Biot-Savart helicity versus physical helicity: A topological description of ideal flows

TL;DR

This paper explores the topological differences between Biot-Savart helicity and physical helicity in ideal fluid flows, revealing their independence and dependence on domain topology, and extends the analysis to magnetohydrodynamic flows.

Contribution

It introduces a topological framework distinguishing Biot-Savart and physical helicities and analyzes their behavior in complex domains, including multiple tori and MHD flows.

Findings

Biot-Savart and physical helicities are independent constants in ideal flows.

In simply connected domains, the two helicities coincide.

Helicity difference depends only on harmonic knot components of the fields.

Abstract

For an isentropic (thus compressible) flow, fluid trajectories are considered as orbits of a family of one parameter, smooth, orientation preserving and nonsingular diffeomorphisms on a compact and smooth-boundary domain in the Euclidian 3-space which necessarily preserve a finite measure, later interpreted as the fluid mass. Under such diffeomorphisms the Biot-Savart helicity of the pushforward of a divergence-free and tangent to the boundary vector field is proved to be conserved and since these circumstances present an isentropic flow, the conservation of the "Biot-Savart helicity" is established for such flows. On the other hand, the well known helicity conservation in ideal flows which here we call it "physical helicity" is found to be an independent constant with respect to the Biot-Savart helicity. The difference between these two helicities reflects some topological features of the domain as well as the velocity and vorticity fields which is discussed and is shown for simply connected domains the two helicities coincide. The energy variation of the vorticity field is shown to be formally the same as for the incompressible flow obtained before. For fluid domains consisting of several disjoint solid tori, at each time, the harmonic knot subspace of smooth vector fields on the fluid domain is found to have two independent base sets with a special type of orthogonality between these two bases by which a topological description of the vortex and velocity fields depending on the helicity difference is achieved since this difference is shown to depend only on the harmonic knot parts of velocity, vorticity and its Biot-Savart vector field. For an ideal MHD flow three independent constant helicities are reviewed while the helicity of magnetic potential is generalized for non-simply connected domains by inserting a special harmonic knot field in the dynamics of the magnetic potential.