English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Cauchy invariants are now viewed as a powerful tool for investigating theLagrangian structure of three-dimensional (3D) ideal flow (Frisch &Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina etal., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariantswith the modern tools of differential geometry and of geodesic flow on thespace SDiff of volume-preserving transformations (Arnold, Ann. Inst. Fourier,vol. 16, 1966, pp. 319-361), all manners of generalisations are here derived.The Cauchy invariants equation and the Cauchy formula, relating the vorticityand the Jacobian of the Lagrangian map, are shown to be two expressions of thisLie-advection invariance, which are duals of each other (specifically, Hodgedual). Actually, this is shown to be an instance of a general result, whichholds for flow both in flat (Euclidean) space and in a curved Riemannian space:any Lie-advection invariant p-form which is exact (i.e. is a differential of a(p-1)-form) has an associated Cauchy invariants equation and a Cauchy formula.This constitutes a new fundamental result in linear transport theory, providinga Lagrangian formulation of Lie advection for some classes of differentialforms. The result has a broad applicability: examples include themagnetohydrodynamics (MHD) equations and various extensions thereof, discussedby Lingam et al. (Phys. Lett. A, vol. 380, 2016, pp. 2400-2406), and includealso the equations of Tao (2016, arXiv:1606.08481 [math.AP]), Euler equationswith modified Biot-Savart law, displaying finite-time blow-up. Our main resultis also used for new derivations, and several new results, concerning localhelicity-type invariants for fluids and MHD flow in flat or curved spaces ofarbitrary dimension.

Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces