Despite the background information coming from another body besides the Earth, the terms I would like to understand are historically geophysical in nature.


I just read today's news "Saturn's moon Dione harbors a subsurface ocean" at Phys.org and this seems to bring the number of sub-surface oceans in the Solar System to about six (the number might vary depending on the level of certainty).

This news about Saturn's moon Dione comes from a model which uses gravitational data from recent flyby's of the Cassini spacecraft. The model is simultaneously applied to another one of Saturn's moons Enceladus.


My question is about the terms in the abstract of the recently published paper on the subject in Geophysical Research Letters and my question is limited to these terms which I believe come from Earth Science and apply to the understanding of the Earth's crust floating on it's magma "ocean" interaction with the hot and slightly viscous mantle below.

What is an isostatic model and what is deviatoric stress in the context of geophysics?

Enceladus' gravity and shape have been explained in terms of a thick isostatic ice shell floating on a global ocean, in contradiction of the thin shell implied by librations. Here we propose a new isostatic model minimizing crustal deviatoric stress, and demonstrate that gravity and shape data predict a 38 ± 4km-thick ocean beneath a 23 ± 4km-thick shell agreeing with – but independent from – libration data. Isostatic and tidal stresses are comparable in magnitude. South polar crust is only 7 ± 4km thick, facilitating the opening of water conduits and enhancing tidal dissipation through stress concentration. Enceladus' resonant companion, Dione, is in a similar state of minimum stress isostasy. Its gravity and shape can be explained in terms of a 99 ± 23km-thick isostatic shell overlying a 65 ± 30km-thick global ocean, thus providing the first clear evidence for a present-day ocean within Dione.

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    $\begingroup$ Earth's crust does not float on a magma ocean. There is no magma ocean in the Earth. Earth's mantle is overwhelmingly solid. This is a very common misconception. $\endgroup$ – Gimelist Oct 6 '16 at 18:42
  • $\begingroup$ @Michael thanks for that! I've made an edit to demonstrate the correction. $\endgroup$ – uhoh Oct 6 '16 at 23:31

An isostatic model describes elevation differences either due to variations in density (Pratt's Isostasy model) or mass (Airy's Isostasy model). In planetary studies, variations in topography may be known, but not much else about the state of the crust. Through topography variations, variations in mass (rather: thickness) and/or density in the crust can be inferred using an isostatic model.

Pratt's model (left), Airy's model (right) (source: stanford.edu)

Deviatoric stress is, as you mentioned, the total stress minus hydrostatic stress. In this context the hydrostatic stress is supplied, in case of Enceladus by the ocean below the crust/shell (the image shows the case of the Earth, where the crust sits on the Asthenosphere). Imposing a minimum deviatoric stress constraint is a way to force the crust to be close to isostatic equilibrium (the idea is that high deviatoric stress lead to internal deformation in the crust that reduce the deviatoric stress).

  • $\begingroup$ OK very helpful, thanks! ! After searching and reading more about isostasy I've learned "every mountain has a downtain" and the Himalaya's deflected a pendulum laterally by only 1/3 the (originally) expected amount due to subduction of the lower density material below the surface. I can appreciate the quote in the phys.org article now; "As an additional principle, we assumed that the icy crust can stand only the minimum amount of tension or compression necessary to maintain surface landforms", said Mikael Beuthe, lead author of the new study. and am ready to tackle the Geophys. Res. letter $\endgroup$ – uhoh Nov 2 '17 at 1:38
  • $\begingroup$ I found this sites.ualberta.ca/~unsworth/UA-classes/210/notes210/B/… and also this seismo.berkeley.edu/~rallen/eps122/lectures/L16.pdf helpful. $\endgroup$ – uhoh Nov 2 '17 at 1:46
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    $\begingroup$ that quote contains another good point: the relation between deviatoric stress and landforms. those links contain some good introduction, for some further reading you could try this: onlinelibrary.wiley.com/doi/10.1029/JB086iB09p07801/full $\endgroup$ – ye-ti-800 Nov 2 '17 at 1:53

Isostatic means the equilibrium between the components of the crust in regards to their vertical movement owing to differences in density.

Deviatoric stress is the kinda sideways stress that particles in a larger system under stress, impose on each other. It's what keeps objects from crumbling to dust when exert vertical stress on them.

So, there model simulates the vertical stress on the crust more than it does the horizontal stresses. Enceladus is a puzzle because it's squished down at the poles much more than any other known moon or planet.

  • $\begingroup$ Thanks for your answer! are you sure that Deviatoric stress is strictly a kind of sideways stress in the crust? It seems to be just the total stress minus the isotropic part (hydrostatic pressure). $\endgroup$ – uhoh Oct 6 '16 at 23:41
  • $\begingroup$ Is Deviatoric stress really "sideways stress"? $\endgroup$ – uhoh Oct 8 '16 at 0:07

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