Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-13T01:40:46.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Laws for third-order moments in homogeneous anisotropic incompressible magnetohydrodynamic turbulence

Published online by Cambridge University Press:  31 July 2008

J. J. PODESTA
J. J. PODESTA*
Affiliation:
Space Science Center, University of New Hampshire, Durham, NH 03824, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known that Kolmogorov's four-fifths law for statistically homogeneous and isotropic turbulence can be generalized to anisotropic turbulence. This fundamental result for homogeneous anisotropic turbulence says that in the inertial range the divergence of the vector third-order moment 〈|δv(r)|2δv(r)〉 is constant and is equal to -4ϵ, where ϵ is the dissipation rate of the turbulence. This law can be extended to incompressible magnetohydrodyamic (MHD) turbulence where statistical isotropy is often not valid due, for example, to the presence of a large-scale magnetic field. Laws for anisotropic incompressible MHD turbulence were first derived by Politano and Pouquet. In this paper, the laws for vector third-order moments in homogeneous non-isotropic incompressible MHD turbulence are derived by a technique due to Frisch that clarifies the relationship between the energy flux in Fourier space and the vector third-order moments in physical space. This derivation is different from the original derivation of Politano and Pouquet which is based on the Kármán–Howarth equation, and provides some new physical insights. Separate laws are derived for the cascades of energy, cross-helicity and magnetic-helicity, the three ideal invariants of incompressible MHD for flows in three dimensions. These laws are of fundamental importance in the theory of MHD turbulence where non-isotropic turbulence is much more prevalent than isotropic turbulence.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 609 , 25 August 2008 , pp. 171 - 194
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Antonia, R. A., Ould-Rouis, M., Anselmet, F. & Zhu, Y. 1997 Analogy between predictions of Kolmogorov and Yaglom. J. Fluid Mech. 332, 395–409.CrossRefGoogle Scholar
Biskamp, D. 2003 Magnetohydrodynamic Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Hill, R. J. 1997 Applicability of Kolmogorov's and Monin's equations of turbulence. J. Fluid Mech. 353, 67–81.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR, 32, 16–18, trans. in Proc. R. Soc. Lond. A 434, 15–17 (1991).Google Scholar
Matthaeus, W. H., Ghosh, S., Oughton, S. & Roberts, D. A. 1996 a Anisotropic three-dimensional MHD turbulence. J. Geophys. Res. 101, 7619–7630.CrossRefGoogle Scholar
Matthaeus, W. H., Bieber, J. W. & Zank, G. P. 1996 b Anisotropic turbulence in the solar wind. Solar Wind Eight (ed. Winterhalter, D., Gosling, J. T., Habbal, S. R., Kurth, W. S. & Neugebauer, M.). AIP Conf. Proc. vol. 382, pp. 256–259.Google Scholar
Meneguzzi, M., Frisch, U. & Pouquet, A. 1981 Helical and nonhelical turbulent dynamos. Phys. Rev. Lett. 47, 1060–1064.CrossRefGoogle Scholar
Monin, A. S. 1959 Theory of locally isotropic turbulence. Dokl. Akad. Nauk. SSSR 125, 515–518.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT Press.Google Scholar
Montgomery, D. & Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825–831.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1982 Two-and-a-half-dimensional magnetohydrodynamic turbulence. Phys. Fluids 25, 345CrossRefGoogle Scholar
Oughton, S., Priest, E. R. & Matthaeus, W. H. 1994 The influence of a mean magnetic field on three-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 280, 95–117.CrossRefGoogle Scholar
Podesta, J. J., Forman, M. A. & Smith, C. W. 2007 Anisotropic form of third-order moments and relationship to the cascade rate in axisymmetric magnetohydrodynamic turbulence. Phys. Plasmas 14, 092305.CrossRefGoogle Scholar
Politano, H. & Pouquet, A. 1998 a Dynamical length scales for turbulent magnetized flows. Geophys. Res. Lett. 25, 273–276.CrossRefGoogle Scholar
Politano, H. & Pouquet, A. 1998 b Von Kármán–Howarth equation for magnetohydrodynamics and its consequences on third-order longitudinal structure and correlation functions. Phys. Rev. E 57, 21.Google Scholar
Politano, H., Gomez, T. & Pouquet, A. 2003 Von Kármán–Howarth relationship for helical magnetohydrodynamic flows. Phys. Rev. E 68, 026315.Google ScholarPubMed
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525–547.CrossRefGoogle Scholar
Taylor, J. B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741–763.CrossRefGoogle Scholar