Geology Reference
In-Depth Information
from place to place result in tilting of the Earth's
surface. Because geomorphic processes are
largely driven by gravity and because the com-
ponent of gravity operating parallel to the
surface is dictated by the slope of the surface,
process rates will be affected by displace-
ment gradients. Of course, as we have
already discussed, in particular instances, direct
connections between geomorphic and tectonic
processes exist, including, for example,
seismically induced mass wasting. Because tec-
tonic and geomorphic processes are so tightly
coupled, and their results so tightly intertwined
in the landscape, one must often resort to
numerical modeling of landscapes in order to
extract the tectonic signal. In fact, if the tectonic
signal is known, such tectonically active
landscapes can be “inverted” to obtain the
geomorphic process rates.
We will discuss in this chapter a wide variety
of models of landscape evolution that have been
developed largely within the last couple of dec-
ades. These models will serve to illustrate the
couplings within the landscape system and to
demonstrate both how tectonic and geomorphic
systems conspire to generate particular land-
forms and how documentation of the shapes of
these landforms can be used to infer tectonic
process rates. We hope to impress upon the
reader that the generation of numerical
landscape evolution models and the collection
of relevant field data are intimately coupled
intellectual exercises.
We will focus on numerical models that rely
on solving equations on a grid of points.
Although such models are more common in the
computer age, it is well worth noting that there
are still solutions to differential equations for
the physics (and sometimes chemistry) involved.
In certain restricted but important cases, these
equations can be solved analytically (with paper,
pencil, and brain), which allows very rapid
assessment of the dependence of the model
solution on one or another process or process
rate. These analytic solutions also form important
tests of numerical codes. Unfortunately, the real
world is complex in that (i) several processes
are acting simultaneously, (ii) some of these
processes are nonlinear, (iii) the tectonic and
climatic forcing of the system is non-uniform in
both space and time, and (iv) the geometric
boundaries of the features are complex. In
general, these complexities preclude analytical
solutions to the problems and require that we
turn to numerical models.
Numerical models come in several flavors:
finite difference, finite element, and boundary
element. Finite difference models operate on a
discretized space, and solve for the change in
some property of each cell in the space (e.g., its
elevation) by approximating the differential
equation at finite (as opposed to infinitesimal)
temporal steps. For example, the differential
equation
z
=
∂ ∂
∂ ∂
could be written as
z
aA
t
x
−
zz
z A
−
Approaches
Δ=
i
i
1
d
t
i
d
x
The approach one takes to the incorporation of
both tectonic and geomorphic processes in a
landscape evolution model depends on the
questions being asked of the landscape. Is the
goal one of deducing the paleoseismic record,
or of deducing long-term changes in slip rates
on a fault? Is the exercise a generic one, in which
a class of landforms is being addressed, or is it
site-specific, in which the attributes of a particu-
lar site are being used as a means of assessing
the “fit” of a particular model.
where d
x
is the node spacing in the
x
direction,
d
t
is the time step, and
i
is the index of the
node. The change in the elevation of node
i
is
calculated, and then the new elevation
Z
i
is
obtained by summing the old elevations with
the changes in elevation.
Boundary element and finite element methods
differ in essence by the characterization of the
boundaries and of the region of interest (see
Crouch and Starfield, 1983). Mathematically,
this translates into finite element solutions that
