Abstract
We re-analyze the Olesen arguments on the self-similarity properties of freely evolving, nonhelical magnetohydrodynamic turbulence. We find that a necessary and sufficient condition for the kinetic and magnetic energy spectra to evolve self-similarly is that the initial velocity and magnetic field are not homogeneous functions of space of different degree, to wit, the initial energy spectra are not simple powers of the wavenumber with different slopes. If, instead, they are homogeneous functions of the same degree, the evolution is self-similar, it proceeds through selective decay, and the order of homogeneity fixes the exponents of the power laws according to which the kinetic and magnetic energies and correlation lengths evolve in time. If just one of them is homogeneous, the evolution is self-similar and such exponents are completely determined by the slope of that initial spectrum which is a power law. The latter evolves through selective decay, while the other spectrum may eventually experience an inverse transfer of energy. Finally, if the initial velocity and magnetic field are not homogeneous functions, the evolution of the energy spectra is still self-similar but, this time, the power-law exponents of energies and correlation lengths depend on a single free parameter which cannot be determined by scaling arguments. Also in this case, an inverse transfer of energy may in principle take place during the evolution of the system.
Highlights
The Olesen solutionIt is well known that the MHD equations [10], under the scaling transformations x → x, t → 1−h t, admit a solution of the type [4]
It is well known that the MHD equations [10], under the scaling transformations x → x, t → 1−h t, admit a solution of the type [4]z( x, 1−ht) = h z(x, t), (1)where > 0 and h are real parameters, and z stands for the velocity of bulk motion, v, or for the magnetic fieldB
6 Case 4 in the case where at the initial time both v and B are homogeneous functions, but with different degrees, no self-similar solutions of MHD equations exist in the turbulence regime
Summary
It is well known that the MHD equations [10], under the scaling transformations x → x, t → 1−h t, admit a solution of the type [4]. The above “Olesen solution”, is valid provided that the dissipative parameters ν (the kinematic viscosity) and η (the resistivity) scale as ν( 1−ht) = 1+h ν(t) and η( 1−ht) = 1+h η(t) Differentiating the latter equations with respect to , and putting = 1 afterwards, we get ν(t) ∝ η(t) ∝ t(1+h)/(1−h). Let us take, for example, h = −1, so that the MHD equations are invariant under the re-scaling L → L and t → 2t of lengths L and times t This means that if, for example, we progressively double any physical size L inside the system taking = 1, 2, 4, 8, ..., the latter will appear the same if we look at it, respectively, at the times t∗, 22t∗, 42t∗, 82t∗,.
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