Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

The dispute on whether the three-dimensional (3D) incompressible Euler equations develop an infinitely large vorticity in a finite time (blowup) keeps increasing due to ambiguous results from state-of-the-art direct numerical simulations (DNS), while the available simplified models fail to explain the intrinsic complexity and variety of observed structures. Here, we propose a new model formally identical to the Euler equations, by imitating the calculus on a 3D logarithmic lattice. This model clarifies the present controversy at the scales of existing DNS and provides the unambiguous evidence of the following transition to the blowup, explained as a chaotic attractor in a renormalized system. The chaotic attractor spans over the anomalously large six-decade interval of spatial scales. For the original Euler system, our results suggest that the existing DNS strategies at the resolution accessible now (and presumably rather long into the future) are unsuitable, by far, for the blowup analysis, and establish new fundamental requirements for the approach to this long-standing problem.

Figure 1

(a) Inverse maximum vorticity, $1/{\omega }_{\max }$, as a function of time; the inset displays an amplified segment very close to the blowup time $t_b=15.870$. The graph shows deterministic chaotic oscillations; it is not smooth, because the vorticity maximum jumps between nodes of the 3D lattice with increasing time. (b) Evolution of maximum vorticity in log-scale, demonstrating chaotic oscillations around the power law $\propto (t_b-t{)}^{-1}$, and wave number $k_{\max }$ corresponding to the vorticity maximum, following in average the power law $(t_b-t{)}^{-\gamma }$ with $\gamma =2.70$. (c) The energy spectrum, $E(k)=\frac{1}{2\mathrm{\Delta }}\sum _{k\le |\mathbf{p}|<\lambda k}|\mathbf{u}(\mathbf{p}){|}^2$ with $\mathrm{\Delta }=\lambda k-k$, in log-scale at different renormalized times $\tau =-\log (t_b-t)$. As $t\to t_b$ corresponding to $\tau \to \infty$, the spectrum develops the power law $E\propto k^{-\xi }$ with $\xi =3-2/\gamma \approx 2.26$.

Figure 2

Absolute value of the third component of renormalized vorticity $|{\tilde{\omega }}_3|$, as a function of two positive wave numbers $k_1>0$ and $k_2>0$ (in log scale) at four different instants $\tau$. The third wave number is fixed at the node nearest to $k_3=e^{\gamma \tau +6}\propto (t_b-t{)}^{-\gamma }$. Values below 0.1 are plotted in white. In the left figure, the small box bounds the square region $|k_{1,2}|\le 4096$, which would be accessible for the high-accuracy DNS with resolution $8192^3$. See also the Supplemental 3D video [45].

Figure 3

Evolution of a small perturbation of vorticity, $\underset{\mathbf{k}}{\max }|\delta \tilde{\mathit{\omega }}|$, in renormalized variables. Solutions deviate exponentially with the Lyapunov exponent ${\lambda }_{\max }\approx 9.18$.

Figure 4

Statistical isotropy: Left panel shows the $\tau$ average of $|{\tilde{\omega }}_3|$ from Fig. 2 in a comoving reference frame ${\eta }^{\prime }=\eta -\gamma \tau$. Right panel shows analogous result for the average of $|{\tilde{\omega }}_2|$ on plane $({\tilde{k}}_1,{\tilde{k}}_3)$. Planes of the two figures are related by the 90° rotation about the ${\tilde{k}}_3$ axis. Similar results are obtained for other elements of the rotation symmetry group ${\mathsf{O}}_{\mathsf{h}}$.

Figure 5

Evolution of $\log \log \mathrm{\Omega }$ and $\log \log (50{\omega }_{\max })$ for the enstrophy and maximum vorticity; the factor 50 is used to avoid complex values of the logarithm. The dash-dotted line indicates the blowup time. The vertical solid line estimates the limit $t_{\mathrm{DNS}}\approx 9$ that would be accessible for the state-of-the-art DNS with the grid $8192^3$. Until $t_{\mathrm{DNS}}$, both $\mathrm{\Omega }$ and ${\omega }_{\max }$ demonstrate the growth not greater than double exponential.