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Keywords:

  • accretion, accretion discs;
  • instabilities;
  • MHD;
  • turbulence

ABSTRACT

This is the first of a series of papers aimed at developing and interpreting simulations of protoplanets interacting with turbulent accretion discs. In this first paper we study the turbulent disc models prior to the introduction of a perturbing protoplanet. We study cylindrical disc models in which a central domain is in Keplerian rotation and unstable to the magnetorotational instability. Models of varying disc size and aspect ratio H/r are considered with magnetic fields having zero net flux. We relate the properties of the turbulent models to classical viscous disc theory. All models were found to attain a turbulent state in their Keplerian domains with volume averaged stress parameter α∼ 5 × 10−3. At any particular time the vertically and azimuthally averaged value exhibited large fluctuations in radius. However, an additional time average over periods exceeding three orbital periods at the outer boundary of the Keplerian domain resulted in a more smoothly varying quantity with radial variations within a factor of 2 or so.

The vertically and azimuthally averaged radial velocity showed much larger spatial and temporal fluctuations, requiring additional time averaging for at least seven to eight orbital periods at the outer boundary of the Keplerian domain to limit them. Comparison with the value derived from the averaged stress using viscous disc theory yielded schematic agreement for feasible averaging times but with some indication that the effects of residual fluctuations remained.

The behaviour described above must be borne in mind when considering laminar disc simulations with anomalous Navier–Stokes viscosity. This is because the operation of a viscosity, as in classical viscous disc theory with anomalous viscosity coefficient, cannot apply to a turbulent disc undergoing rapid changes due to external perturbation. The classical theory can only be used to describe the time averaged behaviour of the parts of the disc that are in a statistically steady condition long enough for appropriate averaging to be carried out.