Fig. 4: Statistical universality of the vorticity profile.
From: From the butterfly effect to spontaneous stochasticity in singular shear flows
The vorticity profile within the mixing layer is measured in terms of the one-point statistical quantity \(p_1(y,t)\), which is a function of the distance \(y\) to the initial discontinuity and of the time \(t\). a The main figure shows the vorticity profile \(p_1(y,t)\) as a function of the rescaled variable \(y/\ell(t)\) at various times logarithmically spaced between \(t = 0.02\) and \(t=0.51\) for the Birkhoff–Rott dynamics with the highest number of point-vortices \(N_b\). The inset shows the same quantity but for the Navier–Stokes dynamics with the lowest value of viscosity \(\nu\). In both cases, the collapse of the datasets (up to numerical noise) reveals the asymptotic convergence of the vorticity profile towards an asymptotic profile \(P_1(y/\ell)\). b The same graph, now at fixed time \(t=0.5\) for both the Navier–Stokes and point-vortex simulations, revealing universality of the self-similar profile \(P_1\) with respect to both regularizations.
