Statistical theory of reversals in two-dimensional confined turbulent flows

It is shown that the truncated Euler equation (TEE), i.e., a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined two-dimensional flow obeying Navier-Stokes equation (NSE) with bottom friction and a spatially periodic forcing. The random reversals of the NSE large-scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation. We demonstrate that these NSE bifurcations are described by the related TEE microcanonical distribution which displays transitions from Gaussian to bimodal and broken ergodicity. A minimal 13-mode model reproduces these results.

Figure 1

Flow transitions: Time series of ${\widehat{\psi }}_{1,1}$ obtained from the DNS of the (a) Navier-Stokes equation (NSE) for different values of $\mathrm{Rh}$ and $\mathrm{Re}=5000/{\pi }^2$, and (b) truncated Euler equation (TEE) for different values of $k_c$. ${\widehat{\psi }}_{1,1}^{\text{NSE}}$ and ${\widehat{\psi }}_{1,1}^{\text{TEE}}$ have been divided by ${10}^4$ and ${10}^2$, respectively. (c) Plot of $k_c=\sqrt{\mathrm{\Omega }/E}$ versus $\mathrm{Rh}$ from the DNS of the NSE for two different Reynold's numbers $\mathrm{Re}=3000/{\pi }^2$ (blue circles) and $\mathrm{Re}=5000/{\pi }^2$ (red crosses). Inset: Log-linear plots of the PDFs of ${\widehat{\psi }}_{1,1}^{TEE}$ for different values of $k_c$. (d) Log-linear plots of the PDFs of ${\widehat{\psi }}_{1,1}$ for different values of $k_c$ obtained from the finite-mode minimal model based on the TEE; the lines on top of these PDFs indicate the estimation from our analytical method.

Figure 2

Reversals: Log-linear plot of the mean waiting time $\tau$ between successive reversals versus $\mathrm{Rh}$, obtained from our DNSs of NSE for $\mathrm{Re}=5000/{\pi }^2$ (blue circles). Inset: Plot of the reversal frequency $1/\tau$ versus $k_c$ from the DNSs of TEE; it shows that the reversal frequency decreases linearly with $k_c$ with a critical $k_c^{*}\simeq 4.06$, below which the reversals are not observed for the integration time.