Research ArticleGEOPHYSICS

Can mantle convection be self-regulated?

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Science Advances  19 Aug 2016:
Vol. 2, no. 8, e1601168
DOI: 10.1126/sciadv.1601168


  • Fig. 1 Numerical simulation results for stagnant lid convection with decaying internal heat production.

    (A) The evolution of average temperature as a function of the reduced time t′, when starting with the initial temperature higher than the equilibrium temperature. The equilibrium temperature (black dashed line) is the temperature corresponding to the internal heat production according to the theoretical heat flow scaling law. Four cases with different (initial) Tz values are shown: 1 (blue), 3 (green), 10 (red), and 30 (magenta). Solutions from parameterized convection models, that is, solutions of Eq. 2 with the theoretical scaling are shown as colored dashed lines. (B) Same as (A), but starting with the initial temperature lower than the equilibrium temperature. (C) Covariation of average temperature and surface heat flux from the runs shown in (A) and (B). All cases are plotted right on top of each other and also on the theoretical scaling (black dashed line). Results from steady-state solutions are shown as solid squares (see Materials and Methods). Their temporal fluctuations are smaller than the size of the symbol. Note that in (A) and (B), the temperature deviation for the case of Tz = 30 starts to increase slightly after a t′ of ~0.6, because dq*/dTa* decreases at lower temperatures (C), making the actual Tz value lower than 30. The slight nonlinearity of the actual heat flow scaling results in small temporal variations in Tz.

  • Fig. 2 Similar to Fig. 1, but with a dynamical system characterized by a negative value of dq*/dTa*.

    (A and B) The equilibrium temperature increases with time, whereas internal heating decreases with time, because of the negative dependency of surface heat flux on internal temperature. Three cases with different Tz values are shown: −1 (blue), −3 (green), and −10 (red). (C) Only the cases of Tz = −1 and −10 are shown for clarity.

  • Fig. 3 Heat flux scaling and the Tozer number.

    (A) Some representative heat flow scaling laws for convection in Earth’s mantle are shown as a function of mantle potential temperature. Solid curves denote calculations based on the scaling of plate tectonics with pseudoplastic rheology (27), with an activation energy of 300 kJ mol−1, a reference viscosity of 1019 Pa s at 1350°C, and a viscosity contrast due to dehydration stiffening of 1 (that is, no melting effect; red) and 100 (blue). The effective friction coefficient is set to 0.025 (red) and 0.02 (blue) to reproduce the present-day average convective heat flux [~75 mW m−2 (10)] at the present-day mantle potential temperature of 1350°C (51). The black dashed line represents the scaling for stagnant lid convection (25), using the same viscosity parameters as stated above, and the pink dashed line shows the effect of mantle melting (26). (B) The Tozer number variations corresponding to the scaling laws shown in (A). A half-life of 2.5 Gy is used to calculate the effective decay constant. The potential temperature Tp is equated with the average temperature Ta, neglecting the effect of a thin, cold thermal boundary layer. The e-folding time scale for approach to or departure from thermal equilibrium is also denoted on the right. (C) The Tozer number variations as a function of planetary mass. Earth mass is used here as a reference. Different curves represent different heat flow scaling laws [classical, plate tectonics (PT), or stagnant lid] or different potential temperatures, as indicated by the labels. Mantle depth, mantle density, and surface gravity, which are used to calculate heat flux, are varied with planetary mass (50). The chemical composition of the mantle is assumed to be the same as that of Earth’s mantle; thus, the effective decay constant is the same as in (B). Similarly, the viscosity parameters are the same as those used in (A) and are assumed to be insensitive to planetary mass. However, even with the same chemical composition, more massive planets may give rise to higher average viscosity because of a pressure effect (52), which would lower surface heat flux; the decreasing trend of Tz with increasing planetary mass would thus be enhanced further.

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