(class project, UW Geophys
501, Autumn 2001)
(a little simplistic, but hey, it was first year. plus it's
fun anyway...)
PROPOSAL
Numerical Modeling to Verify the Linear Diapir Hypothesis
for the Formation of Ice Double-Ridges on Europa
Andy Ganse, APL-UW, 18 Dec 2001
Introduction & Background
I propose to develop and explore a numerical model to
further investigate the linear diapir hypothesis described
in Head et al. 1999, for the formation of the ice
double-ridges on Europa, one of the moons of Jupiter. This
model will be compared to actual 3D data of Europan ridges
estimated from photoclinometry as part of the evaluation of
the model. Ideally the results will offer quantitative
evidence to support or reject the linear diapir hypothesis,
described only qualitatively in the above article, and
provide constraints on the thickness of the ice layers
covering the suspected ocean on Europa. Also, I hope that
it will serve as a catalyst for similar modeling to explore
the several other double-ridge formation hypotheses (see
Pappalardo et al. 1999).
Europa is one of the four largest satellites of Jupiter,
called the Galilean satellites (discovered by Galileo in
1610). Spectroscopy shows that Europa's bright white
surface is predominantly water ice, and orbital and
gravitational measurements yield a density profile of the
moon in which an iron core and silicate mantle are
enveloped by a 100km thick surface layer of water (Anderson
et al. 1998). From telescopes and space probe photos it is
visually clear that the outer, exposed surface of this
water layer is frozen. But because the various phases of
water all have similar densities it is not clear at this
point whether the entire 100km water layer consists of ice
(and of what phase), liquid water, or some combination of
these. It is the hint of an immense liquid water ocean on a
world besides Earth, and the possibility of
extraterrestrial life in that ocean, which entices many
scientists in the exploration of Europa. A Europa-specific
orbiter and a landing probe are planned for the
indeterminate future; in the meantime we are relegated to
analyzing photos and spectral and magnetic measurements for
clues about the structure and formation of the surface
water layer. Investigating surface features such as these
double ridges is interesting in its own right, but in
particular it contributes to the indirect exploration of
Europa's interior and hence the search for a Europan ocean
and the possibility of extraterrestrial lifeforms.
Photos returned from the Voyager space probes in the 1970's
and the Galileo space probe in the 1990's through the
present show numerous large cracks in Europa's surface ice
bounded with large ridges on either side (Figure 1). These
ice cracks are arguably the largest and most noticeable
feature on the surface of Europa, significantly larger than
ice cracks or ridges on Earth (some as large as 300m high
and 2.6km wide), and this double-ridge shape is rarely
found on Earth. Consequently these Europan ice ridges have
inspired a number of hypotheses about processes which form
them and about how these processes constrain the thickness
of the surface ice of Europa.
Figure 1. Picture PIA00589 of
Europan ice ridges from the JPL Planetary Photojournal
Database. This photo, taken from the Galileo space probe in
1997, shows an area of about 14 kilometers by 17
kilometers. The main ridge in the picture is approximately
2.6 kilometers wide and 300 meters high. (Courtesy of
NASA/JPL/Caltech.)
Existing hypotheses for processes forming the double-ridges
include the following (Pappalardo et al. 1999):
1.) Volcanism, in which liquid water is explosively ejected
through fractures and repeatedly deposited in growing
linear debris piles along the fractures.
2.) Tidal squeezing, in which liquid water or warmer ice
from below is racheted up through tidal stress fractures in
the surface layer of cold, brittle ice during each tidal
cycle to form growing linear debris piles along the
fractures.
3.) Linear diapirism, in which frictional heating from a
tidal stress fracture causes an increased buoyancy in a
warmer, ductile layer of ice below. This pushes up the
cold, brittle, surface ice on either side along the length
of the fracture resulting in tilted slabs that form current
double ridge shapes after mass wasting.
4.) Compression, in which tidal compression deforms a
warmer, ductile layer of ice below more than the cold,
brittle, surface ice. This pushes up the surface ice along
the length of a fracture, resulting in tilted slabs that
form current double ridge shapes after mass wasting.
5.) Wedging, in which liquid water from below flows into
tidal stress fracture, freezes, and creates a wedge against
which tidal compression pushes, deforming the surface ice
upward and outward into double ridges along the fracture.
These hypotheses involved little or no numerical work and
some of the authors have even suggested that numerical
modeling needs to be done to help substantiate or discredit
the hypotheses. The aim of this proposal is to conduct
numerical modeling for the linear diapirism hypothesis.
Turtle et al. have created a finite element model for the
wedging hypothesis (Turtle et al. 1998); future modeling of
the remaining other hypotheses would allow for a better
comparison of the various ridge formation hypotheses.
Description of Model
A key point of the linear diapirism hypothesis is that it
addresses a number of morphological features connected with
the double ridges. The double ridges themselves are
generally linear, long, and quite consistent in their
dimensions along the length of the ridge. From space probe
photos they have been measured to reach up to 300 meters in
height, two and a half kilometers in width, and thousands
of kilometers in length, having a side slope as steep as 12
degrees. There are dips on either side of the double
ridges, called marginal troughs, which on 200m high ridges
have been estimated by photoclinometry to be 10-20 meters
deep. Just outside these marginal troughs are often found
thin fractures that run roughly parallel to the ridge. The
slopes of the ridges show evidence of mass wasting, with
chutes where material has sloughed off. Also seen on the
slopes are areas where the background ridges, overlaid with
a new ridge, continue up the side of the new ridge.
Examples of all these features can be seen in Figure 1.
Head et al. compares these features to smaller geologic
features on Earth caused by salt diapirism, and points out
that this diapirism on Europa would explain other Europan
surface features including elliptical pits and domes. He
suggests a two-layer model in which a cold, brittle layer
of ice rests on top of a warmer, ductile ice layer. Stress
fractures induced by diurnal tidal stresses and
nonsynchronous rotation tidal stresses may reach into the
ductile layer, and the friction along these fractures heats
the surrounding ductile ice, which becomes more buoyant and
pushes up the brittle surface ice slabs along the fracture.
Rim synclines, folding caused by lateral forces from the
initial opening of the crack and topography of the ridge,
and flexure from the ridge load are all explanations
pointed out for the marginal troughs under this model.
Finally, during the deformation processes mass wasting of
the steep ridge slopes would be expected as would the
existence of previous ridges on the new slabs being lifted
and deformed.
While an intricate numerical model that exactly fits the
measured topography may not be appropriate at this early
stage when so little is known about Europa's environment, a
rough model based on the physics of the linear diapir
hypothesis would be useful whether it turned out to match
the measured topography or not. A match would support the
linear diapir formation hypothesis, or if there is a poor
match, this may suggest the hypothesis might require
rethinking. In order to describe the features of the
proposed numerical model, let us first look at a simple
version and a few results from it.
Figure 2 shows a simple cantilever plate suspended with
both its top and bottom surfaces free of normal and shear
stresses. There is no gravity and no contacting material to
apply stress to the plate; the only external force acting
on this cantilever is the edge force pushing up at the very
end of the plate. We can set up a mathematical description
for this system in MKS units and solve for the displacement
of the plate, relying on the assumption that the thickness
of the plate is significantly smaller than its length.
Figure 2. A very simple model
for plate deformation to aid in discussing modeling
approaches to substantiate the linear diapir hypothesis for
Europan ice ridge formation.
First we obtain a force moment from the upward edge force P
at the end of the plate:
ML = Px
Next, if we assume that the plate is elastic with flexural
rigidity D then we can describe the force moment in a cross
section of the plate in terms of the place's displacement:
We relate these two by a
balance of moments, ML + Mt = 0, and obtain:
given the following definition
for D with respect to the elastic modulus M, Young's
modulus E, Poisson's ratio v, and thickness t of the plate.
We can integrate the second
order differential equation above twice to obtain the
displacement u(x) of the plate in response to the upward
edge force, a cubic equation:
Then we can plug in reasonable
values for E, v, P, and t, and plot it as seen in Figure 3
as seen below.
Figure 3. Plot of the
solution for plate displacement u vs range x, given values
of E=9.33*109 Pa, v=.325,
P=5*107Pa,
t=35m
The small model here is of course an extremely simplistic
one. We do see that this plate bends upward in response to
the edge force, with a height and slope of the order
described in Head et al. 1999. The addition of a
gravitational component may add a response in the
displacement curve that matches the marginal troughs,
similar to the response seen on the ocean floor as it rises
first before descending into a subduction zone. All the
variables except the edge force are based on “reasonable”
values - Young's modulus E and Poisson's ratio v were taken
from Chuang (2001) for polycrystalline ice at -16ºC, and
ridge length L and thickness t are set to match the
dimensions discussed in Head et al. 1999 regarding the
linear diapir hypothesis. The upward edge force of 5*107
Pa, on the other hand, was merely chosen to make the
displacement in the plot comparable to that in the measured
ridge, and seems rather large. But as a single point force,
presumably it stands in for a spread-out series of more
realistically valued buoyancy forces caused by heat
diffusion emanating from the fracture.
This heat diffusion would create both horizontal and
vertical gradients in the buoyancy forces, significantly
complicating the model. Other factors that would add
insight to the model include exploring the effects of
viscoelastic creep and analyzing the stability of the ridge
slope (that is, at what points in its curvature it breaks,
limiting which curvature solutions would remaining
standing). In any case, keeping the model physically based,
rather than attempting to merely fit it to measured data,
is key.
The resulting model's displacement curves, given various
tunings of its parameters, will then be compared to
estimated 3D data of several Europan ridges, compiled by
NASA/JPL/Caltech via photoclinometry of space probe photos.
A successful model that matches the linear diapir formation
hypothesis as well as the measured topography data would
provide a platform for exploring parameters such as ice
layer thickness and ice rigidity. If the model does not
make for a good match between the linear diapir formation
hypothesis and the estimated 3D data, then this information
is also important as it helps in the narrowing down of
applicable formation hypotheses. Ideally all the formation
hypotheses might be explored numerically in the
intermediate future, in separate projects, so that they can
be better compared and evaluated.
References
Anderson, J. D., G. Schubert, R. A. Jacobson, E. L. Lau, W.
B. Moore, W. L. Sjogren, Europa's Differentiated Internal
Structure: Inferences from Four Galileo Encounters,
Science, 281, 2019-2022, 1998.
Chuang, F.C., P. H. Figueredo, R. L. Kirk, and R. Greeley,
Europa's 'Mitten': Estimate of Ice Thickness Using an
Elastic Plate Model, Lunar and Planetary Science XXXII,
2001.
Gaidos, Eric J. and Francis Nimmo, Tectonics and water on
Europa, Nature, 405, 637, 2000.
Head, James W. and Robert T. Pappalardo, Europa:
Morphological characteristics of ridges and triple bands
from Galileo data (E4 and E6) and assessment of a linear
diapirism model, J. Geophys. Res., 104, 24223-24236, 1999.
Hobbs, Peter V., Ice Physics, Clarendon Press, Oxford,
1974.
NASA Planetary Photojournal, NASA/JPL/Caltech,
http://photojournal.jpl.nasa.gov
Pappalardo, R. T., et al., Does Europa have a subsurface
ocean? Evaluation of the geological evidence, J. Geophys.
Res., 104, 24015-24055, 1999.
Turtle, E. P., H. J. Melosh, and C. B. Phillips, Tectonic
modeling of the formation of Europan Ridges, (abstract),
Eos Trans. AGU, 79, suppl., F541, 1998.