Erratum to: Benchmarking the unified nonlinear transport theory for Goldreich–Sridhar turbulence
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Erratum to: Astrophys Space Sci (2013) 344:187–191 DOI 10.1007/s10509-012-1298-9
In Shalchi (2013) the test-particle simulations performed by Sun and Jokipii (2011) were compared with the so-called Unified Non-Linear Transport (UNLT) theory developed in Shalchi (2010). These numerical and analytical results were obtained by employing a turbulence model based on Goldreich and Sridhar (1995) scaling. It was shown that the UNLT theory agrees very well with the simulations. Therefore, it was concluded that UNLT theory is an accurate analytical theory for particle transport across the mean magnetic field.
Sun and Jokipii (2011) listed parallel and perpendicular diffusion coefficients whereas the UNLT theory contains the corresponding mean free paths. These two parameters are related to each other via λ_{∥}=3κ_{∥}/v and λ_{⊥}=3κ_{⊥}/v, respectively. The mean free paths λ_{∥} and λ_{⊥} have length units and, therefore, these parameters are usually normalized with respect the a characteristic scale of the turbulence L (see Shalchi 2013 for details). The parameter v denotes the particle speed. In Shalchi (2013) the approximation v≈c (where c is the speed of light) was employed to convert the simulations to mean free paths corresponding to the assumption that all particles move with relativistic speeds. This assumption was incorrect and, thus, we provide the correct table and a revised figure in this erratum.
Kinetic energies and the correct products vL obtained from Sun and Jokipii (2011). Here we have used cL=45×10^{20} cm^{2}/s
Run | E_{ kin } in MeV | v/c | vL in 10^{20} cm^{2}/s |
---|---|---|---|
1 | 1.0 | 0.046 | 2.08 |
2 | 3.16 | 0.082 | 3.68 |
3 | 10.0 | 0.14 | 6.52 |
4 | 31.6 | 0.25 | 11.39 |
5 | 100 | 0.43 | 19.27 |
6 | 316 | 0.66 | 29.87 |
7 | 1000 | 0.88 | 39.38 |
Run | R_{ L }/L | (δB/B_{0})^{2} | λ_{∥}/L | λ_{⊥}/L | λ_{⊥}/λ_{∥} |
---|---|---|---|---|---|
1 | 0.019 | 1.0 | 1.5111 | 0.0467 | 0.0309 |
2 | 0.034 | 1.0 | 1.7961 | 0.0525 | 0.02923 |
3 | 0.061 | 1.0 | 2.2107 | 0.0626 | 0.02832 |
4 | 0.109 | 1.0 | 3.2971 | 0.0714 | 0.021655 |
5 | 0.198 | 1.0 | 4.2487 | 0.0835 | 0.01965 |
6 | 0.37 | 1.0 | 6.7682 | 0.1191 | 0.01759 |
7 | 0.759 | 1.0 | 12.4819 | 0.1790 | 0.01434 |
One can easily see from Fig. 1 that the agreement between UNLT theory and the simulations is still remarkable. The only difference between Fig. 1 of Shalchi (2013) and the new figure is that the simulations are now closer to the analytical results obtained for a^{2}=1 rather then a^{2}=1/3. Therefore, the conclusions of Shalchi (2013) are still correct. UNLT theory agrees very well with test-particle simulations performed for Goldreich–Sridhar turbulence.
Footnotes
- 1.
Here we have used Astronomical Units which are related to metric units via 1 AU≃1.496×10^{13} cm≡1.496×10^{11} m.
Notes
Acknowledgements
Andreas Shalchi acknowledges support by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
References
- Goldreich, P., Sridhar, S.: Astrophys. J. 438, 763 (1995)ADSCrossRefGoogle Scholar
- Shalchi, A.: Astrophys. J. 720, L127 (2010)ADSCrossRefGoogle Scholar
- Shalchi, A.: Astrophys. Space Sci. 344, 187 (2013)ADSCrossRefGoogle Scholar
- Sun, P., Jokipii, J.R.: In: 32nd International Cosmic Ray Conference, Beijing (2011)Google Scholar