Christina Morency, Caroline Dumoulin, Maxim Ballmer, Neil Ribe, and you will Letter. Asaadi quickly taken care of immediately concerns. A few anonymous reviewers plus the associate editor Thorsten Becker promote comprehensive of good use reviewers. This research was a student in area backed by NSF grants Ear-0406658 and Ear-0909319. The second grant is actually financed beneath the American Recuperation and Reinvestment Act away from 2009 (ARRA) (Social Law 111–5).
Appendix An excellent:: Expansion so you’re able to Nonlinear Viscosity
The worries rate from inside the, like within the (3), get believe deviatoric worry into nth power. My derivation deal through when (3) is indeed modified [ Sleep, 2011 ]. Nonlinear convection which have shear are complicated because relevant worry are next invariant of fret tensor , and that firmly couples convection having shear. We qualitatively talk about the around three subjects from my papers.
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 Heat-Centered 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 stagnant-lid meilleur site de rencontre pour détenus 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.
Starting with skin temperatures circulate, its value of 21.5 mW yards ?2 once 2473 Ma brings guess off long-term average convective temperatures flow on the bottom of your own lithosphere; 110% from predict steady state temperatures move, which will imply that the fresh new obvious multiplicative lingering inside (8) are 0.517 as opposed to 0.47. Solomatov and you may Moresi gotten 0.53 with the ongoing appearing you to (8) remains essentially right.
not, the fresh new later solid temperature move excursion on ?1300 Ma has reached 139% of one’s predicted value (Figure 5). My personal numerical data render specific understanding of new travel. Convection is actually well organized toward step 3 strong upwellings from the 1310 Ma and you may messy from the 1473 Ma. A cover cooler than just five hundred°C hats convection on both times (Profile B1). The heat assortment across the positively flowing area ?500°C in order to 1300°C is a big the main complete heat diversity, breaking the requirement your rheological boundary level is slim opposed on lithosphere. Brand new requirement away from edge layer theory there are adiabatic places ranging from downwellings remains came across. The cold places having downwellings still entertain just a tiny fraction of the horizontal the amount of model.