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

Abstract

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 = ω · ∇θ) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector B = ∇q × ∇θ satisfies the same type of stretching and folding equation as that for the vorticity field ω 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 D = ω × u, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector (̟ := curl D) turns out to possess similar stretching and folding properties to that of the B-vector.

Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

“…The local conservation law form for the 'potential vorticity' is equivalent to the formula (3.9) derived using a direct method, with Θ ≡ F. The potential vorticity was used to study quasi-conservation laws for compressible Navier-Stokes equations in Gibbon & Holm (2012a), with the mass density ρ used in the place of the arbitrary function F. From the physical point of view, it is interesting to note that, in the latter paper, it was observed that, in the resulting conservation law, the pressure and other thermodynamic terms cancel completely, without any further simplification, such as the barotropic approximation. The dynamics of the gradient of the potential vorticity was considered in Gibbon & Holm (2010, 2012b; possible applications to atmospheric analysed data at the tropopause were discussed.…”

On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models

“…Remark 2.13. A dual version of theorem 2.12, stating in our notation that the cross product between the gradients of two constants of motion must be a symmetry, has been applied to fluid models in [16][17][18][19]. Also, a result analogous to theorem 2.11, stating that the Lie derivative of a constant of motion along a symmetry is a constant of motion, is well known [6] and has been applied to fluid equations recently [20].…”

Stretching and Folding Processes in the 3D Euler and Navier-Stokes Equations

“…Likewise in later sections we also discuss the compressible case in the same spirit. This paper aims to marry the ideas on stretching and folding processes in incompressible flows developed by the authors in [13,14] with their work on compressible flows [15]. Here, two forms of the projection of the vorticity ω onto the gradient of the mass density ρ(x, t) are discussed in §2.…”

“…Namely, the projections within q = ω · ∇ρ and q = ω · ∇(ln ρ) cannot be transported 1 by the pseudo-velocity U q across level sets of the mass density, ρ. These equations may also be useful in the study of stretching and folding in compressible fluid flows, as investigated in the atmospheric context in [13,14].…”