Energy spectra and scaling relations in numerical turbulence with laboratory and astrophysical applications

Abstract

In this work we have focused on the statistical properties of turbulence.

This has been done in two different settings; one with neutral gas (the first four papers) and the other with ionized gas (the last four papers). Regarding the work on the neutral gas, we have looked at four different aspects;

1. Is the mean energy dissipation rate, Cє, independent of Reynolds number for large Reynolds numbers? This is one of the fundamental questions in turbulence, and one believe the answer will be yes, but this is as yet not conclusive. In Paper 1 we demonstrate that the value of Cє is highly sensitive to the method used to measure it. This might explain the discrepancies in the values of Cє found by previous authors. We also show how one can find Cє for a spread of Reynolds numbers from a single simulation.

2. Is there a “bottleneck” in the energy spectrum between the inertial range and the dissipative range? Such a bottleneck is extremely weak - or totally absent, in wind tunnel experiments. In large numerical simulations however the bottleneck is pretty clear. In Paper 2 we show that this discrepancy is due to the physical nature of the one-dimensional energy spectra found in wind tunnels and the three dimensional energy spectra found in numerical simulations.

3. In order to achieve larger Reynolds numbers we investigate the possible errors introduced by using hyper viscosity instead of normal viscosity in Paper 3. Our conclusion is that while hyper viscosity increase the hight of the bottleneck and shortens the dissipative range, it does not otherwise have any significant effect on the energy spectrum, or the structure functions. The inertial range and the large scales are the same both with normal viscosity and hyper viscosity.

4. In decaying turbulence one can find relations under which the Navier- Stokes equations are scale invariant. Using these relations it has recently been suggested by Ditlevsen et al.[1] that the energy spectrum for decaying hydrodynamical turbulence can be described by a scaling function with only two arguments. This has previously been shown both analytically and experimentally, and in Paper 4 we also confirm this in numerical experiments.

For the ionized gas we have focused on five different aspects;

1. What does the large Reynolds number energy spectra look like? Are the kinetic and magnetic energy spectra similar? The results are as yet not conclusive because the Reynolds numbers are still too small, but it seems that what at first looked like a k−3/2 inertial range is actually the bottleneck in a k−5/3 inertial range. Furthermore we have in Paper 5 found that the peak of the magnetic energy spectrum is not proportional to the resistive scale, but to the forcing scale.

2. As intermittency is still an unresolved topic we have looked at the different structure functions of the MHD dynamo. In Paper 6 the longitudinal structure functions based on the Elsasser variables are found to scale like in the model of She & Leveque[2], and the magnetic field is more intermittent than the velocity field. The Elsasser variables have been shown to have a linear scaling of the third order structure function. We do not, however, find the same linear scaling for the individual structure functions of the magnetic and the kinetic field.

3. In Paper 6 we investigate the growth rate of the magnetic field as a function of magnetic Reynolds number, and we find the critical magnetic Reynolds number as a function of magnetic Prandtl number.

4. How is the dynamo altered when one imposes an external large scale magnetic field? In Paper 7 we find that an imposed field tend to suppress the dynamo activity on all scales if the field is large enough. For an imposed magnetic field of the same size as the rms velocity field equipartition is found between magnetic and kinetic energy spectra.

5. Will there be dynamos in supersonic media? One could envisage that the supersonic shock swept up and dissipated the magnetic fields before they got time to grow. Numerical simulations in Paper 8 seem to show that as one increases the Mach number toward unity the critical magnetic Reynolds number increases, but as the Mach number grows even more the critical magnetic Reynolds number stays approximately constant.