Stretching & folding diagnostics in solutions of the three-dimensional Euler & 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 vor- ticity (q = ! � r�) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector B = r qrsatisfies the same type of stretching and folding equation as that for the vorticity field ! in the incompressible Euler equations (Gibbon & Holm, 2010). The quantitymay be chosen as the potential temperature for the strat- ified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discus- sion 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 D = ! � u, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector ($ := curlD) turns out to possess similar stretching and folding properties to that of the B-vector.