Odd viscosity suppresses intermittency in direct turbulent cascades

TL;DR

The paper investigates how odd viscosity, a non-dissipative parity-breaking term, affects intermittency in direct turbulent cascades. Using DNS and a two-channel helical shell model, it shows that odd viscosity suppresses intermittency at small scales by breaking multiple scale invariances via parity-breaking waves with frequencies $\omega_\pm(\bm k)=\pm \nu_{\rm odd} k_z |\bm k|$. It develops a generalized framework with generalized odd viscosity ($\omega_{\pm}(\bm k)=\pm c_{\rm odd} k_z |\bm k|^{\alpha-1}$), predicting a wave-affected cascade with crossover scale $k_{\rm odd}$ and dissipation scale $k_c$, and demonstrates pattern formation and extended inertial ranges. The results indicate a route to designing turbulent flows with tunable intermittency by controlling parity-breaking wave dynamics.

Abstract

Intermittency refers to the broken self-similarity of turbulent flows caused by anomalous spatio-temporal fluctuations. In this Letter, we ask how intermittency is affected by a non-dissipative viscosity, known as odd viscosity (also Hall or gyro-viscosity), which appears in parity-breaking fluids such as magnetized polyatomic gases, electron fluids under magnetic field and spinning colloids or grains. Using a combination of Navier-Stokes simulations and theory, we show that intermittency is suppressed by odd viscosity at small scales. This effect is caused by parity-breaking waves, induced by odd viscosity, that break the multiple scale invariances of the Navier-Stokes equations. Building on this insight, we construct a two-channel helical shell model that reproduces the basic phenomenology of turbulent odd-viscous fluids including the suppression of anomalous scaling. Our findings illustrate how a fully developed direct cascade that is entirely self-similar can emerge below a tunable length scale, paving the way for designing turbulent flows with adjustable levels of intermittency.

Figures (8)

• Figure S1: $K_8=S_8/S_4^2$ in DNS (a) and for the shell model (b).
• Figure S2: Wavelength selection in odd turbulence. (a) Rescaled energy spectrum $E/E_0$ ($E_0$ is the spectrum for $\nu_{\rm odd}=0$) vs wavenumber in DNS for $\nu_{\rm odd}=3\times 10^{-4},6\times 10^{-4},1.2\times 10^{-3}, 2.4\times 10^{-3}$. (b) Rescaled energy spectrum vs wavenumber in the shell model for $\nu_{\rm odd}= 2.6\times 10^{-7}, 1.0\times 10^{-6}, 4.1\times 10^{-6}, 1.6\times 10^{-5}, 6.6\times 10^{-5}$. (c) Rescaled energy spectrum estimated from the bootstrap method for $\nu_{\rm odd}=16, 256, 4096, 65536$, $\nu=1$ and $\epsilon = 1$. In (a-c), stars indicate condensation scale $k_c\sim \nu_{\rm odd}^{-1/4}$.
• Figure S3: Simulation results with extended inertial range. The value of $\nu$ is reduced when odd viscosity is present, such that the dissipation scale is unchanged. (a) Energy flux in DNS for $\nu_{\rm odd}=0, 2.0\times 10^{-2}$. (b) Probability distribution of $x$-direction vorticity in DNS (renormalized by the standard deviation), for same $\nu_{\rm odd}$ values as in (a). Gaussian distribution is shown in dashed line for comparison. (c) Kurtosis $K$ in DNS, for same $\nu_{\rm odd}$ as in (a). (d) Energy flux in the shell model for $\nu_{\rm odd}=0, 2.6\times 10^{-7}, 1.0\times 10^{-6}, 4.1\times 10^{-6}, 1.6\times 10^{-5}, 6.6\times 10^{-5}$. (b) Probability distribution of ${\rm Re}(u_{15}^+)$ in the shell model (renormalized by the standard deviation), for same $\nu_{\rm odd}$ values as in (d). Gaussian distribution is shown in dashed line for comparison. (c) Kurtosis $K$ in the shell model, for same $\nu_{\rm odd}$ as in (d).
• Figure S4: Comparison between DNS, modified Parisi-Frisch and shell model results, for same scale separation as that in DNS. (a) and (c) Energy spectrum and kurtosis in DNS for $\nu_{\rm odd}=0, 3\times 10^{-4}, 6\times 10^{-4}, 1.2\times 10^{-3}, 2.4\times 10^{-3}$, same as that in Fig. 3 of the main text. (d) Kurtosis in the modified Parisi-Frisch theory for $\nu_{\rm odd}=0, 1.2\times 10^{-2}, 2.5\times 10^{-2}, 4.9\times 10^{-2}, 9.9\times 10^{-2}$. (b) and (d) Energy spectrum and rescaled kurtosis in the shell model for $\nu_{\rm odd}=0, 3\times 10^{-4}, 6\times 10^{-4}, 1.2\times 10^{-3}, 2.4\times 10^{-3}$. In (e) kurtosis is rescaled with $\widetilde{K} = 2.4 K$, which ensures that $\widetilde{K} \approx 3$ at the forcing scale.
• Figure S5: Flux loop for small-scale forcing. (a) Energy flux in DNS for $\nu_{\rm odd}= 2.4\times 10^{-3}$. (b) Energy flux in the shell model for $\nu_{\rm odd}= 6.6\times 10^{-5}$. Forcing is applied on $n=13$.