From that point of view the question of chromospheric and coronal driving can in principle be reduced to a question of being able to understand and estimate the Poynting flux passing through the solar surface. In practice, this is not an easy task, and one can furthermore convincingly argue that the magnitude of the Poynting flux indeed depends as much on the sink (the chromosphere and corona) as it depends on the source (the subsurface layers of the convection zone).
If one assumes, for example, that one aspect of the subsurface driving is to twist the magnetic field that passes through the surface, then if there is only a weak resistance to the twist, so the twisting motion can proceed with almost no counteracting force, then very little work is performed, and consequently the Poynting flux through the solar surface must be small. Conversely, if there is a large resistance to the twisting motion, and strong counteracting forces develop, that must correspond to a large Poynting flux through the solar surface.
The key factor that regulates the magnitude of the Poynting flux is the angle that the magnetic field lines make relative to the surface, or relative to the motions that attempt to twist the field lines. If there is a strong resistance to the twisting motion the angle is changed – the field becomes twisted and provides a counteracting force, while if there is almost no resistance the field lines remain straight, and the magnetic field hardly transmits any Poynting flux.
Apart from the twist angle, the Poynting flux is also proportional to the energy density of the magnetic field and to the velocity with which the field is being transported. A knowledge of the these three factors on a boundary is in principle enough to compute the Poynting flux through the boundary. Scaling formulae that express these relations were presented by Parker (1983) and van Ballegooijen (1986). The main uncertainty that one encounters when trying to estimate actual heating rates is the angle factor. Galsgaard and Nordlund (1996) found that simple experiments with boundary driving adhere closely to a scaling law that may be interpreted to mean that the twist angle is always of a size that allows the magnetic field lines to twist around their neighbors about once, from end to end. This rule of thumb is consistent with the stability properties of twisted flux tube, which tend to become unstable (to kinking and similar phenomena) when twisted more than about once.
An important consequence of the scaling is that increased resistivity leads to a decrease of the energy dissipation, in that a larger resistivity allows magnetic field lines to diffuse more quickly, straighten out, and hence become less tilted at the driving boundary (Parker, 1988). Conversely, the work and dissipation obtained for a given resistivity is a lower bound on the work and dissipation obtained at very low resistivities. It is this property that gives hope that numerical experiments can provide accurate predictions of chromospheric and coronal heating.
Indeed, Gudiksen and Nordlund (2005a,b) were able to construct a 3D corona model where sufficient heat was provided exclusively by this mechanism. Synthetic diagnostics from the model (Peter et al., 2005, 2006) fits observed values very well.
The models by Gudiksen and Nordlund (2005a,b) had an artificial velocity field, but with statistical properties consistent with the observed solar surface velocity field, from granular to supergranular scales. Using a fully self-consistent subsurface velocity field, evolved simultaneously with the chromospheric and coronal dynamics requires higher numerical resolution, since one must ideally simultaneously resolve scales from sub-granular sizes up to supergranular or active region sizes. The first steps towards this ultimate goal have been taken (Hansteen and Gudiksen, 2005; Hansteen et al., 2007; Abbett, 2007). Similar studies concentrated on the chromosphere have been initiated by Wedemeyer et al. (2003); see also Schaffenberger et al. (2005) and Schaffenberger et al. (2006).
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