Pre-filtering Turbulent Vector Fields in the Geodynamo

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Pre- filtering of turbulent vector fields in the geodynamo
Patrick R¨
udiger∗Christopher Weber†Hiroaki Matsui‡Eric Heien‡Louise H. Kellogg‡
Bernd Hamann†Hans Hagen*
Figure 1: Image sequence showing emergence of turbulent magnetic field behavior. The highly turbulent region showing as a ”ring” aligned
with the Earth’s equator is emphasized in these volume visualizations via our new filter approach. Several ”columns” are visible as well,
indicating turbulent behavior in directions being orthogonal to the equator and tangential to the Earth’s surface
ABSTRACT
This paper introduces a new and effective approach for visualiz-
ing complex magnetic fields. The paper defines a local approach
for measuring the degree of local directional change of magnetic
field vectors and uses this measure to generate volume visualiza-
tions that emphasize areas with highest values of locally integrated
directional change of a vector field. The presented method was mo-
tivated by the need for having more effective tools supporting inter-
active exploration of the intricate magnetic field behavior captured
in extremely large simulated data of the Earth’s magnetic field.
Of particular interest is a better understanding of the causal rela-
tionship between the Earth’s convection phenomena (geodynamo)
and the induced magnetic field, with high turbulent behavior in the
Earth’s core. The introduced local filter-based approach for visu-
alization leads to a significant reduction in the amount of scientifi-
cally relevant data to be focused on in a rendering, and it therefore
accelerates the overall scientific exploratory process. The viability
of the method is demonstrated by specific examples.
1 INTRODUCTION
The Earth’s magnetic field is generated by the motion of liquid iron
alloy, so-called geodynamo. Numerical simulations of the geo-
dynamo, other planets, and solar dynamos, present an enormous
∗TU Kaiserslautern, Computer Graphics & HCI Group, Germany,
email:ruediger@rhrk.uni-kl.de
†UC Davis, Dept. of Computer Science
‡UC Davis, Dept. of Earth and Planetary Sciences
computational and visualization challenges. In geodynamo simula-
tions, we need to represent a vast length scale form thickness of the
boundary layer (0.1m) to outer core geometry (1000km) and time
scale from rotation of the Earth (1 day) to magnetic field reversals
(105years) of the convection and magnetic fields. Consequently,
geodynamo simulations requires extremely high spatial resolution
and long time integrations. For this purpose, geodynamo simula-
tions have been performed on some of the world’s fastest comput-
ers. The geodynamo represents a major visualization challenge,
as the output consists of time-varying vector and scalar fields rep-
resenting turbulent convection in the Earth’s core and the coupled
magnetic field generated by that flow. The number of fields to be
studied, the resolution required, and the long-time series makes ex-
traction of features very challenging. Moreover, the observation
used to compare against simulations is the magnetic field at and
above the Earths surface far from the outer core which is modeled
in the simulations, and we can only observe large scale magnetic
fields generated in the outer core.
The results from the simulations show that we have a highly turbu-
lent magnetic field in the inside (outer core) which stabilizes going
further away from the center. It is assumed that the field itself is
mainly induced by two main drivers, namely the α- and ω- effect
[2].
The simulation itself is run on a large grid, such that visualizing the
whole vector field as in Fig. 2, is lacking detailed insights of the
emergence of the magnetic field. Increasing the problem size fur-
ther to unsteady data, filtering comes more and more into account.
In our first draw we fetch these effects visually, deriving a filter
that is applied on the magnetic field (Section 2, Angular Direction
Changing Rate), resulting in a less complex scalar field which then
can be visualized with common techniques. This reduced complex-
ity then leads the way for further analysis and visualization tasks.
Our results are showing, that we can use our filter for example to

generate a ”negative” picture of the highly turbulent magnetic field,
thus visualizing the areas of the vector field that are only barely af-
fected by one of the above mentioned effects.
One of the main challenges from the application scientist we want
to tackle with our method described here, is to have a fast, reli-
able and easily interpretable visual output. Such that it enables us
to visually perceive the influence of changes in the simulations pa-
rameter settings.
Figure 2: Streamline visualization using and extended domain, con-
taining the simulation domain of the outer core and a potential field
for the magnetic field outside the simulation domain. Using 100
seed points placed in sphere around the center for streamline trac-
ing. Showing of the dipole structure of the magnetic field as also
known from physical experiments.
1.1 Related Work
First attempts for visualizing geomagnetic data were made by Ohno
et al. [7] and especially for the flow in the mantle by Schr ¨
oder et al.
[8]. A good overview on basic techniques of vector field visualiza-
tion was provided by Salzbrunn et al. [10], Brambilla et al. [1] and
Zhenmin-Peng et al. [9]. They showed and provided examples of
the most common techniques covering feature-based and partition-
based flow field visualization. We have combined a couple of these
techniques in the visualization of our data of interest, using user-
defined features and filters for a kind of partitioning of the data
without doing a full blown topological analysis. Marchesin et al.
[5] showed an interesting approach using streamlines in complex
datasets. They precompute a random set of streamlines and select
the shown streamlines depending on the actual view. When nec-
essary their method seeds new lines in empty regions. Although it
could help to identify regions of interest, we need to have more user
control for the complex vector fields in our field of application. A
different approach for dealing with crowded scenes was presented
by Mehran et al. [6], which is also mainly tailored to flow visu-
alization. Clearly, there exists a need for a more specific/adapted
approach. Our approach extends the one from Daniels et al. [3],
which also covers the problem of interactive feature processing, for
the use in geophysics visualization and our special needs. Finally
the visualization system we used is based on the system introduced
by Kreylos et al. [4].
2 METHOD DESCRIPTION
To reduce the overall complexity we first want to depict the
macroscopic behavior. But our simulation domain is limited to the
core and mantle. To visualize whats happening outside the domain
we can use a precomputed potential field as used in Fig. 2, leading
to an exponentially increasing size of the data set (We have 4 ∗106
vertices to describe only the outer core with an respective diameter
of 3,471 km compared to 6,371 km to only fetch the magnetic field
near the Earth’s surface, we would need an additional 2.5 times of
vertices or 10 times the number of voxels). To avoid this massive
scaling we are using the spherical harmonics analysis to determine
Figure 3: Depiction of the cell structure of the underlying data.
An unstructured grid is used for the description. The size of the
cells vary with increasing radius. As well when going from top
to bottom. In itself the structure is regular, which means the size
changes linear with the components (in spherical coordinates) and
finer in radial direction near the boundaries. The dataset covers
only a shell of a sphere, because the inner core is excluded from the
simulation.
the value of the magnetic field at any given location beyond
the Earth’s surface (see Equations 1 and 2). Whereas a single
evaluation of a point can be done quite efficiently, the evaluation
of multiple ones as needed for streamline visualization is getting
inefficient. Thus we are interested in drawing the minimum number
of streamlines to depict the overall structure. Therefor we are
deriving a filter that fetches areas of same behavior. Due to the fact
that magnetic field lines cannot penetrate each other, the separation
can be reduced to finding only the boundaries of these areas, as
seeding streamlines in between would reveal the same behavior.
B=−∇W,(1)
where Wis the magnetic potential. Wis described by a series of
Gauss coefficients gm
land hm
lby
B=
l
∑
l=1
l
∑
m=0
ReRe
rl+1
[gm
lcos(mφ) + hm
lsin(mφ)]Pm
l(θ),(2)
where Reis the respective earth radius and Pm
lare the normalized
Legendre polynomials. φ,θand rare the spherical coordinates.
2.1 Filter Definition: Angular Direction Changing Rate
The main idea behind this filter applied onto the magnetic field, is
to catch the effect of the main drivers, such it can be used to vi-
sualize only those magnetic field lines that are inducing the main
behavior. As we do not know directly how the α- and ω- effect is
defined, we focus on fetching the areas that are most probably ef-
fected by these or visa versa. This can be achieved by finding those
areas, that vary significantly from the rest. In our case we depict
the variation as the difference in the directions of the vector field
for a specific range around each vertex in the dataset. We used a k-
nearest-neighbor (kNN) approach for the range to avoid errors that
could arise through the chosen interpolation method. Our feature is
defined as:
ADCR(v) = ∑wi∗acos(<v,ni>
||v||||ni|| )(3)
where vis the current vertex we want to determine the variation
for, the nidepict the knearest neighbors, <0,0>denotes the scalar
product and ||∗|| the Euclidean norm for vectors. The differences of
the knearest neighbors are weighted with respect to their distances

using least squares.
wi=dist(v,ni)
∑k
j=0dist(v,nj), with (4)
dist(v,ni) = ||(ni−v)|| (5)
As we want to filter out those regions that are significantly different
from the others, we modeled this significance with the following
function:
f(ADCR) = eλ(ADCR−π)(6)
Following this approach, we have a smooth function which em-
phasizes high changes and discriminate the lower ones. Thus the
resulting value increases the more the angular differences reaches
the maximum of πand additionally norms the value range to [0,1],
which makes it easier for visualization and interpretation. We can
now choose λ, such that we fetch only the desired changing rates
and separate the regions of interest, see Fig. 4 to determine the λ
value.
Figure 4: Effect of the significance function on the resulting ADCR
value. αrepresents the calculated angle difference. λis chosen
according to the desired angle threshold. It is modified until the
desired effect can be visually distinct.
2.1.1 Efficiency
We briefly discuss the complexity of the proposed filter. The data it-
self is a subset of the original one, with only 107vertices, compared
to 1010. The dataset is arranged as an unstructured grid covering a
shell of a sphere, namely the Earth’s liquid outer core (see Fig. 3).
Looking at the mathematical definition there are two main drivers
for our complexity examination, the number of vertices in the se-
lected subset, now denoted as nand the number of neighbors taken
into account k. The computation itself is parted in five steps. First
we need to get the knearest neighbors. Second we compute the
acos. Third we compute the weights for each neighbor difference,
which mainly is a distance computation. Fourth we need to sum
up the results. Lastly we need to do this for all vertices in the sub-
set. For step (1) we precomputed a kd-tree, so that we can take an
average complexity of O(logN). We assume that the kd-tree is com-
puted for the whole dataset with Nvertices, because in most cases
we will use it for more then one subset, amortizing the initial costs.
Regarding the large number of vertices we have in total, we need
to keep in mind, that the kd-tree itself will also need a significant
amount of memory capacity. Step (2) and (3) are computed ktimes
per vertex. Assuming (2) and (3), (4) can be done efficiently we get
an average complexity for Step (5) of O(k∗n). Our experiments
have shown that choosing kin the range of [5,10]is sufficiently
enough to get valuable results. Thus the complexity of Step (5) can
be simplified to O(n). The overall complexity for the computation
of the ADCR is therefore O(logN) + O(n). Step (5) can be par-
allelized easily (splitting up the computation for multiple smaller
subsets of vertices) reducing the computation time significantly.
Figure 5: Full sized visualization of the magnetic field using ADCR
for seeding in all topological relevant areas including ADCR visu-
alization using threshold volume. View from equator.
3 RE SULT S
The most valuable result for the geophysics is shown in Fig. 5 and
Fig. 6d, turbulent areas are highlighted by the ADCR and can be
found in the equatorial plane and the area where he magnetic field
lines go to higher latitude. There are two important process in the
dynamo, namely the ω-effect and α-effect. The ω-effect is the ex-
tension of the magnetic field lines by the differential rotation. The
α-effect is the process in which magnetic field lines are twisted
by the helical flow (such as the convection columns). With typi-
cal methods for seeding and visualization it was very hard to depict
those effects. Fig. 6 provides a good comparison. First we tried
simple approaches for seeding streamlines. With exploratory seed-
ing based on a point location, the limits of a structural perception
reached their limits very fast. With a high resolution line source
we made some first progress in fetching interesting behaviors (see
Fig. 6a and Fig. 6b). As already known by the geophysics the
effects seem to appear in the equatorial regions and normal to it
(rotation axis). Using these insights, it was still hard to perceive
structural patterns in the magnetic field. The next approach was to
use the areas where vorticity’s z-component is vanishing (see Fig.
6c). This revealed some first patterns, as one can see that the field
lines are arranging nearly parallel and seem to have no significant
turbulence in this areas. We then introduced the ADCR to depict
areas of high turbulence to compare them against areas with low
turbulence. Using these two counterparts as input for our seeding
strategy we come up with a visualization that reveals a structural
pattern, that was not visible before. Fig. 6d and Fig. 5 shows how
our method helped to gain insight in a structural pattern that is cor-
related by the geophysics with the α-effect.

(a) Equatorial axis parallel (b) Rotation axis parallel (c) Vanishing vorticity z-component (d) ADCR
Figure 6: Comparing different strategies for seeding streamlines to visualize the magnetic field. All seeding is done by a high resolution line
source. Fig. 6a and Fig. 6b using regularly spaced lines parallel to the respective axis in the middle of the domain. Fig. 6c generated by
placing the lines inside the vanishing z-component areas. Fig. 6d seeding in high turbulent areas and in encapsulated low turbulent areas
(wholes in Fig. 5).
Regarding the large amount of unstructured data we were work-
ing with, our system approach tackles this problem efficiently. We
provide an approach that is highly concurrent. At first we are us-
ing our filter the angular direction changing rate from Section 2.1
to explore regions of interest. Afterwards we are using a volume
threshold to depict high turbulent areas and to analyze the impact
in all three dimensions. We completed the visualization of turbu-
lence with a multilayer isosurface. To further study the correlation
of the ADCR with the overall magnetic field we exploit the struc-
ture of the volume threshold visualization to seed streamlines for
the magnetic field and thus get an impressive output of the field’s
behavior with reduced complexity for further analysis tasks. As the
proposed method for the ADCR is highly concurrent, we are able
to reduce the time for analysis for the geophysics dramatically.
3.1 Application Scientists Insights
In the present visualization, intense ADCR mainly locates lower
latitude near the CMB. And, the magnetic field lines starting from
this area stay in the low latitude without going out from the higher
latitude. We consider that the the magnetic field in the turbulent
flow in the lower latitude is extended in the longitudinal direction by
the ω-effect in the area where ADCR is weak, and twisted towards
the higher latitude by the α-effect at the convection column with
larger ADCR. We expect that the large ADCR in the mid-latitude
represents the area where the magnetic field is twisted by the α-
effect because the dipolar magnetic field (or intense z-component)
is generated by the α-effect. In Fig. 2 one can see, that the equa-
torial plane induces a structural separation of the magnetic field,
which shows that the magnetic field line crossing the equatorial
plane is twisted to the zonal direction by the differential rotation.
In this study, we treat the magnetic field by ADCR, but this method
is applicable for the arbitrary vector field. We expect that we can
extract effects of the turbulence by specifying the length scale to in-
vestigate effects of the magnetic field generation by turbulent flow
in the Earth’s outer core. We also expected if we can extract what
length scale flow and magnetic field contribute the generation of
dipole component of the magnetic field.
Further we are now able to visually perceive changes in the sim-
ulations parameter settings, which were very hard to track before
due to the complex behavior of the vector field. Additionally we
could now study the effect of these parameters separately for the
α-effect. We look forward to find a similar method for the other
effects as well.
4 CONCLUSIONS
We have introduced a filter that effectively reduced the complexity
for visualizing the Earth’s magnetic field. We have shown that our
approaches allow a scientist to rapidly explore a highly complex
simulated magnetic field data set, making it possible to quickly
recognize regions in the field characterized by interesting and
possibly not yet fully comprehended behavior. Our filter should be
a viable tool for geophysicists for a variety of relevant purposes,
including a deeper understanding concerning the influence of a
simulations input parameters, the formulation of new scientific
hypotheses, and the identification of potential flaws in a specific
implementation of a simulation method.
We have shown that the proper use of well established visualization
techniques already provide a good first visualization, but we
have also shown that it lacks of appropriate filters especially for
visualization. Typical flow field approaches do not satisfy this need
and for more advanced magnetic field visualizations, which leads
to the need of new approaches for visualizing 3D vector fields
without motion.
5 FUTURE WORK
We only used a reduced part of the whole simulation here, but we
also want to support the large data sets by the massively parallel
simulation which is 1000 times larger than the present study. Addi-
tionally we want to support the extraction of a specific length scale
of fields in this approach by choosing the range of neighbors taken
into account, such that we can dynamically reduce the complexity
of the data set during visualization. Currently the seeding is done
manually regarding visual detected patterns. We look forward to
automate this seeding procedure and this way to find a possibility to
describe them analytically reducing the overall effort for computa-
tion. We did not cover time series yet, but looking at the emergence
of detected patterns overtime as temporally evolving behavior is of
major interest for the geophysics in the near future.
ACKNOWLEDGMENTS
This work was supported in part by National Science Foundation
awards EAR-0949446 and IIA-1125422.
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