Astron. Astrophys. 337, 962-965 (1998)

3. Theoretical considerations

The 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 R for which no stable steady state solution of the basic equilibrium equations is possible. In this case, it is suggested (Chen et al. 1995, Narayan 1997) that the flow is forced into a time-dependent variable mode, satisfying the required in the mean. This provides further motivation to test the applicability of sandpile-type models. In this section, we concentrate on the model for MHD turbulent energy dissipation in accretion discs presented by Geertsema & Achterberg (1992), because it appears to give rise to SOC. While the link to cataclysmic variable and dwarf nova observations (but not, explicitly, SOC) was made by Geertsema & Achterberg (1992) themselves, we believe that this work has much wider implications: for the general question of SOC in accretion flows; for the connection between SOC and turbulence in general; for the role of SOC in plasma physics, since it represents the first instance where SOC has been observed in a mathematical model derivable from the fundamental equations of MHD; and for the role of SOC in terrestial experimental systems - real sandpiles and ricepiles, as distinct from mathematical idealisations thereof - where uncertainties about its scope remain (Nagel 1992, Feder 1995, Frette et al. 1996, Christensen et al. 1996).

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 k, resembling the removal of material from the edge of a sandpile. The MHD accretion model is, of course, much more complex than the simple sandpile algorithms considered so far in the literature. Two fields, and , each with three components, are involved rather than the single height parameter of conventional sandpile algorithms, and the time evolution is governed by differential equations rather than difference equations. Nevertheless, both the MHD accretion disc and simple sandpile models appear to exhibit similar self-organised, critical behaviour, supporting the claim often made in the sandpile literature that SOC is universal phenomenon shared by large classes of cellular automata.

© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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