Magnetic helicity

Helicity is defined as a qualitative measurement of how a topological configuration is linked, knotted, or twisted. If A is the vector potential of electromagnetism, the quantity then defines magnetic helicity. In order to obtain a nonnull value for K, the condition A (V A) /- 0 must be verified in the volume... 

The quantity in the right-hand side of (21) is called the helicity of the vector b and is used in several contexts, mainly in fluid and in plasma physics. The term was coined by Moffatt in 1969 in a paper on tangled vorticity lines [44] with the velocity of a fluid v and its vorticity oa= V x v as the vectors a and b. The magnetic helicity h [ A BrfV is useful to study the magnetic configurations in astrophysical plasmas and in tokamaks. 

We will write h(X) or simply h if there is no risk of confusion. The helicity (24) is especially useful in two physical contexts (1) in fluid dynamics, where Y is the flow velocity v(r, t), X is the vorticity w = V x v, and h w, D) is called vortex helicity, and (2) in plasma physics, or in general in electromagnetism, under the form of magnetic helicity, dehned as... 

The electric and magnetic fields are invariant under gauge transformations, 4 X(x) i—>A (x) + cM A(x). The effect of these transformations on the helicity has been treated by Marsh [49]. The variation of the magnetic helicity under a gauge transformation 8A = —VA is... 

The relationship between the linking number and the helicity of a self-knotted filamental tube has been studied by Berger and Field [50] and Moffatt and Ricca [51]. We are looking for an expression for the helicity in a filamental tube 7 of strength 5, around a closed magnetic line C that can be self-knotted. In this case, the magnetic helicity takes the value... 

Using this theorem, we conclude that the magnetic helicity of a filamental tube T whose axe is a closed, self-knotted curve C, is given by... 

The electric and magnetic fields are dual to one another and have the same properties in Maxwell theory in empty space. Given the divergenceless vector field B, we have defined the magnetic helicity as... 

We must consider two currents of helicity. In Ref. 47 the following magnetic helicity current was considered ... 

Given any Maxwell field in vacuum, we define the current density of electromagnetic helicity Jf x as one-half times the sum of the current densities of electric and magnetic helicities (41) and (42) ... 

This equation shows a close relation between the wave and particle aspects of the helicity. On the left side, the wave helicity is the semisum of the electric and magnetic helicities that characterizes the topology of the force lines as a function of the linking number of the pairs of electric lines and of the magnetic lines. On... 

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