Stretching and folding processes in the 3D Euler and Navier-stokes equations

Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the context of the vector B = ∇q×∇θ where, in atmospheric physics, θ is a temperature, q = ω ·∇θ is the potential vorticity, and ω = curlu is the vorticity. These ideas are then discussed in the context of the full compressible Navier-Stokes equations where q is taken in the form q = ω ·∇ f(ρ). In the two cases f = ρ and f = lnρ, q is shown to satisfy the quasi-conservative relation ∂tq+divJ =