Applications of a Cole-Hopf transform to the 3D Navier-Stokes equations

The Navier-Stokes equations written in the vector potential can be recast as the nonlinear Schr¨odinger equations at imaginary times, i.e. the heat equations with a potential term, using the Cole-Hopf transform introduced in Ohkitani(2017). On this basis, we study two kinds of Navier-Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection is taking place, whereas the signature of the nonlinear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magnetohydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions.