Facets and singularities

In recent months, in the conclusion of my research at Caltech, two papers have made it into the world. Although they’re pretty heavy on the maths and don’t feature very many illustrative figures, they both make a simple point: with an approximate model, you can show that a large class of shock waves starting off with smooth wobbles on their surface are guaranteed to develop sharp edges after some time which you can quantify. More precisely, there is a time when the shock geometry develops a singularity.

Both papers are based on a technique used by the well-known mathematician Derek Moore, who showed that similar singularities are guaranteed to form on perturbed vortex sheets. Like his study, ours involves a complicated nonlinear formulation and analysis of. the shock wave development in terms of its Fourier mode coefficients, and seek a time when these coefficients fail to decay at an exponential rate with respect to mode number. Our first paper is limited to planar shocks, and has application to a particular cell-pattern which is often seen in detonation waves. The second paper is relevant to our old friend, converging flows, since it applies to cylindrical shocks: in fact, there we show that singularities are guaranteed to form before the shock collapses to its focal point, no matter what!

The importance of symmetry in inertial fusion means that the second result may be especially significant, given the way that these singularities affect the symmetry of the implosion as a whole. To be sure, the analysis is approximate because of the model we use, and has a few other technical caveats, but as far as we know it’s the only one that attempts to quantify the time it takes for these singularities to form.

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