The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

The three-dimensional incompressible Euler equations with a passive scalar theta are considered in a smooth domain with no-normal-flow boundary conditions . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B=a double dagger qxa double dagger theta, provided B has no null points initially: is the vorticity and q=omega a <...a double dagger theta is a potential vorticity. The presence of the passive scalar concentration theta is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (Phys. Fluids 12:744-746, 2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.