Equivalence of nonequilibrium ensembles in turbulence models

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of nonequilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.

Figure 1

Side by side comparison between the time evolution of several observables in the irreversible shell model $\mathrm{SI}$ (left column) and the reversible ${\mathrm{SR}}_{\mathrm{\Omega }}$ model (right column). Shown, from top to bottom are the energy $E$, viscosity $\nu$ (${\alpha }_2$ for ${\mathrm{SR}}_{\mathrm{\Omega }}$), enstrophy $\mathrm{\Omega }$, and energy dissipation $\varepsilon =2\nu \mathrm{\Omega }$ ($2{\alpha }_2\mathrm{\Omega }$ for ${\mathrm{SR}}_{\mathrm{\Omega }}$). All quantities are normalized by their average value. We used $N=15$ shells and $\nu ={10}^{-5}$, corresponding to the energy-cascade regime. Notice that ${\alpha }_2$ is not positive definite: The occurrence of negative values, highlighted with a thick red line, corresponds to instances in which the dissipative terms inject energy into the system.

Figure 2

Phenomenology of the irreversible shell model. The energy spectrum $E_n=\langle |u_n{|}^2\rangle$ is shown for three values of the viscosity $\nu$ with $N=20$ shells. With $\nu ={10}^{-6}$ the characteristic spectrum of the energy cascade $E_n\sim k_n^{-0.72}$ (see the dashed line) appears. For $\nu ={10}^{-12}$ the energy spectrum is roughly at equipartition, namely, the regime of quasiequilibrium (see the text). For $\nu ={10}^{-10}$ a mixed behavior is observed.

Figure 3

Mean values of ${\alpha }_2/\nu$ for simulations of the ${\mathrm{SR}}_{\mathrm{\Omega }}$ model at different Reynolds number $R$ with $N=20$ shells. The $R$ dependence of the ${\mathrm{SR}}_{\mathrm{\Omega }}$ model is intended in the sense that it is initialized with an initial enstrophy $\tilde{\mathrm{\Omega }}$ equal to $\langle \mathrm{\Omega }\rangle$ measured in a run of the $\mathrm{SI}$ model with (fixed) viscosity $\nu =R^{-1}$. The large error bars reported for high $R$ can be ascribed to the limited statistics, due to the high cost of the numerical integration in that range of parameters.

Figure 4

Test of the equivalence for the ${\mathrm{SR}}_{\mathrm{\Omega }}$ model. The second moment (closed symbols) and fourth moment (open symbols) of a velocity field component pertaining to the (a) large scales $n=2$ and (b) small scale $n=10$ are shown as functions of the Reynolds number $R$ for both the $\mathrm{SI}$ and ${\mathrm{SR}}_{\mathrm{\Omega }}$ models with $N=20$ shells. The $R$ dependence of the ${\mathrm{SR}}_{\mathrm{\Omega }}$ model is intended in the sense that it is initialized with an initial enstrophy $\tilde{\mathrm{\Omega }}$ equal to $\langle \mathrm{\Omega }\rangle$ measured in a run of the $\mathrm{SI}$ model with (fixed) viscosity $\nu =R^{-1}$. Errors are smaller than or of the order of the symbol size.

Figure 5

Comparison of several spectral observables between the $\mathrm{SI}$ and the ${\mathrm{SR}}_{\mathrm{\Omega }}$ models in both situations of energy cascade $[\nu ={10}^{-6}$ $(•)$ and $\langle {\alpha }_2\rangle \simeq {10}^6$ $(\circ )]$ and quasiequilibrium $[\nu ={10}^{-12}$ $\left(▴\right)$ and $\langle {\alpha }_2\rangle \simeq {10}^{12}$ $\left(▵\right)]$: (a) energy spectra, (b) energy flux (10), (c) skewness ${\mathcal{S}}_n$ (17) (the inset shows the same plot with the logarithmic $y$ axis), and (d) flatness ${\mathcal{F}}_n$ (18). Errors are the order of or smaller than the symbol size. The dashed line labeled $k_n^{-0.72}$ in (a) represents the scaling behavior in the manner of Kolmogorov plus intermittency correction. The dashed line labeled $k_n^{-1}$ in the inset of (c) represents a dimensional prediction valid at quasiequilibrium; indeed, since $\langle \left|u_n\right|\rangle$ and $\langle {\mathrm{\Pi }}_n^E\rangle$ do not depend on the wave number $k_n$, at least in a certain range of scales, as shown in (a) and (b) respectively, one has that ${\mathcal{S}}_n\sim k_n^{-1}$ [Eq. (17)]. The dashed line labeled $k_n^{0.06}$ in (d) shows a best fit of the curves in the cascade regime. Finally, the horizontal dashed line in (d) displays the value ${\mathcal{F}}_n=2$, which is expected for complex Gaussian variables. In these figures, and in some of the following ones, to ease the identification of the various curves and avoid the superposition of different symbols, not all data points have been marked by a symbol.

Figure 6

Test of the equivalence for the ${\mathrm{SR}}_E$ model. The second moment (closed symbols) and fourth moment (open symbols) of a velocity component at (a) large scales $n=2$ and (b) small scales $n=10$ as functions of the effective Reynolds number $R$ for the $\mathrm{SI}$ and ${\mathrm{SR}}_E$ models with $N=20$ shells. The $R$ dependence of the ${\mathrm{SR}}_E$ model is intended in the sense that it is initialized with an initial energy $\tilde{E}$ equal to $\langle E\rangle$ measured in a run of the $\mathrm{SI}$ model with (fixed) viscosity $\nu =R^{-1}$. Errors are smaller than or of the order of the symbol size.

Figure 7

Energy spectra $E_n$ of the $\mathrm{SI}$ and ${\mathrm{SR}}_E$ models in the regime of quasiequilibrium ($N=20$ and $\nu ={10}^{-12}$). Error bars are smaller than or of the order of the symbol size.

Figure 8

Energy spectra $E_n$ for several reversible models, compared with the irreversible one ($\blacksquare$) (with $N=20$ and $\nu ={10}^{-6}$). All the reversible model simulations are initialized with the same distribution of initial energy in the range $0\le n<15$, but the models conserve different invariants $O_{\chi }$ [see Eq. (14)]. Error bars are smaller than or of the order of the symbol size.

Figure 9

Probability density function of the time-dependent viscosity ${\alpha }_2$ for the ${\mathrm{SR}}_{\mathrm{\Omega }}$ model in three different cases: a situation of energy cascade $[N=20$ and $\mathrm{\Omega }\sim {10}^4$ $(•)]$, a situation of quasiequilibrium $[N=10$ and $\mathrm{\Omega }\sim {10}^8$ $\left(\blacksquare \right)]$, and a case in between $[N=15$ and $\mathrm{\Omega }\sim {10}^4$ $\left(▾\right)]$. The insets on the left show the corresponding typical time evolutions of ${\alpha }_2$. The insets on the right show the corresponding energy spectra.