Topological Torsion and Spin in Plasma Reconnection Processes

In the space-time realm of plasma physics it is little appreciated that magnetic helicity, (A.B), is but the fourth component of a third rank tensor field. This field, defined as Topological Torsion, has four components, [E x A + B.phi; (A.B)], and a four-divergence proportional to the second Poincare invariant, (E.B). When the second Poincare invariant vanishes, the closed integrals of Topological Torsion are deformation invariants (hence, topological properties) for any evolutionary process described by a singly parameterized vector field. This statement does not mean that the volume integral of helicity is conserved, necessarily, for there exist plasma states where the helicity vanishes, but the Topological Torsion tensor does not. Similarly, it is little appreciated that there exists another topological quantity in a plasma, a third rank tensor density defined as Topological Spin, with four components [A x H + D.phi; (A.D)]. Topological Spin generates a topological property of the fields which is distinct from the properties of Topological Torsion. The four divergence of the Topological Spin is proportional to the first Poincare invariant, and is equal to the Lagrangian of the field including interactions between the currents and the potentials: ((B.H) - (D.E)) - ((A.J) - rho.phi). When the first Poincare invariant vanishes the closed integrals of Spin are evolutionary deformation invariants (hence, topological properties). Both conserved closed integrals have values whose ratios are rational (quantized). However, topological spin does not require the existence of topological torsion. There exist plasma solutions to Maxwell’s equations where Torsion is zero, but Spin is not, solutions where Spin is zero but Torsion is not, and solutions where both are zero or both are not zero.
Applications of these concepts with analytic examples of plasma states which support both Topological Spin and Topological Torsion are given (1). In particular, a time dependent analytic solution to Maxwell’s equations is given (2) which starts with a set of initial data dominated by magnetic fields and evolves into another different final state, again dominated by magnetic fields. However, during the evolutionary process, the magnetic field lines appear to go through a cut and re-connect process. In certain regions, the magnetic field diminishes with time, but the electric field grows, preserving the total electromagnetic flux. After the re-connection takes place, the electric field decreases, and the magnetic field increases. Initially, the value of the second Poincare invariant is zero, but this value goes through a time dependent pulse such that (E.B) is not zero and is maximized at the re-connection time. Ultimately (E.B) returns to a zero value. In the examples, it would appear to be necessary that finite values of (E.B) are associated with, if not the cause of, the topological re-connection process.



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Last update 12/01/2000
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