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 proportional to exp( - gamma T), 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.