Vortex filament

Beltrami fields have been advanced [4] as theoretical models for astrophy-sical phenomena such as solar flares and spiral galaxies, plasma vortex filaments arising from plasma focus experiments, and superconductivity. Beltrami electrodynamic fields probably have major potential significance to theoretical and empirical science. In plasma vortex filaments, for example, energy anomalies arise that cannot be described with the Maxwell-Heaviside equations. The three magnetic components of 0(3) electrodynamics are Beltrami fields as well as being complex lamellar and solenoidal fields. The component is identically nonzero in Beltrami electrodynamics if is so. In the Beltrami... 

Quoting Sir Horace Lamb [6], p. 210 There is an exact correspondence between the analytical relations above developed and certain formulae in Electro-magnetism. Hence, the vortex-filaments correspond to electric circuits, the strengths of the vortices to the strengths of the currents in these circuits, sources and sinks to positive and negative poles, and finally, fluid velocity to magnetic force. ... 

Figure 8. Dissected diagram of the vector configuration of a pair of Beltrami vortex filaments formed in the current sheath of the plasma focus (v — flow velocity, B = local magnetic field, j = current density, (>) — vorticity, Bo — background magnetic field).

With this expression for B, we find that the field fines for the solution assume a key geometric relationship in addition to the previously considered axisymmetric helicoidal solutions exemplified in the vortex filaments of the plasma focus. Several researchers [43,44] have termed P the poloidal solution and T the toroidal solution. Accordingly, if the equation for B is expressed in terms of... 

Consequently, much like the Beltrami vortex filaments discussed earlier in conjunction with the magnetostatic FFMF, the Beltrami vector relations associated with nonluminal solutions to the free space Maxwell equations, are directly related to physical classical field phenomena currently unexplainable by accepted scientific paradigms. For instance, such non-PWS of the free-space Maxwell equations are direct violations of the sacrosanct principle of special relativity [72], as well as exhibit other counterintuitive properties. Yet, even more extraordinary, these non-PWS are not only theoretical possibilities, but have been demonstrated to exist empirically in the form of the so-called evanescent mode propagation of electromagnetic energy [72-76]. 

Fig. 5.8.9. Variation of the supetfluid order parameter with distance from the centre of a quantized vortex filament as given by the Ginsburg-Pitaevskii equation (5.8.16). The same curve describes the variation of the tilt angle in the core region of the smectic C disclination. ...

Computational previews of stable particle-like organizing centers have been completed using excitable media with electrophysiological rather than chemical mechanisms [23, 30, 31]. There seems no reason to expect qualitatively different behavior in the Oregonator mechanism or the Belousov-Zhabotinsky reagent, but this expectation has thus far been tested both computationally and in the laboratory only for the dynamical consequences of twist on an arc of initially straight filament (see Sections 4 and 5 below). How are we to quantitatively inquire into the dynamics of vortex filaments The only approach undertaken to date is based on the Local Geometry Hypothesis . 

Only one 3D laboratory experiment has appeared in which Belousov-Zhabotinsky vortex filaments bear non-zero twist. Though it does not attempt to measure the twist or curvature or the filament motion featured in Henze et al. [30] (Section 5), it does indicate the possibilities of Belousov-Zhabotinsky reagent as a medium for experiments in this area. 

In 1984-1988 several potentially quantitative theories were published which addressed the effect of twist on vortex rotation rate. For reasons of symmetry, only even functions of twist can be involved and if derivatives of twist along the filament are involved they can enter only at odd order [23]. In the works of Mikhailov et al. [42, Figure 4a], Davydov et al. [49, 57], Brazhnik [58], w enters as while in the works of Mikhailov et al. [42, Equation (3)], Keener [35], and Biktashev [59] it is or a series of even powers as in Keener and Tyson [52, 53, 39]. Around the same time, it became possible to measure local twist in numerical vortex filaments [23]. Both chemical... 

Think of an uncurved vortex filament as a stack of normal two-dimensional rotors, each radiating a spiral of wavelength Aq in the horizontal plane. Twist the stack uniformly by one full turn in altitude Ittw. The diagonal separation of outgoing wave fronts. A, then satisfies... 

Vortex filaments presumably behave as in Figure 2 if uncurved and twist-free. With uniform curvature or twist this picture may be deformed. 

In three dimensions the rotor becomes a vortex filament or ring and so loses the 2D distinction between clockwise and anti-clockwise enantiomers. 

Henze, C., Vortex Filaments in Three Dimensional Excitable Media, Thesis, University of Arizona (1993). 

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