Helicity and the Călugăreanu invariant

Helicity spectra and topological dynamics of vortex links at high Reynolds numbers

“…For the Gaussian kernel case, it is (Leonard 1985; Winckelmans & Leonard 1993) There are some important features: by employing the smoothing kernel, we transform line vortices to vortex tubes. This is very much in agreement with the derivation of the Moffatt & Ricca formula (Moffatt & Ricca 1992) where both writhe and twist are forms of self-linking involving the vortex lines within single tubes. Pfister & Gekelman (1990), Moffatt & Ricca (1992) and Scheeler et al.…”

“…Employing ribbons for theoretical illustration, they argued that becomes during reconnection. We show here that, in our vortex-dynamical model, becomes , which is something similar, since Pfister & Gekelman (1990) and Moffatt & Ricca (1992) indicate how twist (formed by helical winding of one ribbon edge about the other) can be transformed to writhe (corresponding to self-intersections of the ribbon centreline in a two-dimensional projection of it) via continuous transformations. This interchangeability of and is implied by the topological nature of , and can even be realized when topological calculations that refer to the same geometry employ different view angles (Klenin & Langowski 2000).…”

“…Moffatt (1969, 2014) and Ricca & Moffatt (1992) indicated that this topological change is directly related to helicity, by deriving an intriguing relation between , a physical quantity, and topological measures of a system of vortex tubes: where is the circulation of tube , is the Gauss linking number of tubes and , and and are, correspondingly, the writhe and twist of tube . The sum of and is the Cǎlugǎreanu self-linking number , which measures the degree of knottedness of single closed tubes, and, according to Cǎlugǎreanu's theorem (Cǎlugǎreanu 1961; Moffatt & Ricca 1992), is a topological invariant. Helpful discussions of these topological concepts in the contexts of physics applications are available in Scheeler et al.…”

Godbillon-Vey helicity and magnetic helicity in magnetohydrodynamics

“…1982; Salmon 1982, 1988; Sagdeev et al. 1990; Moffatt & Ricca 1992; Tur & Yanovsky 1993; Padhye & Morrison 1996 a , b ; Holm et al. 1998; Kats 2003; Cotter et al.…”

“…Elsässer 1956; Kruskal & Kulsrud 1958; Woltjer 1958; Moffatt 1978; Berger & Field 1984; Finn & Antonsen 1985; Bieber et al. 1987; Finn & Antonsen 1988; Moffatt & Ricca 1992; Low 2006; Longcope & Malanushenko 2008; Webb et al. 2010; Low 2011; Prior & Yeates 2014; Webb et al.…”