Enhanced dissipation in a vector convection diffusion equation

An important characteristic of turbulent flow is that the rate of energy dissipation becomes independent of viscosity in the limit of vanishingly small viscosity, a phenomenon known as anomalous dissipation. Whether there exists a family of simple (steady or time-periodic) solutions to the Navier-Stokes equations that exhibit anomalous dissipation remains an important open problem in fluid mechanics. In this work, we address this problem by considering a passive vector equation, which is a more flexible variant of the Navier-Stokes equations. Designing solutions that exhibit anomalous dissipation is quite challenging even for this flexible equation. In this work, we construct a family of solutions to the passive vector equation where the rate of energy dissipation is proportional to the viscosity raised to the power of one-third. Notably, this rate of dissipation is significantly faster than what is observed in pure diffusion which implies enhanced dissipation. Our result is established through a design based on convection rolls and a variational principle for the rate of energy dissipation.