On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

Widespread evidence for heterogeneous accretion of the terrestrial planets and planetisimals

The abundance and relative proportion of highly siderophile elements (HSEs) in Earth’s mantle deviate from those predicted by low-pressure equilibrium partitioning between metal and silicate during formation of the core. For many elements, high-pressure equilibration in a deep molten silicate layer (or ‘magma ocean’) may account for this discrepancy [1], but some highly siderophile element abundances demand the late addition, a ‘late veneer’, of extraterrestrial material (i.e. heterogeneous accretion) after core formation was complete [2]. Siderophile elements in smaller asteroidal bodies will not be affected by high-pressure metal-silicate equilibration and so, with highly efficient core formation [3] and if a ‘late veneer’ is absent, significant differences in the proportions of HSEs can be anticipated. Here we present new HSE abundance and 187Os/188Os isotope data for basaltic meteorites, the HEDs (howardites, eucrites and diogenites thought to sample the asteroid 4 Vesta), anomalous eucrites (considered to be from distinct Vesta-like parent bodies) angrites and aubrites (from unidentified parent bodies) and SNCs (thought to be from Mars). Our data, taken with those for lunar rocks [4], demonstrate that these igneous meteorites all formed from mantle sources that possessed chondritic (i.e. primitive solar system) elemental and isotope compositions, indicating that late accretion is not unique to Earth, but is a common feature of differentiated planets and asteroidal bodies. Variations in the total HSE abundance suggest that the proportion of ‘late veneer’ added is a simple consequence of the size of each body (cross-section and/or gravitational-attraction), and may account for the volatile element budget, and the oxidationstate of Earth, Mars, the Moon and Vesta

Constructing sonified haptic line graphs for the blind student: first steps

Line graphs stand as an established information visualisation and analysis technique taught at various levels of difficulty according to standard Mathematics curricula. It has been argued that blind individuals cannot use line graphs as a visualisation and analytic tool because they currently primarily exist in the visual medium. The research described in this paper aims at making line graphs accessible to blind students through auditory and haptic media. We describe (1) our design space for representing line graphs, (2) the technology we use to develop our prototypes and (3) the insights from our preliminary work

A Detailed Investigation of Staged Normal Injection into a Mach 2 Flow

A study of the staged injection of two jets of air behind a rearward facing step into a Mach 2 flow was performed using the SPARK 3-D Navier-Stokes code. Calculated mole fraction distributions were compared with an extensive set of planar mole fraction measurements made with a laser induced iodine fluorescence technique. A statistical measure, the standard deviation, was used to help assess agreement between calculation and experiment. Overall, good agreement was found between calculated and measured values. Generally, agreement was better in the far field of the injectors. The effect of grid resolution was investigated by calculating solutions on grids of 60,000, 200,000, and 450,000 points. Differences in the solutions on the two finer grids were small. However, the mole fraction distributions were distinguishable. The effect of turbulence modeling was investigated by employing three different algebraic models for the jet turbulence: the Baldwin-Lomax model, the Prandtl mixing length model, and the Eggers mixing length model. Overall, the Eggers mixing length model was found to be superior for this case. Finally, the effect of the jet exit conditions was examined. A recently proposed Mach number distribution at the jet exit was found to slightly improve agreement between measurement and calculation

Rotational quenching rate coefficients for H_2 in collisions with H_2 from 2 to 10,000 K

Rate coefficients for rotational transitions in H_2 induced by H_2 impact are presented. Extensive quantum mechanical coupled-channel calculations based on a recently published (H_2)_2 potential energy surface were performed. The potential energy surface used here is presumed to be more reliable than surfaces used in previous work. Rotational transition cross sections with initial levels J <= 8 were computed for collision energies ranging between 0.0001 and 2.5 eV, and the corresponding rate coefficients were calculated for the temperature range 2 < T <10,000 K. In general, agreement with earlier calculations, which were limited to 100-6000 K, is good though discrepancies are found at the lowest and highest temperatures. Low-density-limit cooling functions due to para- and ortho-H_2 collisions are obtained from the collisional rate coefficients. Implications of the new results for non-thermal H_2 rotational distributions in molecular regions are also investigated

Westbrook's Molecular Gun: Discovery of Near-IR Micro-Structures in AFGL 618

We present high-sensitivity near-IR images of a carbon-rich proto-planetary nebula, AFGL 618, obtained with the Subaru Telescope. These images have revealed ``bullets'' and ``horns'' extending farther out from the edges of the previously known bipolar lobes. The spatial coincidence between these near-IR micro-structures and the optical collimated outflow structure, together with the detection of shock-excited, forbidden IR lines of atomic species, strongly suggests that these bullets and horns represent the locations from which [\ion{Fe}{2}] IR lines arise. We have also discovered CO clumps moving at $> 200$ km s$^{-1}$ at the positions of the near-IR bullets by re-analyzing the existing $^{12}$CO $J=1-0$ interferometry data. These findings indicate that the near-IR micro-structures represent the positions of shocked surfaces at which fast-moving molecular clumps interface with the ambient circumstellar shell.Comment: 2 figures. To appear in the ApJ Letter