# How viscous is the Earth's mantle?

I'm posting here to get some more expert information based on this question.

If the Earth were to stop rotating, removing the centrifugal force causing the equatorial bulge, how long would it take for the mantle and crust to return to hydrostatic equilibrium? That is, immediatly, thousands of years, or millions of years?

Never mind that changing the rotation in reality would involve forces and timescales that would not really provide this condition. if matter were arranged in such a shape, how long (orders of magnitude) would it take to settle?

• obviously the oceans and atmosphere would adjust first and quite quickly. That would be quite the disaster if such an event actually happened. The Earth is still rebounding from the ice ages, some 12,000 years, but this bulge would be significantly greater and so the adjustment period could be faster as a result. It's fun to think about but I don't know if anyone will have a real answer to this. I'd be curious to know how quickly the earth would start to move towards a sphere, if it would be visibly obvious in as little as 1 year or even a month. – userLTK Mar 18 '16 at 22:42
• I thought about the ice-age rebounding initially, but that's a superficial effect in the crust. Or does the bouancy of the plate in the mantle still give an indication of the scale? Anyway, I wondered if the effect of the overall mantle thickness would be on a substantially faster time scale. – JDługosz Mar 19 '16 at 1:10
• Wikipedia mentions the mantle's movement early in their post glacial rebound article. en.wikipedia.org/wiki/Post-glacial_rebound I don't know how much the land rises and falls with each ice age, 10 CM per century or so currently, but perhaps 100 feet since the receeding of the last ice age as a guess. Some of that may well be crustal, but everything moves with that much weight on it, if somewhat slowly. Some of it has to be mantle, I would think. – userLTK Mar 19 '16 at 3:29
• @userLTK -- the adjustment would of course start immediately. We would see the greatest movement at the beginning, with the velocity of the adjustment decreasing over time (I would expect the decrease to be exponential, but can't give a good argument for it). – Wolfgang Bangerth Mar 20 '16 at 17:22
• Related (not a dupe): earthscience.stackexchange.com/questions/2236/… – Semidiurnal Simon May 27 '16 at 14:07

The mantle viscosity is likely to be non-linear, e.g., it could be as low as $10^{18} \textrm{Pa}\cdot\rm s$ (over shorter time scales) or as high as $10^{21} \textrm{Pa}\cdot\rm s$ (over longer time scales). In any case the values reported in the literature are somewhere between $10^{18}-10^{21} \textrm{Pa}\cdot\rm s$ and these are based on studies from earthquakes, glacial rebound etc. So you can calculate the relaxation time(s) using these two end member values.

E.g., if you assume the average i.e., $10^{19.5} \textrm{Pa}\cdot\rm s$ then we get a relaxation time of $10^{19.5}/150\cdot10^9$ (viscosity/mu) which is $2.1082\cdot10^8$ seconds or ~$6.6$ years.

• Interesting! If the relaxation time is on this short scale, why is it taking thousands of years for North America to rebound from the ice age? – JDługosz Mar 23 '16 at 16:41
• Keep in mind that relaxation time is the time required for an exponentially decreasing variable to drop from some initial value to 1/e (0.368) times the initial value. – stali Mar 23 '16 at 17:26
• DrGC's answer is 3 orders of magnitude longer, and he has a citation for the values, and it agrees with the observation that glacier rebound takes thousands (not tens) of years. – JDługosz Mar 24 '16 at 13:58
• I can't do the math, but intuitively a 40 mile high crust bump (some 4-5 times the density of a glacier and some 25-30 times as high), so 100-150 times the displacement force, logically that should move more quickly than a glacial displacement of ice, about 1.5 miles thick. Intuitively I would think this question's scenario would correct itself faster than ice-age rebound. – userLTK Mar 25 '16 at 2:39

The viscosity of the mantle varies largely with depth (because depth primarily controls temperature, pressure, and composition), but from the details of your question you seem particularly interested in the deformation time-response to large-scale changes in the stress distribution. The closest analogue to your imaginary scenario is the so called post-glacial rebound, and a huge body of studies now show that the response time of topographic recovery after deglaciation is in the order of a few thousand years. That's why Scandinavia is still uplifting today at rates of cm/yr, even if there is no ice sheet left there. The viscosities of the underlying fluid asthenosphere corresponding to such relaxation times are in the order of $10^{22} \textrm{Pa}\cdot\rm s$ to $10^{23} \textrm{Pa}\cdot\rm s$. These values result from adopting a layered model with constant viscosities, so it can give you an average value, or an effective value for your problem, but keep in mind that in reality the asthenosphere (upper mantle) properties are very heterogeneous.

Since the Earth is about 22 km wider at the equator than at the poles due to rotation, suppressing the centrifugal forces would imply a much larger effect than post-glacial rebound (1 order of magnitude larger). But the time-scales would be similar, since they are controlled by the asthenospheric viscosity.

Remember that the asthenosphere is defined as the fluid (in geological time-scales) underlying the rigid lithosphere (the rigid plates). It is part of the upper mantle.