2013 Theses Doctoral

Zonal flow driven by convection and convection driven by internal heating

Goluskin, David

In the first part, Rayleigh-Benard convection is studied in a two-dimensional, horizontally periodic domain with free-slip top and bottom boundaries. This configuration encourages mean horizontal flows of zero horizontal wavenumber, which we study as an idealization of zonal flows in tokamaks, planetary atmospheres, and annular cylindrical convection experiments. These systems often satisfy free-slip conditions on at least one boundary and are approximately two-dimensional. Stable steady states with zonal flow are found for Prandtl numbers up to 0.3. Stable and unstable steady states with horizontal periods up to six times the layer height are computed for a Prandtl number of 0.1 and Rayleigh numbers, Ra, up to 2*10⁵. Concurrently stable states with and without zonal flow are found where the state without zonal flow convects heat over 10 times faster. Steady zonal flow arises subcritically whenever the horizontal period is not forced to be narrow, contrary to most prior predictions by truncated models.

Steady states and their bifurcations are studied in a truncated model that does predict subcriticality. Direct numerical simulations are performed with a horizontal period twice the layer height, Prandtl numbers between 1 and 10, and Ra between 5*10⁵ and 2*10⁸. Zonal flow arises subcritically as Ra is raised but is seen in all quasi-steady states at large Ra. The fraction of the total kinetic energy comprised by zonal flow approaches unity as Ra grows. At a Prandtl number of 1, vertical convective heat transport occurs in temporal bursts, nearly vanishing in between, and is non-monotonic in Ra. At Prandtl numbers of 3 and 10, convective transport at no time nearly vanishes, and time-averaged Nusselt numbers scale as Ra⁰·⁰⁷⁷ and Ra⁰·¹⁹, respectively. Both growth rates are below the range accepted for Rayleigh-Benard convection without zonal flow. In the second part, two-dimensional direct numerical simulations are conducted for convection sustained by uniform internal heating in a horizontal fluid layer.

Top and bottom boundary temperatures are fixed and equal. Prandtl numbers range from 0.01 to 100. A control parameter, R, that is similar to the usual Rayleigh number is varied up to 5*10^5 times its critical value at the onset of convection. The asymmetry between upward and downward heat fluxes is non-monotonic in R. In a broad high-R regime, dimensionless mean temperature scales as R^-1/5. We discuss the scaling of mean temperature and heat-flux-asymmetry, which we find to be better diagnostic quantities than the conventionally used top and bottom Nusselt numbers.