Abstract
A shell model for magnetohydrodynamics (MHD) is derived directly from the dynamical system driving the evolution of three helical modes interacting in a triad. The use of helical modes implies that two shell variables are required for the velocity as well as for the magnetic field. The advantage of the method is the automatic conservation of all the ideal quadratic MHD invariants. The number of coupling constants is however larger than in traditional shell models. This difficulty is worked around by introducing an averaging procedure that allows to derive the shell model coupling constants directly from the MHD equations. The resulting shell model is used to explore the influence of a helical forcing on the global properties of MHD turbulence close to the onset of the dynamo regime.
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References
Frisch U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995)
Gletzer E.B.: System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dokl. 18, 216–217 (1973)
Lorenz E.N.: Low order models representing realizations of turbulence. J. Fluid Mech. 55, 545–563 (1972)
Yamada M., Ohkitani K.: Lyapounov spectrum of a chaotic model of three-dimensional turbulence. J. Phys. Soc. Jpn. 56, 4210–4213 (1987)
L’vov V.S., Podivilov E., Pomyalov A., Procaccia I., Vandembroucq D.: Improved shell model of turbulence. Phys. Rev. E 58, 1811–1822 (1998)
L’vov V.S., Podivilov E., Procaccia I.: Hamiltonian structure of the Sabra shell model of turbulence: exact calculation of an anomalous scaling exponent. Europhys. Lett. 46, 609–612 (1999)
Gloaguen C., Léorat J., Pouquet A., Grappin R.: A scalar model for MHD turbulence. Physica D 17, 154–182 (1985)
Frick P., Sokoloff S.: Cascade and dynamo action in a shell model of magnetohydrodynamic turbulence. Phys. Rev. E 57, 4155–4164 (1998)
Stepanov R., Plunian F.: Fully developed turbulent dynamo at low magnetic prandtl numbers. J. Turbul. 7(39), (2006)
Lessinnes, T., Verma, M., Carati, D.: Energy transfers in shell models of magnetohydrodynamic turbulence. Phys. Rev. E arxiv:0807.5076 (2008)
Stepanov R., Plunian F.: Phenomenology of turbulent dynamo growth and saturation. Astrophys. J. 680, 809–815 (2008)
Benzi R., Biferale L., Kerr R.M., Trovatore E.: Helical shell models for three-dimensional turbulence. Phys. Rev. E 53, 3541–3550 (1996)
Waleffe F.: The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4, 350–363 (1992)
Lessinnes, T., Carati, D.: Helical shell models for MHD turbulence. In: Proceeding of the 7th Int. PAMIR Conf. vol. 2, pp. 513–524 (2008); Lessinnes, T., Carati, D.: Helical shell models for MHD turbulence. Magnetohydrodynamics 45(2), 193–202 (2009)
Plunian F., Stepanov R.: A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence. New J. Phys. 9, 294–319 (2007)
Ottinger J.L., Carati D.: Shell models for plasma turbulence. Phys. Rev. E 48, 2955–2965 (1993)
Giuliani P., Carbone V.: A note on shell models for mhd turbulence. Europhys. Lett. 43(5), 527–532 (1998)
Gailitis A., Lielausis O., Platacis E., Dement’ev S., Cifersons A., Gerbeth G., Gundrum T., Stefani F., Christen M., Hänel H., Will G.: Detection of a flow induced magnetic field eigenmode in the riga dynamo facility. Phys. Rev. Lett. 84, 4365–4368 (2000)
Gailitis A., Lielausis O., Dement’ev S., Platacis E., Cifersons A., Gerbeth G., Gundrum T., Stefani F., Christen M., Will G.: Detection of a flow induced magnetic field eigenmode in the riga dynamo facility. Phys. Rev. Lett. 88, 3024–3027 (2001)
Müller U., Stieglitz R.: The Karlsruhe dynamo experiment. Nonlinear Process. Geophys. 9, 165–170 (2002)
Benzi R., Ciliberto S., Tripiccione R., Baudet C., Massoaioli F., Succi S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48(1), R29–R32 (1993)
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Lessinnes, T., Plunian, F. & Carati, D. Helical shell models for MHD. Theor. Comput. Fluid Dyn. 23, 439–450 (2009). https://doi.org/10.1007/s00162-009-0165-y
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DOI: https://doi.org/10.1007/s00162-009-0165-y