Vorticity alignment results for the three-dimensional Euler and Navier-Stokes equations

We address the problem in Navier-Stokes isotropic turbulence of why the vorticity accumulates on thin sets such as quasi-one-dimensional tubes and quasi-two-dimensional sheets. Taking our motivation from the work of Ashurst, Kerstein, Kerr and Gibbon, who observed that the vorticity vector omega aligns with the intermediate eigenvector of the strain-matrix S. we study this problem in the context of both the three-dimensional Euler and Navier-Stokes equations using the variables alpha = <(xi)over cap>. S<(xi)over cap> and chi = <(xi)over cap> x S<(xi)over cap> where <(xi)over cap> = w/omega. This introduces the dynamic angle phi(x,t) = arctan(chi/alpha), which lies between omega and S omega. For the Euler equations a closed set of differential equations for alpha and chi is derived in terms of the Hessian matrix of the pressure P = (p,ij). For the Navier-Stokes equations. the Burgers vortex and shear-layer solutions turn out to be the Lagrangian fixed-paint solutions of the equivalent (alpha, chi) equations with a corresponding angle phi = 0. Under certain assumptions for more general Rows it is shown that there is an attracting fixed point of the (alpha, chi) equations which corresponds to positive vortex stretching and for which the cosine of the corresponding angle is close to unity. This indicates that near alignment is an attracting state of the system and is consistent with the formation of Burgers-like structures.