Introduction
Flexure of the lithosphere is a frequently observed processes by which loads
bend the elastic outer shell of Earth or other planets . The sources of these loads are wide-ranging (Fig. ),
encompassing volcanic islands and seamounts
, mountain-belt-forming thrust sheets and their
associated subsurface loads , sedimentary
basins , continental ice sheets
, lakes , seas and
oceans , extensional tectonics (negative
loads) , erosion (negative loads) ,
mantle plumes (basal buoyant and therefore negative loads)
, and more.
Theory to describe deflections of the lithosphere under loads has evolved
significantly over the past 160 years . The development of
this theory started with simple approximations of perfect buoyant
compensation of loads by a lithosphere with no strength overlying a mantle of
known density . These approximations allowed
surveyors to explain the observed lack of significant gravity anomalies
around large mountain belts cf.,. While this theory,
called isostasy, revolutionized the way topography was viewed on the Earth,
more realistic solutions for isostatic deflections of the surface of Earth
take into account the bending, or flexure, of a lithospheric plate of nonzero
but finite strength. This strength may be defined as the elastic thickness, the effective thickness of a flawless plate of the equivalent
strength, or as the flexural rigidity that is characteristic of a plate
of a given thickness (see Eq. ). By bending over
distances of several tens to hundreds of kilometers, the lithosphere low pass filters a
discontinuous surface loading field into a smoothed solid-Earth response.
Even though the early geological theories of and
focused on simple buoyancy, the differential equation basis
for solving lithospheric bending already existed at that time.
and and earlier work developed
the first differential-equation-based theories for plate bending.
reviewed the prize that Germain won in 1811 for her work
on elastic plate flexure, and, on realizing an error in the lumping of terms
due to Germain's incorporation of an incorrect formula by ,
corrected it and produced the first
complete flexure equation see reviews by. Around the same time, and
better connected the theory of elasticity to plate
bending problems. These works predated , who developed
the classical or Kirchhoff–Love plate
theory that remains in use today . While many further
advances have been made e.g., especially
for structural and aeronautical engineering, it is the
Kirchhoff–Love plate theory that has been used most widely for
geological applications e.g.,.
tested classical Kirchhoff plate theory, which is a thin-plate
theory that simplifies the plate geometry and therefore the mathematics
required to solve for it, against a thick-plate theory of
lithospheric flexure. While this thick-plate theory relaxes several
approximations, its solutions are very similar to those for thin-plate
flexure .
Flexural isostasy can be produced in response to a range of
geological loads.
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In the first half of the twentieth century, and applied analytical
solutions of the plate theory of to geological
problems. They employed analytical solutions that relate the curvature of the
bending moment of a plate of uniform elastic properties to an imposed surface
point load, line load, or sinusoidal load. These load solutions could be used
to compute flexural response to any arbitrary sum of individual loads in
either the spatial or spectral domain, due to the linear nature of the
biharmonic flexure equation (Eqs. and
), and may be combined with a variety of boundary
conditions .
Flowchart for gFlex as either a standalone model with configuration and input files, a Python module
or coupled component in a modeling framework, or a GRASS GIS component.
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Computational advances allowed discretized models to replace purely
analytical solutions. These models fall into one of several categories. Many
take advantage of the linear nature of the flexure equation for constant
elastic thickness to superimpose analytical solutions of point loads (in the
spatial domain) or sinusoidal loads (in the wavenumber domain) in order to
produce the flexural response to an arbitrary load . Other models produce numerical solutions to the thin plate
flexure equation by solving the local derivatives in plate displacement with
numerical (mostly finite difference) methods e.g.,. Models in this latter category allow for
variations in the elastic thickness of the plate, a factor of growing
importance as variations in elastic thickness through space and time are
increasingly recognized, measured, and computed e.g.,. In
spite of these efforts, the community currently lacks a robust, easy-to-use,
generalized tool for flexural isostatic solutions that can be used by
modelers and data-driven scientists alike.
Here I introduce a broadly implementable open-source package of solutions to
flexural isostasy. This package, called gFlex (for GNU flexure), advances and
makes more accessible an earlier model, generically called flexure
. gFlex has been released under the GNU General
Public License (GPL) version 3 and is made available to the public at the
University of Minnesota Earth-surface GitHub organizational repository, at
https://github.com/umn-earth-surface/gFlex, and through the Python
Package Index (PyPI). This allows for rapid collaborative editing of the
source code and easy automated installation. It is written in Python
e.g., for easy interoperability with a range of
other programming languages, models, and geographic information systems (GIS)
packages, and to take advantage of the numerical packages for Python that
allow for much more rapid matrix solutions than would be typical with a more
basic interpreted language . See Section for further
information on obtaining and running gFlex.
gFlex can solve plate flexure in two major ways (Fig. ).
First, it can produce analytical solutions to flexural isostasy generated by
superposition of local solutions to point loads in the spatial domain (i.e.,
as a sum of Green's functions) e.g.,. These use
biharmonic equation for plate flexure with uniform elastic properties (Eqs.
and ) . Second,
it can compute finite difference solutions for both constant and arbitrarily
varying lithospheric elastic thickness structures. These solutions follow the
work of , and hence , except that gFlex
does not incorporate terms for end loads but does include a wider range of
implementable boundary conditions (Table ). gFlex can be run as a
standalone program with an input
file, as a component of the in-development Landlab landscape modeling
framework and by extension as a
component within the Community Surface Dynamics Modeling System (CSDMS)
, or as a pair of add-ons to
the Geographical Resources Analysis Support System (GRASS) Geographic Information System (GIS) . The GRASS GIS implementation is
particularly important, as it provides pre-built and standardized
command-line and graphical interfaces and the ability to directly pull inputs
from and compare solutions against field data in their native coordinate
systems.
Boundary conditions. Names provided here are the same as those used
in the model, and b.c. stands for boundary condition. The first five can be
selected for numerical solutions. The final one, NoOutsideLoads, is the
outcome of superposition of analytical solutions, which allows the entire
space to respond to local loads as if the 0-deflection boundaries were
infinitely far away. In this notation, the subscript b indicates the
boundary, generically. Where 0 and n are included as subscripts, i.e.,
for the mirror and periodic boundary conditions, these indicate boundaries at
the first and last node of the model domain along a particular axis.
Subscript x, which is a stand-in for x or y, is a variable distance to
indicate the symmetry across a mirror
boundary. Each of these boundary conditions requires a corresponding boundary
condition for flexural rigidity.
Name
Mathematical
Description
Rigidity b.c.
0Displacement0Slope
wb=0
no displacement at boundaries
d2Dbdx2=0
0Moment0Shear
d2wbdx2=d3wbdx3=0
broken plate with a free cantilever end
d2Dbdx2=0
0Slope0Shear
dwbdx=d3wbdx3=0
free displacement of a horizontally clamped boundary
d2Dbdx2=0
Mirror
wb=n-x=wb=n+x
plane of mirror symmetry at boundary
Db=n-x=Db=n+x
Periodic
wb=n=wb=0
wrap-around boundary: infinite tiling of model domain
Db=n=Db=0
NoOutsideLoads
w∞=0
produced by analytical solutions with uniform D
dDbdx=0
Methods and model development
Two solution types for flexural isostasy are provided in gFlex, and these are
formulated for both one-dimensional (line load, assumed to extend infinitely
in an orientation orthogonal to the line along which the equation is solved)
and two-dimensional (point load) cases. The derivation that forms the basis
for both of these is provided in Appendix Sect. , and
similar approaches to this derivation may be found in the work of
and . The analytical and finite
difference approaches are compared and shown to approximate each other well
in Fig. .
Numerical (FD) and analytical (SAS) solutions in one dimension
(a) and two dimensions (c) and
their differences (b, d) in response to a 100 km (long/in diameter)
central line/circular load. These differences are due primarily to the
NoOutsideLoads boundary condition of the analytical solution and the
0Displacement0Slope boundary condition of the numerical solution. This can be
seen in panel (b) where the example with a lower elastic thickness
is less offset due to the greater number of flexural wavelengths between the
load and the boundary, and in the greater agreement between the solutions on
the longer diagonal boundaries in (d). The offset in the middle,
visible as a small bump in (b) and a blue diamond surrounded by red
petals in (d), is due to the difference between approximating the
load as a sum of point impulses (analytical) and as the solution to a
rectangularly gridded matrix equation based on the same theory (numerical).
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Superposition of analytical solutions
The first solution type takes advantage of the linear nature of the
analytical solution for flexure of a plate of constant thickness and elastic
properties when subjected to a point or line load. These solutions may be
superposed (i.e., summed) in space to compute the full flexural response. The
second approach is to solve the equation for lithospheric flexure as a matrix
equation by employing a finite difference scheme. This employs a sparse
matrix elimination solver e.g.,. The primary gFlex
finite difference solution follows the approach of to
permit computations with steep gradients in flexural rigidity
(Appendix Sect. A2), but gFlex also offers the discretization of
.
The analytical solution imposes the assumption that scalar flexural rigidity,
D, is uniform. This leads to biharmonic expressions for plate bending in
one and two dimensions, respectively:
Dd4wdx4+Δρgw=q,D∇4w=D∂4w∂x4+D∂4w∂y4+2-D∂4w∂x2∂y2+Δρgw=q.
Here, w is vertical deflection of the plate, q is the applied surface
load, and Δρ=ρm-ρf is the density of the mantle minus
the density of the infilling material; see Fig. for a
diagrammatic description of all variables. The Δρ term represents
the feedback by which flexural subsidence can lead a depression to be filled
by material, which leads to additional flexural subsidence. This can occur,
for example, in a system that is fully underwater (e.g., an underwater
volcano load) or one in which the depression is completely filled with
sediments. If this infilling material is not uniform in density and/or
spatial extent – for example, due to onlap or offlap of water along a
shoreline – then one may solve this feedback instead via iteration, by
solving for ρf=ρair≈0 and adding water (or another
load) to regions that match certain conditions after every cycle of the
iteration.
The above equations are linearizable, and therefore can be solved by
superposition of analytical solutions. In gFlex, this is done in the spatial
domain on both structured grids and as a response to an arbitrarily placed
set of point loads. Spectral solutions are possible and efficient using fast Fourier transform algorithms
cf., but have not been implemented. The one- and
two-dimensional solutions for lithospheric flexure take the form of an
exponentially damped sinusoid. In one dimension, this is represented by the
following expression:
wi=qα1-D38Dex-xiα1-Dcosx-xiα1-D+sinx-xiα1-D.
Here, the i subscript indicates that this is the response to a line load at
a single x position, xi. α1-D is the one-dimensional
flexural parameter, defined by
following:
α1-D=4D(Δρ)g1/4.
The significance of the flexural parameter is that the flexural wavelength,
λα is related to the flexural parameter as λα=2πα. The distance from a point load to the first flexural bulge
(forebulge) that it creates around its local depression, for example, is
a flexural half-wavelength, πα. This nature of plate bending as an
exponentially decaying periodic function can be seen most easily in the
one-dimensional analytical (constant Te) solution in Eq. ().
derived that the exponentially damped sinusoid due to a
point load in two dimensions should be expressed by kei
, which is the zeroth-order Kelvin function that
satisfies the equation ker(r)+ikei(r)=K0reπi/4, where K0 is the zeroth-order modified Bessel function of the
second kind. This function was defined by Lord Kelvin to solve for electrical
current density in a circular wire with an applied oscillating (alternating)
current Appendix 5, and its solution has been broadly
applied to the two-dimensional bending of a plate e.g.,.
wi,j=qα2-D22πDkei(x-xi)2+(y-yj)2α2-D
α2-D=D(Δρ)g1/4
The subscripts i,j indicate that this is the flexural response to a single
point load at the x and y positions xi and yj. The two-dimensional
flexural parameter, α2-D, contains D instead of 4D in the
numerator because it does not need to include implicit loads and deflections
along the y orientation that are required in the one-dimensional line-load plate
bending case.
Lithospheric flexure calculated by superposition of analytical solutions can
be represented as a simple sum across all line loads ql or point loads
qp:
w=∑qlwi(1-D),w=∑qpwi,j(2-D).
For a given elastic thickness, each flexural response to a line or point load
is similar in shape, but different in amplitude. Therefore, I optimize
solution speed by pre-calculating the flexural response to a unit load in the
center of a template array. This pre-calculated unit deflection array has
twice the linear dimensions of the solution array, and is subsampled and
re-scaled to compute the distributed response to each cell in the grid that
contains a load. This technique works for all for rectilinear grids with
uniform x and y grid spacing, though the x and y grid spacing do not
have to be equal to one another. A similar optimization is possible for
one-dimensional solutions, but these are so rapid that this has not been
found to be necessary. Within gFlex, this solution type is termed SAS,
which stands for superposition of analytical solutions.
The analytical solution response to point or line loads can also be computed
for a scattered set of loads and a scattered (and not necessarily the same)
set of points at which the flexural response is calculated. This solution
type is termed SAS_NG, which stands for, superposition of analytical solutions: no grid. Because it lacks the grid uniformity that permits the a
solution template to be used, its computational time is not optimized in this
way (Sect. ).
Finite difference solutions
Finite difference solutions in one and two dimensions employ Eqs. ()
and (), respectively. For these solutions,
dx and dy may differ from one another, but each must be constant. First,
for the one-dimensional solution, the expansion of Eq. () is
D∂4w∂x4+2∂D∂x∂3w∂x3+∂2D∂x2∂2w∂x2+Δρgw=q.
The two-dimensional solution is based on an expansion of Eq. ()
:
D∂4w∂x4+D∂4w∂y4+2-D∂4w∂x2∂y2+2∂D∂x∂3w∂x3+∂2D∂x2∂2w∂x2+2∂D∂y∂3w∂y3+∂2D∂y2∂2w∂y2+ν∂2D∂y2∂2w∂x2+ν∂2D∂x2∂2w∂y2+2∂D∂x∂3w∂x∂y2+2∂D∂y∂3w∂x2∂y+2(1-ν)∂2D∂x∂y∂2w∂x∂y+Δρgw=q.
These equations are discretized using a second-order accurate centered finite
difference approximation Table 1.
Finite difference solutions in two dimensions may also be generated following
the solution and discretization of , which produces
solutions for a more limited range of flexural rigidity variations.
The finite difference solution is computed as a linear matrix equation,
AW=Q,
where A is a sparse matrix of operators from a linear
decomposition of Eqs. () or (), W
is a vector of deflections (typically unknown), and Q is a vector of
imposed loads (typically known). It is solved directly by using the sparse
LU (lower upper) factorization package UMFPACK
(unsymmetric-pattern multifrontal package) or, at the
user's choice, iteratively with one of the many solvers that are available
with the SciPy (Scientific Python) package .
Boundary conditions
gFlex supports a number of boundary conditions, and these are summarized in
Table and schematically drawn in Fig. . The
finite difference (sparse matrix) numerical solutions can freely define any
combination of no-displacement-and-no-slope (0Displacement0Slope),
no-bending-moment-and-no-shear (0Moment0Shear), no-slope-and-no-shear
(0Slope0Shear), and mirror boundaries. Periodic boundaries may be mixed with
any combination of the aforementioned boundary conditions, with the
requirement that they exist on both sides of the deflection array, as having
(for example) deflections at the west end of the array sensitive to loading
and deflections to the east, but with those on the east not, in turn, sensitive to the
west, being nonsensical. Superposition of
analytical solutions naturally produce a 0-displacement boundary at infinite
distance from each point load (NoOutsideLoads). This can be seen by solving
Eqs. () and () as x→∞ and y→∞. Each of these boundary conditions can be related to
geological processes or locations that one may wish to model
(Fig. ).
Schematics of boundary condition types allowed in the finite
difference solutions to gFlex.
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Example runs of gFlex with varying elastic thickness and boundary
conditions. (a) depicts a long north–south mountain belt and
foreland basin under uniform elastic thickness. (b) provides a
contrived field of variable elastic thickness. (c) is similar to
(a) except in that it uses a mirror boundary for a symmetrical
mountain belt over a continuous lithospheric plate instead of a broken-plate
solution, and that the plate has the variable elastic thickness structure
given in (b). (d) depicts the flexural interaction of two
mountain belts on the same variable-elastic-thickness lithosphere shown in
(b) and has mirror boundary conditions at all edges.
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The 0Displacement0Slope (or clamped) boundary condition (Fig. a)
may be used to approximate a NoOutsideLoads case
for the finite difference solutions (Fig. ). When
placed one flexural wavelength away from a point or line load, the surface
displacement should, for a plate of constant elastic thickness, be
∼ 0.2 % of that at the point of maximum deflection, which is negligible compared to
most sources of geological error. It is conceivable that a difference in
elastic thickness in a continuous plate may exist that is so great that the
thicker plate can be approximated to not bend; a 0Displacement0Slope boundary
condition may also be used to simulate this, though one must debate whether
to do this or to compute the flexural response across a plate with a
prescribed elastic thickness variability.
The 0Moment0Shear boundary condition (Fig. b) means
that the edge of the plate is completely free to flex, like the cantilevered
end of diving board. This is appropriate for places in which the elastic
thickness of the lithosphere goes to zero. Such broken-plate boundary
conditions have been used in analytical solutions to simulate flexure of the
lithosphere beneath the Hawaiian volcanoes, where heating significantly
weakens the lithosphere . This approximates the (zero-dimensional) single
point discontinuity of a hotspot as a (one-dimensional) line boundary condition. A
broken-plate solution has also been used for zones beneath mountain
ranges where sufficient deformation may weaken the lithosphere
, and may be best suited for continental rift zones
, as these closely approximate a linear discontinuity in an
otherwise thick lithosphere. In all three of these cases, the lithosphere
should lose strength as it approaches the boundary condition. For this
reason, 0Moment0Shear is implemented only for the finite difference
solution, which allows for spatial variations in elastic thickness.
The 0Slope0Shear boundary condition (Fig. c) may be
considered to be a flat clamp on the boundary of the plate that may be freely
moved upwards or downwards. While it may require creative thought to uncover
a geological process that holds a plate edge flat but allows it to move
freely in the vertical, this boundary condition can also be used at an
appropriate distance away from the load(s) to approximate a
NoOutsideLoads boundary for a finite difference solution, though
typically the 0Displacement0Slope boundary provides a closer match.
The periodic boundary condition (Fig. d) wraps one
side of the model around to the other side such that they form an infinite
loop. To visualize this, one may imagine taking a paper map and taping either
the east and west sides together or the north and south sides together, such
that the flexure induced by loads on one edge is continuous with load-induced
flexure on the opposite edge. Elastic thickness and loads both wrap around
this boundary, making it possible to, if one is not careful, create sudden
jumps in elastic thickness at the edge of the model. This takes somewhat
longer to solve (Fig. c), but can be useful to compute
a flexural response to the load of a long mountain belt by modeling just a
limited region perpendicular to the strike of the range crest and allowing
this slice to infinitely repeat in the range-crest-parallel orientation; at
the limit of a very narrow slice of model space, this approaches the one-dimensional line-load solution. If a future model of lithospheric flexure relaxes the current
assumption in gFlex that dx and dy may be different but must be constant
in space, the periodic boundary condition should enable a finite difference
flexural model to be employed on a closed surface, such as a sphere, enabling
full global modeling. This is, to the best knowledge of the author, the first
time that a periodic boundary condition has been implemented for lithospheric
flexure.
Model benchmarking. (a) The ungridded superposition of
analytical solutions (SAS_NG) computation time is proportional to the number
of cells with loads present, as the solutions are calculated once for each of
these positions. (b) The gridded superposition of analytical
solutions (SAS) scales with the total number of grid cells times the number
of cells with loads, as this is the total number of computations that must be
made. (c) Finite difference solutions are computed with sparse
matrices with dimensions equal proportional to the grid dimensions, squared,
and therefore scale with the number of total grid cells. All of the solution
time relationships are close to linear except for the two-dimensional finite
difference solutions, due to the added complexity of their finite difference
stencil. Many fits are to a subset of the data to avoid those solutions that
are so rapid that the amount of time required for the non-solver portions of
the code becomes significant. All marker symbols are semi-transparent,
meaning that darker symbols than those that appear in the legend imply
additional data points underneath.
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The mirror boundary condition (Fig. e) reflects the
elastic thickness and load structure across a plane of symmetry at the
boundary. This may be used to speed a solution where a plane of mirror
symmetry may be implied, which is important for large grids or where gFlex is
used as part of a coupled set of numerical models e.g., through
CSDMS:. Example usage
cases include topographic unloading by erosion of a symmetrical mountain
range (Fig. c, and d), isostatic
adjustment under a symmetrical ice cap, and emplacement of a volcanic load.
The latter two cases often have fully radial symmetry, and therefore may be
placed at the corner of the solution array with mirror boundary conditions on
both adjacent sides to further limit the needed computational area. This is
also to the best knowledge of the author the first application of a mirror
boundary condition to modeling of lithospheric flexure, which is surprising
considering its potential utility.
The names of the boundary conditions are based on their effects on
deflections, w, but solutions also require boundary conditions to be placed
upon the flexural rigidity, D; these are listed in Table . For
the 0Displacement0Slope, 0Slope0Shear, and 0Moment0Shear deflection boundary
conditions, a 0-curvature flexural rigidity boundary condition has been
chosen. This allows for near-boundary gradients in flexural rigidity to be
assumed to continue outside the computational domain. As noted above, mirror
and periodic boundary conditions are applied to the rigidity field as well.
For the analytical solutions, the approximation is an infinite plate of
constant elastic thickness.
In two-dimensional solutions, boundary conditions meet at corners. Where a
boundary condition meets another of the same boundary conditions at the
corner, the two generate a continuous boundary condition that includes the
corner of the array. This is always the case for the analytical solutions
with implicit NoOutsideLoads boundary conditions. Where mirror or periodic
boundary conditions meet themselves at corners, these produce doubly
reflecting or doubly periodic boundaries; if every boundary is mirror or
periodic (necessary in the latter case as periodic boundary conditions must
always exist as pairs on opposite sides), these generate an infinite
tessellated plane of loads and elastic thicknesses. Some boundary conditions
in gFlex can work harmoniously with others. Periodic and mirror boundary
conditions propagate 0Moment0Shear, 0Slope0Shear, and 0Displacement0Slope
boundary conditions that exist orthogonally to them. Where mirror and
periodic boundary conditions intersect at a corner, the periodic boundary
condition will propagate the mirror boundary to ±∞. Those boundary
conditions that do not reflect or repeat the effects of the other boundary
conditions do not share the corners equally: in gFlex, 0Displacement0Slope
boundary conditions dictate all corners where they meet other boundary
conditions, forcing them to remain fixed at 0; physically, this means that
the clamp of the 0Displacement0Slope boundary condition continues
through the edges of the perpendicular boundaries. 0Moment0Shear boundary
conditions were chosen to control the corners where they meet 0Slope0Shear boundary
conditions, as the 0Moment0Shear boundary condition has been recognized in
geological work e.g.,, while
the 0Slope0Shear boundary condition has not.
Discontinuities and limit as <inline-formula><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>→</mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:mrow></mml:math></inline-formula>
Two notable issues inherent to the finite difference solutions and the
treatment of a continuous plate become apparent as Te→0. The
first is that a region of Te=0 must have a width of at least three cells
to produce the expected local isostatic equilibrium; this is a result of
numerical diffusion in the central difference discretization provided in
Eqs. () and (). The second is that
because any region of 0 elastic thickness will enter isostatic equilibrium
with its local loads and not be affected by nonlocal effects; if this region
lies along the edge, it will ignore all boundary conditions. If a Te=0
region along a boundary is ≥3 cells wide, it imposes a
0Displacement0Slope boundary condition on the interior cells; smaller regions
of Te=0 will allow some information on the ultimate boundary condition to
leak through via numerical diffusion.
These issues are important to note, but unlikely to be important in most
cases of gFlex. First, discontinuous transitions to zones of Te=0 may
also be modeled by segmenting the inputs into multiple arrays, running gFlex
for each array with a 0Moment0Shear (broken-plate) boundary condition applied
to the model domain edges representing the discontinuities, and then
recombining the outputs into a continuous displacement field. Second, and
more importantly, the conditions for broken-plate or Te=0 solutions to
be required are rare on Earth. Elastic thickness of 0 implies that there is
no elastic lithosphere, and a broken-plate solution implies that there is
no shear between adjacent lithospheric
blocks. These conditions are most likely to be met in rift zones, though even
these have some nonzero thickness of brittle crust. End loads, which are not
currently included in gFlex, could be used in combination with a
0Moment0Shear boundary condition to better parameterize faults
e.g., and expand the utility of gFlex. However, the
typical case for which gFlex is designed involves glacial-isostatic
adjustment, large-scale water loads, sedimentary basin development,
large-scale erosional unloading, and other processes that extend across a
swath of heterogeneous lithosphere that may contain many faults. In these
cases, it has been found to be sufficient to simply characterize a variable
field of finite elastic thickness across the domain, where elastic thickness
falls around fault zones e.g.,.
Model benchmarking
A set of tests was performed to measure the speed at which gFlex computes
solutions. In these tests, an elastic plate that is 1000 km long
(one-dimensional and two-dimensional) and 1000 km wide (two-dimensional) is
subjected to a square load at its center that ranges from 100 km to the full
1000 km on each side. This load places a normal stress of 9 702 000 Pa on
the surface, which is equal to 300 m of mantle material
(3300 kg m-3). In these scenarios, there is no assumed infilling
material (ρf=0). gFlex computed solutions for uniform rectilinear
grids of increasing size using gridded and ungridded superposition of
analytical solutions (SAS and SAS_NG, respectively) and finite difference
(FD) methods. All boundary conditions (Table and
Fig. ) were tested, though not in combination. The
finite difference solutions include scenarios with both constant (25 km) and
variable (10–40 km) effective elastic thickness, with the latter varying
sinusoidally over a wavelength of 500 km such that the plate contains two
full Te cycles. In the two-dimensional case, Te varies in both
dimensions to produce a smoothed checkerboard pattern of elastic thickness.
Finite difference solutions reported employ the direct solver UMFPACK
, as it has been better tested in gFlex than the iterative
solution methods and is therefore the default solver.
Fig. displays computation time for all of the
benchmarking tests, and Fig. is a
comparison of the SAS_NG, SAS, and FD solution techniques for the case in
which every point at which the solution is calculated also contains a nonzero
load. These solution times do not account for file input or output or
graphics generation. They do include the initialization time for the solution
steps of gFlex; therefore, a number
of the power-law fits to solution time do not include the times calculated
with the smallest arrays, for which initialization time is a significant
fraction of the total model runtime.
Comparison between solution methods where every cell in the domain
contains a load. The ungridded superposition of analytical solutions
(SAS_NG) scales best but in these tests is the slowest. It can, however, be
faster when fewer cells contain loads. Some fits are to a subset of the data
to avoid those solutions that are so rapid that the amount of time required
for the non-solver portions of the code becomes significant. All marker
symbols are semi-transparent, meaning that darker symbols than those that
appear in the legend imply additional data points underneath.
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The factors that determine computation time are the solution method and the
inclusion of periodic boundary conditions. While the SAS_NG method scales
the best with increasing grid size, it is so much slower than the other
methods for standard model-run grid sizes that it will not often exceed their
speed. The finite difference method is the fastest if every cell contains a
load, but can become slower than the analytical methods if only a few cells
contain loads, as analytical methods must make one set of calculations across
the grid per load. Standard runtimes are between a fraction of a second and a
few minutes on a personal laptop computer (Dell XPS 13 Developer Edition
running Ubuntu 14.10) (Figs. and
).
Model interfaces and coupling
Some users of gFlex may want to run a single calculation, while others may
want to produce many solutions as part of a larger-scale numerical modeling
exercise, such as an inversion for elastic thickness or coupling with another
model. Therefore, four different methods to use gFlex have been prepared:
standalone, with input files;
as part of a Python script;
driven by GRASS GIS to simplify integration of geospatially registered data with the lithospheric flexure
model;
as a component for the CSDMS
framework , including its tight integration into
Landlab, a CSDMS-led Python-based Earth-surface modeling framework that is currently being developed .
GRASS GIS integration is also possible for model coupling using Python,
including efforts that use the Landlab framework.
Standalone with input files
Some users may want to employ gFlex as a single calculation, for example to
calculate the flexural response to a set of loads generated by a sedimentary
deposit that was measured in the field. The user prepares an input file of
model settings, an input ASCII grid of loads, and, should the elastic
thickness be nonuniform, an input ASCII grid of lithospheric elastic
thicknesses. Outputs from this mode of running gFlex include an ASCII grid of
surface deflections and a set of plots of surface deflections and loads.
As part of a Python script
gFlex may also be imported as a Python module to be run either as a
standalone simulation or as a
component in a multi-model integration effort. This allows it, for example,
to be a part of a flexural backstripping toolchain or a model of
glacial-isostatic adjustment. Backstripping calculations may be performed by
simply removing the sedimentary load , or, in the case of
a foreland basin, by inverting for the mountain belt loading history and
lithospheric elastic thickness that would be required to produce the basin
. A programmatic approach is also useful for scenarios in
which material infills a depression, but not over the whole domain and/or not
with uniform density. While the flexure equations require that ρf be
constant, a more flexible way to solve for the effect of infilling material
is to compute flexural response with ρf=ρair≈0,
add loads based on some set of rules, and then re-calculate flexure
iteratively until convergence is achieved. This can occur in regions with a
complex set of sedimentary deposits see also
and/or to be used for seawater loading across a shoreline see
also.
Driven by GRASS GIS
gFlex is also prepared for integration with the open-source geospatial
software GRASS GIS as two add-ons or extensions
named r.flexure and v.flexure, which are raster and vector
operations, respectively. As GRASS GIS is a map-based application, r.flexure
and v.flexure employ two-dimensional solutions (both analytical and finite
difference), though future extensions of these modules to compute flexure
from line loads along chosen one-dimensional profiles would be possible.
r.flexure can use the finite difference or SAS solution methods, whereas
v.flexure exclusively uses the SAS_NG solution method to take advantage of
its ability to produce solutions for an arbitrary scatter of point loads.
Advantages of GRASS GIS include
full integration within a geospatially registered environment,
meaning that data can be used directly as model inputs, and that model outputs may be compared against
data;
a documented and standardized command-line interface;
a pre-built and standardized graphical user interface (GUI).
The graphical user interface is incorporated into the GRASS GIS wxPython GUI
, and this is particularly helpful for
researchers, who are not as accustomed to command-line interaction with
computers to use gFlex with their data. For computer modelers, the GRASS GIS
coupling may be used to support broader model coupling and data–model
integration efforts see, for example,.
Modeling frameworks
CSDMS (broadly) and Landlab (in particular) both include methods for
integrating modular blocks of code as part of their respective efforts
towards the community-wide goal to make modeling of Earth systems less
time intensive and more streamlined . gFlex is
included as a modular component of the still-in-development Landlab
Earth-surface modeling framework . Landlab integration provides wrapping with the CSDMS Basic Model Interface (BMI) and Component Model Interface (CMI) using the CSDMS Standard
Name construction conventions . The standard interfaces
provided by both of these modeling frameworks will streamline model coupling
that uses gFlex and help to prevent duplication of effort in building plate
bending models. Furthermore, the inclusion of gFlex in Landlab will allow
numerous Earth-surface systems to be modeled more precisely
(Fig. ).
This coupled model run for a hypothetical extent of the Iceland ice
cap shows the influence of a variable elastic thickness structure
(i). The areal extent of the three ice caps is nearly identical
(a, d, g) in this small-scale and largely topographically controlled
example. Flexural isostasy with a constant 3.7 km elastic thickness
(c) following reduces ice cap extent
and causes some interior ice thickening when compared to the case without
flexure (b), as the ice cap conforms to the bowl-shaped depression
that it creates. Deformation in the case with variable elastic thickness
(f) is focused along the ridge and extends farther on the
southwestern side that has greater elastic thickness, and modifies the
topography of western Iceland (low elastic thickness) to produce spatially
variable ice thickness changes (e, h).
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Application example: Iceland
As a first example to utilize both the ability of gFlex to generate solutions
with variable effective elastic thickness and its incorporation into GRASS
GIS, gFlex is used along with a simple and efficient GIS-enabled glacier and
ice cap model modified from the work of to model a
hypothetical expansion of the Iceland Ice Cap. While the importance of flexural isostasy in ice dynamics
modeling has long been well-known cf., the author knows
of no dynamic ice model that runs with a variable elastic thickness
lithosphere, making this possibly the first such exercise. Earth's crust at
Iceland has been built by the unique intersection of the Iceland hotspot and
the Mid-Atlantic Ridge, which together produce a weak lithosphere with
spatially variable elastic thickness, resulting in short-wavelength
variability in the solid-Earth
response to loading. Here I test the two-way coupling between ice dynamics
and solid-Earth deformation and the differences in steady-state ice caps that
are produced in a modest climate change and ice cap extent scenario.
This coupled ice dynamics and flexural isostatic model of Iceland requires
four input components: the elastic thickness structure around Iceland, the
modern topography of Iceland, the modern surface temperature field of
Iceland, and modern precipitation rates across Iceland. The ice cap model
used here cf. employs a shallow-ice approximation with
basal sliding as a linear function of driving stress, which is intentionally
much simpler than the modeling approach that
used to model the Last Glacial Maximum (LGM)
Iceland Ice Cap. This is because
the goal here is to show schematically the importance of including lateral
variations in elastic thickness on the reconstructed thickness of an ice cap
for a given paleoclimate, with less emphasis on actually reproducing any
particular extent of the Iceland Ice Cap.
The elastic thickness structure under Iceland, in this schematic example, is
related to the age of the oceanic crust. related elastic
thickness to the age of the lithosphere with the simple equation that results from
the square-root time dependence of lithospheric cooling via thermal
conduction cf.:
Te=(2.70±0.15)Δt.
Here, Te is given in kilometers and the age of the lithosphere, Δt, is
given in millions of years. As continental material also exists within the
computational window, the elastic thickness map of is used for all subaerial landmasses. Across the continental
shelves, the oceanic-crust-based map and the map from
are blended using spline interpolation within GRASS GIS .
The regional age of oceanic crust is provided by , but
their map indicates that even crust at the ridge in Iceland has an age of
6–7 Ma, resulting in a greater computed effective elastic thickness than
would be expected based on the presence of the ridge or from heat flow data
e.g.,. While the structure of Iceland is certainly
more complicated than the simpler parts of the ridge due to the effects of
the hotspot and its tectonic environs e.g.,, the assumption here is that the lithospheric effective elastic
thickness structure due to the ridge is as if young crust continued along the
Mid-Atlantic Ridge through all of Iceland, and the elastic thickness map
(Fig. i) was modified to approximate this for the
sake of this example.
The underlying digital elevation model, GEBCO_08 , includes
the modern ice caps on Iceland, but these are already flexurally compensated
and are small compared to the ice cap modeled here. While their removal would
improve reconstructed ice discharge, they are ignored due to the schematic
nature of this modeling effort.
Modern temperature and precipitation fields are from the Monthly
NOAA-CIRES (National Oceanographic and Atmospheric Administration–
Cooperative Institute for Research in Environmental Sciences)
20th Century Reanalysis (V2) by
for further background on their methods,
see. These provide twentieth century mean conditions on a
2∘ × 2∘ latitude–longitude grid (temperature) or a
94 × 192 Gaussian grid (precipitation). These were cast as point
data and interpolated using splines in GRASS GIS . Prior
to this spline interpolation, temperature was projected to sea level using
the mean cell elevation (with lapse
rate of 4.7 ∘C km-1), following for ice
caps; after interpolation, the resultant temperatures were then interpolated
up to their respective surface elevations using the same
4.7 ∘C km-1 lapse rate. Although not all of the Icelandic
surfaces are covered in ice at present, this rule was prescribed uniformly
for the sake of a schematic model.
Three experiments were run: one with no flexure, one with flexure using a
constant elastic thickness of 3.7 km followingand assuming E= 65 GPa,
and one in which the full spatially variable
flexure was used. In each of these runs, temperature was reduced from its
present value by 5 ∘C and ice expanded to cover an area approximately
equal to the currently subaerially exposed continent, approximately
consistent with the previous modeling results of and with
a temperature change that is much less than the LGM drop of 10–13 ∘C
that was predicted to cause ice to spread onto the continental shelves as
well . Mass balance was simulated by a positive degree-day
melt model. June, July, and August temperatures were used to compute
ablation, with a melt factor of 6 mm d-1 K-1. Precipitation was
held constant and all precipitation was assumed to contribute to positive
mass balance. Each scenario was run for 4000 years to reach full glacial and
isostatic equilibrium, with isostatic equilibrium being assumed to occur
instantaneously to facilitate more rapid computation of the equilibrium
solution.
The results in Fig. summarize the experiments.
Figure c and f show
the modeled flexural isostatic deformation and Fig.
b and e show the deviation from the case with no
isostasy; each of these pairs is for constant and variable elastic thickness,
respectively. Figure h shows that with variable
elastic thickness (Fig. i), ice thickness
variability is concentrated where lithospheric elastic thickness is low.
The example of isostatic response to ice advance in Iceland is just one
possibility of a feedback between an Earth-surface (or other geological)
process and flexural deformation. Further such scenarios involving, for
example, orogenesis and foreland basin formation in settings such as
that studied by, rifting , and river delta
morphologic evolution , will improve our understanding of the
dynamic interactions between Earth's surface and subsurface
e.g.,.