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.

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

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

Supernovae in the Central Parsec: A Mechanism for Producing Spatially Anisotropic Hypervelocity Stars

Several tens of hyper-velocity stars (HVSs) have been discovered escaping our Galaxy. These stars share a common origin in the Galactic centre and are distributed anisotropically in Galactic longitude and latitude. We examine the possibility that HVSs may be created as the result of supernovae occurring within binary systems in a disc of stars around Sgr A* over the last 100 Myr. Monte Carlo simulations show that the rate of binary disruption is ~10^-4 yr^-1, comparable to that of tidal disruption models. The supernova-induced HVS production rate (\Gamma_HVS) is significantly increased if the binaries are hardened via migration through a gaseous disc. Moderate hardening gives \Gamma_HVS ~ 2*10^-7 yr^-1 and an estimated population of ~20 HVSs in the last 100 Myr. Supernova-induced HVS production requires the internal and external orbital velocity vectors of the secondary binary component to be aligned when the binary is disrupted. This leaves an imprint of the disc geometry on the spatial distribution of the HVSs, producing a distinct anisotropy.Comment: 7 pages, 4 figures. Accepted for publication in the Astrophysical Journa

On the accretion mode of the intermediate polar V1025 Centauri

The long white-dwarf spin periods in the magnetic cataclysmic variables EX Hya and V1025 Cen imply that if the systems possess accretion discs then they cannot be in equilibrium. It has been suggested that instead they are discless accretors in which the spin-up torques resulting from accretion are balanced by the ejection of part of the accretion flow back towards the secondary. We present phase-resolved spectroscopy of V1025 Cen aimed at deducing the nature of the accretion flow, and compare this with simulations of a discless accretor. We find that both the conventional disc-fed model and the discless-accretor model have strengths and weaknesses, and that further work is needed before we can decide which applies to V1025 Cen.Comment: 9 pages, 8 figures, To appear in MNRAS, includes low-res figures to reduce siz

Outbursts of Young Stellar Objects

We argue that the outbursts of the FU Orionis stars occur on timescales which are much longer than expected from the standard disc instability model with \alpha_{c} \gtrsim 10^{-3}. The outburst, recurrence, and rise times are consistent with the idea that the accretion disc in these objects is truncated at a radius R_{i} \sim 40 \rsun. In agreement with a number of previous authors we suggest that the inner regions of the accretion discs in FU Ori objects are evacuated by the action of a magnetic propeller anchored on the central star. We develop an analytic solution for the steady state structure of an accretion disc in the presence of a central magnetic torque, and present numerical calculations to follow its time evolution. These calculations confirm that a recurrence time that is consistent with observations can be obtained by selecting appropriate values for viscosity and magnetic field strength.Comment: 13 pages, 7 figures, accepted by MNRA

VLA observations of a sample of galaxies with high far-infrared luminosities

Preliminary results are presented from a radio survey of galaxies detected by the IRAS minisurvey. It was found that the main difference between galaxies selected in the far infrared and those selected in the optical is that the former have higher radio luminosities and that the radio emission is more centrally concentrated. There is some evidence that the strong central radio sources in the galaxies selected in the infrared are due to star formation, the star formation rate divided by the volume in which the star formation is occuring is 100 to 1000 times greater in the galaxies selected in the infrared than in the disks of normal galaxies

Magnetic Cataclysmic Variable Accretion Flows

We have used a magnetic accretion model to investigate the accretion flows of magnetic cataclysmic variables (mCVs) throughout a range of parameter space. The results of our numerical simulations demonstrate that broadly four types of flow are possible: discs, streams, rings and propellers. We show that the equilibrium spin periods in asynchronous mCVs, for a given orbital period and magnetic moment, occur where the flow changes from a type characterised by spin-up (i.e. disc or stream) to one characterised by spin-down (i.e. propeller or ring). 'Triple points' occur in the plane of spin-to-orbital period ratio versus magnetic moment, at which stream-disc-propeller flows or stream-ring-propeller flows can co-exist. The first of these is identified as corresponding to when the corotation radius is equal to the circularisation radius, and the second as where the corotation radius is equal to the distance from white dwarf to the L1 point. If mCVs 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 < 0.5 will be stream-like, and those with Pspin/Porb ~ 0.5 will be ring-like. In each case, some material is also lost from the binary in order to maintain angular momentum balance. The spin to orbital period ratio at which the systems transition between these flow types decreases as the mass ratio of the stellar components increases, and vice versa