Hydrodynamic instability in warped astrophysical discs

Warped astrophysical discs are usually treated as laminar viscous flows, which have anomalous properties when the disc is nearly Keplerian and the viscosity is small: fast horizontal shearing motions and large torques are generated, which cause the warp to evolve rapidly, in some cases at a rate that is inversely proportional to the viscosity. However, these flows are often subject to a linear hydrodynamic instability, which may produce small-scale turbulence and modify the large-scale dynamics of the disc. We use a warped shearing sheet to compute the oscillatory laminar flows in a warped disc and to analyse their linear stability by the Floquet method. We find widespread hydrodynamic instability deriving from the parametric resonance of inertial waves. Even very small, unobservable warps in nearly Keplerian discs of low viscosity can be expected to generate hydrodynamic turbulence, or at least wave activity, by this mechanism.Comment: 17 pages, 7 figures, revised version, to be published in MNRA

The linear stability of dilute particulate rings

Irregular structure in planetary rings is often attributed to the intrinsic instabilities of a homogeneous state undergoing Keplerian shear. Previously these have been analysed with simple hydrodynamic models. We instead employ a kinetic theory, in which we solve the linearised moment equations derived in Shu and Stewart 1985 for a dilute ring. This facilitates an examination of velocity anisotropy and non-Newtonian stress, and their effects on the viscous and viscous/gravitational instabilities thought to occur in Saturn's rings. Because we adopt a dilute gas model, the applicability of our results to the actual dense rings of Saturn are significantly curtailled. Nevertheless this study is a necessary preliminary before an attack on the difficult problem of dense ring dynamics. We find the Shu and Stewart formalism admits analytic stability criteria for the viscous overstability, viscous instability, and thermal instability. These criteria are compared with those of a hydrodynamic model incorporating the effective viscosity and cooling function computed from the kinetic steady state. We find the two agree in the `hydrodynamic limit' (i.e. many collisions per orbit) but disagree when collisions are less frequent, when we expect the viscous stress to be increasingly non-Newtonian and the velocity distribution increasingly anisotropic. In particular, hydrodynamics predicts viscous overstability for a larger portion of parameter space. We also numerically solve the linearised equations of the more accurate Goldreich and Tremaine 1978 kinetic model and discover its linear stability to be qualitatively the same as that of Shu and Stewart's. Thus the simple collision operator adopted in the latter would appear to be an adequate approximation for dilute rings, at least in the linear regime

Viscous overstability and eccentricity evolution in three-dimensional gaseous discs

We investigate the growth or decay rate of the fundamental mode of even symmetry in a viscous accretion disc. This mode occurs in eccentric discs and is known to be potentially overstable. We determine the vertical structure of the disc and its modes, treating radiative energy transport in the diffusion approximation. In the limit of very long radial wavelength, an analytical criterion for viscous overstability is obtained, which involves the effective shear and bulk viscosity, the adiabatic exponent and the opacity law of the disc. This differs from the prediction of a two-dimensional model. On shorter wavelengths (a few times the disc thickness), the criterion for overstability is more difficult to satisfy because of the different vertical structure of the mode. In a low-viscosity disc a third regime of intermediate wavelengths appears, in which the overstability is suppressed as the horizontal velocity perturbations develop significant vertical shear. We suggest that this effect determines the damping rate of eccentricity in protoplanetary discs, for which the long-wavelength analysis is inapplicable and overstability is unlikely to occur on any scale. In thinner accretion discs and in decretion discs around Be stars overstability may occur only on the longest wavelengths, leading to the preferential excitation of global eccentric modes.Comment: 11 pages, 8 figure

Disorder driven destruction of a phase transition in a superconductor

We investigate the effects of disorder on a layered superconductor. The clean system is known to have a first order phase transition which is clearly identified by a sharp peak in the specific heat. The peak is lost abruptly as the strength of the disorder is increased. Hence, for strong disorder there is no phase transition as a function of temperature but merely a crossover which is still detectable in the IV characteristic.Comment: 3 pages REVTeX , 5 figure

Local and global dynamics of warped astrophysical discs

Astrophysical discs are warped whenever a misalignment is present in the system, or when a flat disc is made unstable by external forces. The evolution of the shape and mass distribution of a warped disc is driven not only by external influences but also by an internal torque, which transports angular momentum through the disc. This torque depends on internal flows driven by the oscillating pressure gradient associated with the warp, and on physical processes operating on smaller scales, which may include instability and turbulence. We introduce a local model for the detailed study of warped discs. Starting from the shearing sheet of Goldreich & Lynden-Bell, we impose the oscillating geometry of the orbital plane by means of a coordinate transformation. This warped shearing sheet (or box) is suitable for analytical and computational treatments of fluid dynamics, magnetohydrodynamics, etc., and it can be used to compute the internal torque that drives the large-scale evolution of the disc. The simplest hydrodynamic states in the local model are horizontally uniform laminar flows that oscillate at the orbital frequency. These correspond to the nonlinear solutions for warped discs found in previous work by Ogilvie, and we present an alternative derivation and generalization of that theory. In a companion paper we show that these laminar flows are often linearly unstable, especially if the disc is nearly Keplerian and of low viscosity. The local model can be used in future work to determine the nonlinear outcome of the hydrodynamic instability of warped discs, and its interaction with others such as the magnetorotational instability.Comment: 17 pages, 10 figures, revised version, to be published in MNRA

Thermodynamics of the dead-zone inner edge in protoplanetary disks

In protoplanetary disks, the inner boundary between the turbulent and laminar regions could be a promising site for planet formation, thanks to the trapping of solids at the boundary itself or in vortices generated by the Rossby wave instability. At the interface, the disk thermodynamics and the turbulent dynamics are entwined because of the importance of turbulent dissipation and thermal ionization. Numerical models of the boundary, however, have neglected the thermodynamics, and thus miss a part of the physics. The aim of this paper is to numerically investigate the interplay between thermodynamics and dynamics in the inner regions of protoplanetary disks by properly accounting for turbulent heating and the dependence of the resistivity on the local temperature. Using the Godunov code RAMSES, we performed a series of 3D global numerical simulations of protoplanetary disks in the cylindrical limit, including turbulent heating and a simple prescription for radiative cooling. We find that waves excited by the turbulence significantly heat the dead zone, and we subsequently provide a simple theoretical framework for estimating the wave heating and consequent temperature profile. In addition, our simulations reveal that the dead-zone inner edge can propagate outward into the dead zone, before staling at a critical radius that can be estimated from a mean-field model. The engine driving the propagation is in fact density wave heating close to the interface. A pressure maximum appears at the interface in all simulations, and we note the emergence of the Rossby wave instability in simulations with extended azimuth. Our simulations illustrate the complex interplay between thermodynamics and turbulent dynamics in the inner regions of protoplanetary disks. They also reveal how important activity at the dead-zone interface can be for the dead-zone thermodynamic structure.Comment: 16 pages, 16 figures. Accepted in Astronomy and Astrophysic

A novel bidding method for combined heat and power units in district heating systems

We propose a bidding method for the participation of combined heat and power (CHP) units in the day-ahead electricity market. More specifically, we consider a district heating system where heat can be produced by CHP units or heat-only units, e.g., gas or wood chip boilers. We use a mixed-integer linear program to determine the optimal operation of the portfolio of production units and storages on a daily basis. Based on the optimal production of subsets of units, we can derive the bidding prices and amounts of electricity offered by the CHP units for the day-ahead market. The novelty about our approach is that the prices are derived by iteratively replacing the production of heat-only units through CHP production. This results in an algorithm with a robust bidding strategy that does not increase the system costs even if the bids are not won. We analyze our method on a small realistic test case to illustrate our method and compare it with other bidding strategies from literature, which consider CHP units individually. The analysis shows that considering a portfolio of units in a district heating system and determining bids based on replacement of heat production of other units leads to better results