Rapidly rotating plane layer convection with zonal flow

The onset of convection in a rapidly rotating layer in which a thermal wind is present is studied. Diffusive effects are included. The main motivation is from convection in planetary interiors, where thermal winds are expected due to temperature variations on the core-mantle boundary. The system admits both convective instability and baroclinic instability. We find a smooth transition between the two types of modes, and investigate where the transition region between the two types of instability occurs in parameter space. The thermal wind helps to destabilise the convective modes. Baroclinic instability can occur when the applied vertical temperature gradient is stable, and the critical Rayleigh number is then negative. Long wavelength modes are the first to become unstable. Asymptotic analysis is possible for the transition region and also for long wavelength instabilities, and the results agree well with our numerical solutions. We also investigate how the instabilities in this system relate to the classical baroclinic instability in the Eady problem. We conclude by noting that baroclinic instabilities in the Earth's core arising from heterogeneity in the lower mantle could possibly drive a dynamo even if the Earth's core were stably stratified and so not convecting.Comment: 20 pages, 7 figure

A two-fluid single-column model of the dry, shear-free, convective boundary layer

This is the final version. Available on open access from Wiley via the DOI in this record.A single-column model of the dry, shear-free, convective boundary layer is presented in which nonlocal transports by coherent structures such as thermals are represented by the partitioning of the fluid into two components, updraft and environment, each with a full set of prognostic dynamical equations. Local eddy diffusive transport and entrainment and detrainment are represented by parameterizations similar to those used in Eddy Diffusivity Mass Flux schemes. The inclusion of vertical diffusion of the vertical velocity is shown to be important for suppressing an instability inherent in the governing equations. A semi-implicit semi-Lagrangian numerical solution method is presented and shown to be stable for large acoustic and diffusive Courant numbers, though it becomes unstable for large advective Courant numbers. The solutions are able to capture key physical features of the dry convective boundary layer. Some of the numerical challenges posed by sharp features in the solution are discussed, and areas where the model could be improved are highlighted.Natural Environment Research Council (NERC

Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms.Comment: To be published in Physica D, nonlinear phenomena (accepted January 2010

Asymptotic stability of solitary waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, In particular, we study the case where f(u)=u p+1 / (p+1) , p =1, 2, 3 (and 30, with f ∈ C 4 ). The same asymptotic stability result for KdV is also proved for the case p =2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values of p between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameter p increases beyond the critical value p =4.) The solution is decomposed into a modulating solitary wave, with time-varying speed c(t) and phase γ( t ) ( bound state part ), and an infinite dimensional perturbation ( radiating part ). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. As p →4 − , the local decay or radiation rate decreases due to the presence of a resonance pole associated with the linearized evolution equation for solitary wave perturbations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46489/1/220_2005_Article_BF02101705.pd

Organizing to support internal diversification

http://deepblue.lib.umich.edu/bitstream/2027.42/35563/2/b140717x.0001.001.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/35563/1/b140717x.0001.001.tx

Minimally Invasive Spinal Surgery with Intraoperative Image-Guided Navigation

We present our perioperative minimally invasive spine surgery technique using intraoperative computed tomography image-guided navigation for the treatment of various lumbar spine pathologies. We present an illustrative case of a patient undergoing minimally invasive percutaneous posterior spinal fusion assisted by the O-arm system with navigation. We discuss the literature and the advantages of the technique over fluoroscopic imaging methods: lower occupational radiation exposure for operative room personnel, reduced need for postoperative imaging, and decreased revision rates. Most importantly, we demonstrate that use of intraoperative cone beam CT image-guided navigation has been reported to increase accuracy