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Relativistic generation of vortex and magnetic field a)
2. So called “Casimir invariants” represent the singularity (topological defect) of the Poisson bracket. Helicity is a typical example of Casimir invariants, which allows existence of vorticity in an ideal Hamiltonian mechanics; for example see V. Arnold and B. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag, Berlin, 1998).
3. R. G. Charney, J. Meteorol. 4, 135 (1947).
10. “Ideal mechanism” in this paper denotes a process in which the entropy is a function of temperature so that the heat= becomes an exact differential. A “space-time distortion” that special-relativistic effects create, may, for instance covert an exact differential into a “Clebsch form” such as that may have a vorticity . We note that the vortical field derived from a Clebsch form is helicity free ; creation of helicity does need the help of a flow that can produce another Clebsch component ;
11. Let us denote . Then, . Using Eq. (4) and the boundary condition (by the flux conservation, this boundary condition holds constantly on the surface that moves with the fluid), we observe . Hence, .
12. The helicity may be generalized to produce, for example, Gauss’s linking number of the “vortex line”, the curve tangential to ; for example, see H. K. Moffatt, Magnetic field generation in electrically conducting fluids (Cambridge University Press, Cambridge, 1978).
14. L. D. Landau and E. M. Lifshitz, Hydrodynamics (Science, Moscow, 1986);
14. D. I. Dzhavakhrishvili and N. L. Tsintsadze, Sov. Phys. JETP 37, 666 (1973).
15. The notion of “helicity” parallels the total “charge” of a four-vector; if is conserved, i.e., , we obtain the charge (helicity) conservation . Defining the dual , we set K. Then, using 0, the antisymmetry of , and , we observe . We note that all these relations parallels the well-known facts of EM field theory with replacing by and by ; see Ref. 13.
16. The displacement-current term in Ampere’s law will appear as on the left-hand side of Eq. (21). In the time scale ( is the length scale of the structures), this term may be neglected with respect to , and the D’Alembert operator collapses to the elliptic operator, eliminating the EM waves. The displacement-current term may not be neglected when one compares the divergence of Ampere’s law with the mass conservation law. But this is not pertinent to the present calculations.
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