Coupling iceshell thickness variations with heat flow on Europa

Introduction

Figure 1.  Canamara Region on Europa depicting Chaos and Microchaos

My project would be to model the interior heat flow of Europa with ice shell thickness variations. Heat flow variations have been extensively modeled on Europa with constant ice shell thicknesses. (Showman & Han, 2004) These ice shell thickness values typically range from 0.2 to 2km (Billings & Kattenhorn, 2005) and even 15 to 50km. (Showman & Han, 2004) Previous models have used a constant ice shell thickness when making heat flow models due to studies showing that the ice shell is of constant thickness globally due to low heat flux from the interior, lateral ice flow at the base of the shell, or convection. (Nimmo, Thomas, T, & Moore, 2007) (Nimmo, Giese, & Pappalardo, Estimates of Europa's ice shell thickness from elastically-supported topography, 2003) My model would incorporate a varied thickness that better includes the addition of latent heat that previous models exclude. Additionally, this model will be informed by current work that seeks to determine a possible preferred stress orientation on Europa. This has been done by finding the front azimuth of the features and finding the mean angle. To determine if these values are influenced by other neighboring features, front azimuth angles were compared between features.

Figure 3. Convection of Europa with an ice shell thickness
of 50 km (showman & Han, 2004).

Figure 2. Hypothesized layers of Europa

Budget

The budget for this project would include $81,000  in order to support a graduate student for three years. Deepthought II at the University of Maryland would potentially be used in order to run higher resolution models. The price for using deepthought2 would vary since allocation is only given to tenure/tenure track faculty and then passed down to graduate students. Thus since Dr. Laurent Montesis has access to Deepthought II, I would also be granted access with potentially no cost.  $5,000 would be allocated to provide a position for an undergraduate student/summer intern. Their job would be to collect data on ridges, bands, and microchaos on Europa to better constrain parameters that would be used in the models.

Figure 4. Rose Diagram of Microchaos front azimuth angles on Europa.

Broader Impact

The work completed in this project would be used to create a summer workshop for underrepresented girls in low income areas of Washington D.C. This workshop would expose girls to the research that is being done in the outer solar system, an area not well publicized. The summer workshop would include introduction to mathematics, coding in Python and Mathematica, and mentorship from a women of color. This summer camp would allow girls to participate in a field that is difficult to get a start in. Additional outreach would be completed in partnership with the Posse Foundation D.C. This outreach would consist of mentoring Posse Scholar awardees that intend to go to college in STEM the experience of partaking in geophysical research at an entry level.

References

Behounkova, M., Tobie, G., Choblet, G., & Cadek, O. (2010). Coupling mantle convection and tidal dissipation : Applications to Enceladus and Earth-like planets. Journal of Geophysical Research.

Billings, S. E., & Kattenhorn, S. A. (2005). The great thickness debate: Ice shell thickness models for Europa and comparisons with estimates based on flexure at ridges. Icarus, 397-412.

Kalousova, K., Soucek, O., Tobie, G., Choblet, G., & Cadek, O. (2014). Ice melting and downward transport of meltwater by two phase flow in Europa's ice shell. Journal of Geophysical Research, 532-549.

Nimmo, F., Giese, B., & Pappalardo, R. T. (2003). Estimates of Europa's ice shell thickness from elastically-supported topography. Geophysical Research Letters.

Nimmo, F., Thomas, P. C., T, P. R., & Moore, W. B. (2007). The global shape of Europa: Constrains on lateral shell thickness variations. Icarus, 183-192.

Showman, A. P., & Han, L. (2004). Numerical simulations of convection in Europa's ice shell: implications for surface features. Journal of Geophysical Research.

Detectable Gravitational Anomalies from Lunar Lava Tubes

θ = tan− 1(x/h)

Gravitational anomaly created by an idealized lunar lava tube.

Deflection of Plates

 D = the flexure rigidity of a plate, dependent on the Young's Modulus (stiffness of material), Poisson's Ratio (describes the expansion a material perpendicular to the direction of compression), and the thickness of the plate (h in the diagram below).

q(x)= downward force per unit area

P =  A horizontal force, per unit length in the z- direction (out of the page in the diagram below)

w = the vertical deflection of the plate

By relating the observed flexure/ bending of the lithosphere to known surface loads, we can deduce the elastic properties and thicknesses of the plates.

Very important to deriving the equation above:

The deflection of a plate can be determined by requiring it to be in equilibrium under the action of all the forces and torques exerted on it. 


With this condition you can write a force balance equation for all of the vertical forces acting on the plate:

This equation tells you that the change of the net shear force in the x direction is equal to the negative of the downward force per unit area.

Lastly we need to consider all of the net torques on the plate. The first is the bending moment, M. The bending moment is the reaction of the plate when the load is applied to it. When the moments are combined it results in a counterclockwise torque dM. The shear force exerts a net clockwise torque, Vdx. The horizontal force P exerts a net counterclockwise torque -Pdw. This results in a toruqe equation of:

which can be simplified to

which puts the torque balance equation in terms of the gradient in moment along the distance and the slope of the plate.

Further simplification allows us to write this equation into,

Another way to define the the bending moment M, is by the flexure rigidity of the plate divided by its curvature which gives us the equation from the beginning. Ultimately we are able to relate the the plate rigidity, curvature of the plate, to the forces loaded on to the plate.


This equation is important to models that help determine the radius of plates on Europa because the ridges on Europa can be treated as a load.

Darcy's Law: Understanding Fluid (melt) Movement Through Porous Media

Equation: Darcy's law shown here, which is implemented by Avizo (ThermoFisher Scientific) to calculate permeability of porous material. 

Figure 1: Conceptual model of how Darcy's law is implemented in Avizo. 

Mass Transfer from Fluid Infiltration Events

All equations and images from Penniston-Dorland and Ferry, 2008

Electrodynamic Navier-Stokes in a Rotating Frame

Circular Statistics: Determining if Individual Cloudina are Randomly Oriented based on Measurements of their Growth Direction

Figure 1: Individual Cloudina with growth direction indicated by arrows.

Figure 2: A rose diagram displaying the growth orientation of 193 individual Cloudina. The orange arrows represent the resultant vector.

Six Equations: The first four equations relate to the resultant vector, r. The fifth equation relates to circular variance, V. The sixth equation relates to the circular standard deviation, S.