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


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Erratum

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

L. Nocera 1 and M.S. Ruderman 2

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

  5. The quantity [FORMULA] has to be substituted for the quantity [FORMULA] 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 [FORMULA] is [FORMULA]."

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

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

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

    [EQUATION]

    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 [FORMULA] and R conditions (79) are rewritten as

    [EQUATION]

    where [FORMULA] are obtained by inverting the relations

    [EQUATION]

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

    [EQUATION]

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

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

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

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

  10. The interval of [FORMULA] 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 [FORMULA] instead of [FORMULA].

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

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

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

    • on line 6: [FORMULA] instead of [FORMULA];

    • on line 9: [FORMULA] instead of [FORMULA];

    • on line 11: [FORMULA] instead of [FORMULA];

    • on line 18: [FORMULA] instead of [FORMULA].

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

  14. The new Fig. 3 is:

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

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

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

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


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© European Southern Observatory (ESO) 2000

Online publication: February 23, 19101
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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001

helpdesk.link@springer.de