## 3. Theoretical considerationsThe identification of mechanisms responsible for the dissipation of
shear flow energy within an accretion disc is an active field of
research, because of its importance in determining transport across
the disc and onto the compact object. If, as has been suggested (Bak
et al 1988, Mineshige et al. 1994), the combined global effects of
local transport physics result in SOC, observational signatures of the
type described above will emerge. It is widely accepted (see, for
example, Longair (1994) and Narayan (1997)) that anomalous viscosity
caused by MHD turbulence probably plays an important role in the flux
of angular momentum within accretion discs. MHD turbulence can arise
naturally in accretion discs, see for example the instability
mechanisms proposed by Tagger et al. (1990), Vishniac et al. (1990),
and Balbus & Hawley (1991). It has also been pointed out (Chen et
al. 1995) that there exist ranges of accretion rate
and disc radius Before considering the model of Geertsema & Achterberg (1992) in greater detail, let us turn to its results. Fig. 12 of Geertsema & Achterberg (1992) shows the calculated times series of energy dissipation events within the disc. We note that this is qualitatively very similar to the observed time series of energy dissipation measured in an experimental ricepile displaying SOC, Fig. 2c of Frette et al. (1996), and in a related mathematical model of Dendy & Helander (1997, Fig. 3, and 1998, Fig. 5). More quantitatively, Fig. 13 of Geertsema & Achterberg (1992) shows the power spectrum of energy dissipated by MHD disc turbulence, which displays the dependence characteristic of SOC; compare, for example, Fig. 3 of Frette et al. (1996) and Fig. 3 of Christensen et al. (1996) which show measured spectra of energy dissipation and particle transit times, respectively, in SOC ricepiles. Given the clear indications of SOC emerging from the MHD turbulence model of Geertsema & Achterberg (1992), it will be of interest to establish how it has arisen. A full explanation must await diagnostic analysis of the code runs generated by this model. Pending this, we conclude the present section by seeking to identify some of the relevant salient features. In outline, the model of Geertsema & Achterberg (1992) is constructed as follows. The accretion disc is regarded as a differentially rotating turbulent MHD fluid, and is modelled by a reduced system of equations reflecting the most important features of three-dimensional MHD turbulence. The disc is assumed to be thin in comparison with its diameter, and the flow is taken to be subsonic and hence incompressible. The possible existence of a large-scale magnetic field is neglected, and the only turbulent structures considered have a length scale shorter than the height of the disc, making the turbulence essentially three-dimensional. In a coordinate system rotating with the disc angular velocity at radius , the simplified MHD equations for the flow velocity and the normalised magnetic field vector are where and are the local radial and azimuthal coordinates respectively, is the resistivity, and is the viscosity coefficient. is the total scalar pressure, while the term contains the off-diagonal terms of the MHD stress tensor. As these equations are still too complex for a numerical treatment over a sufficiently large dynamic range in scales, they were simplified further following a suggestion by Desnyanski & Novikov (1974) in the context of hydrodynamics, and later applied to MHD by Gloaguen et al. (1985). In this approximation, the space of wave vectors is discretized into a finite set of , and the non-linear interaction between different components of the Fourier transforms of and is described by the set of equations In the three-dimensional generalisation of this approximation scheme, the discretisation is made by dividing the -space into into spherical shells, thus discarding the information regarding the direction of . This allows a greatly simplified system of nonlinear equations to be written down, analogous to that of Gloaguen et al. (1985), which are supplemented with additional terms to account for the effects of differential rotation. These equations, which are taken to model the turbulent cascade of
MHD, were solved numerically by Geertsema & Achterberg (1992).
They found that the turbulent shear stress can be very large, and has
large, chaotic fluctuations on time scales of a few rotation periods.
Perhaps the most striking feature of the simulations is, however, that
the dissipation of energy at the smallest scales of the turbulent
cascade is very intermittent. Energy is released in avalanches with a
wide range of sizes, and the power spectrum of the dissipation rate
obeys a power law over nearly two orders of
magnitude in intensity, see their Fig. 13. This behaviour is, as
already stated, similar to that of simple mathematical sandpile
models. While a power law is to be expected from any scale-free model,
in particular a Kolmogorov-type one in the inertial range, we note
that the similarities between this system and mathematical sandpiles
apparently extend further. Both are fundamentally governed by
nearest-neighbour interactions between a discrete number of nodes. In
the MHD accretion model, these reside in -space,
so the interaction is between wave modes with similar wavelengths
rather than between adjacent regions in real space. The dissipation is
provided by viscosity and resistivity at large © European Southern Observatory (ESO) 1998 Online publication: August 27, 1998 |