Timing of water plume eruptions on Enceladus explained by interior viscosity structure

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1 SUPPLEMENTARY INFORMATION DOI: /NGEO2475 Timing of water plume eruptions on Enceladus explained by interior viscosity structure Marie Běhounková 1, Gabriel Tobie 2, Ondřej Čadek 1, Gaël Choblet 2, Carolyn Porco 3 & Francis Nimmo 4 1 Charles University in Prague, Faculty of Mathematics and Physics, Department of Geophysics, V Holešovičkách 2, Praha 8, Czech Republic. 2 Université de Nantes, CNRS, Laboratoire de Planétologie et Géodynamique de Nantes, UMR-6112, 2 rue de la Houssinière, Nantes Cedex. 3 CICLOPS, Space Science Institute, Boulder CO Dept. Earth and Planetary Sciences, University of California Santa Cruz, CA NATURE GEOSCIENCE 1

2 1 Interior model and viscosity structure Enceladus s mass M kg ice density ρ ice 925 kg m 3 water density ρ water 1000 kg m 3 core density ρ core varying outer radius of the ice shell r t km ocean width ice shell thickness L km far zone L lithospheric thickness L 20 km reduced lithospheric thickness around south pole L SP km extent of L SP (characteristic width) δ 20 width of high-to-low viscosity transition σ 5 eccentricity e viscosity η Pa s shear modulus µ Pa Andrade parameter α 0.3 Andrade parameter ζ 1 time steps per period N step number of periods (length of simulation) N per 10 horizontal resolution (maximum degree) j max 60 vertical resolution (number of layers) n l 96 Table S1: List of reference parameters. 2

3 2 Tensile stress along the stripes and the stress representations Figure S1 displays a comparison between the NAS and FOF representations, for a given interior structure, and for three different values of the threshold parameter, σ t. This shows that the modeled stress curves are more sensitive to the threshold value for a 120 regional sea than for a global ocean, especially for the FOF representation. This is explained by the fact that the amplitude of the stress along the faults is smaller for a regional sea than for a global ocean. For the largest value of the threshold parameter, the initiation of the activity can be offset. However, the time (true anomaly) at which the maximum is reached remains identical irrespective of the stress representation and ocean extent. 3

4 a NAS representation, global ocean b FOF representation, global ocean c NAS representation, = 120 d FOF representation, = 120 Figure S1: Sensitivity to the threshold value σ t ; η = Pa s, L = 60 km, L SP = 2.5 km, L = 20 km, δ = 20. 4

5 3 Comparison with other approaches In the case of a purely elastic shell with a global ocean, the numerical solution has been compared with the approach by Nimmo et al. 1 and with the results of the SatStress code based on the method of Wahr et al. 2 (Figure S2 and S3). Figures S2a and S2c show the time evolution of stress tensor components at two specific locations 1,3. Note that the sign of component σ ϑϕ differs from the one in Nimmo et al. 1 and Smith-Konter and Pappalardo 3 due to different bases (east-positive longitude is used here). Figures S2b and S2d display the evolution of the normal stress. The evolution of all stress components and normal stresses agrees very well for all three approaches. The Love numbers h 2 and l 2 needed in Nimmo et al. 1 were obtained by SatStress and the two approaches are thus not independent. Figure S3 shows the average stress along the faults as a function of time for ice shell thickness of 60 km and 40 km. 5

6 a b c d Figure S2: Comparison of the stress tensors for different approaches corresponding to 60 km thick ice layer and h 2 = and l 2 = at Alexandria tiger stripe 70 S, 165 W with fault orientation 43 1 (a b) and at Damascus tiger stripe 80 S, 315 W with fault orientation 40 3 (c d). a b Figure S3: Comparison between analytical solution 1 (green) and numerical solution for elastic ice shell and global ocean (blue) and threshold stress σ t = 0 Pa; a analytical solution for h 2 = , l 2 = and numerical solution for ice shell thickness of 60 km, b analytical solution for h 2 = and l 2 = and numerical solution for ice shell thickness of 40 km. 6

7 4 Sensitivity tests We performed a systematic exploration of the model parameter space to determine which parameters primarily control the time evolution of the stress along the SPT faults. These sensitivity tests showed that the main controlling parameters are the ocean angular width,, the minimum viscosity, η, and the lithospheric thickness beneath the SPT, L SP, and to a lesser extent the total ice shell thickness, L. The lithospheric thickness away from the SPT plays a role only in the case of a global ocean. Other details on the viscosity structure, such as the width of the transition between the low/high viscosity regions and the maximum viscosity in conductive regions, are found to have a minor influence. The main results of these sensitivity tests are detailed below. Combined effect of the ocean extent and the extent of the low viscosity region Both the ocean width and the viscosity structure influence the response of the ice shell and the time evolution of the stress along the faults. In order to separate the interplay between the viscosity structure and the boundary conditions (ocean/ice: force equilibrium or silicate/ice: no-slip), we performed a first set of simulations for a purely elastic shell and various ocean widths (Figure S4a-b). These simulations showed that the elastic shell response with a hemispheric ocean ( = 180 ) is similar to the global ocean case, and that the stress curves are delayed compared to the global ocean solution when the ocean width is further reduced. The delay reaches a maximum value of 2.5 hours for the smallest ocean width tested ( = 60 ), and depends slightly on the ice shell thickness. However, this delay is not sufficient to explain the observed time lag and an additional delay due to viscous response is required. For a global ocean, only the viscous response of the ice shell can explain the delay. As illustrated on Figure S4c-d, the delay is proportional to the extent and thickness of the viscosity region. Maximum delay of 4 hours is obtained for a global viscosity structure ( v = 360 ) and a thick ice shell (60 km). The delay is also more sensitive to the extent of the viscosity region for L = 60 km than for L = 40 km. For models with a regional sea, the low viscosity region can exist only above the ocean, as high temperature can be maintained only there. The combined effect between the ocean extent and the extent of the low viscosity zone makes the 7

8 Elastic interior a L = 40 km b L = 60 km Viscoelastic interior with a global ocean ( o = 360 ) and varying viscosity structure ( v = ) c L = 40 km d L = 60 km Viscoelastic interior with varying ocean and viscosity structure ( = o = v ) e L = 40 km f L = 60 km Figure S4: Separating the influence of the weak interior and the localized ocean, the NAS representation for the viscoelastic (η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 ) and the elastic interior; a b elastic ice shell, changing extent of the internal ocean o, note the overlapping curves for global ocean and o = 180 c d viscoelastic ice shell, viscosity structure defined by v, global ocean assumed ( o = 360 ), e f viscoelastic ice shell, viscosity structure and ocean defined by = o = v. 8

9 a NAS representation b FOF representation Figure S5: Sensitivity to the minimum viscosity η for both representations; = 120, L = 60 km, L SP = 2.5 km, L = 20 km, δ = 20. a NAS representation b FOF representation Figure S6: Sensitivity to the south pole lithospheric thickness L SP for both representations; η = Pa s, = 120, L = 60 km, L = 20 km, δ = 20. evolution of the delay less straightforward than in the case of a global ocean. The delay reaches a minimum value for = 180 (Figure S4e-f) due to the decreasing viscous response. For smaller ocean widths ( < 180 ) the delay again increases due to increasing influences of the restricted ocean boundary. For the combined ocean and viscosity effects, a maximum delay of 5.5 hours is obtained for = 60. For both global ocean and regional sea models, a delay greater or equal to 4 hours can be obtained if a low viscosity zone is considered. However, for global ocean models, a thicker and wider low viscosity region is required, which has very different implications for the thermal evolution of Enceladus. 9

10 a NAS representation, global ocean b FOF representation, global ocean c NAS representation, = 120 d FOF representation, = 120 Figure S7: Sensitivity to the extent of the south polar lithosphere δ; η = Pa s, L = 60 km and L = 20 km. Influence of the viscosity structure beneath the SPT The viscosity structure beneath the SPT is controlled by the minimum viscosity η, the thickness and extent of the thinned lithosphere, L SP and δ, and the total ice shell thickness L. As illustrated with Figure S5 and Figure S6 for a 120 regional sea, the delay increases with decreasing minimum viscosity and SPT lithospheric thickness. A delay 4 hours can be obtained only for a very thin SPT lithosphere (L SP < 5 km) and a low viscosity (< Pa s). We also tested how the extent of the thin lithosphere around the South Pole, defined with the width parameter δ, affects the viscoelastic solution (Figure S7). These tests showed that as long as the area with a minimal lithospheric thickness corresponds to a significant portion of the SPT (δ > 20 ), the solution remains unchanged. For a lower value (δ = 10 ), the delay is significantly reduced as the thickness on the external part of the SPT is larger. This effect is equivalent to an increase of the averaged thickness illustrated in Figure S6. This outcome is independent of the extent of the ocean (global ocean or = 120 ) and of the representation 10

11 a NAS representation, global ocean b FOF representation, global ocean c NAS representation, = 120 d FOF representation, = 120 Figure S8: Sensitivity to the ice shell thickness L for both representations; η = Pa s, L = 60 km, L SP = 2.5 km, L = 20 km, δ = 20. (NAS or FOF). Finally, we tested the influence of the total ice shell thickness (Figure S8). For a 120 regional sea, the influence is minor, confirming that the delay in this case is not very sensitive to the volume of dissipative ice. In contrast, for models with a global ocean, the delay is very sensitive to the thickness, which appears to be one of the main controlling parameters. In summary, global ocean models require a relatively thick ice shell (L 60 km) to produce significant time lag, while models with a regional sea can lead to significant time lag independently of the ice shell thickness, as long as the average viscosity and lithospheric thickness in the SPT are sufficiently small (η Pa s, L SP 5 km). Influence of the viscosity structure outside the SPT Besides the parameters defining the ocean and the viscosity structure beneath the SPT, i.e., δ, η, L and L SP, we tested the parameters defining the viscosity structure away from the SPT: the far zone lithospheric thickness 11

12 L, the maximum viscosity η max and the width of the transition between minimum/maximum viscosity σ. All these parameters appears to be of minor importance, except the far zone lithospheric thickness in the case of a global ocean. Far zone lithospheric thickness L Figure S9 depicts the dependence of the predicted curve on the far zone lithospheric thickness L. For models with a regional sea (S9c d), the influence is negligible as both the global viscosity structure and the lithospheric thickness below SPT remain identical. In the case of a global ocean (S9a b), the increase of the far zone lithospheric thickness leads to the stiffening of the entire ice shell and the predicted curve is thus advanced compared to curves predicted for low L. The case L = L (S9a b, magenta curve) is a limit of our parameterization corresponding to a case where convection occurs only below SPT the conduction prevails elsewhere. This example shows that the convection in the case of a global ocean cannot be limited only to the SPT in order to explain the observed shift in the plume activity. Maximum viscosity η max Figure S10 shows that predicted curves are not sensitive to the chosen value of maximum viscosity as long as η max Pa s. Width of the lateral transition σ between the low/high viscosity regions We also explored the influence of the width of the lateral transition σ between the low and the high viscosity regions (see structures in Figure S11). This effect is shown to be very limited unless the ocean is strongly localized ( = 60 ). Again, we interpret this last result as an effective increase of the lithospheric thickness beneath the SPT owing to a wider transition zone (σ = 20, Figure S12). 12

13 a NAS representation, global ocean b FOF representation, global ocean c NAS representation, = 120 d FOF representation, = 120 Figure S9: Sensitivity to the far zone lithospheric thickness L ; η = Pa s, L = 60 km, L SP = 2.5 km, δ =

14 a NAS representation, global ocean b FOF representation, global ocean c NAS representation, = 120 d FOF representation, = 120 Figure S10: Sensitivity to the maximum viscosity η max ; η = Pa s, L = 60 km, L SP = 2.5 km, δ =

15 = 120 = 60 σ = 5 σ = 20 σ = 5 σ = 20 Figure S11: Example of two different structures corresponding to localized ocean and different width of transition zone; L = 60 km, L SP = 2.5 km, η = Pa s and L = 20 km. a NAS representation b FOF representation Figure S12: Sensitivity to the width of the transition zone σ; η = Pa s, L = 60 km, L SP = 2.5 km, δ = 20 ; solid line σ = 5, dashed line σ =

16 5 Fitting procedure, successful models Tables S2, S3 and S4 show the overview successful models. When a zero threshold value is considered for stress (Table S3), the lowest misfit is observed for the NAS representation and the largest set of successful models is found for VIMS without 2005 and ISS datasets. Additionally, whereas for the NAS representation the possibility of a global ocean cannot be excluded, we found no succesfull model with a global ocean for the FOF representation. Threshold dependence (Table S4) shows that low threshold (< 1000 Pa) values are favored for the NAS representation. For the FOF representation higher threshold values (< Pa) produce lower misfits than for cases without any threshold especially for VIMS dataset. Note, however, that the misfit defining very good models is lower for the NAS representation rather than for the FOF representation, independent of the threshold considered. 16

17 a) VIMS data with year 2005 b) VIMS data without year 2005 c) ISS data Figure S13: Acceptable models for all datasets and FOF representation, red, orange and yellow denotes 5%, 10% and 20% best models. 17

18 dataset m ν C ν M F CM χ 2 νc χ 2 νm,max χ 2 νm,min NAS FOF VIMS VIMS wo ISS Table S2: Datasets and misfits; m number of data points; ν C and ν M degrees of freedom for constant model C and time dependent model M; F CM critical value of F-distribution at the confidence level of 95%; χ 2 νc misfit for constant model C, χ2 νm,max maximum possible value of misfit for acceptable models, χ 2 νm,min minimum misfit for a given representation. datasets VIMS/ISS VIMS wo 2005/ISS representation NAS FOF NAS FOF No. acceptable models No. successful models No. successful models (global ocean) No. successful models (localized ocean) Table S3: Number of acceptable and successful models for both representations. Both cases where year 2005 is included or removed from the VIMS data set are considered. Threshold value is zero. 18

19 No. No. No. No. succ. No. succ. σ t model accep. succ. global local 0 Pa 1000 Pa 2000 Pa 3000 Pa 4000 Pa 5000 Pa NAS FOF NAS FOF NAS FOF NAS FOF NAS FOF NAS FOF Table S4: Overview of the threshold dependence, acceptable and successful models for VIMS without year 2005 and ISS datasets and both representations. Whatever the threshold value, successful models are defined as χ 2 ν,cew < and χ2 ν,i/f < 9.12 for the NAS and χ2 ν,cew < and χ 2 ν,i/f < 9.39 the FOF representations, respectively. 19

20 6 Additional Figures 20

elastic case, global ocean, L = 60 km global ocean, L = 60 km, η = 2 1013 Pa s LSP = 2.5 km, L = 20 km, δ = 20 = 120, L = 60 km, η = 2 1013 Pa s, LSP = 2.

21 elastic case, global ocean, L = 60 km global ocean, L = 60 km, η = Pa s LSP = 2.5 km, L = 20 km, δ = 20 = 120, L = 60 km, η = Pa s, LSP = 2.5 km, L = 20 km, δ = 20 Figure S14: Example of internal structure (first column), maximum tensile stress along the tiger stripes (second column), mean anomaly for which maximum tensile stress is reached (third column). Ice shell thickness 60 km, η = Pa s, variable ocean width, L = 20 km, δ =

22 = 90, L = 60 km, η = Pa s, LSP = 2.5 km, L = 20 km, δ = 20 = 60, L = 60 km, η = Pa s, LSP = 2.5 km, L = 20 km, δ = 20 Figure S14: Continuation 22

23 VIMS without 2005 = 110, L = 60 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 global ocean, L = 60 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 Figure S15: Example of the most successful models with global and localized oceans for both representations. 23

24 VIMS with 2005 = 120, L = 80 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 global ocean, L = 50 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 Figure S15: Continuation 24

25 VIMS without 2005, FOF representation = 80, L = 50 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 VIMS WITH 2005, FOF representation = 90, L = 50 km, η = Pa s, L SP = 2.5 km, L = 20 km, δ = 20 Figure S15: Continuation 25

26 VIMS without 2005, NAS representation L SP = 2.5 km, L = 20 km, δ = 20 L SP = 5 km, L = 20 km, δ = 20 L SP = 7.5 km, L = 20 km, δ = 20 Figure S16: The quality of the fit for the main parameters, different datasets and both representations. Successful models are shown as models with both halves of circle in red and/or orange, gray and white colors denote acceptable and non-acceptable models, respectively. 26

27 VIMS without 2005, NAS representation L SP = 2.5 km, L = 20 km, δ = 20 VIMS WITH 2005, NAS representation L SP = 2.5 km, L = 20 km, δ = 20 Figure S16: Continuation 27

28 VIMS without 2005, FOF representation L SP = 2.5 km, L = 20 km, δ = 20 VIMS WITH 2005, FOF representation L SP = 2.5 km, L = 20 km, δ = 20 Figure S16: Continuation 28

29 References [1] Nimmo, F., Spencer, J.R., Pappalardo, R.T. & Mullen, M.E. Shear heating as the origin of the plumes and heat flux on Enceladus. Nature 447, (2007). [2] Wahr, J. et al.. Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory, Icarus 200, (2008). [3] Smith-Konter, B. & Pappalardo, R.T. Tidally driven stress accumulation and shear failure of Enceladus s tiger stripes. Icarus 198, (2008). 29

Limits of Enceladus s ice shell thickness from tidally driven tiger stripe shear failure

GEOPHYSICAL RESEARCH LETTERS, VOL. 38,, doi:10.1029/2010gl044950, 2011 Limits of Enceladus s ice shell thickness from tidally driven tiger stripe shear failure John G. Olgin, 1 Bridget R. Smith Konter,