Geology Reference
In-Depth Information
are exact at the nodes but approximate
elsewhere, and into boundary element methods
that are approximate at the nodes but exact
elsewhere. Each method has its pros and cons.
Boundary element methods are relatively simple
and are able to handle multiple discontinuities
(each properly influencing the other). Finite
element methods cannot handle discontinuities
(although faults can be simulated by so-called
slippery nodes or “shear zones”), but they are
more amenable to more complex rheological
and thermal states. (One might argue, however,
that once a problem gets that complicated, the
numerical technique loses it pedagogic value,
and the number of model parameters becomes
large enough that it rapidly becomes very
specialized.)
The choice of model scale in a numerical
code - how much that model can resolve in both
time and space - depends on the goals of the
modeling exercise. One must decide whether
the feature being studied is simple enough to
allow characterization and, hence, modeling in
only one dimension - i.e.,
z
(
x
), where the eleva-
tion
z
is the dependent variable being assessed
on a one-dimensional grid of points in
x
- or
whether the model requires two dimensions -
i.e.,
z
(
x
,
y
). We note as an aside that, although
the landscape being explored in these two-
dimensional models is in fact a three-dimensional
object, the model is still strictly speaking two-
dimensional (some call it a two-dimensional
planform), because the vertical dimension is a
dependent variable.
The choice of modeling strategy is not simple.
To understand the choices, one must have in
mind a target feature and know to what degree
this feature is describable in one or two
dimensions. Is the scarp being assessed
essentially a linear feature whose profile is eve-
rywhere the same? Is the tectonic deformation
profile symmetrical in some way (radially or
cylindrically)? Modeling strategies generally
consist of setting up the problem, embedding in
the code the differential equations for the
various processes to be modeled (all of which
will have free parameters, such as the diffusivity
or the rate of regolith generation or the fault
slip rate), and finally sweeping through a set of
model runs to explore the dependence of the
final results on (i) initial conditions, (ii) boundary
conditions, and (iii) various model parameters
that set the relative importance of one or another
process. The complexity and computational
requirements increase many-fold as one moves
to higher dimensions. Such demands, therefore,
limit the degree to which a particular parameter
can be explored, because any exploration
requires many model runs.
For example, we can clearly discriminate
between models whose goal is to explore the
degradation of a single fault scarp, from those
in which an entire mountain range is to be
addressed. The length scales for the former
might be 1-100 m, whereas those of the latter
might be 10-100 km. Obviously, the details of
the single fault being addressed can be treated
in the smaller fault model, whereas decisions
must be made about how to treat the reality
of numerous faults and their geometrical
complexity when modeling at the mountain
range scale. Given the computational limitations,
this restriction might require that the fault scarp
model be capable of resolving meters, whereas
the mountain range will be resolved at 100 m.
Although this distinction seems at first to be a
simple scaling problem, it becomes clear quickly
that the lower resolution limits the detail with
which certain processes can be treated, and it
requires that other processes either be ignored
altogether, or more likely be parameterized in
some manner in the model. This scaling issue
forms a large part of the art of modeling and of
the challenge faced by the modeling community.
The building blocks
We first illustrate the components of a landscape
evolution model, all of which must be linked in
the final model. Please note that this recitation
is by no means an exhaustive catalog of such
model components! Numerical models discre-
tize space and time into small increments, d
x
and d
y
for two-dimensional space, d
x
alone for
one-dimensional space, and d
t
for time. The
numerical landscape (in either finite element
or finite difference cases) lives within this
