Earth and Planetary Science Letters 146 Ž1997. 401–414
Fractal features in mixing of non-Newtonian and Newtonian
mantle convection
Arkady Ten
a
a,b
, David A. Yuen a,) , Yu.Yu. Podladchikov c , Tine B. Larsen d ,
Elizaveta Pachepsky a , Andrei V. Malevsky e
Department of Geology and Geophysics, Minnesota Supercomputer Institute, UniÕersity of Minnesota, Minneapolis, MN 55415-1227, USA
b
Institute of Mineralogy and Petrography, Russian Academy of Science, NoÕosibirsk, 630090, Russia
c
Geologisches Institut, E.T.H., CH-8092, Zurich,
Switzerland
¨
d
Danish Lithosphere Center, DK-1350 Copenhagen K, Denmark
e
Departement de Physique et CERCA, UniÕersite´ de Montreal, Montreal, Que. H3X 2H9, Canada
Received 4 September 1996; accepted 9 December 1996
Abstract
Mixing processes in mantle convection depend on the rheology. We have investigated the dynamical differences for both
non-Newtonian and Newtonian rheologies on convective mixing for similar values of the effective Rayleigh number. A
high-resolution grid, consisting of up to 1500 = 3000 bi-cubic splines, was employed for integrating the advection partial
differential equation, which governs the passive scalar field carried by the convecting velocity. We show that, for similar
magnitudes of the averaged velocities and surface heat flux, the local patterns of mixing are quite different for the two
rheologies. There is a greater richness in the scales of the spatial heterogeneities of the passive scalar field exhibited by the
non-Newtonian flow. We have employed the box-counting technique for determining the temporal evolution of the fractal
dimension, D, passive scalar field of the two rheologies. We have explained theoretically the development of different
regimes in the plot of N, the number of boxes, covered by a range of colors in the passive scalar field, and S, the grid size
used in the box-counting. Mixing takes place in several stages. There is a transition from a fractal type of mixing,
characterized by islands and clusters to the complete homogenization stage. The manifestation of this transition depends on
the scales of the observation, and the initial heterogeneity and on the rheology. Newtonian mixing is homogenized earlier for
long-wavelength observational scales, while a very long time would transpire before this transition would take place for
non-Newtonian rheology. These results show that mixing dynamics in the mantle have properties germane to fluid
turbulence and self-similar scaling.
Keywords: mantle; mixing; convection; fractals; rheology; models
1. Introduction
The process of mixing in fluid flows is of fundamental importance in many areas, ranging from in)
Corresponding author. E-mail: davey@krissey.msi.umn.edu
dustrial and environmental systems to geophysical
and astrophysical processes. Mixing problems are
still studied case by case. The effects of shear-thinning rheology on mixing have been studied within
the context of time-periodic forcing on cylindrical
flows w1x for industrial applications. Some significant
0012-821Xr97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved.
PII S 0 0 1 2 - 8 2 1 X Ž 9 6 . 0 0 2 4 4 - 0
402
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
effects were found in the chaotic advection of a
passive tracer in the shear-thinning fluids, in which
the viscosity decreases with increasing strain rate w2x.
Our interests stem from mantle dynamics and here
one must consider the mixing phenomenon in the
presence of rheological fluid motions driven by thermal convection. We are interested in mixing from a
geophysical standpoint, with mantle convection being the centerpiece. Since upper mantle rheology is
likely to be non-Newtonian, we may not avoid the
issue of mixing in rheological fluids. In spite of the
seeming ubiquity of non-Newtonian rheology in the
upper mantle, there is a dearth of studies on mixing
in non-Newtonian convection.
Recent work by us w3x has shown the possibilities
for using a very dense grid for mapping out the
complex patterns of passive scalar field driven by
time-dependent convection. These passive scalar
fields exhibit characteristics of turbulent flows at
high Reynolds number. We will present the high-resolution results of the evolution of the passive scalar
fields and compare the differences between Newtonian and non-Newtonian mixing. The high spatial
resolution, well over 10 3 = 10 3 grid points, allows
for a close examination of these issues in mixing
with fractal analysis. We will present a simple model
of mixing based on the evolution of the fractal
dimension and utilize the data from the high-resolution passive scalar fields for testing out the model.
2. Mixing by passive scalar field: details of modelling
Mixing processes can be monitored by a Lagrangian approach using passive tracers w4–7x from
the convective velocity field. The main limitation of
the Lagrangian approach is due to its statistical
nature, which requires the calculation of many particle trajectories and it is also beset by a global
description of the scalar field at large time. We have
taken the alternative Eulerian approach of monitoring a passive scalar field by simulating an advection
partial differential equation, which allows us to have
direct access to the local and global characteristics of
the passive scalar field, as well as to information
about the gradient of the field. Moreover, one can
study by visualization the temporal evolution of this
scalar field. We have here neglected the effects of
diffusion, which determines the homogenization of
the smallest scales of the scalar field.
The mixing of a passive scalar field c in the
absence of chemical reactions and diffusion is governed by the non-dimensional first-order partial differential equation:
Ec
Et
q U P grad c s 0
Ž 1.
where U is a precomputed, time-dependent velocity
field taken from a thermal convection calculation. A
characteristic-based method w8x with fourth-order
spatial and second-order temporal accuracy has been
employed for solving the first-order partial differential equation. This solution of c is simply transported along the characteristic curves with the velocity field U w3x. The set of convection equations used
to calculate the thermal and velocity fields in a
two-dimensional cartesian domain is described elsewhere w9x. They consist of the conservation equations
of mass, momentum and energy with the time-dependence present only in the energy equation, because
of the infinite Prandtl number nature of the mantle.
These solutions were obtained with a similar bi-cubic
spline-based method w8x. The flow dynamics are not
influenced at all by the passive nature of c , which
serves the role of mapping out the structure being
advected by the fluid motion.
There is no mass flux out of the boundaries
because of the impermeable Žvanishing normal velocity. boundary conditions imposed around the box.
Reflecting boundary conditions at the vertical edges
have been imposed on the horizontal velocity and
temperature in the convection simulations w8,9x, as
well as a vanishing vertical velocity at the horizontal
boundaries.
Since the mechanism of mixing produces smallscale features, we have taken the strategy of employing a much denser grid for the scalar c field than for
the temperature and velocity field used in the convection simulation. The need for a much denser grid
for c is consistent with the physics of passive
advection, which is much more chaotic w7x than the
convective flow carrying c . The spline functions
allow for an accurate interpolation of the velocity
field onto a denser grid for integrating Eq. Ž1.. The c
fields presented below have been integrated in an
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
aspect ratio two box with up to 1500 = 3000 unequally spaced grid points, as compared to the temperature and velocity fields which have been integrated with 72 unevenly spaced vertical points by
200 uniform grid points along the horizontal direction. The display of the c fields and the fractal
analysis have been conducted on this same grid. It
should be noted that these simulations and the visualization display have really stretched the present technological capabilities. We had to store and move
about 200 Gbytes of data just for the two runs
reported here. Visualization was done on a wall
consisting of seven million pixels in order to obtain a
global moving view of the complex time-dependent
field, which contains a wide spectrum of scales. The
computations were made in parallel mode using a
SGI cluster of Power Challenge workstations. These
exceedingly high-resolution grid configurations will
unveil the range of multiple scales developed in the
evolution of c and will allow us to conduct a
statistical characterization of the turbulent-like structures by fractal analysis with high fidelity. We note
that laboratory image processing of turbulent mixing
w8x with high-performance CCD cameras has captured up to 2 million pixels.
Because of the enormous computational and human resources which this problem has required, we
have only investigated two rheological cases. These
cases are taken from the extended-Boussinesq convection models in which both adiabatic and viscous
heating terms are included w9x. The first case ŽA. has
a non-Newtonian, temperature-dependent rheology
with a power-law index n s 3 and a temperature-dependent viscosity contrast of 300 across the layer.
The rheological law, appropriate for mantle rocks
w9x, takes the form:
e ij s Aexp Ž BT y CZ . t Žny1.t ij
Ž 2.
where: Z is the depth; A is a material constant;
B s 16.1; C s 6.9; the power-law index n s 3; t ij is
the deviatoric stress tensor; t is the second-invariant
of the deviatoric stress tensor defined by the trace; e ij
is the deviatoric strain-rate tensor. The second case
ŽB. has a Newtonian Žn s 1., temperature-dependent
rheology with B s 5.4 and C s 2.3. Both models
have a temperature-dependent viscosity contrast of
300 across the layer and a depth-dependence in the
viscosity increasing by a factor of 10. These models
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also have a depth-dependent thermal expansivity decreasing by a factor of 1r3 across the layer w10x. The
surface dissipation number, D w9x, and surface temperatures, To w9x, are 0.05 and 0.1, respectively, for
both cases. The volumetrically averaged Rayleigh
numbers are about the same in both cases, namely
Ra s 3 = 10 6 . Similar surface Nusselt numbers,
around 20, are obtained for both cases. We have
included the depth-dependent physical properties to
maintain some degree of realism for application in
mantle convection, while at the same time focusing
on the differences between Newtonian and non-Newtonian rheologies. We have chosen these variable
viscosity cases as possible end-member cases in
studying mixing of strongly time-dependent mantle
convection.
3. Evolution of the passive scalar fields
In Fig. 1 we show the differences between the
passive scalar field, c , and the temperature fields for
the Newtonian case. The initial c field was divided
into three equal parts with red at the top, green in the
middle and blue at the bottom. We have taken as
initial conditions the temperature field from an already convecting solution with a growing instability
at the top thermal boundary layer. This thermal
initial condition is taken from a situation well past
the initial transient. There is an infinite number of
initial conditions for the c field we could have
employed for integrating Eq. Ž1.. We have selected a
relatively simple initial condition for c . The inset
shows a descending plume Žblack. plunging down
and two upwellings Žpurple. along the side boundaries at a time of 13 Myr after the start of the
integration of Eq. Ž1.. We have taken a layer depth
of 2000 km to dimensionalize the time by the thermal diffusion time associated with this depth. A
dimensionless time of 0.001 then corresponds to 127
Myr. Besides having very close Nusselt numbers, the
two rheological models also have nearly the same
root-mean-squared velocity, which lies in the neighborhood of 900 in non-dimensional units. The entrainment of warm Žgreen. material into the cold
Žred. descending plume is clearly revealed in the c
field. Another interesting feature is the appearance of
a secondary rising instability at the right. There are
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Fig. 1. A comparison between the passive scalar field and the temperature field Žinset.. The time is 0.001 after the start of integrating Eq.
Ž1.. The rheology is Newtonian, temperature and depth dependent with a viscosity contrast due to a temperature of 300 and to a depth of 10.
The volumetrically averaged Rayleigh number is around 3 = 10 6 . The spline grid for the temperature field is 72 = 200 points, and
1000 = 1000 points for the c field. For display, the grid has been expanded by interpolation to 1000 = 2000 points. The initial
configuration for c has red at the top, green in the middle and blue at the bottom, all evenly distributed. The initial temperature field has
been taken from an already convecting solution.
many interesting features picked up by the c field,
which reveal much more clearly the complicated
dynamics. This figure shows why a much higher
resolution is required for an accurate description of
the c field.
For comparison of the two rheologies, we show in
Fig. 2 four snapshots of the c field for the Newtonian case and in Fig. 3 four panels portraying the
non-Newtonian case. The first two panels describe
the initial stages, while the third and fourth panels
display the later development. The final times are
approximately the same. In order to comprehend
better the dynamic differences, one has to look at the
animation of the of the c field. The motions of the
c field are really different from the evolution of the
T field. One can see the formation of small eddies in
the bottom two panels in Fig. 2. Newtonian mixing
is primarily dominated by long-wavelength features
with two cells and a significant vertical transport
component. There is a great deal of mixing by the
small eddies. The bottom panel of Fig. 2 shows that,
after nearly four overturns, the system is not completely homogenized but is partitioned into different
domains, separated by the main descending flow in
the middle of the box. In contrast, non-Newtonian
mixing takes place in a helter-skelter and meandering fashion and involves the participation by fastscale jets and large-scale coherent structures Žsee the
bottom two panels in Fig. 3.. Mixing is less efficient
than for Newtonian convection and there are still
many islands of unmixed material Žred and blue. left
even after 3 overturns, indicating some sort of
quenching of the mixing process.
Although both rheological flows have similar
magnitudes of averaged velocities and surface heat
flux, there are significant deviations in the manner of
stirring and stretching of the c field displayed by the
two rheologies. This finding is not so surprising,
because there is firm evidence w13x that non-Newtonian convection tends to generate a jerkier kind of
time-dependent fluctuation and more contrasting spatial heterogeneities in the velocity fields than Newtonian convection. These results are different from
those found in shear-thinning fluid flows w1x forced
by the rotating velocity boundary conditions. There a
decrease in the amount of local stretching was found,
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
405
Fig. 2. Temporal evolution of the c field for Newtonian rheology. Same parameter values are used as in Fig. 1. From top to bottom the
times are: 9 = 10y4 , 1.2 = 10y3 , 3.7 = 10y3 and 4.0 = 10y3 .
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A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
Fig. 3. Temporal evolution of the c field for non-Newtonian rheology. The same temperature and depth dependent viscosity contrasts as for
the Newtonian case and the power-law index of n s 3. Times are: 7 = 10y4 , 1.3 = 10y3 , 3.0 = 10y3 , and 3.8 = 10y3 .
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
whereas in thermal convection the internal dynamics
of the non-Newtonian convection produce the largescale coherent features, giving rise to a somewhat
inefficient route for mixing.
4. Fractal analysis of the passive scalar field
4.1. Fractal model deÕelopment and results
Fractal analysis can shed light on the distribution
of scales in mixing. We have determined the fractal
dimension of the high-resolution images representing
the c field. The grid points along the horizontal and
vertical directions are denoted by nx and ny, respectively. To quantify mixing with time, let us consider
a rectangular strip initially at a depth h with a
vertical thickness D h and extending horizontally
across. Both h and D h are measured in ‘‘grid points
units’’. Typically, nx P ny is ( 10 7, nx P D h ( 10 6 .
This strip is assigned a ‘‘black’’ color, while the rest
of the domain is designated to be ‘‘white’’. As
mixing proceeds, the morphology of the initially
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horizontal strip will become more and more complex. At each time, the resulting black and white
pictures are processed by the box counting technique
w14x. This processing results in a diagram log ŽN. vs.
log ŽS., where N is the minimal w14x number of
boxes of size S needed to cover the black color ŽFig.
4.. The box coverage of a black patch of material is
illustrated on the right hand side of Fig. 5, where the
dotted boxes indicate the presence of some amount
of black material in that box with a grid of width S.
The ‘‘box-counting’’ fractal dimension Žotherwise
known as the Hausdorff or capacity dimension. is
given by the slope of the straight segment on the log
ŽN. vs. log ŽS. plot. Inspection of the actual box
counting results ŽFig. 6. reveals a more complicated
pattern. The log ŽN. vs. log ŽS. curves are composed
of at least three different linear segments that change
their location in time, as mixing proceeds. The trilinear Ž3L. representation of the box-counting results
is sketched in Fig. 5.
At the finest scales the first linear segment lies
below a first crossover scale S1 Ži.e., 1 - S1.. It has
a non-fractal slope of 2 and an intercept nx P D h.
Fig. 4. Schematic diagram showing the various regimes in mixing from a box-counting perspective. The number of boxes with size S needed
to cover an object Žsee black island in the inset. is given by N. Several stages take place with the initial layering, the transient phase and the
steady-state regime. Both the initial state and the state of complete homogenization are non-fractal with D s 2. Fractal mixing is
characterized by a power-law like Žnon-integer exponent. behavior in the log–log plot of N vs. S.
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A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
Fig. 5. Schematic diagram illustrating the decomposition of the box-counting algorithm into the boundary and interior contributions. The
boundary contributions deal with interfaces and increase with time. The interior contributions decrease with time as the object becomes
more convoluted Žsee right panel..
This represents the initial layered state. Initially, this
segment occupies all scales ŽS1 s ny at t s 0.. S1
decreases in time as mixing destroys the initial layering by stirring up the larger scales.
The intermediate linear segment represents the
regime of fractal mixing, which spans scales from S1
to S2, and is decreasing in time with a non-integer
slope 1 - 2. Fractal mixing implies a strongly uneven distribution of heterogeneity, the presence of
‘‘islands’’ of unmixed fluids of any size in between
S1 and S2. The third linear segment is at the coarse
end and spans the scales from a second crossover
scale, S2, to ny. It has the ‘‘non-fractal’’ slope of 2,
the same as the first segment, but with a higher
intercept nx P ny. The third segment represents the
completely mixed state, in which each box contains
at least one black grid point. First, it occurs only at
the largest scales, then its width increases with time,
as S2 decreases. The destruction of the fractal subrange is caused by widening of the segment. It is
natural in fractal scalings to have high and low
cutoffs.
For a better understanding of the box-counting
method, let us introduce N, N b , Ni, which represent
the number of boxes of size S needed to cover the
black color, its boundary, and its interior, respectively. Furthermore, let us assume:
N ( S D , N b ( S D b , Ni ( S Di
Ž 3.
where D, D b , Di are the ‘‘box’’ fractal dimensions
of the black color, its boundary, and its interior,
respectively. From geometrical considerations ŽFig.
5., one may expect the following scenario. First, D b
is increasing from 1 to D inf , 1 - D inf - 2, while
simultaneously D decreases from 2 to D b . Therefore,
the slopes of log ŽN. vs. log ŽS. are function of both
time and scale. At a fixed scale, the evolution of the
local slope of the log ŽN. vs. log ŽS. will start from 2
Žthe initial layering.; then decreases to a minimum,
which at this particular instant can be bigger then or
equal to the fractal dimension of the border. Finally,
it rises back again to D s 2, as a result of complete
homogenization ŽFig. 4..
Fractal dimensions were determined by using different sizes of boxes in the box-counting procedure.
This will give an idea as to the influences of the
sampling size on the time-dependent fractal signatures. Olson et al. w15x have shown numerically for
40,000 particles that the longer wavelength heterogeneities would be well mixed before the smaller
scale inhomogeneities. In Figs. 7 and 8 we show the
time histories of the monofractals for small and large
boxes, respectively. The fractal dimension as a function of time have been constructed from the N ŽS.
curves, shown in Fig. 6. We have also verified the
accuracy of the monofractal dimension D by conducting a multifractal w16x analysis on the same data
w17x.
The boxes used in Fig. 7 have the smallest resolution possible for estimating the fractal dimension.
The area occupied by the two sizes of boxes is at
most 3 = 10y5 times the total area of the image.
This small size yields a very high resolution for
determining small-scale heterogeneities. At these
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
409
Fig. 6. The dependence of N on S in logarithmic format for several times: Ža. Newtonian rheology; Žb. non-Newtonian rheology. A grid with
1500 = 3000 points has been used for the box-counting analysis. Times are given for the beginning and at the termination of the analysis.
Note the erosion with time at the high S end. From the NŽS. curves we then calculate the fractal dimension D with t. The initial thickness of
the heterogeneity, used in the box-counting, is about 0.2 times the total thickness of the layer.
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A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
Fig. 7. Fractal dimension D vs. time t for the two rheologies. A small observational scale, S between 2 and 4, is used. The initial thickness is
the same as in Fig. 6.
Fig. 8. Fractal dimension D vs. t for the two rheologies. A larger observational scale, S, between 16 and 32, is used. The initial thickness is
the same as in Fig. 6.
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
scales there is a marked difference in the evolution
of D between the Newtonian and non-Newtonian
rheologies. The initial D h of the monitored heterogeneity is about 1r5 of the layer. Homogenization is
faster for the Newtonian case, as can also be observed in the spatial patterns by comparing Figs. 2
and 3. At this small scale of observation, mixing
dynamics for both rheologies remain fractal-like for
a long time and no signs are in sight for the transition to the path for complete homogenization.
Increasing the area of observation by a factor of 8
ŽFig. 8. has a much greater impact on the Newtonian
fractal evolution, as we can now reach the valley in
D predicted by the model described above. This
valley is now attainable because, at this coarser
spatial resolution, we have reached a point where
there is a transition from fractal mixing. The rise in
D indicates that, for this scale of observation, complete homogenization is on the way. On the other
hand, even at this relatively large observational scale,
the style of non-Newtonian mixing remains fractallike for a long time.
We have studied the influences of a thinner initial
vertical thickness in Fig. 9, where we reduced D h by
a factor of 3 to about 1r15 of the layer. A thinner
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initial thickness, corresponding to a smaller-scale
initial heterogeneity in the mixing process, causes
the fractal dimension for both rheologies to drop
sharply close to the theoretical limit of 1.35
w11,12,18x. The thinness of the initial layer is responsible for the dominant contribution by the boundary
component of the fractal dimension D b . In this vein
we may propose a kind of ‘‘spatial universality’’ in
that the temporal evolution of D b does not depend on
the initial width D h and the asymptotic limits of
D s 1.35 w11,18x or D s 1.33 w19x for the interface
boundary are reached.
The timescales for reaching the valley D are much
smaller than the timescales for the greater initial
thickness Žcf. Fig. 8.. This would imply that heterogeneities in the high Rayleigh number regime would
drop down precipitously to low values of D, where
the boundary contributions would be at a maximum,
before rising back up to the route of complete homogenization in the interior.
4.2. Box-counting decomposition
Box counting results ŽFig. 6. also show that the
initial layering subrange Žfrom 1 to S1. is being
Fig. 9. Fractal dimension D vs. t for the two rheologies. The observational scale is the same as for Fig. 8, but the initial thickness has been
reduced by a factor of 3.
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
412
destroyed by very fast mixing and is reduced to the
first couple of scales. Therefore, as a rough approximation, we may assume a two-segment representation of the plot log ŽN. vs. log ŽS.. In this case, the
fractal portion is given by:
Ns
n P Dh
SD
, S min - S 2
Ž 4.
whereas the segment characterizing complete mixing
or homogenization is given by:
nxny
N s 2 , S 2 - S max
Ž 5.
s
where n x and n y are the grid points along the two
directions. Next, we can employ the intersection of
these two lines from Eq. Ž3. and Eq. Ž4.; that is, the
crossover from fractal mixing to complete homogenization, as a quantitative measure of mixing. This
condition then determines a ‘‘mixing scale’’, S mix ,
which is:
1
S mix s S 2 s
ny
žD /
Ž
2y D
.
h
Ž 6.
Thus, the mixing scale S mix decreases in time due to
a decrease in D, and it also scales inversely with the
initial ‘‘concentration’’ D hrny. Thus, a bigger initial inhomogeneity would be mixed earlier.
4.3. Rheological uniÕersality
Our two model runs display a remarkable similarity in DŽt. for such different rheologies. Their time
scales cannot be directly compared. However, due to
similar ‘‘effective Ra numbers’’ a rough comparison
can still be made. The monotonic decrease, the spatial universality ŽFig. 9., and the value of the limiting
fractal dimension ŽD s 1.35 " 0.05. are essentially
the same. The limiting value of the fractal dimension
is controlled by the geometry for the interfaces. One
suggests here a universal asymptotic Žfor large time.
fractal dimension for any initially planar interface,
advected by strongly time-dependent convection. The
number D s 1.35 for this limiting universal fractal
dimension of the interface does not depend on the
rheology. This rheological independence confirms
that we are dealing with a turbulent advective subrange of scales, in which the mixing is independent
of the rheology and is only affected by advection.
The physics of advection and surface growth are
known to be mostly controlled by whether the processes is conservative or nonconservative w20x.
Sreenivasan et al. w8,15x claimed the value 1.35 Ž2.35
for the 3-D flows. as a universal fractal dimension of
the interfaces for a wide variety of the turbulent
flows. A value of 1.33 has been derived by Procaccia
and Zeitak w19x on different assumptions.
4.4. Ultimate limit of conÕectiÕe mixing
The above discussion suggests that the convection
cannot produce complete homogenization from any
initial heterogeneity. According to Eq. Ž5., and substituting in the universal value D s 1.35, there is
such a limiting scale, S mix :
S mix s
ny
1.54
žD /
h
Ž 7.
At scales larger than S mix , complete homogenization
is possible, but at scales smaller than S mix , there will
always be islands of unmixed material. The exponent
1.54 may contain a ‘‘defect’’ from the power-law
exponent 3r2, if the value of 1.33 from w19x is
employed instead. An obvious candidate for further
mixing is molecular diffusion. Thus, we can give a
conservative estimate of the complete homogenization time, as the time to remove heterogeneities at
the scale S mix by diffusion:
t(
S 2mix
d
Ž 8.
where d is the molecular diffusion coefficient. However, this estimate could be orders of magnitude
inaccurate by underestimating the diffusion rate by
the fractal interface. The phenomenon of molecular
diffusion enhanced by the turbulent diffusion is well
known w18x.
5. Summary and conclusions
We have carried out high-resolution numerical
simulations of the passive scalar fields carried by
velocities developed in non-Newtonian and Newtonian convection. The passive scalar fields exhibit
characteristics of high Reynolds number turbulent
flows. Visualization of these complex fields has
A. Ten et al.r Earth and Planetary Science Letters 146 (1997) 401–414
revealed that there are substantial differences in the
spatial patterns between the two rheologies, with the
Newtonian flow displaying faster rates of homogenization and the non-Newtonian configuration retaining islands and clusters of heterogeneities and signs
of nonlinear quenching of mixing.
We have quantified mixing dynamics by using the
box-counting technique for evaluating the fractal dimension of the highly resolved passive scalar field.
We have developed a model for time-dependent
mixing based on the log–log plot of N, the number
of the number of boxes covered by a given color, vs.
S, the grid size used in the box-counting. The temporal evolution of mixing can be portrayed by the
intersection of two segments in this plot. A transition
is predicted for a fractal-type of mixing, characterized by the presence of islands and clusters, to
homogenization. We have verified this model with
numerical simulations. The style of mixing also depends on the scale of observation. We have found
that long-wavelength Newtonian heterogeneities
would reach this path toward complete mixing much
earlier than for large-scale non-Newtonian inhomogeneities. We would expect these rheological differences in mixing to exist also in 3-D, as toroidal
excitation would be greater for non-Newtonian rheology w21x.
This model makes a number of predictions, which
warrant further studies. Among them are the concepts of ‘‘spatial’’ and ‘‘rheological’’ universality.
The first idea is concerned with the changes in the
scale of mixing, S mix , with time Žsee Eq. Ž6.. and the
second proposal deals with the asymptotic limit of D
and its independence from rheology. If these statements are corroborated in the future, then we will
have demonstrated the important link between fractal
features and mixing. Multifractal analysis should be
carried out, because it will shed light on the presence
of higher fractal dimensions and their predictions on
the transport properties in mixing. We must await the
next generation of massively parallel computers to
make any inroad into this type of mixing problem in
three dimensions.
Acknowledgements
We thank Bobby Bolshoi and Minye Liu for
encouragement and discussions and Bill Newman for
413
an enlightening review. E. Pachepsky was supported
by a summer internship program of the Minnesota
Supercomputer Institute. We are grateful for the
technical assistance provided by Y. Itoh and D.M.
Reuteler. This research was supported by Cray Research Inc., Ocean Sciences Program of N.S.F., the
Danish Research Council, the Geosciences of D.O.E.
and the Universitair Stimulerings Fonds of the Vrije
Universiteit, Amsterdam, Netherlands. Much of the
computer resources and visualization assistance came
from the Laboratory for Computational Science and
Engineering ŽLCSE. at the University of Minnesota.
[RV]
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