Evolution of Magnetic Fields in Freely Decaying Magnetohydrodynamic Turbulence

Abstract

We study the evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence. By quasi-linearizing the Navier-Stokes equation, we solve analytically the induction equation in quasinormal approximation. We find that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time respectively as EB ∝ t −2(1+p)/(3+p) and ξB ∝ t 2/(3+p) , where p is the index of initial power-law spectrum. In the helical case, the magnetic helicity is an almost conserved quantity and forces the magnetic energy and correlation length to scale as EB ∝ (log t) 1/3 t −2/3 and ξB ∝ (log t) −1/3 t 2/3 .PACS numbers: 52.30.Cv, 98.62.EnThe origin of presently-observed large scale magnetic fields throughout the universe is still unclear [1]. Essentially, there are two possible classes of mechanisms to produce cosmic fields depending on when they are generated: Astrophysical mechanisms acting during or after large-scale structure formation, and mechanisms acting in the primordial universe. Magnetic fields created in the early universe (except those generated during inflation), usually suffer from a "small-scale problem", that is their comoving correlation length is much smaller then the characteristic scale of the observed cosmic fields. However, if magnetohydrodynamic (MHD) turbulence operates during their evolution, an enhancement of correlation length can occur, especially if the magnetic field is helical. As pointed out by Banerjee and Jedamzik [2], the evolution of a magnetic field in the early universe goes through different phases depending on the particular conditions of the primordial plasma. In this paper, we are interested in the case of magnetic fields evolving in the turbulent primordial universe well before recombination epoch and when kinematic dissipative effects are due to diffusing particles. Therefore, we are concerned with the so-called phase of "turbulent MHD". In other phases, such as "viscous MHD" and "MHD with ambipolar diffusion" described in Ref.[2], the dynamics of the magnetic field is very different from that studied here. The problem of determining the evolution properties of magnetic fields in MHD turbulence has been deeply and widely discussed in the literature using different methods and approximations. A direct integration of the full set of MHD equations would allow us to deeply understand the dynamics of freely decaying MHD turbulence. However, MHD equations are quite difficult to handle due to their high non-linearity and it has not been yet brought in a definitive verdict for the evolution laws of magnetic energy and correlation length (for recent numerical studies of freely decaying magnetohydrodynamic turbulence see, e.g., Ref. [2,3,4]).The turbulent MHD equations for incompressible fluids, in the case of non-expanding universe, are [5]:and ∇·v = ∇·B = 0. Here, v is the velocity of bulk fluid motion, B the magnetic field, J = ∇ × B the magnetic current, ν the kinematic viscosity, η the resistivity. The thermal pressure of the fluid, p, is not an independen...

Evolution of helical cosmic magnetic fields as predicted by magnetohydrodynamic closure theory

“…[11], it was shown that Eqs. (10)- (12) remain invariant in the expanding Universe when time derivatives @=@t are replaced by scale factor derivatives ðH 0 =aÞ@=@ ln a, and physical quantities in the integrands of Eqs. (10)-(12) are replaced by their comoving analogues.…”

“…Equations (10)- (12) are a set of well-defined equations, since they ensure conservation of energy, momentum, mass, and helicity density to the lowest nontrivial order in Át.…”

Magnetic fields in turbulent quark matter and magnetar bursts

“…when K d = 0. time evolution of the magnetic field is owing to the MHD turbulence effects accounted for in Eqs. (5)- (8) and (14)- (17). Let us define the scale of the magnetic field as Λ = 1/k.…”

“…The MHD turbulence was found in Ref. [17] to take place when both the Reynolds number, Re = vl/ν, and the magnetic Reynolds number, Re B = vlσ cond , are large. Here v and l are the typical velocity and length scales of the matter motion, ν is the kinematic viscosity, and σ cond is the quark matter conductivity.…”

Effects of theUY(1)Chern-Simons term and its baryonic contribution on matter asymmetries and hypermagnetic fields

“…However, in the inverse cascade mechanism, λ(t) grows faster than a(t) due to the turbulence in the plasma [54]. In this case, the magnetic helicity is approximately conserved but the energy is transferred from small scales to large scales [71], and the spectrum develops with a characteristic scaling law [72]. After recombination, the plasma becomes neutral and the magnetic fields evolve trivially.…”