Optimal bounds with semidefinite programming: an application to stress driven shear flows

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler--Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.Comment: 17 pages; typos removed; extended discussion of linear matrix inequalities in Section III; revised argument in Section IVC, results unchanged; extended discussion of computational setup and limitations in Sectios IVE-IVF. Submitted to Phys. Rev.

Mortgage valuation report forms and the identification of subsidence

This paper examines whether surveyors engaged in mortgage valuation inspections using questionnaire style report forms supplied by lending institutions, are subject to an increased risk of liability in respect of identifying the present and future threat of subsidence to domestic properties. Analysis of the mortgage valuation report forms used by 34 different lending institutions, showed that 20% failed to ask any subsidence related questions, only 6% asked about the geology or soil type of the site, and only 9% asked about the location of trees relative to the building. Evaluation of the report forms showed that the type, quality and quantity of questioning were such that 24 out of the 34 were inadequate and unreliable, leaving the surveyor at an increased risk of litigation

Star formation in normal galaxies

The ways in which recent infrared observations, particularly by the Infrared Astronomy Satellite (IRAS), have influenced ideas about star formation in normal galaxies, are discussed

Atmospheres and Spectra of Strongly Magnetized Neutron Stars II: Effect of Vacuum Polarization

We study the effect of vacuum polarization on the atmosphere structure and radiation spectra of neutron stars with surface magnetic fields B=10^14-10^15 G, as appropriate for magnetars. Vacuum polarization modifies the dielectric property of the medium and gives rise to a resonance feature in the opacity; this feature is narrow and occurs at a photon energy that depends on the plasma density. Vacuum polarization can also induce resonant conversion of photon modes via a mechanism analogous to the MSW mechanism for neutrino oscillation. We construct atmosphere models in radiative equilibrium with an effective temperature of a few \times 10^6 K by solving the full radiative transfer equations for both polarization modes in a fully ionized hydrogen plasma. We discuss the subtleties in treating the vacuum polarization effects in the atmosphere models and present approximate solutions to the radiative transfer problem which bracket the true answer. We show from both analytic considerations and numerical calculations that vacuum polarization produces a broad depression in the X-ray flux at high energies (a few keV \la E \la a few tens of keV) as compared to models without vacuum polarization; this arises from the density dependence of the vacuum resonance feature and the large density gradient present in the atmosphere. Thus the vacuum polarization effect softens the high energy tail of the thermal spectrum, although the atmospheric emission is still harder than the blackbody spectrum because of the non-grey opacities. We also show that the depression of continuum flux strongly suppresses the equivalent width of the ion cyclotron line and therefore makes the line more difficult to observe.Comment: 21 pages, 21 figures; MNRAS; corrected minor typo

The Spin Period of EX Hydrae

We show that the spin period of the white dwarf in the magnetic CV EX Hydrae represents an equilibrium state in which the corotation radius is comparable with the distance from the white dwarf to the inner Lagrange point. We also show that a continuum of spin equilibria exists at which Pspin is significantly longer than \sim 0.1 Porb. Most systems occupying these equilibrium states should have orbital periods below the CV period gap, as observed.Comment: MNRAS, accepte

Atmospheres and Spectra of Strongly Magnetized Neutron Stars

We construct atmosphere models for strongly magnetized neutron stars with surface fields $B\sim 10^{12}-10^{15}$ G and effective temperatures $T_{\rm eff}\sim 10^6-10^7$ K. The atmospheres directly determine the characteristics of thermal emission from isolated neutron stars, including radio pulsars, soft gamma-ray repeaters, and anomalous X-ray pulsars. In our models, the atmosphere is composed of pure hydrogen or helium and is assumed to be fully ionized. The radiative opacities include free-free absorption and scattering by both electrons and ions computed for the two photon polarization modes in the magnetized electron-ion plasma. Since the radiation emerges from deep layers in the atmosphere with \rho\ga 10^2 g/cm$^3$, plasma effects can significantly modify the photon opacities by changing the properties of the polarization modes. In the case where the magnetic field and the surface normal are parallel, we solve the full, angle-dependent, coupled radiative transfer equations for both polarization modes. We also construct atmosphere models for general field orientations based on the diffusion approximation of the transport equations and compare the results with models based on full radiative transport. In general, the emergent thermal radiation exhibits significant deviation from blackbody, with harder spectra at high energies. The spectra also show a broad feature (\Delta E/\Ebi\sim 1) around the ion cyclotron resonance \Ebi=0.63 (Z/A)(B/10^{14}{G}) keV, where $Z$ and $A$ are the atomic charge and atomic mass of the ion, respectively; this feature is particularly pronounced when \Ebi\ga 3k\Teff. Detection of the resonance feature would provide a direct measurement of the surface magnetic fields on magnetars.Comment: 29 pages, 11 figures; corrected factor of 2 in He models: minor changes to figs 4 and 9 as a result; other very minor change

The Accretion Flows and Evolution of Magnetic Cataclysmic Variables

We have used a model of magnetic accretion to investigate the accretion flows of magnetic cataclysmic variables. Numerical simulations demonstrate that four types of flow are possible: discs, streams, rings and propellers. The fundamental observable determining the accretion flow, for a given mass ratio, is the spin-to-orbital period ratio of the system. If IPs are accreting at their equilibrium spin rates, then for a mass ratio of 0.5, those with Pspin/Porb < 0.1 will be disc-like, those with 0.1 < Pspin/Porb < 0.6 will be stream-like, and those with Pspin/Porb ~ 0.6 will be ring-like. The spin to orbital period ratio at which the systems transition between these flow types increases as the mass ratio of the stellar components decreases. For the first time we present evolutionary tracks of mCVs which allow investigation of how their accretion flow changes with time. As systems evolve to shorter orbital periods and smaller mass ratios, in order to maintain spin equilibrium, their spin-to-orbital period ratio will generally increase. As a result, the relative occurrence of ring-like flows will increase, and the occurrence of disc-like flows will decrease, at short orbital periods. The growing number of systems observed at high spin-to-orbital period ratios with orbital periods below 2h, and the observational evidence for ring-like accretion in EX Hya, are fully consistent with this picture.Comment: Accepted for publication in ApJ. 6 figures - included here at low resolutio