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Geophys. J. Int. (1992) 109, 553-564
Diapirism and topography
A. ~oliakovland Y. Podladchikov2
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Hans Ramberg Tectonic Laboratory, Institute of Geology, Uppsala University, Box 555. 751 22 Uppsala, Sweden
Institute of Experimental Mineralogy, Chernogolovka, Moscow District, 142 432, USSR
Accepted 1991 December 4. Received 1991 December 4; in original form 1991 April 13
SUMMARY
A new numerical technique using markers and a deformable Lagrangian mesh is
used t o study the deformation of the surface above a rising diapir. This method
allows modelling of a free-surface boundary in a self-consistent way. The method
makes it possible to investigate many different problems with non-regular geometry.
The codes were verified through comparison with analytical solutions for the initial
stages and with other numerical codes for mature stages.
Our simulations led to the following conclusions. (1) The growth rate of the diapir
is more strongly influenced by the viscosity contrast than the wavelengths. This is
due to the strong growth rate difference during the initial stage. (2) The maximum
elevation above the diapir linearly increases with increasing wavelength and is
approximately the same for different viscosity contrasts. (3) The elevation will be
considerably higher for a low-viscosity diapir at a given depth than for an isoviscous
diapir .
Comparisons for different thicknesses of the buoyant layer showed that the
highest topography is produced when the two layers have equal thickness.
We showed the dependence of the topographic behaviour on the parameter R
(the ratio of the density difference between two layers to the density of the upper
layer). It was found that topography behaves linearly up to R = 0.15. It indicates
that a posteriori estimation of topography for traditional calculations using the
free-slip boundary condition works well within this limit.
Key words: diapirism, finite element method, topography.
1 INTRODUCTION
Salt and granite diapirs, and mantle plumes can exert a
strong influence on the overlying surface topography.
Surface topography can then have a feedback effect on the
development of diapirs. These interrelated processes have
not been explored numerically in a comprehensive manner.
A systematic simulation is required, that will yield
quantitative information about the temporal and spatial
evolution of height, shape and relief above a buoyant body.
The results of these simulations, combined with geophysical
and geological studies of topography, can then be used to
estimate various parameters for natural diapirs such as their
viscosity, size, and depth of origin. A new, more
sophisticated numerical model is therefore required, in
order to directly investigate the interrelated phenomena of
diapirism and topography.
It is first necessary to briefly review previous studies of
topography related to diapirism. Rambeg (1968a, b) derhed
analytical formulae for the Rayleigh-Taylor instability for
the linear stage of diapiric growth and for the rate of
isostatic compensation. He suggested that linear solutions
are valid up to amplitudes of 10 times the initial
perturbation. An analytical solution for the motion of the
surface above a deep rising cylinder or sphere has been
obtained by Morgan (1965) using the method of images.
Talbot & Jarvis (1984) studied the shape of a diapir
extruding onto the surface by using an analytical solution for
flow from a point source. One disadvantage of these
methods is that it is impossible to consider complicated
non-linear interaction between the surface and the rising
diapir.
Laboratory investigation of the surface dynamics of
viscous flows has been conducted using centrifuge
techniques by Ramberg (1968a, b) and Jackson et al. (1988).
Olson & Nam (1986) and Griffiths et al. (1989) studied these
phenomena without a centrifuge using much lower viscosity
materials such as oils. Visualization of the surface motion
was quite difficult in the centrifuge experiments; the
experiments with oils were complicated by surface tension
effects.
Similar problems have been studied numerically. In most
554
A. Poliakov and Y. Podladchikov
cases the surface topography is calculated with a following
procedure (e.g. see McKenzie 1977; Fleitout, Froidevaux &
Yuen 1986; Lliboutry 1987). The subsurface flow is
calculated for a fixed upper boundary (vertical velocity is
zero). Topography is then calculated a posteriori from the
normal stress along this boundary. The relationship between
the surface topography, Ah, and the dynamical normal
stress a,,, to first order, is given by
Coupling between surface topography and subsurface flow
was studied by Myasnikov & Savushkin (1978), Fadeev
(1988) and Pak (1989). If the surface topography is small,
the problem can be treated as two coupled problems and
can be solved iteratively. The technique is based on an
asymptotic decomposition in which the ratio of the thickness
of the hydrodynamic boundary layer to the characteristic
size of the problem is a small parameter. Thus, it is difficult
to use this method for problems with large dynamically
induced surface topography.
Zaleski & Julien (1990) used a top layer with a very low
viscosity and density to represent air or water above the
surface. This allows a simple representation of the free
surface. However, due to the very high viscosity and density
contrast and diffusion between the top layer and the
underlying layers, calculations sometimes become unstable
and give significant errors.
In the present study an alternative method is proposed for
modelling these problems numerically using the Finite
Element Method (FEM). Raleigh-Taylor (RT) instability is
simulated with a moving, free-surface boundary. The
technique is explained below and compared with other
approaches. Several examples are presented.
Surface topography is studied for both the free-surface
boundary condition, using the new technique, and for the
free-slip condition using traditional linearization methods.
Also, the dynamics of surface topography were studied for
different viscosity contrasts and geometries.
2
PHYSICAL A N D NUMERICAL M O D E L
2.1 Raleigh-Taylor instability
The Raleigh-Taylor instability for two layers of viscous fluid
occurs when the density of the upper layer is higher than
that of the bottom layer so that the system is gravitationally
unstable. Any small perturbation on the interface between
the layers grows until finally the lower material rises up and
displaces the upper layer. During the rise of a diapir, the
non-hydrostatical part of pressure (the dynamical pressure)
must be balanced by a rise in the surface above the diapir
producing topography.
The geometry of the problem modelled is shown in Fig. 1.
The two layers are described by their density, viscosity and
thickness; p, p and h respectively. The botton of the mesh
was given a non-slip boundary condition. Along the vertical
sides the free-slip condition was chosen which implies mirror
symmetry. The upper boundary was flexible and had a
free-surface (free stress) boundary condition.
c=o (free surface)
or
vz -0 ,
0 =
0 (free slip)
XZ
height of
diapir
Figure 1. Model geometry, boundary conditions and physical
variables used in the calculations. An initial cosinusoidal
perturbation with a wavelength equal to the characteristic
wavelength and with an amplitude of 4 per cent of the box height is
imposed on the buoyant layer.
2.2 Mathematical formulation
In the present study the following forms of Stokes equations
for slow incompressible fluids was simulated:
where aij is the stress tensor, p is the viscosity, p is the
density, ui the ith component of velocity and aij is the
Kronecker delta. The change in the viscosity and density
fields during each time step were calculated using the
continuity equation: where A is any characteristic material
property (e.g. density, Newtonian viscosity, etc.),
2.3 Comparison of methods
Numerical modelling of the slow flows governed by Stokes
equation is well established. Temam (1977) studied the
application of FEM technique for the solution of this
equation from a mathematical point of view. The main
difficulty in numerical simulation of Stokes equation is to
satisfy the incompressibility condition.
There is a class of methods based on the introduction of a
stream function which satisfies this condition automatically
and gives good results. However, the stress-free boundary
condition causes significant difficulties because it requires
the calculation of third-order derivatives of the stream
function. These algorithms are also not applicable for
irregular meshes. Thus, it is difficult to solve problems in
regions with complicated geometry, and to refine the mesh
in areas of special interest. A further disadvantage is that
the approximation functions are not expressed explicitly in
terms of the nodal variables and an additional system of
linear equations must be solved.
Diapirism and topography
For these reasons we use the pressure-velocity formulation for Stokes equations. This allows us to impose arbitrary
velocity or stress boundary conditions and to work with
arbitrary geometry. However, the two general strategies
(mixed and penalty methods) which are used for satisfying
the incompressibility condition have their own drawbacks.
This is especially true for problems with a free surface,
non-linear rheology or buoyancy forces. Pelletier et al.
(1989) showed that in this case the FEM formulation
reduces to an ill-conditioned system of linear equations.
Erroneous solutions can be obtained for density-driven flow
problems due to a poor choice of pressure discretization.
The main advantage of the integrated method is that
velocity and pressure are computed directly, without
numerical differentiations. However, the system of equations becomes very large and the matrix contains zeros on
the main diagonal, which makes partial pivoting necessary.
For the mixed method Pelletier et al. (1989) proposed
pressure scaling for the stiffness matrix in order to overcome
this ill-conditioning.
There are several different types of penalty methods
including continuous, discrete and iterative methods as well
as the Uzawa algorithm. Cuvelier, Segal & Steenhoven
(1986) have shown that the discrete penalty method (where
the pressure is eliminated after discretization of the
continuity equation and the momentum equation) is
superior to other penalty function methods. The main
advantage of the penalty method is the large reduction of
the system of equations as well as the fact that partial
pivoting is not necessary.
2.4 Choice of elements
Because the system includes both buoyancy forces and
free-surface boundary conditions, basis functions for the
approximation of the pressure and velocity fields must be
chosen carefully.
Many studies have compared different elements for the
solution of Stokes equations (Cuvelier et al. 1986; Crochet,
Davies & Walters 1984; Hughes 1987). Quadrilateral
elements are preferable (Cuvelier et al. 1986), but triangular
elements are more convenient for complex geometries.
Suitable basis functions for velocity and pressure must
satisfy the following requirements. (1) Pressure must be
approximated by interpolation polynomials which are of
order at least one degree less than the polynomials for the
velocity. (2) The number of equations for incompressibility
in the global matrix should not exceed the number of
degrees of freedom for the velocity field. Otherwise, the
system will be overconstrained and the velocity will be
completely determined by the condition of incompressibility
(Crochet et al. 1984). ( 3 ) The constraint ratio, r, introduced
by Hughes (1987) should be approximately equal to 2. r is
the ratio of the number of displacements to the number of
incompressibility conditions per element. If r is less than 2,
the element will be overconstrained and mesh locking will
occur. In other words, if the volume of the element is kept
fixed (incompressibility), the element is not flexible enough
to simulate the velocity field. Following these recommendations and from our own experience, we have chosen
triangular elements with enriched continuous quadratic basis
functions for velocity and discontinuous linear basis
555
Velocity: enriched quadratic (continuous)
(7 nodal points: x)
&
Pressure:(discontinuous) linear (3 nodal points: 0 )
Figure 2. The Crouzeix-Raviart element (after Cuvelier et a!. 1986).
functions for pressure (Fig. 2). Discontinuous functions
were used because continuous functions were found to
produce a 'chessboard' effect (the incompressibility condition was satisfied for the domain as a whole but not for the
individual elements). This element satisfies all three
conditions listed above.
We use this element for both mixed and penalty
techniques depending on the problem.
2.5 Solution procedure
Problems are solved with a time-stepping procedure. At
each time step the velocity field is obtained by solving
steady-state Stokes equations (2)-(4) for a given geometry.
The velocity field is then used to move the material
according to the continuity equation (5).
2.5.1 Velocity calculations
At each time step, our program first calculates the velocity
field based on the current density and viscosity distribution.
The Galerkin method was used to reduce Stokes equations
to a symmetric, non-positive definite system of linear
equations in the case of mixed formulation and a positive
definite system in the case of the discontinuous penalty
method. This system was assembled and solved using the
frontal method (Hood 1976) with double precision (the
relative accuracy of the solution was approximately
and the divergence of velocity in each element was about
for the mixed method and
for the penalty
method. We have used typically about 21 nodal points in the
z-direction and a proportional number of nodes in
x-direction depending on the goemetry.
2.5.2 Method of markers
Moving a density or viscosity field which contains sharp
discontinuities through a discrete mesh is difficult due t o
problems with numerical diffusion (Hirt & Nichols 1981).
These diffusion problems can be overcome by using a
method of characteristics based on marker points (particles).
Weinberg & Schmeling (1991) developed a marker
technique which is very effective for multiphase flow where
each phase has different rheological properties. In their
method, each marker contains information on all material
phases present. This allows the effective material properties
at the nodal points to be estimated correctly. Also,
uncertainties which arise when more than two phases are
present are avoided. Marker points, containing material
property information are distributed throughout the
numerical mesh.
The markers are moved with the numerical mesh
according to the velocity field. In places where the density of
markers becomes too low (as in extentional zones) new,
556
A. Poliakov and Y. Podladchikov
additional markers are added with material properties
interpolated from neighbouring markers. This procedure is
easy to program, computationally cheap and produces a
minimum of diffusion compared with techniques which
redistribute all the markets in order to maintain the same
number of marker particles. Also, sedimentation can be
simulated by introducing new markers on the upper surface.
Erosion can be simulated by removing markers on the
surface. A high-order, 13 point integration formula (Hughes
1987) for calculation of the stiffness matrix was used in order
to conserve detailed information from the marker field in
the coarse FEM mesh. Direct projection from non-regulary
distributed markers to non-regulary located integration
points is a time consuming operation. It is necessary to
search through all markers for each integration point,
looking for the closest markers. Instead of doing this, we
use a two-step algorithm. We introduce a very fine regular
mesh and first interpolate properties from the markers to
this mesh. We then project from this mesh to the integration
points.
We tested our codes for different grid sizes, numbers of
markers and resolution of the fine interpolation grid. In the
cases presented in this work we found it optimal to have 100
markers and a 5 x 5 interpolation fine mesh per
quadrilateral (which contains two triangles). Thus, for a
typical problem with 21 x 21 FE mesh we will have lo4
markers and 2.5 x lo3 fine mesh nodes that use only about
200-300 K of memory.
2.5.3
Technique for moving mesh, boundaries and markers
Markers are usually moved through a fixed, Eulerian mesh.
Unfortunately, this method is very expensive, because it is
necessary to update the locations of markers and material
properties at every time step (or even several times per time
step for Runge-Kutta, or predictor-corrector methods).
These calculations involve finding the element to which each
marker belongs and interpolating the velocity field of the
element to the marker location. Also, the updated material
properties must be projected onto the integration points.
The Langrangian method, where the mesh moves with the
material, is faster. Values of density and viscosity are
conserved in the integration points. Therefore, the searching
and interpolation required for the Eulerian method is not
necessary for every time step. Of course, this method is not
good for extremely large strains because the mesh becomes
too distorted.
Following a suggestion of P. Cundall (personal communication 1991), we combined both the Lagrangian
technique and markers as shown on Fig. 3. At the initial
stage (Fig. 3a), material properties of the different layers are
projected to both the integration points of each element and
to the markers (schematically represented by 'stars' and
'crosses' indicating the different materials). For each
marker, the number of the element which contains the
marker is stored. Also, within each element the Cartesian
coordinates of the markers are converted to local
coordinates (area coordinates for triangular elements).
The mesh is then updated according to the Lagrangian
technique. At each time step new positions for the mesh
grid nodes are calculated from the current velocity field
using an explicit two stage Runge-Kutta method (Fig. 3b).
Figure 3. Explanation of the method for moving the boundaries
and mesh used in the present work. (a) Initial stage; 'crosses' and
'stars' indicate different materials (b) Moving mesh with markers
'chilled in element' until it is necessary to remesh. (3) Mesh and
markers after remeshing.
This Lagrangian movement is very fast because it is onhy
necessary to move the mesh nodes according to the
calculated velocity field. No interpolation is necessary. The
moving mesh automatically tracks evolution of the upper
surface.
When the mesh becomes too distorted (Fig. 3b), it is
necessary to remesh. Since the local coordinates of the
markers remain unchanged during the Lagrangian movement of the mesh, the Cartesian coordinates of the markers
can be obtained by simple interpolation from the nodes of
the elements. Only at this stage is it necessary to interpolate
from the markers to the integration points as is described in
Section 2.4.2. This is in contrast to the Eulerian method
where this interpolation must be performed at every time
step. This procedure is demonstrated in Fig. 3(c), which
show the same stage as Fig. 3(b) but with a new regular
I
557
Dinpirism and topography
mesh. The new mesh is adjusted to fit the deformed upper
boundary.
We found that this Lagrangian approach combined with
markers was much faster then the Eulerian approach and
was much easier to program. This method is probably more
precise, because interpolation is required only for remeshing
so that the total number of interpolations is only about 5-10
for overturn. We feel that this method would also be useful
for heat transfer problems. The moving elements would
eliminate convective terms from the heat transfer equation
leaving only the diffusion term.
h, = 0.3, R = 0.1
(a)
0.025--';
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-.-.-. analytical,(Ramberg1988b)
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= 1.1
analytical, Ramberg,l968b)
- - - . p1/p2= 1 0 , X = 3.0
,
-
'1
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/:
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-
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- pI/~2
= 1, h
50
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.
100
.
,
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,
,
150
,
200
TIME
Numerical results
3.1 Non-dimensionalization and verification of the
method
h, = 0.3, R = 0.1
(b)
All variables were non-dimensionalized on the basis of
characteristic velocity:
where L is the sum of the thickness of both layers, p , is the
dynamic viscosity of the upper layer, Ap = (p, - p,) is the
density contrast between the two layers, g is the acceleration
of gravity, t is time and x and z are coordinates. For
calculations with the free-surface condition, the solution
depends not only on the density Ap as in the free-slip caie
but also on the additional parameter
p, and can be neglected. We will show
where pair<<
(Section 4) that for geophysical applications (R < 0.15), the
results are similar and topography depends linearly on R.
The initial shape of the simulated region was rectangular
with unit height and a length equal to one half of the
wavelength, A. The amplitude of the initial cosinusoidal
perturbation was 0.04 times the height of the region. This
makes possible a comparison with the linear analytic theory
during the initial stages of flow.
We verified our code for the initial linear stages by
simulating Raleigh-Taylor (RT) instability of a viscous
buoyant layer covered by another viscous layer with a free
surface. Non-dimensional parameters for this problem are
the ratio of the densities (parameter R), the viscosities of
the two layers and the thickness of the buoyant layer h, (see
Fig. 4).
The results of our numerical simulations were compared
t o the analytical solutions of Ramberg (1968b) in Fig. 4. The
height of the diapir (Fig. 4a) and the surface elevation (Fig.
4b) are plotted against non-dimensional time. For each
viscosity ratio a characteristic wavelength, A, was also
calculated from the formula of Ramberg (1968b).
We can see that there is satisfactory agreement between
theory and numerical experiments for the height of the
diapir up to a non-dimensional time of 60 for the isoviscous
case and up to 10 for a viscosity of contrast lo3. This is in
agreement with the suggestion of Schmeling (1987) that the
analytical solution is valid up to amplitudes of 10 per cent of
the wavelength or up t o 30 per cent of the height of the
thinnest layer. However, surface elevation above the diapir
E
g
-
0
,
v
, PI/&
6
,
- - -.
-0.6
0
50
, = , 1, h
=
,
1.1
,
,
,
,
analytical, Ramberg,l968b)
pl/p2 = 10 , h = 3.0
analytical,(Ramber ,1968b)
100
150
200
TIME
- pl/p2 =
--'-,
-0.8
0
lo3, X = 3.0
Peter van Keken, (pers. comm.,1991)
20
40
TIME
Figure 4, Comparison of present codes with analytial solution of
Ramberg (1968b) for linear stage (a, b) for free-surface boundary
condition and (c) non-linear stages for free-slip upper boundary
condition.
from the numerical solution starts to deviate from the
analytical solutions two times earlier (Fig. 4).
For the non-linear stage, with a free-slip upper boundary,
we verified our codes by comparison with calculations
performed by U. Christensen for the isoviscous case. We
also compared our codes for a viscosity contrast of lo3 with
the code of Peter van Keken (personal communication 1991)
(see Fig. 4c).
3.2 Dynamics of the upper surface under different
conditions
We have carried out a brief parametric study in order to
determine which factors influence the evolution of surface
topography. The geologic setting in which actual diapirs
develop is, of course, much more complicated than the
simple, two-layer isothermal model we study.
558
A. Poliakov and Y.Podladchikov
3.2.1 Effectof z~iscosity
The viscosity contrast between a diapir and the overburden
significantly influencesthe evolution of the system, including
the surface topography and the height, shape and growth
rate of the diapir. In various geological settings the viscosity
contrast can range over several orders of magnitude due to
compositional, rheological or temperature differences.
Therefore, we will first study the influence of viscosity.
The growth of diapirs is also strongly influenced by the
initial conditions such as the shape, amplitude and
wavelength of the perturbation on the interface between two
materials. It is usually assumed that a wide spectrum of
perturbations is present on the interface between the
buoyant layer and the overburden in natural systems. Since
perturbations with different wavelengths grow at different
rates, the perturbation growing at the fastest rate will
suppress dominate the others. Thus, the diapirs will be
spaced at this fastest growing interval (characteristic
wavelength).
Under certain geologic conditions, such as faulting,
sedimentation or erosion, growth of diapirs at noncharacteristic wavelengths can be induced. Schrneling (1987)
explored, for the isoviscous case, the type of perturbations
necessary to excite diapir growth at non-characteristic
wavelengths. We will consider here cases where the initial
perturbation is equal to, shorter than or longer than the
characteristic wavelength. The initial prescribed perturbation must be strong enough to inhibit the natural tendency
of the system to revert to the characteristic wavelength. In
all cases we set the amplitude of the initial perturbation
equal to 0.04 which was adequate to ensure that the desired
wavelength survived.
Figs 5(c), (d) and (e) show calculations for viscosity
contrasts of 1, 10 and lo3 respectively. The calculations were
given initial perturbations so that each grew at the
characteristic wavelength for that viscosity contrast. The
wavelengths, A, were 1.1, 1.68 and 3.0 respectively. The
velocity field (left column) and the passive strain markers
(right column) are shown at several stages, The values of
time and the maximum velocity (for scaling of the arrows)
are also shown. Strain markers are shown for visualization
of the deformations. Markers were not attached to the
boundaries due to the high strain on the boundaries. When
distortion is too large, some markers were excluded (white
fields). Figs 5(a) and (b) show the elevation directly above
the diapir and the height of the diapir versus nondimensional time (see Fig. 1 for terminology).
It is difficult to compare results for different wavelengths
because of the extra parameter, A. However, the difference
in topography can be explained as follows. For longer
wavelengths, the size of the diapir increases and if the
buoyant body is larger in the vertical direction the buoyant
forces will be larger. Therefore topography will be higher
for larger diapirs in order to compensate this force. U.
Christensen (personal communication 1991) suggested a
comparison between the size of a diapir in the vertical
direction and the difference in topography. We can see that
for the case pl/p, = lo3 that the vertical size of the diapir is
two times higher than for the isoviscous case and that the
topography is also two times higher. This is in good
agreement with the results shown in Fig. 5(a).
We now compare calculations for the same three viscosity
contrasts for A = 3.0, the characteristic wavelength for a
viscosity contrast of lo3 ((~ig. 6) and for h = 1.1, the
characteristic wavelength for the isoviscous case (Fig. 7). To
see the temporal evolution of the surface topography,
exaggerated surface profiles the shape of the diapir are
shown on the same plot for the three different cases (Fig. 6c,
d, e). Each line type corresponds to different time steps for
the same diapir (bottom) and topography (top).
The growth rate of the diapir is strongly dependent on the
viscosity of the bottom, buoyant layer. For a given
wavelength (A = 1.1 or 3.0) the growth rate of the diapir
increases significantly with decreasing viscosity of the
buoyant layer. Also, the wavelength has very little effect on
the diapir growth rate.
Topography, in constrast to diapir growth rate, is very
sensitive to the wavelength. For a fixed wavelength the
maximum value of elevation depends very little on the
viscosity contrast. However, as the wavelength becomes
longer, maximum surface elevation increases.
The dependence of the growth rate on the viscosity
contrast is more pronounced at the earlier stages of diapir
growth. This can be seen from Figs 6(b) and 7(b), where the
slope of the elevation versus time curves are quite different
for early times but become similar at later stages. For
example, at the initial stage the growth rate is five times
faster for a viscosity contrast, p1/p2= 10 and 10 times faster
for the viscosity contrast, pI/p2 = lo3 when compared to the
isoviscous case. By the time the top of the diapir has
(a)
h, = 0.3, R = 0.1
7
0 . 0 2 5 L , ," " ' " " ' 1 " " '
TIME
(b)
- 0 . 8 1 . .
0
h, = 0.3, R = 0.1
.
,
I
50
.
.
.
.
I
100
.
.
.
. , . .
150
,
,
I
200
TIME
Figure 5. Elevation (a) and height of diapir (b) for different
viscosity contrasts at the characteristic wavelengths for each
viscosity contrast. (c, d, e) Passive strain marker field (left columns)
and velocity (right columns). For each part the value of the
maximum velocity and time are also given.
Diapirisrn and topography
velmax= 3.e-02
time35.
Figure %(Continued)
559
560
A. Poliakou and Y. Podladchikou
2I
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Diapirism and topography
I
(a)
h = 1.1, h, = 0.3,R = 0.1
TIME
(b)
-0.81
0
.
.
A
= 1.1,
.
.
I
50
561
h, = 0.3, R = 0.1
. . . . , . . . . , . . .
100
150
200
TIME
Figure 7. As Fig. 6 with A = 1.1.
reached an elevation of -0.4, the growth rate for all three
viscosity contrast are very close.
This observation can be explained by dividing diapir
growth into two stages. During the first stage, the 'bulb' of
the diapir forms from the buoyant layer. During the second
stage the mature diapir rises toward the surface. The first
stage, in which the diapir is formed is most strongly
influenced by the viscosity contrast. After the mature diapir
is formed the time of ascent is very similar for the various
viscosity contrasts. The total time for the ascent of the diapir
is the sum of the two stages. Therefore, in cases where the
contribution of the second stage is dominant, the total time
of ascent will be only slightly influenced by the viscosity
contrast. The second stage will be dominant when the
thickness of the buoyant layer is very thin or the
perturbation is very strong.
In order to isolate the influence of wavelength we present
Fig. 8, where the viscosity contrast was fixed (PI
= 10))
but the wavelength was varied to 1.1, 1.68 and 3.0 (1.68 was
the characteristic wavelength). Fig. 8 shows that the time in
which the diapir rises to the depth -0.05 is very close for all
A. However, the surface topography varies considerably and
is two times greater for A = 3.0 compared to A = 1.1.
Although the maximum elevation of surface topography is
approximately the same for a given h, the evolution of the
growth rate is very different. We now compare surface
topography for different viscosity contrasts for a fixed
wavelength, A = 3. We examine the rate of growth when the
diapir has reached a given depth. For example, when the
diapirs rise to a depth of -0.4, the maximum elevation
above the isoviscous diapir is only 0.005 while the elevation
is 0.016 for the lo3 viscosity contrast case (Figs 6c-e).
This difference can be explained by examining the shapes
of the diapirs at this stage. It can be seen that the diapir with
viscosity one (Fig. 6c, solid line) is more narrow and there is
a layer of the buoyant material at the bottom. The diapir
with a viscosity of
(Fig. 6e, solid line) has coalesced
into one 'bulb'. The buoyant force (or dynamic pressure) is
higher for this configuration. The difference in pressure is
higher for the high-viscosity case because there is no layer of
buoyant material. The same effect could be seen for h = 1.1.
3.2 2
Ef~ec. of the thickness of .he
buoyant layer
We now examine the effect of the relative thickness of the
buoyant layer on the surface topography. The combined
thickness of the two layers is held constant, while the
thickness of the lower layer is set to h, = 0.1, 0.3, 0.5, and
0.7 (Figs 9c-f, respectively). The wavelength of the initial
perturbation is the same for all cases.
562
A. Poliakov and Y. Podladchikov
20
(d)
g
g
40
80
60
100
TIME
TIME
h = 1.68,
0.010
0.000
-0.010
.....
-0.020
-0.030
0.0 0.2 0.4 0.6 0.8 1.0
Figure 8. Elevation (a) and height of diapir (b) for fixed viscosity contrast and at the variable wavelengths. (c, d, e) Contours of diapirs
(bottom) and exaggerated topography (top) for different moments in time.
The surface elevation directly above the diapir is plotted
versus non-dimensional time for these three cases in Fig.
9(a). The height of the diapir is plotted versus time in Fig.
9(b). The maximum height of the topography is highest
when the two layers have equal thickness. This is because
the total potential energy available to drive the flow is
greatest for this configuration and causes the most intensive
flow. The maximum height of the topography differs for
h, = 0.3 and h, = 0.7 due to the asymmetry of boundary
conditions on the surface and bottom. Nonetheless,
potential energies of these systems are similar. The time for
topographic growth increases with decreasing h, as one
would expect.
interior processes. These assumptions are usually met for
large-scale geophysical problems.
For free-slip calculations, topography is linearly dependent on R. In order to compare this prediction with the
results of free surface simulations we made calculations for
various values of R. Fig. 10 shows the elevation of the
surface directly above the diapir for two viscosity contrasts
for the same parameters used in Section 3.1. Horizontal
profiles of topography (scaled by R ) do not differ
considerably with increases of this parameter. Thus, the
relation between topography and R is indeed linear from
R = 0.03 to 0.15. This indicates that the linear theory is valid
for geologically reasonable density contrasts.
4 INFLUENCE OF THE PARAMETER R ON
THE TOPOGRAPHIC B E H A V I O U R
5
Raleigh-Taylor instability is usually modelled with a
free-slip boundary condition on the upper surface. The
topography is then estimated a posteriori from equation (1).
This procedure is valid as long as the topography is small
and adjustment of the surface occurs much faster than
GENERAL CONCLUSIONS
We have presented a new numerical technique using
markers and a deformable Lagrangian mesh to study the
deformation of the surface above a rising diapir. This
method allowed us to model a free-surface boundary in a
self-consistent way. The method makes it possible to
investigate many different problems with non-regular
- -
I
-
Diapirisrn and topography
(a)
0.020
A = 1.1, P,/P, = I ,
563
R = 0.1
-1
I
-1.0
0
200
TIME
I
200
TIME
400
b,
...
I"'..
I
:*. ,
/
!
I
\
'.
\;*
\
I
,'
:
'..
................
----.a,
\
Figure 9. Elevation (a) and height of diapir (b) for different thicknesses of buoyant layer at the fixed wavelength A = 1.1. (c, d, e) Contours of
diapirs (bottom) and exaggerated topography (top) for different moments in time.
geometry. Due to lagrangian updating of the elements,
codes become much faster and are easier to program than
when using the Eulerian approach. Markers are combined
with moving elements and are used only during a few
remeshing procedures.
Our results led to the following conclusions. (1) The
growth rate of the diapir is more strongly influenced by the
viscosity contrast than the wavelengths. This is due to the
strong growth rate difference during the initial stage. In our
cases the growth times is 3-5 times less for a viscosity
contrast of lo3 than for a viscosity contrast of 1 (for a given
A). However, for different A the time of ascent varies by
only 30 per cent (for a given viscosity contrast). (2) The
maximum elevation above the diapir linearly increases with
increasing wavelength and is approximately the same for
different viscosity contrasts. (3) The elevation will be
considerably higher for a low-viscosity diapir at a given
depth than for an isoviscous diapir. This is because the
shape of the diapir depends on the viscosity contrast.
It was found that topography behaves linearly up to
R = 0.15 which indicates that a posteriori estimation of
topography for the free-slip calculations are valid within this
limit.
ACKNOWLEDGMENTS
DENSITY RATIO, R =
P ,-P,/P,
Figure 10. The maximum elevation dependence on the parameter
R, for different viscosity contrasts.
We are grateful to Christopher Talbot for suggesting the
free-surface problem. David Yuen is thanked both for
suggesting comparison with traditional modelling methods
564
A. Poliakov and Y. Podladchikov
and for his critical reading of our manuscript. We thank
Harro Schmeling for introducing us to the marker
technique, and Peter Cundall for suggestion the Lagrangian
approach. Andrei Malevsky and Harro Schmeling gave
many helpful explanations of physical processes. Peter van
Keken and Ulrich Christensen kindly run their codes for a
benchmark comparison of our program. V. Novikov and A.
Lyakhovsky significantly contributed in the beginning of the
present codes. We thank Ethan Dawson for patiently
revising the English of our manuscript. Laura Weyer is
thanked for drawing the pictures. Debbie Downs and Sara
Green established a proper research atmosphere. U.
Christensen and H. Schmeling provided excellent reviews
and gave many ideas on how to improve the paper. We
thank the Minnesota Supercomputer Institute for providing
facilities during the course of this work. A study grant and
computer facilities from the University of Uppsala is also
acknowledged. Y. Podladchikov is grateful to the Th.
Nordstroms fond, Royal Swedish Academy of Sciences for
research grant 'Numerical modelling of Geological
structures'.
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