Volume 503, Number 2, August IV 2009
|Page(s)||589 - 590|
|Published online||09 July 2009|
Drift instabilities in the solar corona within the multi-fluid description
R. Mecheri 1 - E. Marsch2
1 - Queen Mary University of London, Astronomy Unit,
Mile End Road, E1 4NS, London, UK
2 - Max-Planck institut für Sonnensystemforschung, Max-Planck Strasse 2, 37191 Katlenburg-Lindau, Germany
A&A 481, 853-860 (2008), DOI: 10.1051/0004-6361:20079221
The purpose of this erratum is to point out and to correct some unfortunate typographical mistakes, as well as algebraic errors contained in our paper ``Drift instabilities in the solar corona within the multi-fluid description'' published in A&A 481, 853 (2008). We also want to clarify some important issues with respect to the assumed background state on which the drift-wave instability develops.
The first point is that the nonuniformity of the background density
in the corona model considered should be expressed more precisely as
whereby the density increases linearly in the coordinate x with respect to some arbitrary reference location x0, at which the density is assumed to have the constant value and not n0j, which is the previous notation that led to some confusion in the subsequent perturbation expansion about the nonuniform equilibrium background state. Therefore, the density as given by Eq. (3) in the original paper should be understood as describing (like in (1)) its dependence on x, starting from position x0. Because of the above mistake, there is a factor (1+(x-x0)/L)-1 missing and also a typographical sign error (it should read + rather than -) in the expression for the drift velocity (9) in Mecheri & Marsch (2008). However, the right sign has been used throughout their numerical calculations.
Also, the expression of the drift velocity should not contain the
since we assumed an ideal gas law
the mass density. Consequently, the acoustic speed
Csj is given by
as quoted in the original paper. Thus the correct expression of the
drift velocity is
The second major issue is related to the properties of the assumed background plasma state in the paper by Mecheri & Marsch (2008). If the background electric field is considered to be zero, the equilibrium momentum equation is given by
Thus in equilibrium the background pressure (and density) gradient must be across , which has components in both the x- and z-directions. In our original paper, only a pressure gradient in the x-direction was considered. However, consideration of an additional gradient in the z-direction does not change the main equations, (10) to (12), of the paper by Mecheri & Marsch (2008). Moreover, the above system of equations gives
and then from (3) we have
Thus, the insertion of the expression for into the last equation above leads exactly to the expression of the drift velocity given in our present Eq. (2).
However, because of the equilibrium diamagnetic current density in
the background magnetic field must also be
perturbed according to Ampère's law. This subtle issue has not
been investigated and discussed in the original paper, in which it
was assumed (but not mentioned explicitly) that the field
perturbation should be negligible due to the low beta of the coronal
plasma, where the magnetic pressure dominates the gas pressure. To
clarify this point, we now investigate the corresponding equation.
For simplicity, we consider that the background plasma has a
with only one component in the
z-direction and a density and pressure gradient in the
x-direction. Thus for the curl of
which by use of (3) can in turn be written as total pressure equilibrium in the form:
Integration of Eq. (5) with respect to the reference position x0, which has a reference field strength , leads to
It can be clearly seen that, for small plasma beta , the spatial variation of the background magnetic field, induced by the background diamagnetic current density, is very small, and thus . Due to this additional dependence of on x, the principal question arises whether the neglection of the term by comparison with the term in the momentum equation is still justified or not. If we insert expression (6), we obtain from (2) the gradient of the drift speed as
where we have used as obtained from (6). As the is small, we obtain the approximation
which agrees with the approximations adopted in our paper, in the sense that the wavelength of interest, , is much smaller than the non-uniformity length scale L. Indeed the perturbation wavelength in our model is close to the ion inertial length, , i.e., m in typical coronal conditions, and L=1 km.Consequently, the term can be safely neglected in comparison to the term in the momentum equation.
In conclusion, it turns out that the only difference with the original equations of the paper by Mecheri & Marsch (2008) is the missing term (1+(x-x0)/L)-1 in their expression for the drift velocity. This does not affect the result concerning the local perturbation analysis, since those were calculated at the reference location x0, where (1+(x-x0)/L)-1=1. These results were in turn used as initial conditions to solve the ray-tracing equations (as mentioned in Sect. 3.3 of the original paper). However, the results concerning the nonlocal (i.e., ray-tracing) analysis should be recalculated, since (1+(x-x0)/L)-1 is no longer equal to the unity when moving away from the reference location x0.
The ray-tracing recalculation should also take into account an additional variation of the background pressure (and density) in the z-direction, in order to consistently fulfill the equilibrium conditions (3) mentioned previously. Such a variation may also be represented by a linear profile (similar to the one in the x-direction). In this case the results of the local perturbation analysis will again remain unchanged; however, the extent of the region for which the new ray-tracing calculations are to be performed must be restricted to a smaller region (of 10 km), thus satisfying that the density is allowed to increase locally by at most a factor 10 (as mentioned at the end of Sect. 2 of the original paper). Considering this will prevent an increase in the density to unrealistically high coronal values. All this unfortunately was not adequately considered in the original paper.
We thank an unknown referee for his considerable efforts in reviewing our original paper with respect to these critical issues, and we appreciate his valuable comments that stimulated and greatly contributed to this erratum.
Copyright ESO 2009
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