Christina Morency, Caroline Dumoulin, Maxim Ballmer, Neil Ribe, and N. Asaadi rapidly taken care of immediately inquiries. A couple private reviewers plus the member publisher Thorsten Becker promote comprehensive of use reviewers. This research was at part supported by NSF provides Ear canal-0406658 and you can Ear canal-0909319. Aforementioned grant try financed in Western Recuperation and you will Reinvestment Work off 2009 (ARRA) (Personal Laws 111–5).

## Appendix A good:: Extension so you can Nonlinear Viscosity

The tension speed inside the, eg from inside the (3), get rely on deviatoric fret into the nth power. My personal derivation deal as a consequence of when (3) is really so altered [ Sleep, 2011 ]. Nonlinear convection with shear try challenging due to the fact associated stress is the next invariant of stress tensor , and that strongly couples convection having shear. I qualitatively talk about the three subjects regarding 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 Temperature-Mainly based Viscosity

Stagnant-lid convection requires that the temperature contrast across the lithosphere T_{L} be much greater than the scale temperature T_{?}. Solomatov proposed the criterion that T_{L}/T_{?} > 8. The model with T_{?} = 150 K (Figures 5 and B1) bears on the transition from the stagnant-lid regime, as the ratio T_{L}/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 epidermis temperature flow, its value of 21.5 mW meters ?2 shortly after 2473 Ma brings guess away from enough time-term mediocre convective temperatures flow on the bottom of lithosphere; 110% out-of predicted steady state temperature circulate, which could signify new obvious multiplicative constant within the (8) was 0.517 in lieu of 0.47. Solomatov and you will Moresi acquired 0.53 with the constant showing you to (8) continues to be generally best.

not, the fresh new later strong temperatures disperse journey during the ?1300 Ma is at 139% of forecast worth (Profile 5). My numerical computations promote particular insight into the newest excursion. Convection was well organized into step three strong upwellings within 1310 Ma and you may unpleasant on 1473 Ma. A cover cold than five hundred°C limits convection on each other moments (Shape B1). The heat variety along side earnestly streaming area ?500°C so you’re able to 1300°C is a big an element of the full temperature range, breaking this new requirement that the rheological line layer is thin compared toward lithosphere. This new expectations out of border covering theory that there are adiabatic nations anywhere between downwellings is still satisfied. The cold countries having downwellings nonetheless occupy just a tiny fraction of the horizontal the amount of your model.