Singularity formation of vortex sheets in two-dimensional Euler equations using the characteristic mapping method

The goal of this numerical study is to get insight into singular solutions of the two-dimensional (2D) Euler equations for nonsmooth initial data, in particular for vortex sheets. To this end, high resolution computations of vortex layers in two-dimensional incompressible Euler flows are performed using the characteristic mapping method (CMM). This semi-Lagrangian method evolves the flow map using the gradient-augmented level set method. The semigroup structure of the flow map allows its decomposition into submaps (each over a finite time interval), and thus, the precision can be controlled by choosing appropriate remapping times. Composing the flow map yields exponential resolution in linear time, a unique feature of CMM, and thus, fine-scale flow structures can be resolved in great detail. Here, the roll-up process of vortex layers is studied varying the thickness of the layer showing its impact on the growth of palinstrophy and possible blow up of absolute vorticity. The curvature of the vortex sheet shows a singular-like behavior. The self-similar structure of the vortex core is investigated in the vanishing thickness limit. Conclusions on the presence of posssible singularities of two-dimensional Euler equations for nonsmooth initial data are drawn by tracking them in the complex plane.