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

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

Algebraic bounds on the Rayleigh–Bénard attractor

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851Funder: Einstein Visiting Fellow ProgramAbstract: The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations

Algebraic bounds on the Rayleigh–Bénard attractor

Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851Funder: Einstein Visiting Fellow ProgramAbstract: The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations

Ecological Modeling of Aedes aegypti (L.) Pupal Production in Rural Kamphaeng Phet, Thailand

Background - Aedes aegypti (L.) is the primary vector of dengue, the most important arboviral infection globally. Until an effective vaccine is licensed and rigorously administered, Ae. aegypti control remains the principal tool in preventing and curtailing dengue transmission. Accurate predictions of vector populations are required to assess control methods and develop effective population reduction strategies. Ae. aegypti develops primarily in artificial water holding containers. Release recapture studies indicate that most adult Ae. aegypti do not disperse over long distances. We expect, therefore, that containers in an area of high development site density are more likely to be oviposition sites and to be more frequently used as oviposition sites than containers that are relatively isolated from other development sites. After accounting for individual container characteristics, containers more frequently used as oviposition sites are likely to produce adult mosquitoes consistently and at a higher rate. To this point, most studies of Ae. aegypti populations ignore the spatial density of larval development sites. Methodology - Pupal surveys were carried out from 2004 to 2007 in rural Kamphaeng Phet, Thailand. In total, 84,840 samples of water holding containers were used to estimate model parameters. Regression modeling was used to assess the effect of larval development site density, access to piped water, and seasonal variation on container productivity. A varying-coefficients model was employed to account for the large differences in productivity between container types. A two-part modeling structure, called a hurdle model, accounts for the large number of zeroes and overdispersion present in pupal population counts. Findings - The number of suitable larval development sites and their density in the environment were the primary determinants of the distribution and abundance of Ae. aegypti pupae. The productivity of most container types increased significantly as habitat density increased. An ecological approach, accounting for development site density, is appropriate for predicting Ae. aegypti population levels and developing efficient vector control program