Christina Morency, Caroline Dumoulin, Maxim Ballmer, Neil Ribe, and you may N. Asaadi rapidly taken care of immediately questions. One or two private reviewers and the associate editor Thorsten Becker promote comprehensive of good use reviewers. This research was a student in area backed by NSF gives Ear canal-0406658 and Ear-0909319. The latter give try financed under the American Healing and you may Reinvestment Work out of 2009 (ARRA) (Social Laws 111–5).
Appendix Good:: Extension in order to Nonlinear Viscosity
The tension speed when you look at the, such as for example in (3), could possibly get rely on deviatoric worry on nth energy. My personal derivation deal using when (3) is really modified [ Sleep, 2011 ]. Nonlinear convection having shear was complicated as the associated be concerned was another invariant of the be concerned tensor , and this highly partners convection that have shear. I qualitatively discuss the around three topics of my paper.
The derivations apply using (7) to obtain the dependence on T?, providing that ?down is understood to be an effective nonlinear viscosity.
My derivation of scalloping involves forces and flow at the apex of upwellings. Qualitatively, the thicker rheological boundary layer at high n from (7) suppresses scalloping at given T?. Formally, the pressure term in (25) may be modified to ?P n and N replaced with N/n in (26) and (27). Numerical calculations are necessary to obtain a calibrated expression analogous to (28).
Alignment of convection into rolls by plate motions does not appear to be strongly dependent on nonlinear viscosity. van Hunen and Zhong presented numerical calculations with results similar to those of their linear viscosity cases. Mathematically, the thermal time scale of the rheological boundary layer in (29) and the shear velocity in (30) depend on the thickness of this layer at a given heat flow. The derivation of their ratio in (31) applies, again using (7) to obtain the dependence on T?. A further complication is the tendency of the initial convective instability to avalanche from sluggish to high velocity during the initial spin up [e.g., van Hunen et al., 2005 ]. This effect is not important in the realignment of already vigorous convection [ van Hunen and Zhong, 2006 ].
Appendix B:: Weakly Temperature-Situated Viscosity
Stagnant-lid convection requires that the temperature contrast across the lithosphere TL be much greater than the scale temperature T?. Solomatov proposed the criterion that TL/T? > 8. The model with T? = 150 K (Figures 5 and B1) bears on the transition from the web sites de rencontres gay stagnant-lid regime, as the ratio TL/T? = () = 8.7 is near the expected limit. As it is conceivable that mantle viscosity is in fact weakly temperature dependent, I discuss deviation of this from predicted stagnant-lid behavior. I then provide caveats in applying models with weakly temperature-dependent viscosity to the Earth.
Beginning with surface heat flow, its value of 21.5 mW yards ?dos immediately after 2473 Ma will bring guess out-of long-identity average convective temperature move on the bottom of the lithosphere; 110% out of predicted steady state temperatures move, that will signify the apparent multiplicative lingering into the (8) is actually 0.517 in lieu of 0.47. Solomatov and Moresi obtained 0.53 for the constant exhibiting one (8) has been basically proper.
Although not, the brand new late solid temperature disperse excursion on ?1300 Ma is at 139% of your predict well worth (Shape 5). My personal mathematical computations render certain understanding of the fresh travels. Convection is actually well-organized towards the step three solid upwellings at 1310 Ma and you may messy in the 1473 Ma. A cover cold than just 500°C hats convection on both times (Figure B1). The temperature diversity along the actively streaming area ?500°C so you can 1300°C is a significant the main full heat range, breaking the brand new expectations that rheological line coating try thin opposed for the lithosphere. The newest expectations out-of edge layer concept there are adiabatic countries between downwellings has been found. Cold weather regions that have downwellings nevertheless inhabit just a fraction of one’s lateral the quantity of model.
