Long-term memory in experiments and numerical simulations of hydrodynamic and magnetohydrodynamic turbulence

We analyze time series stemming from experiments and direct numerical simulations of hydrodynamic and magnetohydrodynamic turbulence. Simulations are done in periodic boxes, but with a volumetric forcing chosen to mimic the geometry of the flow in the experiments, the von Kármán swirling flow between two counterrotating impellers. Parameters in the simulations are chosen to (within computational limitations) allow comparisons between the experiments and the numerical results. Conducting fluids are considered in all cases. Two different configurations are considered: a case with a weak externally imposed magnetic field and a case with self-sustained magnetic fields. Evidence of long-term memory and $1/f$ noise is observed in experiments and simulations, in the case with weak magnetic field associated with the hydrodynamic behavior of the shear layer in the von Kármán flow, and in the dynamo case associated with slow magnetohydrodynamic behavior of the large-scale magnetic field.

Figure 1

Sketch of the experimental setups. (a) VKS experiment drives a von Kármán swirling flow of liquid sodium in a cylindrical vessel. Data from the experiments reported here were obtained in a configuration with an inner copper shell and an annulus in the midplane. (b) VKG experiment drives a von Kármán flow of liquid gallium in a cylindrical vessel. A pair of lateral Helmholtz coils allows us to apply a transverse magnetic field. Positions of the measurement probes are indicated with the stars.

Figure 2

Sketch of the numerical Taylor-Green cell, and position of the points where magnetic and velocity measurements are made (indicated with stars). The two large circular arrows represent the two counterrotating Taylor-Green vortices.

Figure 3

Numerical simulations of Taylor-Green MHD induction. Top: Power spectrum of the temporal fluctuations of the axial velocity $u_z$ at point 1 (light gray dashed line) and of the axial induced magnetic field (dark gray solid line) in the numerical simulations at low magnetic Prandtl number ($\text{Pm}=0.05$) and low interaction parameter $N=4\times {10}^{-5}$. Frequency axis has been normalized by the forcing frequency $F_0=u_{\mathrm{rms}}/\pi$. The global energy level of the spectra has been arbitrarily set to make them coincide for $f=F_0$. Bottom: Power spectrum of the amplitude of Fourier mode $k=(10,10,10)$ of $u_z$ (light gray dashed line) and of Fourier mode $k=(1,0,0)$ of $u_z$ (dark gray solid line). Note that incompressibility condition imposes the mode $k=(0,0,1)$ of $u_z$ to vanish, while symmetries of the TG flow impose the mode $k=(0,1,0)$ to be identical to the mode $k=(1,0,0)$ shown in the figure. In both panels, several slopes reported before in the VKS and VKG experiments are shown as references, although we are mostly interested in the behavior for $f<F_0$.

Figure 4

Experiments of MHD induction in VKG flow. Power spectrum of the axial velocity (light gray dashed line) and of the induced axial magnetic field in VKG experiment with a weak imposed transverse magnetic field (light gray solid line). Frequency axis has been normalized by the forcing frequency $F_0$ taken as the rotation rate of the impellers $\Omega$ (10 Hz for the shown experiment). The total energy level of each spectrum has been normalized by its value at $f/F_0=1$. Several slopes are shown as references.

Figure 5

Experiments vs simulations (induction case): comparison of the spectra of the induced magnetic field in Figs. 3 (simulation, black dashed line) and 4 (experiment, light gray solid line). The total energy level of each spectrum has been normalized by its value at $f/F_0=1$. Slopes are shown as references. Measurements from the simulation and the experiment agree remarkably well except for very low frequencies.

Figure 6

Numerical simulations. Top: position of the shear layer as a function of time (in simulation units). Bottom: power spectrum of the signal in the top figure. A slope of $-1$ is indicated as a reference.

Figure 7

VKG experiment. Top: proxy of the evolution of the shear layer as a function of time, as indicated by the evolution of the axial induced magnetic field measured at point $3^{\prime }$. By ${\mathcal{R}}_{\pi }$ symmetry around the $Ox$ axis, this component of induced field must be zero. However, it is visible from the measurements that it fluctuates between a high level state (of order 4 gauss) and a low level state (of order $-4$ gauss). We interpret these fluctuations as a signature of the instantaneous breaking of the ${\mathcal{R}}_{\pi }$ symmetry due to the motion of the shear layer. Bottom: power spectrum of the signal in the top figure. A slope of $-1$ is indicated as a reference.

Figure 8

Experiments vs simulations in the dynamo case. Top: power spectra of the self-sustained magnetic field in the VKS experiment (black) and in the simulation (gray). A slope of $-1$ is indicated as a reference. Bottom left: semilog plot of top figure. Bottom right: semilog plot of magnetic spectra measured at point 3 in VKS (from Ref. [13]). The different curves correspond to different magnetic Reynolds numbers as indicated in the legend, with increasing $\text{Rm}$ from light gray to black curves.

Figure 9

Position of the shear layer as a function of time in the numerical simulations of dynamo action, and power spectrum of the signal. A slope of $-1$ is indicated as a reference.