Spectral direct numerical simulations of incompressible MHD turbulence at a resolution of up to 1024 3 collocation points are presented for a statistically isotropic system as well as for a setup with an imposed strong mean magnetic field. The spectra of residual energy, E (anisotropic case, perpendicular to the mean field direction). A model of dynamic equilibrium between kinetic and magnetic energy, based on the corresponding evolution equations of the eddy-damped quasi-normal Markovian (EDQNM) closure approximation, explains the findings. The assumed interplay of turbulent dynamo and Alfvén effect yields E R k ∼ kE 2 k which is confirmed by the simulations.The nonlinear behavior of turbulent plasmas gives rise to a variety of dynamical effects such as self-organization of magnetic confinement configurations in laboratory experiments [1], generation of stellar magnetic fields [2] or structure formation in the interstellar medium [3]. The understanding of these phenomena is incomplete as the same is true for many inherent properties of the underlying turbulence.Large-scale low-frequency plasma turbulence is treated in the magnetohydrodynamic (MHD) approximation describing the medium as a viscous and electrically resistive magnetofluid neglecting additional kinetic effects. Incompressiblity of the flow is assumed for the sake of simplicity. In this setting the nature of the turbulent energy cascade is a central and still debated issue with different phenomenologies being proposed [4,5,6,7,8] (cf.[9] for a review). The associated spectral dynamics of kinetic and magnetic energy, in spite of its comparable importance, has received less attention (as an exception see [10]).This Letter reports a spectral relation between residual and total energy,respectively, as well as the influence of an imposed mean magnetic field on the spectra. The proposed physical picture, which is confirmed by accompanying direct numerical simulations, embraces two-dimensional MHD turbulence, globally isotropic three-dimensional systems as well as turbulence permeated by a strong mean magnetic field.In the following reference is made to two highresolution pseudospectral direct numerical simulations of incompressible MHD turbulence which we regard as paradigms for isotropic (I) and anisotropic (II) MHD turbulence. The dimensionless MHD equationsare solved in a 2π-periodic cube with spherical mode truncation to reduce numerical aliasing errors [11]. The equations include the flow vorticity, ω = ∇ × v, the magnetic field expressed in Alfvén speed units, b, as well as dimensionless viscosity, µ, and resistivity, η. In simulation II forcing is applied by freezing the largest spatial scales of velocity and magnetic field. Simulation I evolves globally isotropic freely decaying turbulence represented by 10243 Fourier modes. The initial fields are smooth with random phases and fluctuation amplitudes following exp(−k 2 /(2k 2 0 )) with k 0 = 4. Total kinetic and magnetic energy are initially equal with also decreases during turbulence decay from 0.7...
The scaling properties of three-dimensional magnetohydrodynamic turbulence with finite magnetic helicity are obtained from direct numerical simulations using 512(3) modes. The results indicate that the turbulence does not follow the Iroshnikov-Kraichnan phenomenology. The scaling exponents of the structure functions can be described by a modified She-Leveque model zeta(p) = p/9+1-(1/3)(p/3), corresponding to basic Kolmogorov scaling and sheetlike dissipative structures. In particular, we find zeta(2) approximately 0.7, consistent with the energy spectrum E(k) approximately k(-5/3) as observed in the solar wind, and zeta(3) approximately 1, confirming a recent analytical result.
Direct numerical simulations of decaying and forced magnetohydrodynamic (MHD) turbulence without and with mean magnetic field are analyzed by higher-order two-point statistics. The turbulence exhibits statistical anisotropy with respect to the direction of the local magnetic field even in the case of global isotropy. A mean magnetic field reduces the parallel-field dynamics while in the perpendicular direction a gradual transition towards two-dimensional MHD turbulence is observed with k −3/2 inertial-range scaling of the perpendicular energy spectrum. An intermittency model based on the Log-Poisson approach, ζp = p/g 2 + 1 − (1/g) p/g , is able to describe the observed structure function scalings. PACS: 47.27Gs; 47.65+a; 47.27Eq; 52.35.Ra Turbulence is the natural state of many plasma flows observed throughout the universe, its statistical properties being essential for the theoretical understanding of, e.g., star-forming regions in the interstellar medium, the convection in planetary and stellar interiors, as well as the dynamics of stellar winds. The solar wind, in particular, represents the only source of in-situ measurements, since laboratory experiments are far from generating fully-developed turbulence at high magnetic Reynolds numbers. A simplified nonlinear model of turbulent plasmas is incompressible magnetohydrodynamics (MHD) [1]. In this framework the kinetic nature of microscopic processes responsible for, e.g., energy dissipation, is neglected when studying the fluid-like macroscopic plasma motions.The spatial similarity of incompressible MHD turbulence is usually investigated by considering two-point statistics of the Elsässer variables z ± = v ± B [2] combining velocity v and magnetic field B (given in Alfvén-speed units). Restricting consideration to turbulence with small cross helicity H C = V dV(v · B), V being the volume of the system, allows to set z + ≃ z − = z. With δz ℓ = [z(r + ℓ) − z(r)] · ℓ/ℓ the longitudinal isotropic structure functions of order p are defined as S p (ℓ) = δz p ℓ , the angular brackets denoting spatial averaging. The structure functions exhibit self-similar scaling S p (ℓ) ∼ ℓ ζp in the inertial range where the dynamical influence of dissipation, turbulence driving and system boundaries is weak.The inertial range has been introduced in Kolmogorov's K41 phenomenology of incompressible hydrodynamic turbulence [3,4] which assumes a spectral energy-cascade driven by the break-up of turbulent eddies. This leads to the experimentally well-verified energy-spectrum E(k) ∼ k −5/3 [5] corresponding to ζ 2 = 2/3. Iroshnikov and Kraichnan (IK) [6,7] included the effect of a magnetic field by founding the energy-cascade on the mutual scattering of Alfvén waves triggered by velocity fluctuations. The IK picture phenomenologically yields E(k) ∼ k −3/2 , i.e., ζ 2 = 1/2.The validity of the two phenomenologies in MHD turbulence is still under discussion. Two-dimensional direct numerical simulations (DNS) support the IK picture [8,9] while three-dimensional simulations exhibit K41 s...
A comprehensive picture of three-dimensional (3D) isotropic magnetohydrodynamic (MHD) turbulence is presented based on the first 5123-mode numerical simulations performed. Both temporal and spatial scaling properties are studied. For finite magnetic helicity H the energy decay is governed by the constancy of H and the decrease of the ratio of kinetic and magnetic energy Γ=EK/EM. A simple model consistent with a series of simulation runs predicts the asymptotic decay laws E∼t−1/2, EK∼t−1. For nonhelical MHD turbulence, H≃0, the energy decays faster, E∼t−1. The energy spectrum follows a k−5/3 law, clearly steeper than k−3/2 previously found in 2D MHD turbulence. The scaling exponents of the structure functions are consistent with a modified She–Leveque model ζpMHD=p/9+1−(1/3)p/3, which corresponds to a basic Kolmogorov scaling and sheet-like dissipative structures. The difference between the 3D and the 2D behavior can be related to the eddy dynamics in 3D and 2D hydrodynamic turbulence.
Decay laws for three-dimensional incompressible magnetohydrodynamic turbulence are obtained from high-resolution numerical simulations using up to 512 3 modes. For the typical case of finite magnetic helicity H the energy decay is found to be governed by the conservation of H and the decay of the energy ratio Γ = E V /E M . One finds the relation (E 5/2 /ǫH)Γ 1/2 /(1 + Γ) 3/2 = const, ǫ = −dE/dt. Use of the observation that Γ(t) ∝ E(t) results in the asymptotic law E ∼ t −0.5 in good agreement with the numerical behavior. For the special case H = 0 the energy decreases more rapidly E ∼ t −1 , where the transition to the finite-H behavior occurs at relatively small values.
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