On differentially dissipative dynamical systems

Dissipativity is an essential concept of systems theory. The paper provides an extension of dissipativity, named differential dissipativity, by lifting storage functions and supply rates to the tangent bundle. Differential dissipativity is connected to incremental stability in the same way as dissipativity is connected to stability. It leads to a natural formulation of differential passivity when restricting to quadratic supply rates. The paper also shows that the interconnection of differentially passive systems is differentially passive, and provides preliminary examples of differentially passive electrical systems

Angular Momentum Transfer and Lack of Fragmentation in Self-Gravitating Accretion Flows

Rapid inflows associated with early galaxy formation lead to the accumulation of self-gravitating gas in the centers of proto-galaxies. Such gas accumulations are prone to non-axisymmetric instabilities, as in the well-known Maclaurin sequence of rotating ellipsoids, which are accompanied by a catastrophic loss of angular momentum (J). Self-gravitating gas is also intuitively associated with star formation. However, recent simulations of the infall process display highly turbulent continuous flows. We propose that J-transfer, which enables the inflow, also suppresses fragmentation. Inefficient J loss by the gas leads to decay of turbulence, triggering global instabilities and renewed turbulence driving. Flow regulated in this way is stable against fragmentation, whilst staying close to the instability threshold for bar formation -- thick self-gravitating disks are prone to global instabilities before they become unstable locally. On smaller scales, the fraction of gravitationally unstable matter swept up by shocks in such a flow is a small and decreasing function of the Mach number. We conclude counterintuitively that gas able to cool down to a small fraction of its virial temperature will not fragment as it collapses. This provides a venue for supermassive black holes to form via direct infall, without the intermediary stage of forming a star cluster. Some black holes could have formed or grown in massive halos at low redshifts. Thus the fragmentation is intimately related to J redistribution within the system: it is less dependent on the molecular and metal cooling but is conditioned by the ability of the flow to develop virial, supersonic turbulence.Comment: 5 pp., 1 figures, to be published by the Astrophysical Journal Letters. Minor corrections following the referee repor

The bar-mode instability in differentially rotating neutron stars: Simulations in full general relativity

We study the dynamical stability against bar-mode deformation of rapidly spinning neutron stars with differential rotation. We perform fully relativistic 3D simulations of compact stars with $M/R \geq 0.1$, where $M$ is the total gravitational mass and $R$ the equatorial circumferential radius. We adopt an adiabatic equation of state with adiabatic index $\Gamma=2$. As in Newtonian theory, we find that stars above a critical value of $\beta \equiv T/W$ (where $T$ is the rotational kinetic energy and $W$ the gravitational binding energy) are dynamically unstable to bar formation. For our adopted choices of stellar compaction and rotation profile, the critical value of $\beta = \beta_{dGR}$ is $\sim 0.24-0.25$, only slightly smaller than the well-known Newtonian value $\sim 0.27$ for incompressible Maclaurin spheroids. The critical value depends only very weakly on the degree of differential rotation for the moderate range we surveyed. All unstable stars form bars on a dynamical timescale. Models with sufficiently large $\beta$ subsequently form spiral arms and eject mass, driving the remnant to a dynamically stable state. Models with moderately large $\beta \gtrsim \beta_{dGR}$ do not develop spiral arms or eject mass but adjust to form dynamically stable ellipsoidal-like configurations. If the bar-mode instability is triggered in supernovae collapse or binary neutron star mergers, it could be a strong and observable source of gravitational waves. We determine characteristic wave amplitudes and frequencies.Comment: 17 pages, accepted for publication in AP

Nonlinear Development of the Secular Bar-mode Instability in Rotating Neutron Stars

We have modelled the nonlinear development of the secular bar-mode instability that is driven by gravitational radiation-reaction (GRR) forces in rotating neutron stars. In the absence of any competing viscous effects, an initially uniformly rotating, axisymmetric $n=1/2$ polytropic star with a ratio of rotational to gravitational potential energy $T/|W| = 0.181$ is driven by GRR forces to a bar-like structure, as predicted by linear theory. The pattern frequency of the bar slows to nearly zero, that is, the bar becomes almost stationary as viewed from an inertial frame of reference as GRR removes energy and angular momentum from the star. In this ``Dedekind-like'' state, rotational energy is stored as motion of the fluid in highly noncircular orbits inside the bar. However, in less than 10 dynamical times after its formation, the bar loses its initially coherent structure as the ordered flow inside the bar is disrupted by what appears to be a purely hydrodynamical, short-wavelength, ``shearing'' type instability. The gravitational waveforms generated by such an event are determined, and an estimate of the detectability of these waves is presented.Comment: 25 pages, 9 figures, accepted for publication in ApJ, refereed version, updated, for quicktime movie, see http://www.phys.lsu.edu/~ou/movie/fmode/new/fmode.b181.om4.2e5.mo

Gravito-inertial waves in a differentially rotating spherical shell

The gravito-inertial waves propagating over a shellular baroclinic flow inside a rotating spherical shell are analysed using the Boussinesq approximation. The wave properties are examined by computing paths of characteristics in the non-dissipative limit, and by solving the full dissipative eigenvalue problem using a high-resolution spectral method. Gravito-inertial waves are found to obey a mixed-type second-order operator and to be often focused around short-period attractors of characteristics or trapped in a wedge formed by turning surfaces and boundaries. We also find eigenmodes that show a weak dependence with respect to viscosity and heat diffusion just like truly regular modes. Some axisymmetric modes are found unstable and likely destabilized by baroclinic instabilities. Similarly, some non-axisymmetric modes that meet a critical layer (or corotation resonance) can turn unstable at sufficiently low diffusivities. In all cases, the instability is driven by the differential rotation. For many modes of the spectrum, neat power laws are found for the dependence of the damping rates with diffusion coefficients, but the theoretical explanation for the exponent values remains elusive in general. The eigenvalue spectrum turns out to be very rich and complex, which lets us suppose an even richer and more complex spectrum for rotating stars or planets that own a differential rotation driven by baroclinicity.Comment: 33 pages, 14 figures, accepted for publication in Journal of Fluid Mechanic

On the Formation and Evolution of Common Envelope Systems

We discuss the formation of a common envelope system following dynamically unstable mass transfer in a close binary, and the subsequent dynamical evolution and final fate of the envelope. We base our discussion on new three-dimensional SPH calculations that we have performed for a close binary system containing a $4\,M_\odot$ red giant with a $0.7\,M_\odot$ main-sequence star companion. The initial parameters are chosen to model the formation of a system resembling V~471~Tau, a typical progenitor of a cataclysmic variable binary. In our highest-resolution calculation, we find evidence for a corotating region of gas around the central binary. This is in agreement with the theoretical model proposed by Meyer \& Meyer-Hofmeister (1979) for the evolution of common envelope systems, in which this central corotating region is coupled to the envelope through viscous angular momentum transport only. We also find evidence that the envelope is convectively unstable, in which case the viscous dissipation time could be as short as $\sim100$ dynamical times, leading to rapid ejection of the envelope. For V~471~Tau, our results, and the observed parameters of the system, are entirely consistent with rapid envelope ejection on a timescale $\sim1\,$yr and an efficiency parameter $\alpha_{CE}\simeq1$.Comment: uses AAS latex macros v4, 36 pages with figures, submitted to ApJ, complete postscript also available at http://ensor.mit.edu/~rasio/paper

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature