Behind the mirror: the hidden dissipative singular solutions of ideal reversible fluids on log-lattices

TL;DR

This work develops a dynamical route to construct singular, potentially dissipative Euler solutions by embedding Navier–Stokes dynamics in a reversible framework on log-lattices and controlling energy storage with a tunable efficiency parameter. It reveals a phase transition between hydrodynamic and singular regimes, with pre-blow-up spectral exponents $h$ spanning from $2/3$ to $1/3$ as the efficiency decreases, and shows that past blow-up regularization via truncation or stochastic friction yields dissipative power-law exponents $h_{pb}$ that align with those observed in ordinary turbulence. The observed exponents on log-lattices reproduce multi-fractal-like spectra, supporting a dynamical mechanism for generating dissipative Euler solutions and connecting time-reversal symmetry breaking to turbulent dissipation. The results provide a computationally tractable platform to study selection mechanisms for singular solutions and dissipative anomalies, with potential implications for the Onsager conjecture and turbulence intermittency, and suggest avenues to translate these ideas to regular Fourier lattices.

Abstract

Empirical observations show that turbulence exhibits a broad range of scaling exponents, characterizing how the velocity gradients diverge in the inviscid limit. These exponents are thought to be linked to singular solutions of the Euler equations. In this work, we propose a dynamic approach to construct concept of these solutions directly from the fluid equations, using a reversible framework and introducing the efficiency $\cal{E}$, a non-dimensional number that quantifies the amount of energy stored within the flow due to an applied force. To circumvent the computational burden of tracking singularities at finer and finer scale, we test this approach on fluids on log-lattices, which allow for high effective resolutions at a moderate cost, while preserving the same symmetries and global conservation laws as ordinary fluids. We observe a phase transition at a given efficiency, separating regular, viscous solutions (hydrodynamic phase), from singular, inviscid solutions (singular phase). The singular solutions experience self-similar blow-ups with exponents corresponding to non-dissipative solutions. By applying a stochastic regularization, we are able to go past the blow-up, and show that the resulting solutions converge to power-law solutions with exponents characterizing dissipative solutions. Overall, the range of scaling exponents observed for log-lattice solutions is comparable to those of ordinary fluids.

Figures (8)

• Figure 1: Example of 2D log-lattice with spacing parameter $\lambda$.
• Figure 2: Convergence of the stochastic protocol for $\mathcal{E} \approx 8.6$. The main panel shows an almost constant $\left<\nu_r\right>$ for $N_* \geq 18^3$. The inset presents the corresponding energy spectra highlighting a good collapse for $N_* \geq 20^3$.
• Figure 3: In each figure markers are associated to different grid size $N$ such that $N = 8^3$, $N = 12^3$, $\color{black}\smallstar$$N = 18^3$, $N = 20^3$, $N = 22^3$, $N = +\infty$ (adaptative resolution, see Section \ref{['sec:LL']}). A different symbol is set for the Euler equations: $\oplus$ Euler ($\cal{E} = +\infty$). The blue colored symbols correspond to stochastically regularized solutions. (a) Adimensionnalized and averaged reversible viscosity as a function of the inverse of the efficiency $1 / {\cal E}$. The dashed line correspond to a master curve $\left<\nu_r\right> \propto (\mathcal{E}^{-1} - \mathcal{E}_*^{-1})^{3}$ , separating the two phases of the system costa2023reversible, where well resolved simulations collapse regardless of the resolution $N$ (full symbols). (b) Snapshots of the energy spectrum taken at three distinct times during the same viscous (RNS) blow-up, illustrating its time evolution. (c) Energy spectrum for truncated RNS - dynamics highlighting thermalization effects. (d) Energy spectrum for a stochastically stabilized blow-up solution. (e) Energy spectra along the master curve of (a).
• Figure 4: Colors and symbols: same as Figure \ref{['fig:RNS_blowup']} with the addition of dark green symbols encoding regularized Euler simulations. (a) Time averaged spectral exponents as a function of $1 / {\cal E}$. In case of thermalization, only the large scale exponent is reported (e.g extracted from the $-1.56$ slope of Figure \ref{['fig:RNS_blowup']}c). The dashed line represent the peculiar value $h_{KG} = 1 / 3$. (b) Adimensionnalized mean energy dissipation as a function of the adimensionalized mean viscosity for various solutions of Figure \ref{['fig:RNS_blowup']}. For comparison purposes (Euler and RNS) we introduce the integral scale $L_I = \int k^{-1}E(k)dk/\int E(k)dk$. The dashed line represents a fit $D_\epsilon = \epsilon_*\mathcal{E}^{3/2}$, where the value $\cal{E}(<\nu>)$ is obtain by inverting the master curve equation (Figure \ref{['fig:RNS_blowup']}a). This fit highlights a possible anomalous dissipation in the inviscid limit which value is compatible with the dissipation found in the regularized Euler simulations (dark green symbols) that slowly converges towards the fit. This value is also compatible with the total energy injection of the regularized RNS solutions (red crosses, $\bm{+}$) defined by summing the standard energy injections and the viscous contributions (that inject, in average, energy as $\left<\nu_r\right>_t < 0$). Small deviations of the red crosses from the dissipative anomaly might be linked to statistical errors as getting really long simulations is difficult, even on LL.
• Figure 5: Resolutions are encoded by the following symbols and colors $N_* = 12^3$, $N_* = 14^3$, $N_* = 16^3$, $\color{orangeplot}\filledstar$$N_* = 18^3$, $N_* = 20^3$, $N_* = 22^3$, $N_* = 24^3$. (a) Check of the validity of Onsager conjecture on log-lattice: the energy dissipation ($N_* = 22^3$) starts as soon as the spectral exponent goes below $h=1/3$, as materialized by the red vertical dotted line. (b) Time series of the rescaled energy for regularized Euler simulations highlighting an increasing dissipation as $N_* \rightarrow \infty$.