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

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          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 (\(q = \bom\cdot\nabla\theta\)) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector \(\bdB = \nabla q \times \nabla\theta\) 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 \(\theta\) 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.

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          Most cited references53

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          Remarks on the breakdown of smooth solutions for the 3-D Euler equations

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            Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

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              On the Evolution of Vorticity and Potential Vorticity in the Presence of Diabatic Heating and Frictional or Other Forces

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                Author and article information

                Journal
                16 December 2010
                Article
                1012.3597
                fd16bfaa-104b-4451-8352-648f6922f6a1

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                To appear in Proceedings of Warwick University Conference, July 2010
                nlin.CD math.AP

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