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\chapter{Gravity and the lithosphere}
The Earth is self-gravitating in as much as it creates its own gravitational field. The
gravitational body force exerted on material of unit volume in this field is given by the
product of its density and the acceleration due to gravity (which is function of the
mass distribution in the Earth). The way mass is distributed in the lithosphere is central
to the notion of {\em isostatic compensation} and the consequences of gravity acting on the
mass distribution in isostatically compensated lithosphere is central to all of
geodynamics.
\section{Isostasy}
It has long been recognized that the surface elevation of the continents is in some way
related to the density distribution in the subsurface. The gravity field at the surface of
the earth reflects the distribution of mass at depth and gravity measurements across
mountain belts show that regions of high elevation generally have a deficiency of mass at
depth. That is, there is some specific depth beneath the mountain range where the rocks
have a lower density compared with rocks at the same depth beneath the low lying regions
flanking the mountain range. The gravity field shows that the deficiency of mass at depth
is, to a first approximation, equal to the excess mass in the mountains, implying that at
some great depth within the earth, termed the depth of {\it isostatic compensation}, the
mass of the overlaying rock is equal and independent of the surface elevation. From the
gravity data alone, it is not possible to determine exactly how the density is distributed
in order to compensate the topography, and two rival isostatic models have been proposed,
referred to as Pratt Isostasy and Airy Isostasy (Figure \ref{isostasy}). These models were
proposed long before the concept of the lithosphere was formalized and in the original
formulation both models considered that isostatic compensation was achieved entirely within
the crust.
\includegraphics{Figures/Chapter_03/Figure_01.jpg}
\begin{figure}[tb]
%\label{isostasy} \vspace{1.25in} \hspace{0.3in}
%\special{epsf=:gdfigs:chap2:Fig2_1.eps scale=0.70}
\caption{\protect\scriptsize{ Pratt
(a) and Airy (b) isostasy assume rather different subsurface density distributions to
compensate the excess mass associated with the additional topography in mountains. Density
distribution is proportional to stipling. Seismic studies beneath mountains such as the
Himalayas show that the density structure approaches that of Airy Isostasy, although it is
now assumed that isostatic compensation occurs beneath the lithosphere and not at the
Moho.}} \end{figure}
While modern seismic methods have shown that the structure of mountain ranges closely
approximates the Airy model, it is important to dispel the notion that compensation is
generally achieved at the bottom of the crust. The logical place for compensation to take
place is beneath the lithosphere, and in this course we will initially assume, and then
attempt to demonstrate, that this is the general case. Indeed the very existence of large
lateral temperature gradients in the oceanic mantle lithosphere leads to significant
horizontal density gradients. These thermally induced density changes lead to corresponding
changes in the surface elevation of the oceanic lithosphere, through a kind of thermal
isostasy. The horizontal force resulting from the ocean floor topography, termed {\em ridge
push}, provides one of the fundamental driving forces for the motion and deformation of the
lithosphere. Isostatic compensation can be achieved because the lithosphere essentially
{\it floats} on a relatively {\it inviscid} substrate: the weak peridotite of the
asthenosphere. Changes in the buoyancy or elevation of the lithosphere are accommodated by
displacement of asthenospheric mantle. However, asthenospheric mantle is not completely
inviscid (that is, its viscosity is not negligible), and its displacement in response to
lithospheric loading or unloading must take a finite length of time, related to its
effective viscosity. An insight into the timescales for the isostatic response of the
asthenosphere to loads is provided by the rebound of continental lithosphere following the
removal of glacial icecaps. The rebound following the removal of the Pleistocene Laurentide
icecap shows that the time scales appropriate to this isostatic response are of the order
of {$10^{4}-10^{5}$} years. Since mountain belts are built and decay on the time scale of
{$10^{7}-10^{8}$} years, the isostatic response is effectively instantaneous.
\section{The flexural strength of the lithosphere}
An important question concerning isostasy is the horizontal length scale on which
isostatic compensation is achieved. Gravity measurements across mountain belts suggest
that, on the scale of several hundred kilometers, the lithosphere often approaches
isostatic equilibrium. However the same measurements show that the mass excess associated
with small-scale topography, for example, individual mountain peaks within mountain
ranges, is generally not compensated. The length-scale of isostatic equilibrium (viz.,
regional versus local isostatic compensation) relates to the strength of the lithosphere:
on small length-scales departures from isostatic equilibrium are supported by the flexural
(or elastic) strength of the lithosphere. We illustrate the interaction between the
flexural strength of the lithosphere and the horizontal length scale of isostatic
compensation by considering a hypothetical infinitely strong lithosphere floating on a
completely inviscid substrate. Subject to a substantial, localized load such as a mountain
range, such a hypothetical lithosphere will be depressed such that the mass of the
displaced asthenosphere is equivalent to the mass of the load. In this case the mass of the
mountains is compensated over the regional horizontal dimension of the lithosphere. Of
course gravity shows us that this is not the case for the earth, because the density
distribution beneath mountains somehow compensates the excess mass of the mountains on a
length-scale of comparable dimension to the mountains (Figure \ref{flexure}b). However, on
the smaller length scale of individual mountain peaks compensation is not achieved; that is
the lithosphere is sufficiently strong to distribute the load of an individual mountain
over a length scale which is large compared to the lateral dimension of the individual
mountain.
\begin{figure}[tb]
\label{flexure}
% \vspace{1.2in} \special{epsf=:gdfigs:chap2:Fig2_2.eps scale=0.570}
\caption{\protect\scriptsize{ The response of the lithosphere (a) to applied
loads depends on its flexural strength (or rigidity). An infinitely strong lithosphere
compensates loads regionally over the lateral dimension of the whole lithosphere (b). At
the other end of the spectrum compensation may occur completely beneath the load, so called
local isostatic compensation (d). The behaviour of the lithosphere, which has some finite
flexural strength, lies somewhere between these two extremes (c), implying that this
flexural response of the lithosphere is an important component of deformation associaed with loading.
For a visco-elastic lithosphere in which the response to loading is time dependent these
figures illustrate schematically the expected temporal evolution following loading as the
elastic response gives way to a viscous response through time. } } \end{figure}
As we shall see, the lithosphere has a finite flexural strength, in as much as it can
support a limited amount of loading without permanent deformation. The effective flexural
strength of the lithosphere is characterized by the thickness of the elastic lithosphere,
which in turn is dependent on the thermal and compositional structure of the lithosphere.
Importantly the flexural response of the lithosphere to an applied load may be time
dependent (as would be expected for a visco-elastic material); after the emplacement of a
load, the effective elastic thickness of the lithosphere may decay with time as the elastic
stresses are relieved by permanent deformation.
\section{Gravitational potential energy } {\it Isostasy is a vertical stress balance!}
Local isostasy requires that at the level of isostatic compensation the mass of overlaying
rock is equalized, independently of the density distribution above this level. Equivalently
the normal stress in the vertical direction {$\sigma_{zz}$} is equalized at the depth of
compensation. Necessarily, the vertical density structure at, and below, the level of
compensation must everywhere be equal. The vertical stress resulting from gravitational
{\em body forces} acting on the mass, or body, of the rock. The vertical stress,
{$\sigma_{zz}$}, at depth $h$ is given by:
\begin{equation} \label{eq:sigma} \sigma_{zz} = g\, \int_{0}^{h} \, \rho\: dz
\end{equation} where {$\rho$} is the density.
Under hydrostatic conditions ($\sigma_{zz} = \sigma_{xx} = \sigma_{yy}$) the lithostatic
pressure, $P$, is equal to $\sigma_{zz}$. Thus for constant density (depth independent)
hydrostatic conditions:
\begin{equation} \label{eq:pressure} P = \sigma_{zz} = g \,\rho\, h
\end{equation}
{\it Isostasy is not a complete stress balance!} A complete stress balance requires that
horizontal stresses as well as vertical stresses must be balanced. In order to achieve this
in the presence of a gravitational field all density interfaces must be horizontal. This is
clearly not the case for the lithosphere which is characterized by significant lateral
variation in the density structure. For example, significant topography characterize the
density interfaces at the Earth's surface (rock-air), the Moho (mantle-crust) and within
the mantle lithosphere. Variations in lateral density structure within the lithosphere
contribute to variations in the lithospheric gravitational potential energy, $U_l$. The
gravitational potential energy of a lithospheric column of unit area is given by the
integral of the vertical stress from the surface of the earth to the base of the
lithosphere (i.e., depth of isostatic compensation):
\begin{eqnarray} \label{pe} U_l &=&
\int_{0}^{h}\, \sigma_{zz}\: dz \nonumber \\
&=& g \, \int \!\!\int_{0}^{h}\, \rho \: dz \nonumber \\
&=& g \int_{0}^{h}\, \left(\rho \, z \right)\: dz
\end{eqnarray}
Because the lithospheric potential energy is defined in terms of a column of
unit area it has dimensions J m$^{-2}$ or equivalently, N m$^{-1}$. Lateral variations in
gravitational potential energy, $\Delta U_l$, induce substantial horizontal forces,
termed {\em buoyancy forces}. The difference in potential energy between two columns, 1 and
2, integrated over the depth $h$, is (Figure \ref{buoyancy}): \begin{eqnarray}
\label{eq:bforce} \Delta U_l &=& \int_{0}^{h}\, \sigma_{zz_{1}}\: dz - \int_{0}^{h}
\sigma_{zz_{2}}\: dz \nonumber \\
& = & g \, \int_{0}^{h}\, \left(\Delta \rho \,z \right)\: dz
\end{eqnarray}
where $\Delta \rho$ is $\left(\rho_1 -\rho_2 \right)_z$.
\footnote{ The term $\int_{0}^{h}\, \left(\Delta \rho \, z \right)\: dz$ is
referred to as the dipole moment of the density distribution.}
Equation \ref{pe} and \ref{eq:bforce} simply state that in the presence of a gravitational
field there is a tendency to reduce fluctuations in gravitational potential energy ({\it
note that these equations apply equally to the forces operating on an ageing brie, as it
spreads to lower its centre of gravity}). While variations in potential energy generally
correlate with surface elevation it is important to realise that this is not necessarily
so, as illustrated in Figure \ref{fig:pe}. \marginpar{\scriptsize{In Chapter 9 we show
that, ignoring contributions due to variations in the thermal structure of the mantle
lithosphere, for Airy isostasy the surface elevation, $h$, varies linearly with crustal
thickness, $z_c$, while potential energy, $U_l$ varies with $z_c^2$ ({\em see also Chapter
14, p 290, in Brown, Hawkesworth and Wilson}).}} In the absence of an externally applied
stress field, regions of high potential energy will experience tension, and regions of low
potential energy will experience compression.
\includegraphics{Figures/Chapter_03/Figure_03.jpg}
\begin{figure} \label{buoyancy}
% \vspace{2.3in} \hspace{0in}
\special{epsf=:gdfigs:chap2:Fig2_3.eps scale=0.640 } \caption{ \protect\scriptsize{The
magnitude of the buoyancy forces arising from variations in potential energy generally
correlate with topographic gradients on density interfaces in the lithosphere is equal to
the area between the {$\sigma_{zz}$} vs. depth curves for both regions. Note that
{$\sigma_{zz}$} is equalized at the level of isostatic compensation, independent of the
overlaying density structure.}} \end{figure}
\begin{figure} \label{fig:pe} \vspace{2.3in} \hspace{0.3in}
\special{epsf=:gdfigs:chap2:Fig2_4.eps scale=0.700 } \caption{ \protect\scriptsize{While
variations in potential energy generally correlate with variations in elevation, they need
not do so as illustrated in this scenario. Clearly column 2 is gravitationally unstable. In
a geological context this scenario may have some relevance to old ocean lithosphere (see
Chapter 7).}} \end{figure}
The buoyancy forces arising from variations in gravitational potential energy are large
(in fact they provide the fundamental driving forces for the horizontal motion and
deformation in the lithosphere). These forces can be sustained because rocks have finite
strength {\em just as an immature brie does}. An understanding of the rheology of rocks,
that is the way in which they respond to applied forces, is fundamental to developing
quantitative tectonic models.
Gravity acts on mass and as mass is distributed throughout the volume of a body the forces
resulting from the action of gravity are termed {\it body forces} (other types of body
forces such as magnetic forces are largely irrelevant in tectonics). As we have seen in the
previous section the effect of gravitational body forces acting on two columns with
different density distributions produces forces acting on the surface between the two
columns. Such {\it surface forces} are dependent on the area the orientation of the
surface. Since the sum of the forces acting on a body are equal to the mass times
acceleration (Newton's second law) it is relatively simple to derive the equations of
motion, relating displacements and forces (see Appendix A.1). The equations of motion are
necessarily couched in terms of the components of the stresses (or, more correctly, the
stress tensor) and therefore they must be rendered useful through combination with
equations relating stresses to displacements. Such equations, termed {\it constitutive
equations}, are material dependent and their study is the stuff of {\it rheology}.
\section{Lithospheric potential energy and geoid anomalies}
The contributions of lateral variations in the density
structure of the lithospheric to observed geoid anomalies
can be directly related to the dipole moment of the density distribution:
\begin{equation}
\Delta N = - \frac{2 \, \pi \, G}{g} \int_{0}^{h}\, \left(\Delta \rho \, z \right)\: dz
\end{equation}
where $\Delta N$ is the geoid anomaly in metres.
Consequently, variations in lithospheric potential energy are reflected in variations
in the geoid, with an anomaly of 5 m corresponding to a variation in potential energy of
about 1 x 10$^{12}$ N m$^{-1}$.