Some topological aspects of fluid dynamics

Abstract

Abstract

Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations

“…Similar to the discussions in [27], we show that the hybrid helicity provides a lower bound for the energy when α = β = S −1 R H . This bound, which was referred to as the Arnold inequality in the case of the magnetic helicity [43,Section 8], shows that non-zero hybrid helicity, as a measure of the knottedness, provides a topological barrier which prevents a hybrid energy defined by B + S −1 R H ω 2 from decaying below a certain value. The conclusion also holds for dissipative flows where the helicity is not conserved.…”

On Moffatt's magnetic relaxation equations

“…where the unknowns are the incompressible velocity vector field u, the magnetic vector field B, and the fluid pressure p. We consider the problem posed on T d = [−π, π] d with d ∈ {2, 3}, and u is taken to have zero mean on T d . The parameter γ ≥ 0 is a regularization parameter of the constitutive law B → u: the case γ = 0 corresponds to a Darcy-type regularization (as was done in [Mof85,Bre14,Mof21]), the case γ = 1 corresponds to a Stokes-type regularization, while the general case γ > 0 may be alternatively used in numerical simulations to smoothen the velocity gradients. This constitutive law may be written as…”

“…The idea is to use the evolution dynamics of the magnetic field to reach an Euler/magnetic equilibria which preserves Kelvin circulation. This concept was developed by Moffatt [Mof85] (see also the excellent recent overview [Mof21]). The magnetic relaxation procedure envisioned by Moffatt preserves the streamline topology of an initial divergence free three-dimensional vector B 0 (x), but abandons the constraint that B(x, t) should remain smooth as t → ∞.…”

“…Moffatt introduced the concept of topological accessibility which is weaker than topological equivalence 1 because it allows for the appearance of discontinuities in the magnetic field (current sheets) as t → ∞. As an example of a constitutive law relating u to B, Moffatt [Mof85,Mof21], also Brenier [Bre14], suggested…”

Helical Flow States in Active Nematics

“…In these cases of long plant cells, helical flow enables significant transport, mixing and enhanced rates of nutrient exchange with the surrounding membrane [62]. Helicity generally plays an important role in characterizing the topological nature of 3D flows [66]. Indeed, in stark contrast to the defect dancing observed in the Ceilidh vortex lattice state, the grinder train does not possess any topological singularities, while the double helix possesses a single disclination line that winds through the two helical flows.…”