Vortex line

Kleinert H.. Gauge Fields in Condensed Matter, Vol. 1 Superflow and Vortex Lines, World Scientific, Singapore,1989. 

An external magnehc held does not create vortex lines because currents induced by gradients of the phase of kA are absent [see Eq. (19)]. To determine the response of the CEL condensate to rotahon we shall consider its free energy in the frame of reference, rotahng with a constant angular velocity lJ. This energy is dehned as follows... 

We note that there is a gradient term in Eq. (30), which is independent of the vector potentials so the rotation of the condensate produces a lattice of neutral vortex lines [12, 19], which simulates the rigid body rotation. We emphasize also that the equations for the current (19) and (20) do not change, because... 

Figure 13.3 Instantaneous isosurfaces of the vorticity magnitude for the square jet [9] at two times ti (a) and (6). 1 — hairpin rib pairs, 2 — vortex rings, 3 — vortex lines, 4 — flow ejection, and 5 — vortex-ring flattening...

The last equation here implies that the directions of a Beltrami field and the direction of its double rotational counterpart form an acute angle. If the A field is solenoidal, V A = 0, then Eq. (93) implies the condition A Vfc = 0 for any kind of k(r, t) function. This results in the fact that current lines and vortex lines are located on surfaces k r, t) = Ct for any given t. 

The counter-intuitive behavior of the single-particle conductance Eq. (3) which increases with decreasing was first predicted by Andreev [10]. Comparing Eq. (3) with the ballistic ( d) expression Eq. (1) we see that disorder with d stimulates the single-particle transport by opening of new single-particle conducting modes that are blocked by Andreev reflections in the ballistic limit. The conductance reaches its maximum when the mean free path decreases down to a, after which the distinction between the usual and the Andreev diffusion is lost and Eq. (3) transforms into Eq. (4) for a dirty wire (see [11] for the particular case of vortex lines). 

The left-hand side represents the advection (or convection) of vorticity by the velocity u, and the second term on the right-hand side represents the transport of vorticity by diffusion (with diffusivity = the kinematic viscosity v). These two terms are familiar in the sense that they resemble the convection and diflusion terms appearing in the transport equation for any passive scalar. A counterpart to the second term does not appear in these transport equations, however. Known as the production term, it is associated with the intensification of vorticity that is due to stretching of vortex lines. It is not a true production term, however, because it cannot produce vorticity where none exists. Indeed, because (10-5) contains to linearly in every term, it is clear that vorticity can be neither created nor destroyed in the interior of an isothermal, incompressible fluid It can only be convected, diffused, or changed in magnitude once it is already present.6... 

The jet-like or elongational flow field in these features can be understood as a local separation from the wall. An analysis of the vorticity field however poses a fundamental problem related to the non-slip condition at the wall since the Helmholtz laws are incompatible with this condition. One of Helmholtz s law states that a vortex line has to be closed or that it has to end at a boundary. Since at the wall (xy-plane) the velocities u and v parallel to the wall are identical zero, the same must hold for the vorticity component J normal to the wall. No vortex line can hence be attached to the wall. The main aspects of this problem are described by LIGHTHILL (1963). 

It is evident that for a description of such flows the viscosity of the fluid has to be introduced. This has been done by LIM, CHONG PERRY (1980) who showed that there exists a linearised solution of the Navier-Stokes equations which they called a viscous tornado or a complex eigenvalue critical point flow. This solution permits a vortex line to be quasi attached to the wall under non-slip conditions in the sense that a versatil spiral flow around a separation streamline arrises. The concept be scetched as follows The flow near a separation point at the wall can be classified by the form... 

In superconductors of the second kind at T< in a mixed state (in a magnetic field between Hei and H 2, where is the lower and H 2 the upper critical field) a new effect in the thermal conductivity is possible phonon and electron scattering by Abrikosov vortex lines penetrating the superconductor (Red ko and Chakalski 1987). 

Only a weak attractive interaction occurs between the pancake vortices of different layers, the strength of which depends on the magnetic field and the temperature. Therefore the pancake vortices are much more flexible than the continuous, relatively rigid vortex lines in conventional superconductors. 

Fig. 4.2-18 Vortex line caused by an external field exhibiting an angle 0 to the al)-planes...

Fig. 4.2-17 Vortex line in YBa2Cu307 (left-hand picture) and pancake vortices in Bi2Sr2Ca2Cu30io (right-handpicture)-, theCu2-planes are drawn with bold lines and the isolating planes with broken lines...

The topological index i counts the number of times that the current density vector 7 rotates completely while one walks counterclockwise around a circle of radius e, so small that 7 has no zeroes inside except the SP at its centre. The topological index i of a saddle (vortex) line is -1 (+1). Both SPs have (r, i) = (2,0). 

The splitting of a SL into several SLs is regirlated by a fundamental topological theorem proved by Gomes [55-57, 93] in the form of an index conservation constraint. Recall that, according to footnote 3, the index of a saddle (vortex) line is — 1 (+1). When an SL of index rp splits into m new lines, the sum of the indices of the SLs which emerge from the branching point must satisfy the condition... 

For instance, a vortex line may bifurcate giving rise to two new vortex lines and one saddle line. This bifurcation conserves the total index +1. 

The primary diatropic vortex SL branches out at two (0, 0) critical points (too far to be seen in Fig. 7.7) into a set of three SLs on the plane of the nuclei a central (blue) saddle SL and two (green) vortical SLs crossing the C-H bonds. These vortex lines become saddle-type at points with coordinates (x = 0, y = 1.05, z = 0.86) bohr, where the i topological index changes from -1 to 1. The saddle SLs merge at... 

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