Electronic Geosciences
ISSN 1436-2511
Electronic Geosciences (1999) 4:1
Visualization and Analysis of Mixing Dynamical
Properties in Convecting Systems with Different
Rheologies
A. A. Ten1, 2, D. A. Yuen1, and Yu. Yu. Podladchikov 3
1 Department of Geology and Geophysics, University of Minnesota Supercomputer
Institute,1200 Washington Avenue S, Minneapolis, MN 55415-1227, USA.
2 Institute of Mineralogy and Petrography, Novosibirsk, 630090, Russia
3 Geologisches Institut, E.T.H., CH-8092, Zurich, Switzerland
Correspondence to:
A. A. Ten (email: arkady@msi.umn.edu)
Received: 12 April 1999 / Revised: 24 August 1999/ Accepted: 30 August 1999
Abstract. We analyzed and compared the mixing properties of 2-D mantle convection
models. Two rheologically different models, Newtonian and non-Newtonian (power-law),
were considered with both the line and field methods. The line method is based on monitoring
of passive particles joined into lines, while the field method relies on the advection of a
passive scalar field. Both visual and quantitative estimates revealed that the efficiency of the
Newtonian mixing is greater than the non-Newtonian. A heterogeneity placed in the
non-Newtonian convection forms horizontal structures, which may persist for at least 1 Ga on
the upper-mantle scale. In addition, the non-Newtonian medium reveals a lesser amount of
stretching of the lines than the Newtonian material. The rate of the Newtonian stretching fits
well with an exponential dependence with time, while the non-Newtonian rheology shows the
stretching rate close to a power-law dependence with time. In the Newtonian medium the
heterogeneity is reorganized into two unstable vertical columns, while the non-Newtonian
mixing favors horizontal structures. In the latter case, these structures are sufficiently stable in
both the temporal and spatial planes to explain the mantle geochemical and geophysical
heterogeneities. Due to the non-linear character of power-law rheology, the non-Newtonian
medium offers a "natural" scale-dependent resistance to deformation, which prevents efficient
mixing at the intermediate length scales.
Key words: Fractal – Heterogeneity – Mantle convection – Mixing – Non-Newtonian
rheology
·
·
·
·
·
·
Introduction
Thermal Convection Model
Monitoring Techniques
Results and Discussion
Conclusions
References
Introduction
Mixing is an important component of many evolutionary processes in the universe. It can be
scaled from unimaginably large galactic distances (Wu et al. 1999) to those of a coffee cup.
The universality of this process extends its importance to both technology and the basic
sciences. One area of application is mixing in the earth's mantle.
The isotopic heterogeneities of the mantle xenoliths and their host basalts indicate that the
mantle preserves huge reservoirs of unmixed material (Hart and Zindler 1989). Whether these
heterogeneities have sources in the mantle (e.g. Wasserburg and De Paolo 1979) or they are
relicts of unmixed protomaterial remains a debatable issue.
Even though the modeling of mantle convection has exhibited great progress over the last
decade, mixing remains poorly understood. Understanding of the mixing rate in the mantle
and its ability to destroy heterogeneities on different scales plays a key role in the
interpretation of the geochemical anomalies (e.g. Ten et al. 1997, Ferrachat and Ricard 1998).
In this paper, we assess how different types of mantle rheologies influence mixing and
investigate the efficiency of upper-mantle mixing as a mechanism of heterogeneity
destruction.
Generally, mixing may be considered as a superposition of three processes (Hoffman and
McKenzie 1985, Ottino 1989):
·
·
·
Diffusional homogenization takes place by molecular diffusion. One may see that a
drop of milk added to coffee quickly loses its sharp boundary . It is almost impossible
to locate exact position of the boundary between milk and coffee even after a few
seconds after the initial contact.
Stretching and folding. Stirring of two immiscible materials will result in stretching
and folding. The interface surface between them grows quickly, while retaining its
original sharpness. Initially the isometric shapes become similar to convoluted threads.
However, due to the negligibly small rate of diffusion the boundary remains distinct.
Folding is important because it generates new scales in the system and increases its
structural complexity.
Breakups. The logical consequence of the stretching is breakup. There are two main
reasons for breakup. First, strong stretching may bring surface tension into play.
Surface tension induces topological instability of the threads and reconnection of
interfaces. Second, the high stresses during the course of severe deformation can cause
rupturing. In any case, breakup involves a topological transformation. For example, a
sphere has one outer surface; if divided into two parts, it then forms two surfaces (e.g.
Constantin et al. 1993).
The diffusion and breakup are purely irreversible processes, while the stretching is kinematic
and purely reversible. Another very important aspect in our understanding of mixing is the
scale. One-centimeter heterogeneity may symbolize an unmixed system on the scale of a cup
of coffee. The same-size feature in the Pacific Ocean would not be considered as a significant
inhomogeneity looking down from a satellite. By geochemical means we would not be able to
resolve a deep-sited heterogeneity less than 10 km in diameter, which is also below the
resolution of seismic tomography of the upper mantle.
All the above aspects of mixing are valid for the mantle. However, we may reasonably
disregard the diffusional homogenization because of its extremely low rate, even at high
temperatures. Diffusion in the mantle is not able to destroy even a 1-km heterogeneity over
the span of the earth's age. In spite of deep-focused earthquakes indicating that the brittle
behavior is present in the upper mantle, and that subducted oceanic crust can be broken into
relatively small segments by the subduction process (Yasuda and Fujii 1998), the importance
of breakup for mixing is small. The surface tension of mantle materials is extremely small,
considering the high viscosity of the mantle rocks and scaling. It starts acting on a scale much
smaller than that under consideration. Thus, stretching is the most important factor in mixing
processes in the upper mantle. This kind of understanding of the mantle "mixing" is similar to
the idea put forth in Hoffman and McKenzie (1985). In this context, the mixing properties of
the system are entirely defined by the patterns of the velocity field and its dynamics. In other
words, we are dealing with kinematic mixing.
Hoffman and McKenzie (1985) analyzed a 2-D Newtonian mantle convection and established
that lateral spreading of heterogeneity is proportional to the square root of the product of the
eddy diffusivity and time. They estimated the time of doubling of material interfaces as
Ra-0.51. Based on these findings the authors concluded that whole-mantle convection would
probably homogenize the earth's mantle on a time-scale of 1 Ga, whereas upper-mantle
convection requires only 200 to 300 Ma for homogenization. Not only the characteristic
velocity but also the style of the convection was recognized as affecting efficiency of the
mixing (Christensen 1989, Hansen et al. 1992). Three-dimensional stationary convection in a
Newtonian fluid with the infinite Prandtl number was found to preserve particle heterogeneity
on a time-scale of 200 Ma for the upper mantle (Schmalzl et al. 1996), consistent with the
2-D estimations. All these results rely on Newtonian rheology, whereas the rheology of the
upper mantle is essentially non-linear (e.g. Karato 1997). It is unclear, however, whether
rheology plays a significant role in the mixing. The visualization of mixing patterns in
Newtonian and non-Newtonian convection (Ten et al. 1996) showed significant differences in
the convection styles of the two rheologies. Such a difference results in different fractal
behavior (Ten et al. 1997) and structural evolutionary behavior (Ten et al. 1997). In spite of a
more chaotic character, non-Newtonian convection inhibits mixing to a greater degree than
with Newtonian rheology.
Although many properties affect the formation of the convective velocity field, we have
concentrated our investigation on a rheological comparison. In other words, we have
analyzed the difference in mixing due to differences in the two rheologies for a similar degree
of convective vigor. We have used both line and field methods for the quantification and
comparison of mixing properties.
Thermal Convection Model
To model upper-mantle convection we used a 2-D, basally heated, time-dependent model.
The Prandtl number is taken to be infinity. The model combines the stream function
formulation with a highly accurate fourth-order spline-based method (Naimark and Malevsky
1987, Malevsky and Yuen 1991 ) . Both non-Newtonian temperature-dependent and
depth-dependent and Newtonian temperature-dependent and depth-dependent rheologies
were considered. Rheology for the non-Newtonian model was close to olivine. This rheology
establishes the power-law dependence between stress and strain rate:
is a strain rate tensor,
is a stress tensor, denotes the second invariant of the stress
tensor, T is temperature, z is depth, A is a material constant, and B and C define temperature
and depth dependencies. The power-law exponent n is equal to 3. The Newtonian rheology
takes the same form of the rheological equation except that the power-law exponent n equals
1. Extensive references on mantle rheology can be found in Poirier (1985) and Karato (1997).
To ensure the correctness of the comparison of the two mixing scenarios, approximately the
same Nusselt number (Nu ~ 20) and all other parameters have been maintained except for the
rheology. The constants B and C have been chosen in order to obtain a
temperature-dependent viscosity contrast of about 300 and a depth-dependent viscosity
increase of about 10 across the layer. The effective Rayleigh numbers are about 3x106 for
both cases, thermal expansivity linearly decreasing by a factor of 1/3 across the layer. The
aspect ratio of the box (width/depth) is 2.
Free-slip boundary conditions have been imposed for the stream function. Dimensionless
temperature has been kept at 0 and 1 on upper and lower boundaries respectively, while left
and right sides of the box maintained symmetrical boundary conditions.
These two models have been used to generate time-dependent velocity fields for mixing
analysis. Since all the effective parameters for the models have been the same, we would
expect that the observed differences in mixing are mainly due to the rheological influences.
Monitoring Techniques
We used two methods to monitor mass advection in the flow. The first is known as the field
method. A passive, horizontally stratified color field (x) is initially placed in a 2D domain
(Fig. 1). The horizontally-stratified field for the initial state of the models has some
geophysical validity. However, a number of other initial states could have been chosen,
probably with equal validity – blob markers from bottom to top (e.g. Hoffman and McKenzie,
1985), a small cube near the bottom (e.g. Schmalzl et al. 1996) or many other variations.
Fig. 1 Initial horizontally stratified color field
The field being advected by convection portrays the flow in the domain. An injection of dyes
of different colors is a common technique used in flow visualization in laboratory
experiments. The field method is a mathematical analogue of this technique. The evolution of
the color field (x, t) is driven by the velocity and governed by a partial differential equation:
(1)
where x is a coordinate vector and t is time. This equation states that there are no mass
sources/sinks and reactions in the system. In other words, the scalar field (x, t) can only be
changed by advection, and the evolution of (x, t) is dictated by the velocity field V(x, t).
However, the straightforward solving of Equation 1 has a distinct drawback. The finite
difference approximation of the equation consists of spatial derivatives of V(x, t) and (x, t).
While velocity derivatives could be estimated and are bounded, the evolution of (x, t) is not
so easily predicted because of its hyperbolic character (Sethian 1996). In general, gradients in
(x, t) should become greater with time. Material points, that were initially far apart, can
approach close to each other, thus resulting in the very high gradients in (x, t) and
consequently increasing the approximation error. The higher the gradient, the greater the
error. An unfortunate outcome is that derivatives of (x, t), which are responsible for an
approximation error, cannot be estimated in advance and do not have bounding values.
Visually, this kind of error is manifested by mass flux between different colors. For instance, a
close contact of blue and red parts results in the artificial changing of some points to green.
Another type of error developing along the sharp gradients is the Gibbs instability, which
involves sharp spatial oscillations of the field.
This problem may be partially resolved with the Lagrangian formulation of the evolutionary
equation (Eq. 1). In this case, the equation may be rewritten as:
(2)
where D/Dt is a substantive derivative.
Mathematically, both equations are equivalent, but in practice there are some differences.
Lagrangian formulation (Eq. 2) may drastically improve the accuracy of a numerical solution.
However, solving Equation 2 requires much computer memory and high computational
speed. Since many time-steps need to be integrated, the price for the accuracy becomes very
high. The middle-ground could be the semi-Lagrangian method. The idea of the
semi-Lagrangian method is to solve the equation using a Lagrangian formulation for only two
(or any limited number) of sequential time-steps (e.g. Malevsky and Yuen 1991, Malevsky
and Thomas 1997). Using the velocity field, we may backtrack the location of a material point
at every node of an Eulerian grid. In other words, we may find the position of the material
point from the previous time-step along with the value of the field in this point and assign that
value to the current node. To perform this operation, we need a velocity field and scalar field
at the previous time step (x, t– t) (see the animation in Fig. 2).
Fig. 2 Characteristic-based method. A value
of the field from previous time-step is
advected to the current time-step. A ray is
emitted backward from a grid node on the
current time-step to locate coordinates and
the value of the field for the previous
time-step
On each node of the Eulerian grid, the following algorithm is applied:
·
·
1. Take the coordinate (t) of the node (i, j) of the current time-step t.
Find the coordinate (t–dt) of the material point currently located in the node (i, j) on
·
·
the previous time-step (t– t).
Find the value of the field in that coordinate [ ( (t– t), t– t)].
Assign the value found to the node (i, j) [ ( (t), t) = ( (t– t), t– t)].
The method is also referred to as the characteristics-based method (Malevsky and Yuen
1991), because finding the coordinate on the previous time-step is similar to launching the
characteristics backward with time:
(3)
We used the fourth-order explicit Runge-Kutta method to integrate Equation 3 forward in
time. This technique separates a temporal approximation error from a spatial approximation
error. The third step is the spatial interpolation, which can be carried out using any convenient
and reasonable method. Although any interpolation method has the approximation error, it is
still much easier to implement a high accuracy interpolation than to solve a high-order
approximation to a differential equation. To interpolate the field at a given point we employed
a bi-cubic spline approximation (Ahlberg et al. 1967, Naimark et al. 1998). The great
advantage of the field method is that it gives a uniform information density over the entire
system (e.g. Zia et al. 1998). The whole computational domain is covered uniformly by a
color field representing deformation in any part of the system. Such a global behavior is not
manifested in particle-based methods.
The Lagrangian tracer (particle)-based method represents another type of tracing method.
It simulates, for example, aluminum particles, which are often used to visualize flows in
laboratory experiments. This is a straightforward method, which requires solving one vector
differential equation:
(4)
where x denotes the vector of particle positions. The method does not have the pitfalls of the
field approach. It solves the same characteristic equation but does not require interpolation.
The accuracy of the method is independent on how close the particles approach each other.
The absence of interpolation and the related errors greatly improves the overall accuracy of
the method. However, it is not a good all-round method. The particles, positioned initially as
blobs, become very dispersed with time, thus making the estimate of the mixing very
approximate. The information density is very uneven because particles cover only a small
fraction of the area being investigated. Such a problem could be partially overcome with a line
method.
The line/contour modification is the logical extension of the particle method. Each line is
represented by a set of up to one million particles (Ten et al. 1998). Larger numbers of
particles on the line allow the resolution of smaller-scale flow features. Insufficient numbers
of particles can cause degradation in the accuracy (Franjione and Ottino 1987). Although the
original particle method does not require interpolation, the line method does. Due to the
highly irregular deformation field, some parts of the lines are stretched much more extensively
than others. Numerical accuracy, which is proportional to the linear density of the particles
along the line, degrades much faster in the stretched areas. As a result, in order to maintain
uniform accuracy, the particles need to be redistributed along the lines. Another reason for
redistribution is that some parts of the line are straight, while others may be strongly bent. A
straight line may be efficiently restored with two known points. The same length but a bent
line requires more points for a proper representation. In general, the number of particles
should be proportional to the local curvature of the line (e.g. Dritschel 1988).
We used the fourth-order implicit Adams-Moulton method with variable time-step for
integrating Equation 4. Particles are distributed over the line according to a density function
that is proportional to the curvature. Although dynamic redistribution induces interpolation
errors, it drastically increases the overall accuracy because the bent section of the line is
represented by many particles.
Line analysis technique. Dynamics of lines show how mass is being transported in a system.
However, the dynamics give us no quantitative estimation for mixing. In order to retrieve
mixing properties from the line dynamics, we employed a line analysis technique. Assume for
now that heterogeneity is an uneven distribution of some marker (particles or field). If a
marker layer or dense cluster of particles is set in a calculation domain, the heterogeneity of
such a system will be the highest possible, since the layer or the cluster is compactly localized
in space. As the system evolves, the marker (akin to the dye movement in laboratory
experiments) is distributed over the entire domain. Therefore the strength of the heterogeneity
degrades because the distribution function of the marker layer tends to the homogeneous
distribution. We employed a unique line-based method to quantify the heterogeneity of a
system. Our marker is represented by a horizontal layer, which is bounded by two lines and
box sides. Its thickness is one-tenth depth of a calculation domain. If we subdivide the box on
horizontal levels and find the fraction of the marker layer hosted by each level, we may draw a
depth distribution function from this data (see right panel of Fig. 3). Peaks on the plot show
the depths where the layer mass is concentrated. The marker layer is deforming, changing its
position, which induces the dynamical behavior of the distribution. Thus, the distribution over
the depth evolves also dynamically. In other words, a new depth distribution for the layer is
generated at each time-step. These distributions assembled along the time-axis build a mixing
surface (or map) on the depth--time plane. High terrains on this map indicate comparatively
high concentration of the marker in corresponding time--depth points. A flat landscape on the
map indicates a homogeneous distribution of the marker over the depth.
Fig. 3 Mass distribution analysis. Mass of
the marker layer is calculated at each
monitoring level. Data represent distribution
of the marker layer over depth (right panel)
Results and Discussion
Fig. 4 Newtonian mixing as monitored by the field
method. The mixing is characterized by a smooth
two-cell motion. Initial color field is shown in Fig.
1(click for animated GIF, 1.9 MB)
Fig. 5 Non-Newtonian mixing as monitored by the
field method. The mixing is erratic. Plume events
essentially affect mixing. During these events,
mixing occurs in narrow localized zones (click for
animated GIF, 1.7 MB)
Field. Figures 4 and 5 show the evolutions of the marker fields in the Newtonian and the
non-Newtonian convection respectively, as they are portrayed by the field method. The field
method is very suitable for visualization, but the presence of high gradient zones makes
qualitative analysis inaccurate. In some places on the figures one may find numerical
instabilities (the Gibbs oscillations) due to the high gradients. We used a grid with up to
3000×1500 nodes for monitoring the field. The actual calculations were performed in spline
space on a 1000x1000 grid. The resolution of animations (Figs. 4 and 5) was reduced in order
to fit with the requirement of this electronic publication. As can be seen from the animations,
the Newtonian system evolves much more smoothly than the non-Newtonian. The
non-Newtonian mixing evolution mainly takes place during occasional plume events. The
smooth cellular-type motion of the Newtonian convection results in different mixing
properties as compared to the chaotic, localized mixing of the non-Newtonian convection.
Figure 6 shows the evolution of the local [in scale (S) and time (t) spaces] fractal dimensions
(D) for the non-Newtonian (left panel) and the Newtonian (right panel) mixing.
Fig. 6 Fractal dimension as a function of the scale
and time. Scale is the size of the box used to
calculate the fractal dimension by the box-counting
method. Left panel presents an evolution of fractal
dimension for the non-Newtonian convection. Right
panel shows the fractal dimension for the
Newtonian convection
The scales represent a measure of the size of the box used in the determination of the fractal
(Ten et al. 1997) and are related to the spatial resolution. Thus, the figure depicts fractal
dimensions on different scales at different times. Both figures are characterized by the two
blue regions. The lower-left blue area represents non-mixed scales, while the upper-right blue
area represents a well-mixed area. The green-to-red transition zone reflects the state of
mixing. A mixing process starts with the blue unmixed stage with a fractal dimension of 2.
Then the convection starts destroying and building new scales which change the fractal
dimension to the lower value of 1.6 to 1.8. After that, the medium becomes mixed and its
fractal dimension reaches a value of 2 again (Ten et al. 1997). As can be seen from the figure,
both the Newtonian and the non-Newtonian convection destroy large scales (top parts of the
figures) very efficiently. It takes a relatively short time (the green–red band) to homogenize
the large scales. However, at smaller scales the efficiency of the mixing is decaying in both
cases. As a result, the smaller scales require more time for the homogenization and the
green--red band becomes wider. Moreover, for the log(S) less than 4 the efficiency degrades
to such a level that it seems not to be capable of homogenizing the smaller scales further.
In spite of some resemblances, the Newtonian mixing differs markedly from the
non-Newtonian mixing. While the Newtonian convection exhibits smooth propagation of
transitions through the various scales with time, the non-Newtonian convection demonstrates
sharp jumps and drops. Such behavior is characteristic of the non-Newtonian convection due
to the highly chaotic nature of this kind of flow (Malevsky et al. 1992, Larsen et al. 1995,
Larsen et al. 1996). Figure 6 indicates also that the green--red band is wider or, in other
words, the non-Newtonian mixing requires more time to mix.
This way of using fractals to characterize the degree of homogenization is also employed in
astrophysics (e.g. Wu et al. 1999). Another method for quantifying the complexity of a fractal
field is proposed by Rodriguez-Iturbe and co-authors (1998) in hydrology. It may be a viable
alternative to the measuring of fractal dimension.
Lines. The lines method delivers a much higher accuracy than the field method. We placed
ten horizontal lines (each with 106 particles) at different depths (from red at the top to blue at
the bottom; Fig. 7).
Fig. 7 Initial positions of lines
This high resolution is capable of resolving very sharp and hardly deformed shear zones (Figs.
8 and 9) without any line crossing, which would indicate inaccuracies in the calculation. We
note that the one million particles per line means that the separation between any two
particles initially is 1.3 m when scaled to the upper-mantle system. This number suggests that
it would be very difficult to obtain a resolution of 1 to 10 km in 3-D mixing, but that
resolution of O(102 km) should be feasible with 5 to 10 million particles distributed over the
whole mantle.
Fig. 8 Newtonian mixing as monitored by lines
(click for animated GIF, 2.9 MB)
Fig. 9 Non-Newtonian mixing as monitored by lines
(click for animated GIF, 1.9 MB)
The analysis (Fig. 3) applied to the lines reveals that the mass of the marker layer is dispersed
within a very short time in the Newtonian convection. We analyzed one marker layer located
at a depth of 0.5 (about 300 km for the upper mantle). A mixing map (Fig. 10) shows how
the mass of the monitoring layer is distributed over the entire depth with time. From our
previous results, the Newtonian convection in the upper mantle makes the marker layer
statistically indistinguishable after less than 300 Ma (Fig. 10). Scaled for the upper mantle, a
dimensionless time of 0.01 corresponds to around 1 Ga.
Fig. 10 Mixing map depicts spatial-temporal
evolution of Newtonian convection. Both surface
(upper panel) and contours (lower panel) are
shown. Data show evolution of marker layer
distribution with time. Higher terrains represent
higher concentration of the layer in depth--time
space. Convection disperses the layer for a short
time
In contrast to common expectations but in accordance with the fractal dimension behavior
(Fig. 6), the non-Newtonian medium is less efficient in mixing and preserves the unmixed
material even on a scale of 1 Ga (Fig. 11). The mass of the marker layer is concentrated in the
upper part of the convection box and remains stable. Even after 1 Ga the significant part (20
to 30% of the mass) of the marker layer is located at a depth of 0.2 (about 120 km) near the
top. This type of analysis is structurally selective, i.e. it is mainly directed towards recognizing
horizontal structures, whereas the vertical structures may be left undistinguished.
Fig. 11 Mixing map of non-Newtonian convection.
Both surface (upper panel) and contours (lower
panel) are shown. Data show an evolution of depth
distribution with time. Higher terrains represent
higher concentration of marker layer in depth--time
space. The marker layer resists mixing. It remains
distinguishable for at least 1 Ga as scaled to the
upper-mantle convection
However, the same technique (Fig. 3) may be applied to the analysis of the vertical structures.
We used the same horizontal marker layer, but the monitoring levels were positioned
vertically. Thus, this modification shows the accumulation of the material of the marker layer
in vertical columns. Analysis of the vertical structures reveals another type of behavior (Figs.
12 and 13). Now the non-Newtonian medium does not display any more significant vertical
heterogeneity (Fig. 13), while the Newtonian convection still does (Fig. 12).
Fig. 12 Mixing map of Newtonian convection.
Analysis of vertical structures. Both surface (upper
panel) and contours (lower panel) are shown. Data
show an evolution of depth distribution with time.
Higher terrains represent higher concentration of
marker layer in depth--time space. Position of
marker layer was the same as for horizontal analysis
(Fig. 10). The Newtonian convection forms stable
vertical structures
Sporadically, the Newtonian convection forms vertical columns. They may last several
hundred million years. These structures are less stable and are not so sharp and distinct
compared to the non-Newtonian medium. Thus, the structural properties of the mixing differ
as well. The Newtonian convection builds up vertically orientated structures, while the
non-Newtonian flow tends to form horizontal heterogeneities. Mixing in the non-Newtonian
medium is less efficient in comparison to the Newtonian medium. One can probably notice
this difference in the mixing from the animations taken from the field approach (Figs. 4 and
5). Similar results were observed in the laboratory experiment for viscoelastic medium
(Niederkorn and Ottino 1993).
Fig. 13 Mixing map of non-Newtonian convection.
Analysis of vertical structures. Both surface (upper
panel) and contours (lower panel) are shown. Data
show an evolution of depth distribution with time.
Higher terrains represent higher concentration of
marker layer in depth--time space. In contrast to
Newtonian convection, non-Newtonian convection
develops no stable vertical structures
Length. Stretching is one of the most important aspects of the mantle mixing. One can easily
monitor the stretching with the line method. The length of the line represents an integral
characteristic of stretching along the Lagrangian interface. It yields more accurate
information, since it represents the naturally averaged deformation, which can be strongly
spatially dependent. The dynamics of the lengths are shown in Figure 14. The lengths evolve
in a very similar way for all the lines (independently of the initial depth). This means that the
mixing is more or less uniform along the depth. The Newtonian case (left panel of Fig. 14)
exhibits more effective stretching than the non-Newtonian (right panel of Fig. 14). The
exponential stretching in the Newtonian model fits with previous observations (Hoffman and
McKenzie 1985, Christensen 1989). Similar mixing properties of the stretching statistics of
the length were observed for granular media (Shinbrot et al. 1999). The non-Newtonian
model exhibits a different rate of extension. It is less efficient and fits with a power-law with
time on the later stages of the evolution. After 1 Ga, the line is stretched by as much as about
100 times longer for the non-Newtonian convection and 300 times longer for the Newtonian
convection. The Newtonian medium is roughly three times more efficient in mixing than the
non-Newtonian medium. We may expect that effective velocity no longer scales as Ra0.51
(McKenzie et al. 1974) for a non-linear medium.
The smoothness of the Newtonian convection results also in a smooth stretching rate,
whereas the non-Newtonian stretching is step-like (left panel of Fig. 14). This is further
evidence of the strongly time-dependent plume-driven nature of the non-Newtonian mixing.
The abrupt and powerful extension events occur during the plume bursts. As a result, the
main mixing motion in the Newtonian convection is of cellular size, while for the
non-Newtonian we may expect the redistribution of the kinetic energy between large-scale
cell motion and smaller-scale plumes. Such a redistribution results in different scale-breaking
properties, and, consequently, in different mixing properties.
Fig. 14 Evolution of line lengths.
Extension rates in Newtonian
convection fit well with exponential
law (left panel), while
non-Newtonian convection exhibits
power-law extension rates (right
panel)
Physical reasoning. It was commonly held (e.g. Hoffman and McKenzie 1985) that mixing in
the Newtonian convection is much less efficient than in the non-Newtonian convection. It was
suggested that the more chaotic behavior of the non-linear system would result in better
mixing. Our results refute this common belief.
The explanation of this contradiction comes from consideration of the velocity spectrum. In
order to maintain a finite amount of energy in the system, it must decay with the horizontal
wave number (Fig. 15) regardless of rheology. In other words, the variation of velocity
becomes smaller over finer scales. On the other hand, the temporal variation of the velocity is
the strain rate. Thus the spectrum reflects the behavior of the strain-rate. The strain-rate
degrades on small scales, as does the stress. It is a key point for the non-Newtonian rheology
whose effective viscosity is stress-dependent. Smaller stress for smaller scales means a higher
effective viscosity (Fig. 13). Consequently, we may expect that the non-Newtonian rheology
would probably resist mixing over smaller scales. However, plume-driven dynamics may
affect mixing efficiency, enhancing it over the smaller scales. The combination of these two
mechanisms would probably result in a scale partitioning of mixing and higher mixing
resistance over the intermediate scales. This is an additional scale-dependent mechanism
affecting only the non-Newtonian mixing. In contrast, the Newtonian viscosity, which does
not depend on the stress, does not have this scale-dependent behavior.
Fig. 15 Hypothetical spectra for non-Newtonian
convecting medium. The velocity spectrum should
decay. Consequently, effective viscosity rate should
increase. However, the tendency of non-Newtonian
media to localize a flow may affect the actual
spectrum
Figure 5 shows that the highest rate of the non-Newtonian convection development occurs
during the plume event. In other words, the mixing in non-Newtonian media is driven by the
plumes in its significant part. Thus, we may expect strong influences from these types of
velocity field patterns on mixing. The cellular type mixing generates certain structural scales,
which are different from those generated by the plume dynamics. The presence of
multiple-scale generators for non-Newtonian convection may result in the multifractal style of
mixing (Fig. 6).
Fast-decaying velocity spectra (Malevsky and Yuen 1993) would imply that on a smaller scale
deformation would become negligibly small. Thus, mixing degrades with the reduction of the
scales (Metcalfe et al. 1995). This simply means that convection destroys the large-scale
heterogeneities very fast, but it requires much more time to destroy the smaller scales.
Three-dimensional convection shows a much stronger decay of kinetic energy spectra
(Malevsky and Yuen 1993). Hence, the efficiency of 3-D mixing might be much lower than
that of 2-D mixing, even with toroidal mixing. The question arises as to whether convection
may efficiently destroy scales of less than 10 km and how much time is required to do so.
Using the kitchen analogy, one cannot slice vegetables to get much smaller-sized pieces than
the knife-edge, but how sharp then is the convective knife-edge?
Conclusions
In this study we have described the various numerical techniques used in the study of mixing
and have displayed the visualizations derived from both the field and line-particle methods.
The following principal conclusions can be drawn:
·
·
·
·
The evolution of the fractal dimension shows that the non-Newtonian convection
remains more fractal-like and lasts for a longer time than for the Newtonian convection.
The Newtonian convection homogenizes the mass heterogeneities much faster and goes
to a fractal dimension of 2, much more readily than for the non-Newtonian convection,
whose fractal dimension lies around 1.6 for a long time.
Stretching of the line indicates a greater amount of deformation for the Newtonian
medium, whose line length grows exponentially with time. On the other hand, the
length associated with the non-Newtonian mixing grows according to a power-law in
time. The length of the line is three times longer for the Newtonian than for the
non-Newtonian mixing, showing a greater amount of stretching.
Horizontal layered structures are formed and are preserved by the non-Newtonian
medium, while for the Newtonian medium there is a tendency for vertically polarized
structures to develop.
Because of the nature of the stress-dependent rheology, mixing at intermediate scales
meets greater resistance in the non-Newtonian rheological medium. This
scale-dependent threshold will limit the efficiency of mixing in non-Newtonian
rheological medium. Toroidal motions in 3D will also meet this fate in non-Newtonian
situations.
Acknowledgement
The fractal dimension surface (Fig. 6) was constructed by Leeza Pachepsky. We thank Julio
Ottino for a stimulating conversation and Maarten A. Koenders and an anonymous reviewer
for constructive reviews. We are grateful for the highly valuable assistance provided by John
D. Clemens. This research has been supported by the EMSP program and the Geosciences
program of the Department of Energy.
References
Ahlberg JM, Nilson EN, Walsh JL (1967) The theory of splines and their applications.
Academic Press, London
Christensen U (1989) Mixing by time-dependent convection. Earth Planet Sci Lett 95:
382–394
Constantin P, Dupont TF, Goldstein RE, Kadanoff LP, Shelley M, Zhou SM (1993) Droplet
breakup in a model of the Hele-Shaw cell. Phys Rev E47: 4169–4181
Dritschel DG (1988) Contour surgery: a topological reconnection scheme for extended
integrations using contour dynamics. J Comp Phys 77: 240–266
Ferrachat S, Ricard Y (1998) Regular vs. chaotic mantle mixing. Earth Planet Sci Lett 155:
75–86
Franjione JG, Ottino JM (1987) Feasibility of numerical monitoring of material lines and
surfaces in chaotic flows. Phys Fluid 30: 3641–3643
Hansen U, Yuen DA, Kroening SE (1992) Mass and heat transport in strongly
time-dependent thermal convection at infinite Prandtl number. Geophys Astophys Fluid Dyn
63: 67–89
Hart S, Zindler A (1989) Constraints on the nature and development of chemical
heterogeneities in the mantle. In: Mantle convection: plate tectonics and global dynamics.
Gordon and Breach Science Publishing, New York, pp. 261–378
Hoffman NRA, McKenzie DP (1985) The destruction of geochemical heterogeneities by
deferential fluid motions during mantle convection. Geophys J R Astron Soc 82: 163–206
Karato S (1997) Phase transformation and rheological properties of mantle minerals. In:
Crossley D, Soward AM (eds) Earth's deep interior. Gordon and Breach, New York, pp.
223–272
Larsen TB, Yuen DA, Malevsky AV, Smedsmo JL (1995) Dynamics of thermal convection
with Newtonian temperature-dependent viscosity at high Rayleigh number. Phys Earth Planet
Inter 89: 9–33
Larsen TB, Yuen DA, Malevsky AV, Smedsmo JL (1996) Dynamics of strongly
time-dependent convection with non-Newtonian temperature-dependent viscosity. Phys Earth
Planet Inter 94: 75–103
Malevsky AV, Thomas SJ (1997) Parallel algorithms for semi-Lagrangian advection. Int J
Numer Methods Fluids 25: 455–473
Malevsky AV, Yuen DA (1991) Characteristics-based methods applied to infinite Prandtl
number thermal convection in the hard turbulent regime. Phys Fluid A(3) 9: 2105–2115
Malevsky AV, Yuen DA (1993) Plume structures in the hard-turbulent regime of
three-dimensional infinite Prandtl number convection. Geophys Res Lett 20: 383–386
Malevsky AV, Yuen DA, Weyer LM (1992) Viscosity and thermal fields associated with
strongly chaotic non-Newtonian thermal convection. Geophys Res Lett 19: 127–130
McKenzie D, Roberts JM, Weiss NO (1974) Convection in the earth's mantle: towards a
numerical simulation. J Fluid Mec. 62: 465–538
Metcalfe G, Bina CR, Ottino JM (1995) Kinematic considerations for mantle mixing.
Geophys Res Lett 7: 743–746
Naimark BM, Malevsky AV (1987) An approximation method for the solution of the problem
of gravitational and thermal instability. Comput Seism 20: 30–52
Naimark BM, Ismail-Zadeh AT, Jacoby WR (1998) Numerical approach to problems of
gravitational instability of geostructures with advected material boundaries. Geophys J Int
134: 473–483
Niederkorn TC, Ottino JM (1993) Mixing of viscoelastic fluids in time-periodic flows. J Fluid
Mech 209: 243–268
Ottino JM (1989) The kinematics of mixing: stretching, chaos, and transport. Cambridge
University Press, New York
Poirier JP (1985) Creep of crystals. Cambridge University Press, Cambridge
Rodriguez-Iturbe I, D'Odorico P, Rinaldo A (1998) Configuration entropy of fractal
landscapes. Geophys Res Lett 25: 1015–1018
Schmalzl J, Houseman GA, Hansen U (1996) Mixing in vigorous, time-dependent
three-dimensional convection and application to earth's mantle. J Geophys Res 10:
21847–21858
Sethian J (1996) The level set methods. Cambridge University Press, Cambridge
Shinbrot T, Alexander A, Muzzio FJ (1999) Spontaneous chaotic granular mixing. Nature
397: 675–678
Ten A, Yuen DA, Larsen TB, Malevsky AV (1996) The evolution of material surfaces in
convection with variable viscosity as monitored by a characteristics-based method. Geophys
Res Lett 23: 2001–2004
Ten A, Yuen DA, Podladchikov Yu Yu, Larsen TB, Pachepsky E, Malevsky AV (1997)
Fractal features in mixing of non-Newtonian and Newtonian mantle convection. Earth Planet
Sci Lett 146: 401–414
Ten A, Podladchikov Yu Yu, Yuen DA, Larsen TB, Malevsky AV (1998) Comparison of
mixing properties in convection with the particle-line method. Geophys Res Lett 25:
3205–3208
Wasserburg GJ, DePaolo DJ (1979) Models of earth structure inferred from neodymium and
strontium isotopic abundances. Proc Nat Acad Sci USA 76: 3594–3598
Wu KKS, Lahav O, Rees MJ (1999) The large-scale smoothness of the universe. Nature 397:
225–230
Yasuda A, Fujii T (1998) Ascending subducted oceanic crust entrained within mantle plumes.
Geophys Res Lett 25: 1561–1564
Zia R, Yuen DA, Cortese TA, Vincent AP, Balachandar S (1998) An approach to a visual
language for convective fluid flows: visualization of geophysical simulations. Electron Geosci
3:1
Online Publication: September 24, 1999
LINK Helpdesk
© Springer 1999