Christina Morency, Caroline Dumoulin, Maxim Ballmer, Neil Ribe, and you may Letter. Asaadi easily responded to concerns. A few anonymous writers in addition to user editor Thorsten Becker give comprehensive helpful writers. This study was in part supported by NSF provides Ear-0406658 and you can Ear canal-0909319. The latter give was funded under the Western Recuperation and you may Reinvestment Act out of 2009 (ARRA) (Personal Legislation 111–5).

Appendix A beneficial:: Extension so you’re able to Nonlinear Viscosity

The tension price inside the, such as for instance for the (3), may count on deviatoric be concerned into the nth strength. My personal derivation sells owing to when (3) is so changed [ Sleep, 2011 ]. Nonlinear convection which have shear is complicated once the associated be concerned was the following invariant of the worry tensor , and that strongly partners convection having shear. We qualitatively talk about the about three subject areas from my report.

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 https://datingranking.net/de/professionelle-dating-sites/ 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 Temperatures-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 temperatures circulate, their worth of 21.5 mW yards ?dos immediately following 2473 Ma provides guess out-of a lot of time-identity average convective temperatures flow on the bottom of the lithosphere; 110% out-of predicted steady-state heat circulate, that would indicate that the brand new obvious multiplicative constant in the (8) is 0.517 rather than 0.47. Solomatov and Moresi obtained 0.53 to your lingering exhibiting one to (8) is still generally best.

not, the newest late solid heat flow trip during the ?1300 Ma is at 139% of one’s forecast worthy of (Figure 5). My mathematical computations offer specific understanding of the brand new travel. Convection try well-organized into step 3 solid upwellings on 1310 Ma and you will unpleasant during the 1473 Ma. A top cooler than simply five hundred°C limits convection at one another minutes (Contour B1). The temperature diversity along side definitely moving region ?500°C so you can 1300°C is a big an element of the complete temperatures range, violating this new requirement the rheological edge level try narrow compared towards the lithosphere. This new expectations off line layer concept there exists adiabatic places ranging from downwellings is still satisfied. Frigid weather regions that have downwellings nonetheless consume just a small fraction of horizontal the quantity of the model.