On Inverse Cascades and Primordial Magnetic Fields

We consider primordial spectra with simple power behaviours and show that in the NavierStokes and magnetohydrodynamics equations without forcing, there exists systems in three dimensions with a subsequent inverse cascade, transferring energy from small to large spatial scales. This can have consequences in astrophysics for the evolution of density fluctuations, for primordial magnetic fields, and for the effect of diffusion. In general, if the initial spectrum is kα, then in the “inertial” range, for α > −3 there is an inverse cascade, whereas for α < −3 there is a forward cascade. Electronic address: [email protected] Recently there has been a considerable discussion of the development of primordial magnetic fields subsequent to the creation of these fields (see e.g. refs. [1], [2], and [3], where further references can be found). In the early universe the Reynold number Re is often quite large. Since Re is of the order the ratio of the nonlinear to the linear terms, this suggest that the nonlinearities could be quite important. This was indeed found in ref. [2] and [3], where an “inverse cascade” was observed. In this note we point out that it is an exact consequence of the Navier-Stokes and magnetohydrodynamics equations without external forces that there exists an inverse cascade, moving the field to larger scales, provided the initial energy is distributed at relatively small scales. This result can be expressed as a scaling relation for the relevant energy. In astrophysics one often considers “primordial” spectra behaving like some power kα. The question we address in the following is how such spectra evolve as consequence of the non-linear equations of motion. We shall find that if the such a system is “left alone” (like in many astrophysics applications), its energy is moved to larger and larger scales, in contrast to a “forced” system (relevant for earthbound hydrodynamics) in three dimensions, where energy is moved from larger to smaller scales. Hence there is a fundamental difference between systems which are forced or left alone. In this note we shall consider hydrodynamics or magnetohydrodynamics (MHD). In hydrodynamics one often considers systems which are under the influence of forces, e.g. at large distances (“stirring”). In these systems there is a forward cascade in three dimensions, which means that energy is transferred from large scales to smaller scales (from “order to chaos”). As mentioned before, cases without forcing may also be of interest. In particular, in astrophysics one may often encounter systems without any essential “stirring”, where the initial “primordial” spectrum is given. It is then quite dangerous to apply results from forced systems. Here we shall show that for a certain initial spectrum there exist an inverse cascade in hydrodynamics (from “chaos to order”), whereby energy is transferred from smaller scales to larger scales, as a consequence of the exact Navier-Stokes or magnetohydrodynamics (MHD) equations. The most rigorous of our results is that if the primordial energy spectrum is given by k at time 0, then at later times it will evolve as k ψ(kt), (1) where ψ is a scaling function (provided the viscosity coefficient is constant in time) with ψ(0) = 1. For MHD an equally rigorous result can be derived for the magnetic energy. These results take into account diffusion. In the “inertial” range, where diffusion can be ignored, a much more general result, relevant for an initial spectrum kα, can be derived. In order to have a finite energy, we must introduce an ultraviolet cutoff to be specified more precisely later, so that k ≤ K. In a more special context the scaling behavior (1) of the energy density has been considered before, first by Heisenberg [4] in his effective diffusion model, and then by Parisi [5] in a continuous version of the cascade (GOY) model [6]. A generalization of the scaling in (1) was then found in an investigation of the inverse cascade in the continuous GOY version of three dimensional relativistic MHD by Brandenburg et al. [3]. Later the scaling (1) has also been generalized to turbulent mixtures [7]. In the following the result (1) and its generalization to MHD are derived from the exact Navier-Stokes and MHD equations, showing that Eq. (1) is much more general than previously thought 1. This result can be used to check if model calculations are in agreement with the Navier-Stokes equation: If forcing is taken to be absent, the scaling (1) should be valid. From this point of view the cascade (GOY) models are excellent, since they have the scaling as an exact solution in the force-free case, as shown in Refs. [5], [2], and [7].