The onset of filamentation on vorticity interfaces in two-dimensional Euler flows

Abstract

Two-dimensional Euler flows, in the plane or on simple surfaces, possess a material invariant, namely the scalar vorticity normal to the surface. Consequently, flows with piecewise-uniform vorticity remain that way, and moreover evolve in a way which is entirely determined by the instantaneous shapes of the contours (interfaces) separating different regions of vorticity -- this is known as `Contour Dynamics'. Unsteady vorticity contours or interfaces often grow in complexity (lengthen and fold), either as a result of vortex interactions (like merger) or `filamentation'. In the latter, wave disturbances riding on a background, equilibrium contour shape appear to inevitably steepen and break, forming filaments, repeatedly -- and perhaps endlessly. Here, we revisit the onset of filamentation. Building upon previous work and using a weakly-nonlinear expansion to third order in wave amplitude, we derive a universal, parameter-free amplitude equation which applies (with a minor change) both to a straight interface and a circular patch in the plane, as well as circular vortex patches on the surface of a sphere. We show that this equation possesses a local, self-similar form describing the finite-time blow up of the wave slope (in a rescaled long time proportional to the inverse square of the initial wave amplitude). We present numerical evidence for this self-similar blow-up solution, and for the conjecture that almost all initial conditions lead to finite-time blow up. In the full Contour Dynamics equations, this corresponds to the onset of filamentation.

Figures (15)

• Figure 1: Illustration of the linear evolution of disturbances to a vorticity interface. Here, $T=4\pi/\omega$ is the linear wave period. Note that the initial disturbance reverses after half a period, then recovers its initial form after one full period. After a quarter of a period, the solution turns from anti-symmetric to symmetric about its centre, and this also reverses after a further half period.
• Figure 2: Time evolution of the wave amplitude $\mathcal{A}$ (left, with real and imaginary parts in blue and red respectively) for a circular vorticity interface on a sphere, together with the corresponding power spectrum $|a_m|^2$ (right) for 5 selected times $\tilde{\tau}$ (increasing downwards). Initially, just $a_2$ and $a_3$ are non-zero. [colour online]
• Figure 3: Time evolution of the spectral slope $q$, obtained by a least-squares fit of $\log|a_m|^2$ to $q\log{m}+c$, between wavenumbers $m=10$ and $M/2=1024$. The inset shows the slope over the last $0.1$ units of time computed (note, the results are not considered reliable beyond $\tilde{\tau}=0.355$ due to insufficient resolution).
• Figure 4: Time evolution of various diagnostics, as labelled, from $\tilde{\tau}=0$ to $0.355$ (the last reliable time). In the figure for ${\rm d}\theta_{\max}/{\rm d}\tilde{\tau}$, 1-2-1 averages were repeated 64 times to remove most of the noise occurring around the maximum near $\tilde{\tau}=0.32$ (endpoint values were replaced by linear extrapolation of the averaged interior data points). This noise arises from the imprecision in locating $\theta_{\rm max}$.
• Figure 5: Time evolution of the maximum wave slope $s(\tilde{\tau})=|\partial\mathcal{A}/\partial\theta|_{\max}$ together with a fit to $\sqrt{c/(\tilde{\tau}_c-\tilde{\tau})}$ (left), and the function $f(\tilde{\tau})=s^2(\tilde{\tau}_c-\tilde{\tau})-c$ (right) which would be zero for a perfect fit. [colour online]