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 = curl u 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 = ln ρ, q is shown to satisfy the quasi-conservative relation ∂t q + div J = 0.