\previousTopic{gd_13.latex}
\appendix
\pagenumbering{roman}
\chapter{}
\section {The equations of motion}
Consider a small rectangular parallelepiped aligned in a cartesian co-ordinate
system, $x$, $y$ and $z$, with sides of length $\delta_{x}$, $\delta_{y}$ and
$\delta_{z}$ respectively, as shown in Figure \ref{balance}. Remembering that force equals
stress by area we begin by decomposing the
surfaces forces into stresses acting on each of the faces.
\begin{figure}
\label{balance}
\vspace{1.9in}
\hspace{1.7in}
\special{epsf=:gdfigs:chap2:Fig2_5.eps scale=0.700 }
\caption{\protect\scriptsize{Co-ordinate system used to derive the equations of motion}}
\end{figure}
Assume that at the point, P,
centred within the parallelpiped the normal stresses in the directions $x$, $y$ and
$z$ are given by $\sigma_{xx}$, $\sigma_{yy}$ and $\sigma_{zz}$. The shear stress in the
direction normal to $x$ and parallel to $y$ is given by $\tau_{xy}$ and parallel to $z$
by $\tau_{xz}$. Similarly the shear stresses normal to $y$ are given by
$\tau_{yx}$ and $\tau_{yz}$, and normal to $z$ by $\tau_{zx}$ and
$\tau_{zy}$. Then the forces acting in the direction $x$ on the faces BB'CC'
and AA'DD' are, respectively:
\[
\left\{ \sigma_{xx} + \frac{1}{2} \frac{\partial{\sigma_{xx} }} {\partial
{x}} \delta_{x}\right\} \delta_{y} \delta_{z}
\]
and
\begin{equation}
-\left\{ \sigma_{xx} - \frac{1}{2} \frac{\partial{\sigma_{xx} }} {\partial
{x}} \delta_{x}\right\} \delta_{y} \delta_{z}
\end{equation}
where the negative sign is due the fact that stresses are treated as positive
in tension and negative in compression.
Across the faces A'B'C'D' and ABCD the forces are
\[
\left\{ \tau_{xy} + \frac{1}{2} \frac{\partial{\tau_{yx} }} {\partial
{y}} \delta_{y}\right\} \delta_{x} \delta_{z}
\]
and
\begin{equation}
-\left\{ \tau_{xy} - \frac{1}{2} \frac{\partial{\tau_{yx} }} {\partial
{y}} \delta_{y}\right\} \delta_{x} \delta_{z}
\end{equation}
Across the faces DCC'D' and ABB'A' the forces are
\[
\left\{ \tau_{xz} + \frac{1}{2} \frac{\partial{\tau_{zx} }} {\partial
{z}} \delta_{z}\right\} \delta_{x} \delta_{y}
\]
and
\begin{equation}
-\left\{ \tau_{yz} - \frac{1}{2} \frac{\partial{\tau_{zx} }} {\partial
{z}} \delta_{z}\right\} \delta_{x} \delta_{y}
\end{equation}
The body force acting on the volume with density, $\rho$, in the direction $x$ is
given by
\begin{equation}
\rho X \delta_x \delta_y \delta_z
\end{equation}
The total force in the $x$ direction is then
\begin{equation}
\left\{ \frac{\partial{\sigma_{xx}}} {\partial{x}} +
\frac{\partial{\tau_{yx}}} {\partial{y}} +
\frac{\partial{\tau_{zx}}} {\partial{z}} + \rho X \right\}\delta_x \delta_y
\delta_z
\end{equation}
If the component of displacement of point P in the $x$ direction is $u$ then
Newton's second law gives
\begin{equation}
\rho \frac{\partial{^2u}}{\partial{^2t}} = \frac{\partial{\sigma_{xx}}}
{\partial{x}} + \frac{\partial{\tau_{yx}}} {\partial{y}} +
\frac{\partial{\tau_{zx}}} {\partial{z}} + \rho X
\end{equation}
Similar results obtain for the $y$ and $z$ directions. In tectonic settings
accelerations can be regarded as neglible, and the only body force is gravity which
acts in the vertical direction, taken to be $z$, and using the convention for
summation over repeated indices, the equations of motion can be reduced to:
\begin{equation}
\label{eq:motion} 0 = \frac{\partial{\tau_{i,j}}} {\partial{x_i}} + a_{i} \rho g
\end{equation}
where $a_i$ is the unit vector $\left(0,0,1\right)$. Note that in the convention for indices adopted
in Eqn \ref{eq:motion} the co-ordinates $x$, $y$ and $z$ are given by $x_1$, $x_2$ and $x_3$,
respectively, while $\sigma_{ii} = \tau_{ii}$. Equation \ref{eq:motion} is general and can be applied
to many problems related to tectonic phenomona. However, since it is couched in terms of the
components of the stress tensor it must be rendered useful through combination with {\it constitutive equations}
relating stresses to displacements.
%\section{Non-dimensionalising the heat equation}
\section{Calculation of ridge-push force}
In
order to solve Eqn. \ref{eq:ridgef}
we need to formulate the density distribution
appropriate to Figure \ref{ocean}a. The appropriate density distribution is (Figure \ref{ocean}b):
\begin{eqnarray}
\rho_{z} = \rho_{m},& t = 0, & 0 < z \nonumber \\
\rho_{z} = \rho_{w}, & t = t_{1}, & 0 < z < w \nonumber \\
\rho_{z} = \rho_{m}\left[1 + \alpha\, \left(T_{m} - T_{z}\right)\right],
&t = t_{1}, & w < z < w + z_{l}
\end{eqnarray}
where $\rho_{m}$ is the density of mantle at $T_{m}$, the temperature of the
asthenosphere, $\rho_{w}$ is the density of water, $\alpha$ is the
volumetric coefficient of thermal expansion of peridotite. The density
distributions defined by Eqn \ref{eq:ridgef} give the following variation
$\left(\sigma_{zz}\right)_{z}$:
\begin{eqnarray}
\left(\sigma_{zz}\right)_{z} = \rho_{m}\, g\, z ,& t = 0, & 0 < z
\nonumber \\
\left(\sigma_{zz}\right)_{z} = \rho_{w}\, g\, z , & t = t_{1}, & 0 < z < w
\nonumber \\
\left(\sigma_{zz}\right)_{z} = \rho_{w}\, g \,w + g\, \int_{w}^{w+z_{l}}
\rho_{z} dz , &t = t_{1}, & w < z < w + z_{l}
\end{eqnarray}
$F_{1}$ and $F_{2}$ are given by:
\begin{equation}
F_{1} = \int_{0}^{w+z_l}\left(\sigma_{zz}\right)_{z} \, dz
= \frac{\rho_{m}\,Êg\,Ê\left(wÊ+Êz_{l}\right)^{2}}{2}
\end{equation}
\begin{equation}
F_{2} = \int_{0}^{w}\left(\sigma_{zz}\right)_{z} \, dz
= \frac{\rho_{w}\,Êg\,Êw^{2}}{2}
\end{equation}
Since in the lithosphere the density is itself a function of depth the third term, $F_{3}$, in
Eqn \ref{eq:ridgef} is given by:
\begin{equation}
F_{3} = \int_{w}^{w+z_{l}}\left(\sigma_{zz}\right)_{z} \, dz \\
\end{equation}
Assuming that the lithospheric geotherm at $t_{1}$ is linear in depth, the
temperature at the surface of the lithosphere $T_{s} = 0^{o}$C, and $\alpha$ is
independent of temperature then:
\begin{displaymath}
T_{z} = T_{m}\, \frac{ z}{z_{l}} , \; \rho_{z} =\rho_{m}
\left(1Ê+Ê\alpha\,ÊT_{m}Ê\left(1-Ê\frac{z}{z_{l}}\right)\right)
\end{displaymath}
then
\begin{displaymath}
g\, \int_{w}^{w+z_{l}}Ê\rho_{z} \,dz = g\, z_{l}\, \rho_{m} +
\frac{gÊ\,z_{l}Ê\,\rho_{m}\,Ê\alphaÊ\,T_{m}}{2}
= g \,z_{l} \,\rho_{m} \left(1Ê+Ê\frac{\alpha \,ÊT_{m}}{2} \right)
\end{displaymath}
and
\begin{equation}
F_{3} = \rho_{w} \,g\, w\, z_{l} + \frac{g\,Êz_{l}^{2}\,Ê\rho_{m}}{2}
\left(1Ê+ÊÊ\frac{\alphaÊ\,T_{m}}{2} \right)
\end{equation}
Thus Eqn \ref{eq:ridgef} is given by
\begin{equation}
F_{R} = \frac{\rho_{m}\,Êg\,Ê\left(wÊ+Êz_{l}\right)^{2}}{2} -
\frac{\rho_{w}\,ÊgÊ\,w^{2}}{2} -
\left(\rho_{w}\,Êg\,Êw\,Êz_{l}ÊÊ+ÊÊ\frac{gÊ\,z_{l}^{2}\,Ê
\rho_{m}}{2}\left(1Ê+ÊÊ\frac{\alpha
\,ÊT_{m}}{2}\right)\right)
\end{equation}
The condition of isostatic compensation at depth, $w\, + \,z_{l}$, requires that
\begin{displaymath}
\sigma_{zz \, \left(z = w + z_{l},\, t = 0\right)} = \sigma_{zz \, \left(z = w +
z_{l}, \, t = 1\right)}
\end{displaymath}
Solving for $z_{l}$ gives:
\begin{displaymath}
z_{l} = \frac{wÊ\left(\rho_{m}-\rho_{w}\right)}{\alpha \,Ê\rho_{m}\,ÊT_{m} }
\end{displaymath}