Compatible finite element spaces for geophysical fluid dynamics

Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties

Estimating eddy diffusivities from noisy Lagrangian observations

The problem of estimating the eddy diffusivity from Lagrangian observations in the presence of measurement error is studied in this paper. We consider a class of incompressible velocity fields for which is can be rigorously proved that the small scale dynamics can be parameterised in terms of an eddy diffusivity tensor. We show, by means of analysis and numerical experiments, that subsampling of the data is necessary for the accurate estimation of the eddy diffusivity. The optimal sampling rate depends on the detailed properties of the velocity field. Furthermore, we show that averaging over the data only marginally reduces the bias of the estimator due to the multiscale structure of the problem, but that it does significantly reduce the effect of observation error

Compatible finite element methods for numerical weather prediction

This article takes the form of a tutorial on the use of a particular class of mixed finite element methods, which can be thought of as the finite element extension of the C-grid staggered finite difference method. The class is often referred to as compatible finite elements, mimetic finite elements, discrete differential forms or finite element exterior calculus. We provide an elementary introduction in the case of the one-dimensional wave equation, before summarising recent results in applications to the rotating shallow water equations on the sphere, before taking an outlook towards applications in three-dimensional compressible dynamical cores.Comment: To appear in ECMWF Seminar proceedings 201

Improved surface quality of anisotropically etched silicon {111} planes for mm-scale integrated optics

We have studied the surface quality of millimeter-scale optical mirrors produced by etching CZ and FZ silicon wafers in potassium hydroxide to expose the $\{111\}$ planes. We find that the FZ surfaces have four times lower noise power at spatial frequencies up to $500\, {mm}^{-1}$. We conclude that mirrors made using FZ wafers have higher optical quality

Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

Relationship Between Quantitative MRI Biomarkers and Patient-Reported Outcome Measures After Cartilage Repair Surgery: A Systematic Review.

Background:Treatment of articular cartilage injuries remains a clinical challenge, and the optimal tools to monitor and predict clinical outcomes are unclear. Quantitative magnetic resonance imaging (qMRI) allows for a noninvasive biochemical evaluation of cartilage and may offer advantages in monitoring outcomes after cartilage repair surgery. Hypothesis:qMRI sequences will correlate with early pain and functional measures. Study Design:Systematic review; Level of evidence, 3. Methods:A PubMed search was performed with the following search terms: knee AND (cartilage repair OR cartilage restoration OR cartilage surgery) AND (delayed gadolinium-enhanced MRI OR t1-rho OR T2 mapping OR dgemric OR sodium imaging OR quantitative imaging). Studies were included if correlation data were included on quantitative imaging results and patient outcome scores. Results:Fourteen articles were included in the analysis. Eight studies showed a significant relationship between quantitative cartilage imaging and patient outcome scores, while 6 showed no relationship. T2 mapping was examined in 11 studies, delayed gadolinium-enhanced MRI of cartilage (dGEMRIC) in 4 studies, sodium imaging in 2 studies, glycosaminoglycan chemical exchange saturation transfer (gagCEST) in 1 study, and diffusion-weighted imaging in 1 study. Five studies on T2 mapping showed a correlation between T2 relaxation times and clinical outcome scores. Two dGEMRIC studies found a correlation between T1 relaxation times and clinical outcome scores. Conclusion:Multiple studies on T2 mapping, dGEMRIC, and diffusion-weighted imaging showed significant correlations with patient-reported outcome measures after cartilage repair surgery, although other studies showed no significant relationship. qMRI sequences may offer a noninvasive method to monitor cartilage repair tissue in a clinically meaningful way, but further refinements in imaging protocols and clinical interpretation are necessary to improve utility

Arthroscopic Anterior Shoulder Stabilization With Incorporation of a Comminuted Bony Bankart Lesion.

Bony Bankart lesions are a common finding in patients with anterior glenohumeral dislocation. Although there are no defined guidelines, small bony Bankart fractures are typically treated arthroscopically with suture anchors. The 2 main techniques used are double- and single-row suture anchor stabilization, with debate over superiority. Biomechanical studies have shown improved reduction and stabilization with the double-row over the single-row suture anchor technique; however, this has not been reported for small or comminuted bony fragments. Both techniques have shown promising preliminary clinical outcomes. In this Technical Note, we describe our preferred technique for arthroscopic instability repair using a single-row all-suture anchor method with the incorporation of a comminuted bony Bankart fragment in the lateral decubitus position

Compatible finite element spaces for geophysical fluid dynamics

This is the final version. Available from Oxford University Press via the DOI in this record.Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the non-linear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarize analytic results about dispersion relations and conservation properties and present new results on approximation properties in three dimensions on the sphere and on hydrostatic balance properties