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On uniqueness of transfer rates in magnetohydrodynamic turbulence

Published online by Cambridge University Press:  11 October 2019

Franck Plunian Open the ORCID record for Franck Plunian [Opens in a new window] ,
Rodion Stepanov  and
Mahendra Kumar Verma
Franck Plunian*
Affiliation:
Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, IRD, IFSTTAR, ISTerre, 38000 Grenoble, France
Rodion Stepanov
Affiliation:
Perm National Research Polytechnic University, Institute of Continuous Media Mechanics, Korolyov 1, Perm, 614013, Russia
Mahendra Kumar Verma
Affiliation:
Department of Physics, Indian Institute of Technology, Kanpur 208016, India
*
†Email address for correspondence: [email protected]
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Abstract

In hydrodynamic and MHD (magnetohydrodynamic) turbulence, formal expressions for the transfer rates rely on integrals over wavenumber triads $(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ satisfying $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$. As an example $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ denotes the kinetic energy transfer rate to the mode $\boldsymbol{k}$, from the two other modes in the triad, $\boldsymbol{p}$ and $\boldsymbol{q}$. However as noted by Kraichnan (Phys. Rev., vol. 111, 1958, pp. 1747–1747), in $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$, what fraction of the energy transferred to the mode $\boldsymbol{k}$ originated from $\boldsymbol{p}$ and which from $\boldsymbol{q}$ is unknown. Such an expression is thus incongruent with the customary description of turbulence in terms of two-scale energy exchange. Notwithstanding this issue, Dar et al. (Physica D, vol. 157 (3), 2001, pp. 207–225) further decomposed these transfers into separate contributions from $\boldsymbol{p}$-to-$\boldsymbol{k}$ and $\boldsymbol{q}$-to-$\boldsymbol{k}$, thus introducing the concept of mode-to-mode transfers that they applied to MHD turbulence. Doing so, they had to set aside additional transfers circulating within each triad, but failed to calculate them. In the present paper we explain how to derive the complete expressions of the mode-to-mode transfers, including the circulating transfers. We do it for kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic energy, magnetic energy and magnetic helicity in MHD turbulence. We find that the degree of non-uniqueness of the energy transfers derived from the induction equation is a priori higher than the one derived from the Navier–Stokes equations. However, separating the contribution of magnetic advection from magnetic stretching, the energy mode-to-mode transfer rates involving the magnetic field become uniquely defined, in striking contrast to the hydrodynamic case. The magnetic helicity mode-to-mode transfer rate is also found to be uniquely defined, contrary to kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates have the same properties as mode-to-mode transfer rates. Finally calculating the fluxes, we show how the circulating transfers cancel in accordance with conservation laws.

Keywords

plasma dynamicsplasma flowsplasma nonlinear phenomena
Type
Research Article
Information
Journal of Plasma Physics , Volume 85 , Issue 5 , October 2019 , 905850507
Copyright
© Cambridge University Press 2019 

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References

Alexakis, A., Mininni, P. D. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys. Rev. E 72 (4), 046301.Google Scholar
Aluie, H. & Eyink, G. L. 2009 Localness of energy cascade in hydrodynamic turbulence. II. Sharp spectral filter. Phys. Fluids 21 (11), 115108.Google Scholar
Aluie, H. & Eyink, G. L. 2010 Scale locality of magnetohydrodynamic turbulence. Phys. Rev. Lett. 104, 081101.Google Scholar
Batchelor, G. K. 1953 Theory of Homogeneous Turbulence. Cambridge University Press.Google Scholar
Carati, D., Debliquy, O., Knaepen, B., Teaca, B. & Verma, M. 2006 Energy transfers in forced mhd turbulence. J. Turbul. 7, 1–12.Google Scholar
Dar, G., Verma, M. K. & Eswaran, V. 2001 Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results. Physica D 157 (3), 207–225.Google Scholar
Domaradzki, J. A. 1992 Nonlocal triad interactions and the dissipation range of isotropic turbulence. Phys. Fluids A 4 (9), 2037–2045.Google Scholar
Domaradzki, J. A. & Rogallo, R. S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2 (3), 413–426.Google Scholar
Kraichnan, R. H. 1958 Irreversible statistical mechanics of incompressible hydromagnetic turbulence. Phys. Rev. 109, 1407–1422.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5 (4), 497–543.Google Scholar
Kumar, A., Chatterjee, A. G. & Verma, M. K. 2014 Energy spectrum of buoyancy-driven turbulence. Phys. Rev. E 90, 023016.Google Scholar
Mininni, P., Alexakis, A. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. II. Kinematic dynamo. Phys. Rev. E 72 (4), 046302.Google Scholar
Mininni, P. D. 2011 Scale interactions in magnetohydrodynamic turbulence. Annu. Rev. Fluid Mech. 43 (1), 377–397.Google Scholar
Müller, U., Stieglitz, R. & Horanyi, S. 2004 A two-scale hydromagnetic dynamo experiment. J. Fluid Mech. 498, 3171.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press.Google Scholar
Stieglitz, R. & Müller, U. 2001 Experimental demonstration of a homogeneous dynamo. Phys. Fluids 13, 561–564.Google Scholar
Teaca, B., Verma, M. K. & Carati, D. 2009a Energy transfers in magnetohydrodynamic turbulence in presence of an ambiant imposed magnetic field. Phys. AUC 19, 22–34.Google Scholar
Teaca, B., Verma, M. K., Knaepen, B. & Carati, D. 2009b Energy transfer in anisotropic magnetohydrodynamic turbulence. Phys. Rev. E 79 (4), 046312.Google Scholar
Verma, M. K. 2004 Statistical theory of magnetohydrodynamic turbulence: recent results. Phys. Rep. 401, 229–380.Google Scholar
Verma, M. K., Ayyer, A., Debliquy, O., Kumar, S. & Chandra, A. V. 2005 Local shell-to-shell energy transfer via nonlocal interactions in fluid turbulence. Pramana 65 (2), 297.Google Scholar
Yokoyama, N. & Takaoka, M. 2017 Integrated analysis of energy transfers in elastic-wave turbulence. Phys. Rev. E 96, 023106.Google Scholar