversesquare distance law. Thus, in the 
54 

limit as $r$ increases; 
55 

\begin{equation} 
56 

\lim_{r\to\infty} U(P) = 0 
57 

\end{equation} 
58 

Provided that the domain being solved is large enough, the potential will decay 
59 

to zero. This stipulates a dirichlet condition where $U=0$ on the boundary of a 
60 

domain. 
61 


62 

This boundary condition can be satisfied when there is some mass suspended in a 
63 

freespace. For geophysical models where the mass of interest may be an anomaly 
64 

inside a host rock, the anomaly can be isolated by subtracting the density of the 
65 

host rock from the model. This creates a ficticious free space model that will 
66 

obey the boundary conditions. The result is that $Y=4\pi\gamma\Delta\rho$, where 
67 

$\Delta\rho=\rho\rho_0$ and $\rho_0$ is the baseline or host density. This of 
68 

course means that the true gravity response is not being modelled, but the 
69 

reponse due to an anomalous mass $\Delta g$. 
70 


71 

For this example we have set all of the boundaries to zero but only one bondary 
72 

point needs to be set for the problem to be solvable. The normal flux condition 
73 

is also zero by default. For a more realistic and complicated models it may be 
74 
necessary to give careful consideration to the boundary conditions of the model, 
necessary to give careful consideration to the boundary conditions of the model, 
75 
which can have an influence upon the solution. 
which can have an influence upon the solution. 
76 


98 
\begin{equation} 
\begin{equation} 
99 
\vec{g} = \nabla U 
\vec{g} = \nabla U 
100 
\end{equation} 
\end{equation} 
101 
This for example as a vertical component $g_{z}$ where 
$\vec{g}$ is a vector and thus, has a a vertical component $g_{z}$ where 
102 
\begin{equation} 
\begin{equation} 
103 
g_{z}=\vec{g}\cdot\hat{z} 
g_{z}=\vec{g}\cdot\hat{z} 
104 
\end{equation} 
\end{equation} 
121 
grav_pot=sol,g_field=g_field,g_fieldz=g_fieldz,gz=gz) 
grav_pot=sol,g_field=g_field,g_fieldz=g_fieldz,gz=gz) 
122 
\end{python} 
\end{python} 
123 


124 

It is quite simple to visualise the data from the gravity solution in Mayavi2. 
125 

With Mayavi2 open go to File, Load data, Open file \ldots as in 
126 

\autoref{fig:mayavi2openfile} and select the saved data file. The data will 
127 

have then been loaded and is ready for visualisation. Notice that under the data 
128 

object in the Mayavi2 navigation tree the 4 values saved to the VTK file are 
129 

available (\autoref{fig:mayavi2data}). There are two vector values, 
130 

\verbgfield and \verbgfieldz. Note that to plot both of these on the same 
131 

chart requires that the data object be imported twice. 
132 


133 

The point scalar data \verbgrav_pot is the gravitational potential and it is 
134 

easily plotted using a surface module. To visualise the cell data a filter is 
135 

required that converts to point data. This is done by right clicking on the data 
136 

object in the explorer tree and sellecting the cell to point filter as in 
137 

\autoref{fig:mayavi2cell2point}. 
138 


139 

The settings can then be modified to suit personal taste. An example of the 
140 

potential and gravitational field vectors is illustrated in 
141 

\autoref{fig:ex10pot}. 
142 


143 

\begin{figure}[ht] 
144 

\centering 
145 

\includegraphics[width=0.75\textwidth]{figures/mayavi2_openfile.png} 
146 

\caption{Open a file in Mayavi2} 
147 

\label{fig:mayavi2openfile} 
148 

\end{figure} 
149 


150 

\begin{figure}[ht] 
151 

\centering 
152 

\includegraphics[width=0.75\textwidth]{figures/mayavi2_data.png} 
153 

\caption{The 4 types of data in the imported VTK file.} 
154 

\label{fig:mayavi2data} 
155 

\end{figure} 
156 


157 

\begin{figure}[ht] 
158 

\centering 
159 

\includegraphics[width=0.75\textwidth]{figures/mayavi2_cell2point.png} 
160 

\caption{Converting cell data to point data.} 
161 

\label{fig:mayavi2cell2point} 
162 

\end{figure} 
163 


164 
\begin{figure}[ht] 
\begin{figure}[ht] 
165 
\centering 
\centering 
166 
\includegraphics[width=0.75\textwidth]{figures/ex10apot.png} 
\includegraphics[width=0.75\textwidth]{figures/ex10apot.png} 
167 
\caption{Newtonian potential with g field directionality.} 
\caption{Newtonian potential with g field directionality.} 
168 
\label{fig:ex10pot} 
\label{fig:ex10pot} 
169 
\end{figure} 
\end{figure} 
170 

\clearpage 
171 


172 
\section{Gravity Well} 
\section{Gravity Well} 
173 
\sslist{example10b.py} 
\sslist{example10b.py} 
174 
Let us now investigate the effect of gravity in three dimensions. Consider a 
Let us now investigate the effect of gravity in three dimensions. Consider a 
175 
volume which contains a sphericle mass anomaly and a gravitational potential 
volume which contains a sphericle mass anomaly and a gravitational potential 
176 
which decays to zero at the base of the anomaly. 
which decays to zero at the base of the model. 
177 


178 
The script used to solve this model is very similar to that used for the gravity 
The script used to solve this model is very similar to that used for the gravity 
179 
pole in the previous section, but with an extra spatial dimension. As for all 
pole in the previous section, but with an extra spatial dimension. As for all 