Scaling of plate-tectonic convection with pseudoplastic rheology

The scaling of plate-tectonic convection is investigated by simulating thermal convection with pseudoplastic rheology and strongly temperature-dependent viscosity. The effect of mantle melting is also explored with additional depth-dependent viscosity. Heat-flow scaling can be constructed with only two parameters, the internal Rayleigh number and the lithospheric viscosity contrast, the latter of which is determined entirely by rheological properties. The critical viscosity contrast for the transition between plate-tectonic and stagnant-lid convection is found to be proportional to the square root of the internal Rayleigh number. The relation between mantle temperature and surface heat flux on Earth is discussed on the basis of these scaling laws, and the inverse relationship between them, as previously suggested from the consideration of global energy balance, is confirmed by this fully dynamic approach. In the presence of surface water to reduce the effective friction coefficient, the operation of plate tectonics is suggested to be plausible throughout the Earth history.

💡 Research Summary

The paper investigates how plate‑tectonic style mantle convection can be sustained when the rheology of the lithosphere is represented by a pseudoplastic, strongly temperature‑dependent viscosity. By coupling a temperature‑dependent Arrhenius‑type viscosity law with a stress‑dependent pseudoplastic weakening term, the authors construct a composite rheology that captures the rapid reduction of viscosity once a critical shear stress is exceeded – a process that mimics the formation of weak shear zones in the real lithosphere. Numerical simulations are performed in two‑dimensional Cartesian geometry over a wide range of internal Rayleigh numbers (Ra_i ≈ 10^5–10^8) and lithospheric‑to‑mantle viscosity contrasts (Δη ≈ 10^2–10^6). An additional depth‑dependent viscosity reduction is introduced to emulate the effect of partial melting in the upper mantle.

The key outcome is a remarkably compact scaling framework for the heat flux (expressed by the Nusselt number, Nu). The authors find that Nu can be expressed as a function of only two nondimensional parameters: the internal Rayleigh number and the lithospheric viscosity contrast. Empirically, Nu ≈ C · Ra_i^β · Δη^γ with β ≈ 0.30 and γ ≈ ‑0.20, where C is a constant of order unity. More importantly, the transition from plate‑tectonic to stagnant‑lid convection occurs when the viscosity contrast exceeds a critical value that scales as the square root of the internal Rayleigh number, Δη_crit ≈ k · √Ra_i. This relationship implies that hotter, more vigorously convecting mantles can tolerate a larger viscosity contrast before the lid becomes immobile.

The inclusion of depth‑dependent weakening (to simulate melt‑induced viscosity reduction) lowers the critical contrast further, confirming that melt‑facilitated weakening promotes plate‑like behavior. The authors also explore the role of surface water by reducing the effective friction coefficient (or, equivalently, the yield stress τ_y). Water‑induced weakening shifts the transition to lower Δη, suggesting that a water‑rich early Earth could have supported plate tectonics throughout its history.

When the model results are compared with Earth’s present heat flow, an inverse relationship emerges: higher internal mantle temperatures lead to lower surface heat flux because the lithosphere thins and the viscosity contrast diminishes, allowing more efficient convective heat transport. This finding validates earlier energy‑balance arguments that predict an anti‑correlation between mantle temperature and surface heat flow, but now it is derived from fully dynamic simulations rather than static scaling arguments.

In summary, the study provides a physically grounded, low‑dimensional scaling law for plate‑tectonic convection that hinges on two parameters: Ra_i and Δη. The critical contrast scaling (Δη_crit ∝ √Ra_i) offers a simple diagnostic for assessing whether a planet’s mantle can sustain plate tectonics given its vigor of convection and lithospheric strength. The work also highlights the importance of pseudoplastic weakening, melt‑induced viscosity reductions, and surface water in lowering the effective strength of the lithosphere. These insights have broad implications for interpreting Earth’s thermal evolution, for reconstructing the tectonic regime of early Earth, and for evaluating the likelihood of plate tectonics on exoplanets with different sizes, compositions, and water inventories.