Fig. 1: Explosive separation between two realizations of the Navier–Stokes dynamics.
From: From the butterfly effect to spontaneous stochasticity in singular shear flows
Explosive separation is observed by measuring the separation energy \({\mathcal{E}}\) as a function of time t between two solutions of the Navier–Stokes equations, which are ε-close at the initial time. The observational scale is half the size of the computational domain \(L=\pi\) Each line corresponds to a different viscosity \(\nu\) and a different initial separation \(\varepsilon \propto \nu\) In the main figure, quantities are reported on a log–log scale and the dashed black line corresponds to the asymptotic linear law \({\mathcal{E}}\approx 0.14\ t\) emerging in the limit of vanishing viscosity and initial separation. The inset highlights the nonuniversal initial exponential stage. It shows in lin–log scales the data delimited within the black rectangle, along with exponential fitting. To highlight the exponential growth, the separation energy is rescaled by the separation at time \(\tau = 0.0044\) and data are shown as a function of \(t-\tau\).
