Cambridge Journals Online

Transient high-Rayleigh-number thermal convection with large viscosity variations

Anne  Davaille a1 and Claude  Jaupart a1 a1 Laboratoire de Dynamique des Systèmes Géologiques, Université de Paris 7 et Institut de Physique du Globe, 4 place Jussieu, 75252 Paris Cedex 05, France

Article author query davaille a     jaupart c

Abstract

The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 10⁶ and 10⁸ and viscosity contrasts are up to 10⁶. Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written \[Q_{\rm s} = - CK_{\rm m} (\alpha g/\kappa \nu_{\rm m})^{\frac{1}{3}} \Delta T^{\frac{4}{3}}_{\rm v}\], where k_m and ν_m are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. ΔT_v, is the ‘viscous’ temperature scale defined by \[\Delta T_{\rm v} = - \frac{\mu (T_{\rm m})}{({\rm d}\mu /{\rm d}T)(T_{\rm m})}\] where μ(T) is the fluid viscosity and T_m the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2ΔT_v).