Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay

Abstract

We perform numerical simulations of decaying hydrodynamic and magnetohydrodynamic turbulence. We classify our time-dependent solutions by their evolutionary tracks in parametric plots between instantaneous scaling exponents. We find distinct classes of solutions evolving along specific trajectories toward points on a line of self-similar solutions. These trajectories are determined by the underlying physics governing individual cases, while the infrared slope of the initial conditions plays only a limited role. In the helical case, even for a scale-invariant initial spectrum (inversely proportional to wavenumber k), the solution evolves along the same trajectory as for a Batchelor spectrum (proportional to k 4 ). [7], galaxy clusters [8], and the early Universe [9,10]. In the latter case, cosmological magnetic fields generated in the early Universe provide the initial source of turbulence, which leads to a growth of the correlation length by an inverse cascade mechanism [11], in addition to the general cosmological expansion of the Universe. In the last two decades, this topic has gained significant attention [12]. The time span since the initial magnetic field generation is enormous, but it is still uncertain whether it is long enough to produce fields at sufficiently large length scales to explain the possibility of contemporary magnetic fields in the space between clusters of galaxies [13].In this Letter, we use direct numerical simulations (DNS) of both hydrodynamic (HD) and magnetohydrodynamic (MHD) decaying turbulence to classify different types by their decay behavior. The decay is characterized by the temporal change of the kinetic energy spec- * Electronic address: brandenb@nordita.org † Electronic address: tinatin@andrew.cmu.edu trum, E K (k, t), and, in MHD, also by the magnetic energy spectrum, E M (k, t). Here, k is the wavenumber and t is time. In addition to the decay laws of the energies E i (t) = E i (k, t) dk, with i = K or M for kinetic and magnetic energies, there are the kinetic and magnetic integral scales,We quantify the decay by the instantaneous scaling exponents p(t) ≡ d ln E/d ln t and q(t) ≡ d ln ξ/d ln t. Thus, we study the decay behaviors by plotting p(t) vs. q(t) in a parametric representation. The pq diagram turns out to be a powerful diagnostic tool. Earlier work [8,14,15] has suggested that the decay behavior, and thus the positions of solutions in the pq diagram, depend on the exponent α for initial conditions of the form E ∼ k α e −k/k0 , where k 0 is a cutoff wavenumber. Motivated by earlier findings [2,11] of an inverse cascade in decaying MHD turbulence, Olesen considered the time-dependent energy spectra E(k, t) to be of the form [15]where ξ(t) ∝ t q , with q being an as yet undetermined scaling exponent, and ψ is a function that depends on the dissipative and turbulent processes that lead to a departure from a powerlaw at large k. Moreover, the slope ψ ′ ≡ dψ/dκ with κ = kξ must vanish for κ → 0. This turns out to be a critical restriction. Olesen then makes use of the f...

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“…(5) Decrease of the chemical potential and saturation of the large-scale chiral dynamo. Although the magnetic field cannot grow any further, the spectrum continues to move to smaller wavenumbers in a shape-invariant fashion (see Brandenburg & Kahniashvili 2017).…”

Advances in mean-field dynamo theory and applications to astrophysical turbulence

“…where ξ M is the magnetic correlation length, φ M is a universal function for the magnetic spectra at all times, and β is an exponent that depends mostly on the physics governing the decay and, in some cases, also on the initial conditions (Olesen 1997). For example, β = 0 in the fully helical case when A · B is conserved, β = 1 when A 2 is conserved, β = 2 when the Saffman integral is conserved, and β = 4 when the Loitsiansky integral is conserved; see Brandenburg & Kahniashvili (2017) for details. Assuming that ξ M (t) ∝ t q with exponent q, we then expect the magnetic energy to decay like…”

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“…The values of β i are believed to depend on the physics that governs a particular case [54]. It is convenient to define and plot instantaneous scaling exponents as p i (t) = d ln E i /dt versus q i (t) = d ln ξ i /dt for i = M and K and discuss the evolution of the point…”