Stretching and folding diagnostics in solutions of the three-dimensional Euler and Navier-Stokes equations

Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity ($q = \bom\cdot\nablaθ$) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector $\bdB = \nabla q \times \nablaθ$ satisfies the same type of stretching and folding equation as that for the vorticity field $\bom $ in the incompressible Euler equations (Gibbon & Holm, 2010). The quantity $θ$ may be chosen as the potential temperature for the stratified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discussion of these S&F-flow diagnostics concerns a numerical test for Euler codes and also includes a connection with the two-dimensional surface quasi-geostrophic equations. The second S&F-flow diagnostic concerns the evolution of the Lamb vector $\bsD = \bom\times\bu$, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector ($\boldsymbol{\varpi} := \bsD$) turns out to possess similar stretching and folding properties to that of the $\bdB$-vector.