1,394 research outputs found
Nonlinear bending waves in Keplerian accretion discs
The nonlinear dynamics of a warped accretion disc is investigated in the
important case of a thin Keplerian disc with negligible viscosity and
self-gravity. A one-dimensional evolutionary equation is formally derived that
describes the primary nonlinear and dispersive effects on propagating bending
waves other than parametric instabilities. It has the form of a derivative
nonlinear Schroedinger equation with coefficients that are obtained explicitly
for a particular model of a disc. The properties of this equation are analysed
in some detail and illustrative numerical solutions are presented. The
nonlinear and dispersive effects both depend on the compressibility of the gas
through its adiabatic index Gamma. In the physically realistic case Gamma<3,
nonlinearity does not lead to the steepening of bending waves but instead
enhances their linear dispersion. In the opposite case Gamma>3, nonlinearity
leads to wave steepening and solitary waves are supported. The effects of a
small effective viscosity, which may suppress parametric instabilities, are
also considered. This analysis may provide a useful point of comparison between
theory and numerical simulations of warped accretion discs.Comment: 15 pages, 2 figures, to be published in MNRA
The non-linear fluid dynamics of a warped accretion disc
The dynamics of a viscous accretion disc subject to a slowly varying warp of
large amplitude is considered. Attention is restricted to discs in which
self-gravitation is negligible, and to the generic case in which the resonant
wave propagation found in inviscid Keplerian discs does not occur. The
equations of fluid dynamics are derived in a coordinate system that follows the
principal warping motion of the disc. They are reduced using asymptotic methods
for thin discs, and solved to extract the equation governing the warp. In
general, this is a wave equation of parabolic type with non-linear dispersion
and diffusion, which describes fully non-linear bending waves. This method
generalizes the linear theory of Papaloizou & Pringle (1983) to allow for an
arbitrary rotation law, and extends it into the non-linear domain, where it
connects with a generalized version of the theory of Pringle (1992). The
astrophysical implications of this analysis are discussed briefly.Comment: 23 pages, 5 figures, to be published in MNRA
The effect of planetary migration on the corotation resonance
The migration of a planet through a gaseous disc causes the locations of
their resonant interactions to drift and can alter the torques exerted between
the planet and the disc. We analyse the time-dependent dynamics of a
non-coorbital corotation resonance under these circumstances. The ratio of the
resonant torque in a steady state to the value given by Goldreich & Tremaine
(1979) depends essentially on two dimensionless quantities: a dimensionless
turbulent diffusion time-scale and a dimensionless radial drift speed. When the
drift speed is comparable to the libration speed and the viscosity is small,
the torque can become much larger than the unsaturated value in the absence of
migration, but is still proportional to the large-scale vortensity gradient in
the disc. Fluid that is trapped in the resonance and drifts with it acquires a
vortensity anomaly relative to its surroundings. If the anomaly is limited by
viscous diffusion in a steady state, the resulting torque is inversely
proportional to the viscosity, although a long time may be required to achieve
this state. A further, viscosity-independent, contribution to the torque comes
from fluid that streams through the resonant region. In other cases, torque
oscillations occur before the steady value is achieved. We discuss the
significance of these results for the evolution of eccentricity in
protoplanetary systems. We also describe the possible application of these
findings to the coorbital region and the concept of runaway (or type III)
migration. [Abridged]Comment: 15 pages, 6 figures, to be published in MNRA
A self-sustaining nonlinear dynamo process in Keplerian shear flows
A three-dimensional nonlinear dynamo process is identified in rotating plane
Couette flow in the Keplerian regime. It is analogous to the hydrodynamic
self-sustaining process in non-rotating shear flows and relies on the
magneto-rotational instability of a toroidal magnetic field. Steady nonlinear
solutions are computed numerically for a wide range of magnetic Reynolds
numbers but are restricted to low Reynolds numbers. This process may be
important to explain the sustenance of coherent fields and turbulent motions in
Keplerian accretion disks, where all its basic ingredients are present.Comment: 4 pages, 7 figures, accepted for publication in Physical Review
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