GEOPHYSICAL RESEARCH LETTERS, VOL. 25, NO. 16, PAGES 3205-3208, AUGUST 15, 1998
Comparison of Mixing Properties in Convection with
the Particle-Line Method
Arkady A. Ten, Yuri Yu. Podladchikov, David A. Yuen, Tine B. Larsen,
Andrei V. Malevsky
Abstract. Spatial resolution in mixing processes is an acute
problem. We propose a line method, akin to the contour
dynamics technique, which is an extension of the particle
method but with the particles redistributed on the line with
time. We have used up to 105 particles per line and ten
lines to investigate the dynamical and structural properties
of mixing for both Newtonian and non-Newtonian temperature-dependent viscosity convection in 2D geometry. The
spatial structures and the time history of the lines formed
in Newtonian convection are different from those produced
in non-Newtonian convection, which has the tendency for
producing long-living horizontal structures. Efficient mixing
in the upper mantle would be inhibited by non-Newtonian
rheology.
Introduction
Mixing processes occur universally with flow processes,
be it in the mantle, magma chambers or in stirred tank reactors. We consider here kinematic mixing in convection,
where effects of diffusion can be neglected in the scale range
under consideration. In this mixing regime, most previous
investigators in mantle convection have employed passive
tracers [Olson et. al., 1984; Hoffman and McKenzie, 1985].
A logical extension of tracers is to connect them up into a
chain, as in the dynamical modelling of phase transitions
by Christensen and Yuen [1984] and in compositional layering by van Keken et al. [1996]. In the atmospheric sciences the dynamical contour methods have reached a high
level of sophistication in the ozone hole problem [Waugh
and Plumb, 1994]. Contours are represented by a set of particles linked together. This method has not been as popular
as the tracer method due to the higher computational demand [Franjione and Ottino, 1987]. With the current shared
memory architecture, memory is becoming less of an issue
and we employ this line method for quantifying mixing and
investigating its evolving spatial structural properties. We
have chosen the line method based on particles because it
is intrinsically more accurate than the field approach used
previously by us [Ten et al., 1996, 1997]. The field approach is preferable for visualization, since it yields all at
once a very rich display of information which we can readily
appreciate. On the other hand, many particles dispersing
about in the late stage of mixing become chaotic and burdensome to the eye. The line-method represents a middleground between the particle and field approaches in terms
of computational costs and visualization. However, the line
method is much more accurate for quantitative analysis.
Copyright 1998 by the American Geophysical Union.
Paper Number 98GL51991.
0094-8534/98/98GL-51991$05.00
Numerical techniques
We have used lines formed by connected tracers, which
are dynamically redistributed at each time step (see the
zoomed-in panel at the left-hand side of Fig. 1). At time t
=0, ten horizontal lines are placed equidistantly at various
depths, allowing for the subsequent sweeping through of the
entire system by these lines. The positions of the tracers are
governed by a vector differential equation:
dx
= V (x, t),
dt
where x is the vector of the particle positions and V is the
velocity field ,taken from the time-dependent incompressible convection calculations [Naimark and Malevsky, 1987;
Malevsky and Yuen, 1991]. In order to insure proper numerical accuracy of the particle trajectories, we have employed a 4th order implicit Adams-Moulton variable timestep method. For maintaining a uniform resolution along the
line, we redistribute the particles on the line at each timestep by using a 3rd order Lagrangian interpolative scheme.
This step is particularly important, because of the highly
variable strain produced by strongly time-dependent convection.
It is well known that tracking of material surfaces associated with kinematic mixing requires a much higher effective
resolution than for the temperature and velocity fields [Metcalfe et al., 1995; Ten et al., 1996]. Here, we have used a
set of ten lines with 105 particles for each .For analyzing the
structural features of mixing we have employed the method,
sketched in Fig. 1, which maps a two-dimensional field (center panel) onto a one-dimensional profile (right panel). The
time-dependent evolution of this profile constitutes the basic thrust of our idea concerning the quantification of local
structures in mixing.
There are two main components in this procedure (1)
marker layer (see central panel) to be monitored (2) the
depth levels (right panel) which registers the accumulation
of the mass at the monitoring level (see central panel). The
marker layer is defined to be an area bounded by two lines,
which will be the subject of analysis. Since the entire mass
of the layer is monitored, it is similar to the layer formed
by the passive field, albeit only two lines are advected. This
represents a distinct advantage of carrying the information
over the particle method (see center panel). Next we perform the projection onto a profile by dividing up the twodimensional mixing domain into horizontal levels where the
monitoring of the marker layer takes place. At each time
step (t = t1 ) the amount of mass trapped inside each level
is calculated and tabulated as a function of depth (see the
curve in right panel). This technique is similar to the tracer
histogram approach by Schmalzl et. al. [1996]. Peaks in
the curve are associated with a large fraction of the marker
layer. From accumulating over successive time-steps, we can
3205
3206
TEN ET AL.: COMPARISON OF MIXING PROPERTIES
Mass of marker layer at t=t1
0.0
0.2
0.4
Top
Mass of marker layer inside monitoring level
Depth
Monitoring level
Tracers
at t=t1
Original position (t=0)
of marker layer.
Marker layer at t=t1
Bottom
Figure 1. Schematic description of processing the data from the particle-line (left panel) results to 1-D profile (right
panel) of the mass of the marker layer. The definitions are given in the respective locations. The dark region represents
the mass inside the monitoring level (central panel).
now construct a mixing surface M(z,t) which can shed light
on the temporal evolution of mixing activities. These surfaces can be found in the last figure (Fig. 4) of this paper.
By taking different initial depths of the marker layer, we
can learn about how the different portions of the system are
being homogenized.
The analysis described in Fig. 1 can only distinguish horizontal structures, because of the original position and horizontal orientation of the marker layer. On the other hand, if
we have heterogeneities with preferred vertical orientations,
we would not be able to detect them with this approach. In
other words, this technique is analogous to having a polarization filter, which allows for the detection of horizontally
layered structures and their subsequent development.
ing included [Larsen et al., 1995]. The surface temperature T0 is 0.1 and the surface dissipation number is 0.05.
The depth-dependence of the viscosity is a factor of ten
across the layer with a thermal expansivity decrease by a
factor of 3. The temperature-dependent viscosity contrast
is near 300 for both non-Newtonian (n = 3) and Newtonian
(n = 1) across the layer, which have an aspect-ratio of two.
The effective Rayleigh numbers of the Newtonian and nonNewtonian cases, as measured by the average viscosity, is
around 3 × 106 with a similar surface heat-flow with a Nusselt number of around 20. The strain- rate dependence in
the rheology is the only significant difference between these
two cases. For additional details, the reader should consult
Ten et al. [1997]. Many time-steps of the convection solution
1.00
Results and discussion
We present and discuss the mixing of two convection
cases, representing Newtonian and non-Newtonian temperature- and pressure-dependent rheologies. These models have
been investigated previously by the field approach [Ten et
al., 1996, 1997], and they are used here in order to show
that the mixing dynamics can also be captured by the different techniques. The time-dependent velocity fields used
in the mixing come from the extended Boussinesq, baseheated convection models with viscous and adiabatic heat-
0.80
1.00
0.60
0.50
0.00
0.00
0.40
0.80
1.20
1.60
2.00
1.30
1.40
1.50
1.60
Figure 2. The mixing after two overturns as portrayed by
Figure 3. Zoomed-in area of the very narrow shear band
the lines. There are ten lines with 105 particles per line.
from boxed region in Fig. 2.
TEN ET AL.: COMPARISON OF MIXING PROPERTIES
must be stored to ensure proper accuracy in the integration
of the particle trajectories. The entire computational task
requires the use of parallel processors and large data storage
involving more than 50 Gbytes.
In Fig. 2 we show the development of the 10 lines developed after two overturns. This situation is taken from the
non-Newtonian case. Lines, because of their inherent structural character, reveal the complexities of the mixing to the
same degree as for the field approach [Ten et al., 1997]. The
accurate integration scheme allows for very sharp resolution in the highly deformed zones associated with the downwellings. The type of resolution displayed here is roughly
equivalent to a grid with 30,000×30,000 grid points and is
greater than those used in meteorology [Waugh and Plumb,
1994]. We note that we are able to resolve the clustering
of lines at the bottom of Fig. 2 and the zoomed-in box
shows the carry over of the more squeezed lines from an
earlier squeezing event. The multiresolution character of
the lines is emphasized next in the zoomed-in shot (Fig.
3207
3) of the box area outlined in Fig. 2. Here one is struck
with the fidelity and sharpness of the clustered lines, which
reveals a localized deformation zone, akin to fault zones.
This extremely narrow downwelling accomodates most of
the deformation and creates tiny sheared bands. This type
of shear-band structure induced by mixing is characteristic of non-Newtonian mixing. Niederkorn and Ottino [1994]
also found that the deformation field is more uniform in the
Newtonian system, as contrasted with their non-Newtonian
model. The ability for non-Newtonian rheology to induce
abrupt jumps in the flow development and to create regions
with sharp differences in flow character has long been recognized [Malevsky et. al., 1992]. In this non-Newtonian case
the two upwellings along the edges are active periodically,
while the downwelling in the middle comes down more or less
continuously. All of these features are reflected in the evolution of the mono-fractal dimension in the mixed field pattern [Ten et al., 1997]. While the magnitudes of the monofractal dimension between Newtonian and non-Newtonian
Figure 4. Spatial-temporal evolution of mixing surfaces Newtonian (a) and non-Newtonian (b). Regions of high concentration are delineated by the bright color. A dimensionless time of 0.001 corresponds to 127 Myr for this upper-mantle
model.
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TEN ET AL.: COMPARISON OF MIXING PROPERTIES
mixing are close to one another, Newtonian rheology does
exhibit a smoother rate of change [Ten et al., 1997]. However, the mono-fractal dimension represents only a global
measure and the presence of any sort of spatial clustering
would require the use of multifractal analysis or the line
method, shown in Fig. 1.
For analyzing the development of the local features in
mixing, we have placed initially the mass distribution as a
single layer with constant amplitude between z =0.45 and
0.55. Two hundred horizontal monitoring levels (see Fig.
1) are used to construct the temporal evolution of the mass
distribution profile. This would correspond to a vertical resolution of around 3 km for an upper-mantle model. Fig. 4
shows the contour levels of M(z,t), where M is the time history of the profile describing the mass of the marker layer.
Both Newtonian (Fig. 4a) and non-Newtonian (Fig. 4b)
cases are considered. In Newtonian mixing, there is a transient regime with anomalies being brought up to the top.
With time there is greater homogenization with some material (M=0.3) trapped in the middle. This picture of efficient
mixing after several overturns accords with previous constant viscosity investigations [Hoffman and McKenzie, 1985;
Christensen and Yuen, 1984; Hansen et al., 1992; Schmalzl
et al., 1996]. In contrast, in Fig. 4b we observe that in nonNewtonian convection the system remains quite heterogeneous for a long time, as shown by the persistence of islands
with M greater than 0.3. During the entire evolution the
marker layer is not at all well mixed and a large fraction
of the anomaly lies in the top half of the system between
z =0.1 and 0.4. The mixing map in Fig. 4b reveals these
events as upwellings and downwellings. Thus mixing in nonNewtonian takes place primarily by means of these strongly
time-dependent vertical flows. On the other hand, the Newtonian system is mixed by large-scale cellular motions.
This technique may be applied for 3D mixing analysis.
In this case, the lines should be replaced with surfaces consisting of many tracers.
Concluding remarks
We have developed a high-resolution method for portraying in close details local features and for analyzing the evolution of local structures in mixing. We have shown that
the signatures of the initial horizontal structures can persist
for a long time in non-Newtonian convection. Statistically
this implies that the original chemical heterogeneities can
remain unmixed for a long period and may be concentrated
at certain depths. Translating into physical timescales for
the upper mantle, we suggest that non-Newtonian convection would not mix up efficiently the initial heterogeneity on
timescales on the order of around Ga. These timescales are
comparable to previous estimates for whole mantle circulation [Hoffman and McKenzie, 1985; Schmalzl et al., 1996].
Thus the non-linear nature of upper-mantle rheology may,
in fact, help to sustain the chemical heterogeneity of the
upper mantle.
Acknowledgments. We thank stimulating discussions
with Dr. Minye Liu from CRAY-SGI Inc. and Prof.
F.H.Busse. This research has been supported by the geosciences program of the Dept. of Energy and CRAY-SGI
Inc. A.Ten acknowledges support as a visiting scholar from
the Minnesota Supercomputer Institute.
References
Christensen, U.R. and D.A. Yuen, The interaction of a subducting lithospheric slab with a chemical or phase boundary, J.
Geophys. Res., 89, 4389-4402, 1984.
Franjione, J.G. and J.M. Ottino, Feasibility of numerical monitoring of material lines and surfaces in chaotic flows, Phys.
Fluid, 30, 3641-3643,1987
Hansen U., D.A. Yuen and S.E. Kroening, Mass and heat transport in strongly time-dependent thermal convection at infinite
Prandtl number, Geophys. Astophys. Fluid Dynamics, 63, 6789, 1992.
Hoffman N.R.A. and D.P. McKenzie, The destruction of geochemical heterogeneities by differential fluid motions during mantle
convection, Geophys. J. R. Astron. Soc., 82, 163-206, 1985.
Larsen T.B., D.A. Yuen and A.V. Malevsky, Dynamical consequence on fast subducting slabs from a self-regulating mechanism due to viscous heating in variable viscosity convection,
Geophys. Res. Lett., 22, 1277-1280, 1995.
Malevsky A.V. and. D.A. Yuen, Characteristics-based methods
applied to infinite Prandtl number thermal convection in the
hard turbulent regime, Phys. Fluids A (3), 2105-2115, 1991.
Malevsky A.V., D.A. Yuen and L.M. Weyer, Viscosity and thermal fields associated with strongly chaotic non-Newtonian
thermal convection, Geophys. Res. Lett., 19, 127-130, 1992.
Metcalfe G., C.R. Bina and J.M. Ottino, Kinematic considerations for mantle mixing, Geophys. Res. Lett., 22, 743-746,
1995.
Naimark, B.M. and A.V.Malevsky, An approximation method
for the solution of the problem of gravitational and thermal
instability, Comput. Seism., 20, 30- 52, 1987.
Niederkorn T.C. and J.M. Ottino, Chaotic mixing of shearthinning fluids, AICHE J., 40, 11, 1782-1793, 1994.
Olson P., D.A. Yuen and D.S. Balsiger, Mixing of passive heterogeneities by mantle convection, J. Geophys. Res., 89, 425-436,
1984.
Schmalzl J., G.A. Houseman and U. Hansen, Mixing in vigorous,
time-dependent three-dimensional convection and application
to Earth’s mantle, J. Geophys. Res., 10, 21847-21858, 1996
Ten, A., D.A. Yuen, T.B. Larsen, A.V. Malevsky, The evolution
of material surfaces in convection with variable viscosity as
monitored by a characteristics-based method, Geophys. Res.
Lett., 23, 2001-2004,1996
Ten, A., D.A. Yuen, Yu. Yu. Podladchikov, T.B. Larsen, E.
Pachepsky, A.V. Malevsky, Fractal features in mixing of nonNewtonian and Newtonian mantle convection, Earth Planet.
Sci. Lett., 146, 401-414,1997
Waugh D.W. and R.A. Plumb, Contour advection with surgery:
A technique for investigating fine-scale structure in tracer
transport, J. Atmos. Sci., 51, 530-540, 1994.
Van Keken P., S. Karato and D.A. Yuen, Rheological control of
oceanic crust separation in the transition zone, Geophys. Res.
Lett., 23, 1821-1824 1996.
Arkady A. Ten, Univ. of Minnesota Dept. of Geology
and Geophysics, Univ. of Minnesota Supercomputer Institute,
1200 Washington ave. S., Minneapolis, MN 55415 (e-mail:
arkady@msi.umn.edu) Permanent address: Inst. of Mineralogy
and Petrography, Novosibirsk, 630090, Russia
Yuri Yu. Podladchikov, Geologisches Institut, E.T.H., CH8092, Zurich, Switzerland (e-mail: yura@erdw.ethz.ch)
David A. Yuen, Univ. of Minnesota Dept. of Geology
and Geophysics, Univ. of Minnesota Supercomputer Institute,
1200 Washington ave. S., Minneapolis, MN 55415 (e-mail:
davey@krissy.msi.umn.edu)
Tine. B. Larsen, National Survey and Cadastre, Geodynamics Dept., DK-2400 Copenhagen NV, Denmark (e-mail:
tbl@kms.min.dk)
Andrei. V. Malevsky, Departement de Physique et CERCA,
Universite de Montreal, Montreal, Que, H3X 2H9, Canada, (email: malevsky@CERCA.UMontreal.CA)
(Received February 27, 1998; revised May 12, 1998;
accepted June 3, 1998.)