# Publications by Steven Balbus

## Evolution of relativistic thin discs with a finite ISCO stress: I. Stalled accretion

Monthly Notices of the Royal Astronomical Society Oxford University Press **489** (2019) 132–142-

We present solutions to the relativistic thin disc evolutionary equation using an α-model for the turbulent stress tensor. Solutions with a finite stress at the innermost stable circular orbit (ISCO) give rise to bolometric light curves with a shallow power-law time dependence, in good agreement with those observed in tidal disruption events. A self-similar model based on electron scattering opacity, for example, yields a power-law index of −11/14, as opposed to −19/16 for the case of zero ISCO stress. These solutions correspond to an extended period of relaxation of the evolving disc which, like the light curves they produce, is not sustainable indefinitely. Cumulative departures from the approximation of exact circular orbits cause the power-law index to evolve slowly with time, leading eventually to the steeper fall-off associated with traditional zero ISCO stress models. These modified solutions are discussed in detail in a companion paper.

## Evolution of relativistic thin discs with a finite ISCO stress: II. Late time behaviour

Monthly Notices of the Royal Astronomical Society Oxford University Press **489** (2019) 143–152-

We present solutions to the relativistic thin disc evolutionary equation using a modified description of the mean fluid flow within the disc. The model takes into account the effects of sub-circular velocities in the innermost disc regions, and resolves otherwise unsustainable behaviour present in simple finite innermost stable circular orbit (ISCO) stress disc models. We show that the behaviour of a relativistic thin disc evolving with a finite ISCO stress is comprised of three distinct stages which join the ordinarily distinct finite and vanishing ISCO stress solutions into a fully continuous model parametrization. The most important prediction of our model is the existence of an intermediate stage of ‘stalled accretion’, controlled by a single dimensionless parameter. The hallmarks of this evolutionary phase appear to have been seen in General Relativistic MHD simulations as well as in the late time X-ray observations of tidal disruption events, but dedicated simulations and extended observations are needed for a deeper understanding.

## The evolution of Kerr discs and late-time tidal disruption event light curves

Monthly Notices of the Royal Astronomical Society Oxford University Press **481** (2018) 3348–3356-

An encounter between a passing star and a massive black hole at the centre of a galaxy, a so-called tidal disruption event or TDE, may leave a debris disc that subsequently accretes onto the hole. We solve for the time evolution of such a TDE disc, making use of an evolutionary equation valid for both the Newtonian and Kerr regimes. The late time luminosity emergent from such a disc is of interest as a model diagnostic, as it tends to follow a power law decline. The original simple ballistic fallback model, with equal mass in equal energy intervals, produces a −5/3 power law, while standard viscous disc descriptions yield a somewhat more shallow decline, with an index closer to −1.2. Of four recent, well-observed tidal disruption event candidates however, all had fall-off power law indices smaller than 1 in magnitude. In this work, we revisit the problem of thin disc evolution, solving this reduced problem in full general relativity. Our solutions produce power law indices that are in much better accord with observations. The late time observational data from many TDEs are generally supportive, not only of disc accretion models, but of finite stress persisting down to the innermost stable circular orbit.

## The general relativistic thin disc evolution equation

Monthly Notices of the Royal Astronomical Society Oxford University Press **471** (2017) 4832-4838

In the classical theory of thin disc accretion discs, the constraints of mass and angular momentum conservation lead to a diffusion-like equation for the turbulent evolution of the surface density. Here, we revisit this problem, extending the Newtonian analysis to the regime of Kerr geometry relevant to black holes. A diffusion-like equation once again emerges, but now with a singularity at the radius at which the effective angular momentum gradient passes through zero. The equation may be analysed using a combination of WKB, local techniques, and matched asymptotic expansions. It is shown that imposing the boundary condition of a vanishing stress tensor (more precisely the radial-azimuthal component thereof) allows smooth stable modes to exist external to the angular momentum singularity, the innermost stable circular orbit, while smoothly vanishing inside this location. The extension of the disc diffusion equation to the domain of general relativity introduces a new tool for numerical and phenomenolgical studies of accretion discs, and may prove to be a useful technique for understanding black hole X-ray transients.

## Demonstration of a magnetic Prandtl number disc instability from first principles

Monthly Notices of the Royal Astronomical Society Oxford University Press **472** (2017) 3021–3028-

Understanding what determines the strength of MHD turbulence in accretion discs is a question of fundamental theoretical and observational importance. In this work we investigate whether the dependence of the turbulent accretion disc stress (α) on the magnetic Prandtl number (Pm) is sufficiently sensitive to induce thermal-viscous instability using 3D MHD simulations. We first investigate whether the α-Pm dependence, found by many previous authors, has a physical or numerical origin by conducting a suite of local shearing-box simulations. We find that a definite α-Pm dependence persists when simultaneously increasing numerical resolution and decreasing the absolute values of both the viscous and resistive dissipation coefficients. This points to a physical origin of the α-Pm dependence. Using a further set of simulations which include realistic turbulent heating and radiative cooling, and by giving Pm a realistic physical dependence on the plasma temperature and density, we demonstrate that the α-Pm dependence is sufficiently strong to lead to a local instability. We confirm that the instability manifests itself as an unstable limit cycle by mapping the local thermal-equilibrium curve of the disc. This is the first self-consistent MHD simulation demonstrating the Pm instability from first principles. This result is important because a physical Pm instability could lead to the global propagation of heating and cooling fronts and a transition between disc states on timescales compatible with the observed hard/soft state transitions in black hole binaries.

## When is high Reynolds number shear flow not turbulent?

JOURNAL OF FLUID MECHANICS **824** (2017) 1-4

## An accretion disc instability induced by a temperature sensitive alpha parameter (vol 441, pg 681, 2014)

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **464** (2017) 4544-4544

## Surprises in astrophysical gasdynamics.

Reports on progress in physics. Physical Society (Great Britain) **79** (2016) 066901-

Much of astrophysics consists of the study of ionized gas under the influence of gravitational and magnetic fields. Thus, it is not possible to understand the astrophysical universe without a detailed knowledge of the dynamics of magnetized fluids. Fluid dynamics is, however, a notoriously tricky subject, in which it is all too easy for one's a priori intuition to go astray. In this review, we seek to guide the reader through a series of illuminating yet deceptive problems, all with an enlightening twist. We cover a broad range of topics including the instabilities acting in accretion discs, the hydrodynamics governing the convective zone of the Sun, the magnetic shielding of a cooling galaxy cluster, and the behaviour of thermal instabilities and evaporating clouds. The aim of this review is to surprise and intrigue even veteran astrophysical theorists with an idiosyncratic choice of problems and counterintuitive results. At the same time, we endeavour to bring forth the fundamental ideas, to set out important assumptions, and to describe carefully whatever novel techniques may be appropriate to the problem at hand. By beginning at the beginning, and analysing a wide variety of astrophysical settings, we seek not only to make this review suitable for fluid dynamic veterans, but to engage novice recruits as well with what we hope will be an unusual and instructive introduction to the subject.

## The Goldreich-Schubert-Fricke instability in stellar radiative zones

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **460** (2016) 338-344

## Simplified derivation of the gravitational wave stress tensor from the linearized Einstein field equations.

Proceedings of the National Academy of Sciences National Academy of Sciences (2016)

A conserved stress energy tensorfor weak field gravitational waves propagating in vacuum is derived directly from the linearized general relativistic wave equation alone, for an arbitrary gauge. In any harmonic gauge, the form of the tensor leads directly to the classical expression for the outgoing wave energy. The method described here, however, is a much simpler,shorter, and more physically motivated approach than is the customary procedure, which involves a lengthy and cumbersome second-order (in wave-amplitude) calculation starting with the Einstein tensor. Our method has the added advantage of exhibiting the direct coupling between the outgoing wave energy flux and the work done by the gravitational field on the sources. For nonharmonic gauges, the directly derived wave stress tensor has an apparent index asymmetry. This coordinate artifact may be straightforwardly removed, and the symmetrized (still gauge-invariant) tensor then takes on its widely used form. Angular momentum conservation follows immediately. For any harmonic gauge, however, the stress tensor found is manifestly symmetric from the start, and its derivation depends, in its entirety, on the structure of the linearized wave equation.

## The radiative zone of the Sun and the tachocline: stability of baroclinic patterns of differential rotation

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **457** (2016) 1711-1721

## Differential rotation and radiative equilibrium in the Sun: is the tachocline spreading?

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **448** (2015) 2077-2084

## INEFFICIENT DRIVING OF BULK TURBULENCE BY ACTIVE GALACTIC NUCLEI IN A HYDRODYNAMIC MODEL OF THE INTRACLUSTER MEDIUM

ASTROPHYSICAL JOURNAL **815** (2015) ARTN 41

## An accretion disc instability induced by a temperature sensitive a parameter

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **441** (2014) 681-689

## On the high-frequency spectrum of a classical accretion disc

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **444** (2014) L54-L57

## Dynamical, biological and anthropic consequences of equal lunar and solar angular radii

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES **470** (2014) ARTN 20140263

## Axisymmetric and non-axisymmetric magnetostrophic MRI modes

PHYSICS OF THE EARTH AND PLANETARY INTERIORS **223** (2013) 21-31

## On the behaviour of the magnetorotational instability when the Rayleigh criterion is violated

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **423** (2012) L50-L54

## The dynamics of inner dead-zone boundaries in protoplanetary discs

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **424** (2012) 1977-1990

## The stability of stratified, rotating systems and the generation of vorticity in the Sun

MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY **426** (2012) 1546-1557