Lunar and Planetary Science XLVIII (2017)
1676.pdf
Convection and Dichotomies within Enceladus' Ice Shell: Effects of Variable Surface Temperatures. M. B.
Weller, L. Fuchs, T. W. Becker, and K. M. Soderlund, Institute for Geophysics, Jackson School of Geosciences The
University of Texas at Austin, Austin, TX ([email protected], [email protected], [email protected], [email protected])
Introduction: Despite its relatively small size observations of Enceladus reveal it as one of the more
geologically active bodies in the solar system. Its surface is heavily deformed and includes ridges, grooves,
grabens, rifts, and folds that cover a significant fraction
of the planet [1-3]. Perhaps most notably, there is evidence of a hemispheric dichotomy between the south
(the South Polar Terrain – SPT), and the remaining
bulk of the planet. The extent, cause, and duration of
the SPT have spurred much debate. It has been suggested that convection [4-10] or a local ocean [10-12]
coupled with local tidal dissipation may allow for activity in the SPT. However, to date convection in full
3D spherical experiments have required a priori assumptions of an underlying weakness [e.g., 7], variations in basal boundary geometry [e.g., 8], or grain size
reduction and random placement [e.g., 9] for localization and placement. In this work, we address the effects
of latitudinally variable surface temperature (through
solar insolation), and consequently of variable (nearsurface) ice shell geometry, on the convective vigor
and planform of Enceladus' ice shell.
Numerical Methods: The governing equations of
mass, momentum, and energy conservation, assuming
infinite Prandtl number and Boussinesq fluid approximation, are given in non-dimensional form by:
u i ,i = 0 (1)
((
− P,i + η u i, j + u j ,i
))
,j
+ RaT δ ir = 0 (2)
T ,t + u i T ,i = T , ii + Q (3)
where u is the velocity, P dynamic pressure, η the viscosity, Ra the Rayleigh number, T the temperature, δij
the Kronecker delta, Q the heat production rate, i and j
represent spatial indices, r is a unit vector in the radial
direction, t is time, and the form X ,y represents the
derivative of X with respect to y. Repeated indices imply summation. Eqs. (1-3) are solved using the well benchmarked finite element code CitcomS [e.g., 13, 14].
Our models include temperature-dependent viscosity as follows:
  1
−
η (T ) = η 0 exp  A
 T + 1
1

2  
(4)
Here, η0 is the pre-exponential factor and is set to the
reference viscosity at the base of ice shell, estimated to
be ~1013 to ~1015 Pa s [e.g., 15, 16]. The rheological
parameter, A, is chosen to give a variation in viscosity
of 3·104 for T [1:0]. The Rayleigh number at the base
of the ice shell is 3·105, and internal heating rates are
varied from ~49 to ~2 (corresponding to heating rates
between ~10-11 and 2.3·10-10 W Kg-1). Internal heating
rates are uniform in the model domain. Runs start at
maximum levels of heating (Q = 49), and decrease with
each subsequent run. Each model uses the thermal output of the proceeding model as the initial starting condition. Total ice shell thickness is taken as 100 km as
an endmember (giving a core fraction of 0.603 for
these models). Note, this is slightly greater than estimates of the current thickness of ~60 km [e.g., 17].
The surface temperature (Ts) of Enceladus varies
from solar insolation from approximately 73 K at the
equator, to ~23 K at the poles, and the base of the ice
shell may reach temperatures ~273 K assuming a pure
water ocean [18]. We have modified CitcomS to allow
for continuous variation in the surface temperature
boundary conditions from pole to equator (a basal nondimensional temperature of unity, equatorial temperature of 0.165, and a polar temperature of zero). Both
sets of models are initialized independent of each other, and allowed to evolve in their respective paths.
Results: First we consider models of uniform surface temperature as the reference case (left column
Figure 1). Convection occurs under a thick stagnantlid, with convective planforms evolving from high
temperature cells, to stable multi-plume structures as
internal temperatures decrease (through a reduction in
heating rate). Unsurprisingly, convection does not exhibit any form of hemispheric dichotomy.
Next, we consider variable surface temperature
(right column Figure 1). These conditions favor the
development of initially thicker boundary layers along
the poles, as compared to the equatorial region (likely
due to a reduction in viscosity from the higher surface
temperatures), in general agreement from Ts variations
for tidally locked exoplanets [19]. As the system develops, an unstable planform of convection begins to
form (a hemisphere slightly dominated by upwellings,
and one slightly by downwellings – as can be seen in
Figure 1; Q = 49, right hand side). With decreasing
heating, a hemisphere dominated by a large, degree 1
upwelling, and one fully dominated by slower, colder
downwellings emerges (shown by a depression of colder iso-surfaces, right hand side Figure 1; Q = 10). The
active region contains ice near the melting point (T >
0.9 or > 246 K), and a reduced boundary layer thick-
Lunar and Planetary Science XLVIII (2017)
ness (indicating larger heat flow in the active regions).
The remaining ~80-95% of the surface is cold and relatively inactive.
Surface velocities can range over several orders of
magnitude, with highs focused on regions of upwellings and downwellings (Figure 2). Surface strain rates
for models with fixed Ts are inferred to be on the order
of 10-16 to 10-18 s-1 for all heating rates considered (an
effective global stagnant-lid regime; see Figure 2 caption for description). Globally variable Ts models are in
a sluggish-lid regime. Strain rates show a strong dichotomy, and are inferred to be on the order of 10-14 (in
the local mobile area, purple region bottom of Figure
2) to 10-16 s-1 (across the stagnant or “inactive” remainder), in agreement with estimated strain rates for
Enceladus [15, 16].
In our models, the degree 1 structure is stable at the
equator, the region of the highest surface temperatures.
This is a configuration currently not witnessed on Enceladus. However, such a large and persistent dynamic
structure would likely lead to reorientation of the satellite [e.g., 20]. It is currently unclear what the effects of
reorientation will be on the internal dynamics and driving forces of these systems (e.g., change in orientation
of Ts variations), but will be considered in the future.
Additional future work will explore different shell
thicknesses (e.g. ~60 km), yielding, and additionally
employ strain tracking, which will allow for a strain
accumulation, or “memory”, within the ice shell.
Figure 1: Iso-temperature surfaces (red = 0.9; blue = 0.3) for variable Q. The left-hand side is for fixed Ts (0), while the right is for
pole to equator variation in Ts (0.165). Poles are oriented with the zaxis (top to bottom of plots). Note: Q =10 for variable Ts is rotated
along the equator to illustrate the plume structure.
1676.pdf
Figure 2: Non-dimensional surface velocities from convection
models (Q = 10). Note, scales vary between models (maximum of
2.33 for fixed Ts and 123.3 for variable Ts). Fixed Ts models are in
an effective stagnant-lid (low surface velocity), with the ratio of
surface to internal velocity (Mobility) = 2.12·10-2 – 1.08·10-4 (average of 1.06·10-2). Variable Ts models are in a sluggish-lid (global
average), local mobility occurs in the region of the upwelling along
the equator, effective immobility occurs in the remainder (Mobility
= 0.967 – 4.79·10-3, average of 0.626). The effective global Ra,
(calculated at shell mid-depth) is 4.0·104 for fixed Ts, and 9.1·103
for variable Ts. For variable Ts, the effective Ra ranges from ~ 105
(degree one region), to ~ 103-102 (remainder). In general, Mobility ≥
1 indicates mobile-lid, where a Mobility ≤ 0.1 indicates a stagnantlid. Orientation follows from Figure 1.
References: [1] Kargel, and Pozio (1996) Icarus 119,
385-404. [2]. Porco, et al. (2006) Science 311, 1393-1401.
[3] Barr and Preuss (2010) Icarus 208, 499–503. [4] Barr,
(2008) JGR 113, doi:10.1029/2008JE003114. [5] Roberts,
and Nimmo (2008) GRL 35. [6] O’Neill and Nimmo (2010)
Nature Geoscience. [7] Han et al., (2012) Icarus 218, 320330. [8] Showman et al., (2013) Geophys. Res. Lett. 40. [9]
Rozel et al., (2014) JGR 19, 416–439. [10] Stegman et al.,
(2009) Icarus. 202:669-680. [11] Collins and Goodman
(2007) Icarus 189, 72-82. [12] Tobie et al. (2008) Icarus.
[13] Moresi and V.S. Solomatov (1995), Physics of Fluids,7
(9), 2154-2162. [14] Zhong et al. (2000), JGR, 105, 11063–
11082. [15] Durham, and Stern (2001) Annu. Rev. Earth
Planet. Sci. 29, 295–330. [16] Barr and McKinnon (2007)
GRL 34. [17] Beuthe et al., GRL, 2016. [18] Glein et al.,
GCA, 2016. [19] Van Summeren et al., (2011), Astrophys. J.
Lett., 736, L15. [20] Nimmo and Pappalardo (2006), Nature,
441, 614–616.