Internal heating driven convection at infinite Prandtl number

We derive an improved rigorous bound on the space and time averaged temperature $$ of an infinite Prandtl number Boussinesq fluid contained between isothermal no-slip boundaries thermally driven by uniform internal heating. A novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non-singular background profile, yielding the estimate $\geq 0.419[R\log(R)]^{-1/4}$ where $R$ is the heat Rayleigh number. The analysis relies on a generalized Hardy-Rellich inequality that is proved in the appendix

Topics in Geophysical Fluid Dynamics.

The dynamical evolution of fluids appears in many areas of science. Theoretical understanding and reliable computational description of complex and turbulent flows are one of the grand challenges of science. Although much has been accomplished recently, there is a significant amount of work remaining to get an accurate and effective description of fluid dynamics. This thesis uses two different approaches on different topics in geophysical fluid dynamics. First, rigorous theoretical bounds on the transport of heat are discovered for convection driven both by an internal heat source, and convection driven by an enforced temperature gradient. For stress-free vertical boundaries it is shown that at arbitrary Prandtl number in two dimensions (or at infinite Prandtl number in three dimension) the enhanced heat transport due to convection is bounded as Nu < Ra^(5/12) where Ra is a measure of the strength of the driving force. For these same type of boundaries (and under the identical assumptions on the Prandtl number and dimension) with internal heating, the spatially and temporally averaged temperature is bounded from below by H^(12/17) where H is the strength of the internal heating. For no-slip boundaries at infinite Prandtl number the temperature is bounded by H^(3/4) log(H)^(-1/4)$. Second, methods from numerical analysis and physical intuition are used to test the numerical models intended to describe the evolution of the earth's climate and weather. A stability analysis is carried out to test the numerical stability of divergence damping (a form of numerical dissipation meant to model unresolved sub-grid processes) applied on a latitude-longitude grid. The analysis yields sharp stability constraints, and highlights some of the issues inherent to the choice of grid. A test is also proposed to consider the consistency between the integration of the primitive equations, and the advection of passive tracers in a atmospheric dynamical core. Potential voriticity is used to examine the level of inconsistency between dynamics and tracers in the four dynamical cores present in the National Center for Atmospheric Research's Community Atmosphere Model (CAM5.0).Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91464/1/jaredwh_1.pd

"Ultimate state" of two-dimensional Rayleigh-Benard convection between free-slip fixed temperature boundaries

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $Nu \leq 0.2295 Ra^{5/12}$ uniformly in the Prandtl number Pr. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection.Comment: 4 page

A Bound on the Vertical Transport of Heat in the \u27ultimate\u27 State of Slippery Convection at Large Prandtl Numbers

An upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. a variation of the background method is implemented that allows negative values of the quadratic form to yield \u27small\u27 (O.1=Pr/) corrections to the subsequent bound. for large (but finite) Prandtl numbers this bound is an improvement over the \u27ultimate\u27 Ra1=2 scaling and, in the limit of infinite Pr, agrees with the bound of Ra5=12 recently derived in that limit for stress-free boundaries. © 2013 Cambridge University Press

Stability of Vortex Solutions to an Extended Navier-Stokes System

We study the long-time behavior an extended Navier-Stokes system in $\R^2$ where the incompressibility constraint is relaxed. This is one of several "reduced models" of Grubb and Solonnikov '89 and was revisited recently (Liu, Liu, Pego '07) in bounded domains in order to explain the fast convergence of certain numerical schemes (Johnston, Liu '04). Our first result shows that if the initial divergence of the fluid velocity is mean zero, then the Oseen vortex is globally asymptotically stable. This is the same as the Gallay Wayne '05 result for the standard Navier-Stokes equations. When the initial divergence is not mean zero, we show that the analogue of the Oseen vortex exists and is stable under small perturbations. For completeness, we also prove global well-posedness of the system we study.Comment: 24 pages, 1 figure, updated to add authors' contact information and to address referee's comment