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\part{The continents} \chapter{Continental Deformation}
The structural geometry of both convergent and extensional continental
orogens at the outcrop scale is very (some would say - horribly) complex.
More than anything else, this complexity reflects the very strong mechanical
anisotropy of crustal rocks; that is, the structural geometry resulting from
the deformation is largely a consequence of the inherited structure and is
only weakly coupled to the nature of the forces driving the deformation.
Indeed this complexity begs the question as to whether it is possible, or
even useful to attempt, to evaluate the parameters governing the geodynamic
evolution of continental orogenic belts. However, some of the most
impressive large scale features of orogenic belts are much more regular than
their internal structural geometry. For example, the topography of orogenic
belts, while very fragmented (fractal) on small scales, is very regular at
the scale of the orogenic belt. Indeed, just as an understanding on the
control on topographic variation in the ocean basins provides fundamental
insights, understanding the controls on topography provides very important
insights into the mechanics of continental orogens.
We begin by examining
the controls and some consequences of topography and potential enegrgy using
simple calculations based on the assumption of local isostatic equilibrum.
This assumption is only likely to be valid for thermally mature orogenic
systems which have started to develop plataeus (e.g., Tibet) or, in
extension, wide basins, in which the deformation of the lithsophere is
induced by forces applied as {\em end loads}. The margins of orogenic belts involving wedge shaped
thrust belts are certainly not in isostatic equilbrium and so the topographic
variation in such circumstances need to be evaluated using a differnet
set of {\em boundary conditions}. To tackle the mechanics of the frontal parts of
mountain belts, and accretionar wedges, we need to investigate the dynamics
of {\em critical wedges}, where the driving forces are imparted to the deforming
crust as {\em basal tractions} along some kind of master thrust or
decollement.
\section{Deformation of the lithosphere subject to an end load}
Any consideration of the
behaviour of the continental lithsophere during deformation needs to
recognise the contrasting influence of the crust and the mantle lithosphere
in mediating both the thermal and isostatic response: \begin{itemize}
\item{\em The crust} represents the buoyant part of the lithosphere as well
as the part where lithospheric heat production is concentrated. Thickening
of the crust therefore increases the buoyancy and potential energy of the
lithospheric column, as well as steepening the geotherm through the
increase in the heat production in the thickened column.
\marginpar{\scriptsize{Note that any increase in the geotherm accompanying
thermal equilibration of the thickened lithosphere may take
considerable time following deformation (i.e, 50 Ma).}}\item{\em The mantle lithosphere}
represents the dense negatively buoyant part of the lithosphere, which is
relatively devoid of heat producing elements. Thickening of the mantle
lithosphere therefore reduces the buoyancy and potential energy of the isostatically equilibrated
column as well suprressing the geotherm in the overlying crust
by reducing the heat flow into the base of the crust.
\end{itemize}
Because of this contrasting influence it is useful to consider the effects
of the crust and mantle lithosphere, independently. This can be achieved by
describing the deformation of the lithosphere by the ratio of the changes in
thickness of the crust, $f_c$, and mantle lithosphere, $f_m$,
\marginpar{\scriptsize{In terms of these parameters the change in thickness of the
total lithosphere $f_l$ as used by Sandiford \& Powell, 1990 etc. is given by
$ f_l = f_m + \psi \left( f_c - f_m \right)] $
where $\psi = z_{c0}/z_{l0}$.}} where these are
defined at any stage of the deformation by: \[ f_c = \frac{z_c}{z_{c0}} \]
\[ f_m = \frac{z_m}{z_{m0}} \] where $z_c$ and $z_m$ are the deformed
thickness of the crust and mantle lithosphere, respectively, and $z_{c0}$ and
$z_{m0}$ are the initial thickness of crust and mantle lithosphere prior to
deformation.
\subsection*{Airy Isostasy and crustal thickening}
Assuming Airy isostasy then crustal
thickening results in both an increase
in surface elevation, $c_h$, and the
the development of the crustal root, $c_r$.
\begin{figure}[h,t,b] \label{cthick}
\includegraphics{Figures/Chapter_09/Figure_01.jpg}
% \vspace{3.0in} \hspace{0.0in}
%\special{epsf=:gdfigs:chap9:Fig9_1.eps scale=0.73}
\caption{\protect\scriptsize{ Isostatic effects of crustal thickening.
}}
\end{figure}
For a crust of constant density, $\rho_c$,
overlying mantle of density $\rho_m$, the ratio of the change in surface
elevation $c_h$ to the thickness of the crustal root $c_r$ is given by
\[ \frac{c_h}{c_r} = \frac{\rho_m - \rho_c}{ \rho_c}\]
The change in crustal thickness is given by the sum of $c_h + c_r$:
\[ f_c = \frac{c_h + c_r + z_{c0}}{z_c0}
\]
giving
\[ c_r = f_c \,z_{c0}- c_h - z_{c0} \] Therefore
\[c_h = z_{c0}\left(1 -f_c \right) \left(\frac{\rho_c-\rho_m}{\rho_c} \right)
\]
Note that the change in surface elevation is linear in the change
in crustal thickness;
that is, it is linear in $f_c \, z_{c0}$.
The change in potential energy $U_c$ for this scenario is given by:
\[ U_c = \frac{g \, \rho_c}{2}\left(\left(z_{c0}\,f_c\right)^2
-z_{c0}^2 - 2\, c_r^2\right) - \frac{g \,\rho_m}{2}c_r^2 \]
Note that potential energy changes with the square of the crustal thickening.
The effects of crustal thinning (i.e., $f_c < 1$) are exactly opposite crustal
thickening, that is it results in subsidence and a reduction in potential energy
(see Chapter 10).
\subsection*{Airy Isostasy and the mantle lithosphere}
See Sandiford \& Powell (1990) {\em EPSL} \marginpar{\scriptsize{
Earth and Sanitary Appliance Letters}}
\section{Deformation within the lithosphere due to basal tractions}