Horizontal length of finite-amplitude thermal convection cells with temperature-dependent viscosity

Temperature-dependent viscosity convection is investigated for various horizontal wavelengths of the convective cells. Finite-amplitude steady solutions are obtained by the Newton method in a two-dimensional layer for various values of the Rayleigh number and strength of temperature-dependence of viscosity, and their stability is examined through numerical time integrations. The viscosity $\eta$ of the model varies with temperature $T$ as $\eta \propto exp\left(-\mathrm{\gamma T}\right)$, where the parameter $\gamma$ denotes the strength of the temperature-dependency of $\eta$. Although approximately square convection cells are stable when $\gamma$ is small, the stable convective structure elongates horizontally as $\gamma$ increases in the middle range of $\gamma$ less than about 10. When $\gamma$ exceeds that range, the stable convection approaches a square cell.

Scaling relations for the Nusselt number that include the effect of the horizontal wavelength are developed. The results obtained by the numerical steady solutions are well explained by the proposed novel scaling relations. When the solutions with the maximum Nusselt number are traced using the scaling relations for various $\gamma$, we find that the convective cells elongate gradually as $\gamma$ increases until $\gamma <8.6$, and then the convection becomes narrower. The most elongated convection is expected to appear at the threshold with a horizontal length $\lambda$ of $6.6$, which may not depend on the Rayleigh number. Our results suggest that rocky exoplanets (such as super-Earths), which will be studied in detail in the future, may have surface plates with various horizontal scales.