Self-similar regimes of turbulence in weakly coupled plasmas under compression

Turbulence in weakly coupled plasmas under compression can experience a sudden dissipation of kinetic energy due to the abrupt growth of the viscosity coefficient governed by the temperature increase. We investigate in detail this phenomenon by considering a turbulent velocity field obeying the incompressible Navier-Stokes equations with a source term resulting from the mean velocity. The system can be simplified by a nonlinear change of variable, and then solved using both highly resolved direct numerical simulations and a spectral model based on the eddy-damped quasinormal Markovian closure. The model allows us to explore a wide range of initial Reynolds and compression numbers, beyond the reach of simulations, and thus permits us to evidence the presence of a nonlinear cascade phase. We find self-similarity of intermediate regimes as well as of the final decay of turbulence, and we demonstrate the importance of initial distribution of energy at large scales. This effect can explain the global sensitivity of the flow dynamics to initial conditions, which we also illustrate with simulations of compressed homogeneous isotropic turbulence and of imploding spherical turbulent layers relevant to inertial confinement fusion.

Figure 1

Evolution of turbulent kinetic energy $\mathcal{K}$ as a function of the compression parameter $L$ for both DNSs at resolution ${512}^3$ and EDQNM simulations of a HIT compression case with ${\text{Re}}_0=250$ and ${\text{Cp}}_0=0.1$. The critical compression parameter $L_M$ corresponds to the kinetic-energy maximum before the beginning of the viscous phase.

Figure 2

Top row: distribution of turbulent kinetic energy $\mathcal{K}$ in the DNS at four instants of the evolution, indicated in the kinetic energy curve of Fig. 1. The scaling of the box corresponds to the moving frame of reference. Bottom row: Associated energy spectra $E\left(\tilde{k}\right)$ for both DNS and EDQNM simulations, at the same instants.

Figure 3

Isocontour map of the critical compression parameter $L_M$ corresponding to maximum of kinetic energy, as a function of the initial Reynolds number ${\text{Re}}_0$ and the compression number ${\text{Cp}}_0$. Results from EDQNM simulations. The black circle corresponds to the parameters used in Figs. 1 and 2, and the red triangle to that of Fig. 8. The red line is the contour line at $L_M=1$.

Figure 4

Evolution of kinetic energy $\mathcal{K}$ normalized by its initial value as a function of the compression parameter $L$ for the three typical cases indicated on the parametric map of Fig. 3. The spectrum in region (a) illustrates the immediate decay regime, that of region (b) the intermediate regime, and region (c) shows the cascade regime.

Figure 5

Evolution of (top) the kinetic energy $\mathcal{K}$ and (bottom) the integral scale ${\ell }_I$ as a function of the compression parameter $L$ using EDQNM simulations. Solid line: Batchelor initial condition ($s=4$). Dashed line: Saffman initial condition ($s=2$). The scaling laws corresponding to the self-similar solutions derived in Sec. 3a are also shown.

Figure 6

Evolution of (top) the Reynolds number $\text{Re}$ and (bottom) the compression number $\text{Cp}$ as a function of the compression parameter $L$ using EDQNM simulations. Solid line: Batchelor initial condition ($s=4$). Dashed line: Saffman initial condition ($s=2$). The different scaling laws corresponding to the self-similar solutions derived in Sec. 3a are also shown.

Figure 7

Kinetic-energy spectra $E\left(k\right)$, corresponding to EDQNM simulations of Figs. 5 and 6, taken at different values of the compression parameter $L$. Left: Saffman initial condition ($s=2$). Right: Batchelor initial condition ($s=4$).

Figure 8

Evolution of turbulent kinetic energy $\mathcal{K}$ as a function of the compression parameter $L$ for both ${1024}^3$ DNS and EDQNM simulations corresponding to the case of the spherical turbulent layer. Solid line: DNS; dashed line: EDQNM simulation. Saffman and Bachelor initial conditions with respectively ($s=2$) and ($s=4$) are used, as indicated in the legend.

Figure 9

Distribution in a plane cut of turbulent kinetic energy $\mathcal{K}$ in the DNS at different instants indicated in Fig. 8. Top row: the Batchelor case ($s=4$); and bottom row: the Saffman case ($s=2$). The scaling corresponds to that of the moving frame.

Figure 10

Evolution of (top) the Reynolds number $\text{Re}$ and (bottom) the compression number $\text{Cp}$ as a function of the compression parameter $L$ for the spherical turbulent layer case extracted from DNS (solid line) and EDQNM (dashed line) simulations. Black lines: Batchelor initial condition ($s=4$). Red lines: Saffman initial condition ($s=2$).

Figure 11

Evolution of the integral scale ${\ell }_I$ (DNS and EDQNM) and the turbulent layer size ${\ell }_{\text{MZ}}$ (DNS) as a function of the compression parameter $L$ corresponding to the case of the spherical turbulent layer with (red lines) Saffman and (black lines) Batchelor initial conditions.

Figure 12

Energy spectra as a function of wave number $E\left(k\right)$ corresponding to the spherical turbulent layer case both in DNS (solid line) and EDQNM (dashed line) and taken at different values of the compression parameter $L$. Right: Batchelor initial condition ($s=4$). Left: Saffman initial condition ($s=2$).

Figure 13

Spherically integrated kinetic-energy profile extracted from DNS as a function of radius $\tilde{r}$ and taken at different values of the compression parameter $L$. Left: Saffman initial condition ($s=2$). Right: Batchelor initial condition ($s=4$).