model results should diminish. If these parameters and conditions were included, the model may
approach the actual behavior of the natural
process more closely. On the other hand, the additional parameters and conditions may be less well
constrained by actual data. A master of this kind
of analysis, Art Lachenbruch, describes this
dilemma in the following way:
In selecting a formal model there is a trade-off; as its
complexity increases, we can usually conclude less
from it with more confidence. In this paper we select a
simple model and attempt to conclude a lot from it,
while recognizing that the literal application of these
conclusions is questionable, but that the insight is
likely to be useful (Lachenbruch, 1973).
It is relatively easy, given the power of modern
computers and the sophisticated tools available
for modeling complex mechanical systems, to
create a model of a geological process that is itself
too complex to understand, or that is too poorly
constrained by data. In hindsight it is clear that
Gilbert’s piston laccolith was simple enough to
give him insight, and practical in the sense that
the necessary field data could be gathered to test
the hypothesis derived from the model.
In general terms the procedure we have
described in this section is not new. Indeed, it can
be traced back to the reductionist methodology
first articulated by Descartes in 1619 (Davis and
Hersh, 1986). It is one of the cornerstones of
modern scientific research.
12.2 Selection of general boundary
conditions
General boundary conditions define the context
of a problem, so the appropriate theoretical principles, fundamental laws, governing equations,
and computational tools can be brought to bear
on finding a solution. The selection of general
boundary conditions is not a simple matter, and
no easily written prescription can address all the
variations and complexities one is likely to
encounter. This procedure requires an understanding of both the field observations that serve
to characterize the structures, and the physical
principles and scaling relationships that underlie
possible models for these structures. Here we give
examples, all based upon the structure that
Gilbert set out to investigate over a hundred years
ago in the Henry Mountains, and show how different questions, posed in the field about the
same structure, lead to different choices for the
context of modeling, that is to different general
boundary conditions.
Because tectonic processes are multifaceted,
often involving solid and fluid deformation, heat
and mass transport, and chemical reactions, rarely
is there a single choice of general boundary conditions that enables one to address all the interesting
questions that arise from field observations.
Furthermore, we are limited in our understanding
of how these various processes are coupled
together, and the solution methods for coupled
problems may be difficult to implement. Finally,
the solutions to coupled problems may be so
complex that they are difficult to understand and
therefore are not readily applied to the simple
questions asked in the field. For all of these reasons
we choose to break up tectonic processes into component parts following the methodology described in the quote at the beginning of this chapter
(Crick, 1988). Each part is analyzed according to a
particular choice of general boundary conditions.
Thus, the selection of general boundary conditions
is influenced by practical necessity and guided by
an underlying reductionist philosophy of science.
Once we have made the choice to work within
a particular area of continuum mechanics such as
plate theory, elasticity theory, viscous fluid
mechanics, or heat conduction, the most general
differential equations for that area usually are
simplified by eliminating certain variables or
terms from these equations. This elimination may
be based on field observations that justify, for
example, the reduction of a three-dimensional
structure to a two-dimensional approximation for
that structure. Or, for example, the elimination
may be based on a choice to ignore the initial and
final phases of deformation during which a flow
field changes in time, so a steady-state approximation can be used to describe the well-developed
flow field that existed between these phases. In
other cases dimensional analysis is used to quantify the relative importance of terms in the governing equations and eliminate those that are
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MODEL DEVELOPMENT AND METHODOLOGY
approach the actual behavior of the natural
process more closely. On the other hand, the additional parameters and conditions may be less well
constrained by actual data. A master of this kind
of analysis, Art Lachenbruch, describes this
dilemma in the following way:
In selecting a formal model there is a trade-off; as its
complexity increases, we can usually conclude less
from it with more confidence. In this paper we select a
simple model and attempt to conclude a lot from it,
while recognizing that the literal application of these
conclusions is questionable, but that the insight is
likely to be useful (Lachenbruch, 1973).
It is relatively easy, given the power of modern
computers and the sophisticated tools available
for modeling complex mechanical systems, to
create a model of a geological process that is itself
too complex to understand, or that is too poorly
constrained by data. In hindsight it is clear that
Gilbert’s piston laccolith was simple enough to
give him insight, and practical in the sense that
the necessary field data could be gathered to test
the hypothesis derived from the model.
In general terms the procedure we have
described in this section is not new. Indeed, it can
be traced back to the reductionist methodology
first articulated by Descartes in 1619 (Davis and
Hersh, 1986). It is one of the cornerstones of
modern scientific research.
12.2 Selection of general boundary
conditions
General boundary conditions define the context
of a problem, so the appropriate theoretical principles, fundamental laws, governing equations,
and computational tools can be brought to bear
on finding a solution. The selection of general
boundary conditions is not a simple matter, and
no easily written prescription can address all the
variations and complexities one is likely to
encounter. This procedure requires an understanding of both the field observations that serve
to characterize the structures, and the physical
principles and scaling relationships that underlie
possible models for these structures. Here we give
examples, all based upon the structure that
Gilbert set out to investigate over a hundred years
ago in the Henry Mountains, and show how different questions, posed in the field about the
same structure, lead to different choices for the
context of modeling, that is to different general
boundary conditions.
Because tectonic processes are multifaceted,
often involving solid and fluid deformation, heat
and mass transport, and chemical reactions, rarely
is there a single choice of general boundary conditions that enables one to address all the interesting
questions that arise from field observations.
Furthermore, we are limited in our understanding
of how these various processes are coupled
together, and the solution methods for coupled
problems may be difficult to implement. Finally,
the solutions to coupled problems may be so
complex that they are difficult to understand and
therefore are not readily applied to the simple
questions asked in the field. For all of these reasons
we choose to break up tectonic processes into component parts following the methodology described in the quote at the beginning of this chapter
(Crick, 1988). Each part is analyzed according to a
particular choice of general boundary conditions.
Thus, the selection of general boundary conditions
is influenced by practical necessity and guided by
an underlying reductionist philosophy of science.
Once we have made the choice to work within
a particular area of continuum mechanics such as
plate theory, elasticity theory, viscous fluid
mechanics, or heat conduction, the most general
differential equations for that area usually are
simplified by eliminating certain variables or
terms from these equations. This elimination may
be based on field observations that justify, for
example, the reduction of a three-dimensional
structure to a two-dimensional approximation for
that structure. Or, for example, the elimination
may be based on a choice to ignore the initial and
final phases of deformation during which a flow
field changes in time, so a steady-state approximation can be used to describe the well-developed
flow field that existed between these phases. In
other cases dimensional analysis is used to quantify the relative importance of terms in the governing equations and eliminate those that are
462
MODEL DEVELOPMENT AND METHODOLOGY