Transition from dissipative to conservative dynamics in equations of hydrodynamics

We show, by using direct numerical simulations and theory, how, by increasing the order of dissipativity $\left(\alpha \right)$ in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case $\alpha \to \infty$ [U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of $\alpha$ greater than a crossover value ${\alpha }_{\mathrm{crossover}}$. We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows.

Figure 1

Solutions of the HBE, zoomed around $x_{\mathrm{s}}$, for various values of $\alpha$ (see legend) at $t=1.5$ showing oscillations at $x_{\mathrm{s}}$ with increasing amplitude as $\alpha$ increases (see text). Inset: Solution of the HBE with no oscillations at $x_{\mathrm{s}}$ for small $\alpha =20$.

Figure 2

The solution of the HBE for $\alpha =100$, with the solution for the ordinary Burgers subtracted out, zoomed around $x_{\mathrm{s}}$. A clear symmetric bulge, similar to those seen in inviscid, conservative truncated systems [9], is seen.

Figure 3

Solution of the HBE, for $\alpha =250$ at $t=1.5$, showing the presence of a tyger [9] at $x_{\mathrm{s}}$. Inset: $u\left(x\right)$ for $\alpha =250$ at a later time $\left(t=5.0\right)$ confirming that for $\alpha \gtrsim {\alpha }_{\mathrm{crossover}}$ the system eventually thermalizes.

Figure 4

The solution $\tilde{u}$ of the linearized equation (5) with initial conditions (I2) at time $t=2.0$ (blue or darker curve) and $t=2.5$ (red or lighter curve) for $\alpha =100$ (see text). Inset: $\tilde{u}$ with initial conditions (I1) at time $t=10.0$.