Astron. Astrophys. 364, 931-932 (2000)

Erratum

On the steady state of nonlinear quasiresonant Alfvén oscillations in one-dimensional magnetic cavity

L. Nocera ¹ and M.S. Ruderman <sup>2

1</sup> Institute of Atomic and Molecular Physics, National Research Council, Via Giardino 7, 56127 Pisa, Italy
² Department of Applied Mathematic, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

Astron. Astrophys. 340, 287 (1998)

Send offprint requests to: M.S. Ruderman (sheffield.ac.uk)

The following corrections should be instroduced in the paper:

1. In Sect. 3, lines preceeding Eq. (22), it should be instead of n.

2. The sign of the right-hand sides of Eqs. (34), (36) and of the first sum on the right-hand side of Eq. (44) has to be changed.

3. There must be `+' instead of `-' between two terms in the large brackets in Eq. (49).

4. The quantity has to be substituted for the quantity in Eqs. (58), and (64), and on the left-hand side of Eq. (65). The quantity has to be substituted for the quantity on the right-hand side of Eq. (65).

5. The quantity has to be substituted for the quantity in Eqs. (73), (75), (76) and (82).

6. The second sentence in the last paragraph on p. 294 has to be: "The discriminant of the cubic equation is ."

7. The text starting from the second paragraph on p. 295 has to be:

   . There are three real roots to Eq. (77). It is obvious that when either or , while otherwise. Since , it follows that , so that always . Since (), we obtain that , while when , and when . Since , it follows that when either and , or and . Otherwise .

   Summarizing the analysis we state that there are three roots to Eq. (74) when

   

   Otherwise there is only one real root to Eq. (74). Examples of unique and triple solutions are shown in Figs. 4a and 5a respectively.

   In terms of and R conditions (79) are rewritten as

   

   where are obtained by inverting the relations

   

   Note that the dependences of on R are given by the asymptotic formulae

   

   valid for . The upper bound of R is necessary since the quasiresonant perturbation scheme adopted in Eq. (19) requires .

   It can be shown that the equation for has only one positive real root for all values of R when . The dependencies of on , and and on R for , are shown in Figs. 2 and 3 respectively.

8. The inequality on the line preceeding Eq. (85) has to be .

9. The equality on the line preceeding Eq. (88) has to be .

10. The interval of variation in Sect. 5.2 has to be [0.15,0.21] instead of [0.20,0.23].

11. Sect. 6:

    • On the the first line it has to be instead of .

    • On the eighth line it has to be instead of .

    • On the third line of the third paragraph it has to be instead of .

12. In Sect. 7, p. 298, the second colomn, it has to be

    • on line 6: instead of ;

    • on line 9: instead of ;

    • on line 11: instead of ;

    • on line 18: instead of .

13. Fig. 2 remains qualitatively the same, but varies from 0 till 0.70.

14. The new Fig. 3 is:

    Fig. 3. The dependence of and on R for . Unique (triple) solutions exist in the shaded (non-shaded) area.

15. Fig. 6 is practically the same, but and at the horisontal axis has to be substituted by and respectively.

16. Fig. 7 is practically the same, but in the caption it has to be instead of .

17. Fig. 8 is practically the same, but in the caption it has to be instead of .

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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