Non-invariance of both magnetic fluxes within flux tubes and global helicities in an ideal plasma and numerical demonstrations of generalized self-organization theory

KONDOH Yoshiomi, SERIZAWA Shunsuke, TAKAHASHI Toshiki1), James W. Van Dam 2)

1)Dept. of Electronic Engineering, Gunma University, Kiryu, Gunma 376-8515, Japan
2) Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA.

It is proved that when we correctly use the full generalized Ohm's law instead of the simplified Ohm's law, both magnetic fluxes within flux tubes and global helicities (including the magnetic helicity and the self-helicity) are not invariant, even in an ideal MHD plasma. The identity of a flux tube cannot be maintained in a flowing turbulent plasma, even if the plasma itself is ideally conducting. The result is that topological quantities are not preserved during the turbulent flow of an ideally conducting plasma. This removes the physical basis for self-organization theories based on the concept of frozen-in magnetic field lines. It is also showed that the time rate of change of the self-helicity Ks and the time rate of change of the magnetic and kinetic energies Wm+k have exactly the same power-law dependence on the fluctuation wave number. This result implies that selective decay between Ks and Wm+k does not occur. Hence, theories based on helicity invariance have questionable relevance to relaxation and self-organized states in experimental plasmas.
It is presented three typical applications of the new general theory extended from [1] and originated from [2,3] for how to find self-organized states in order to demonstrate that its applicability to various dissipative nonlinear dynamical systems. The three applications are (1) Korteweg-deVries solitons with dissipation, (2) two-dimensional incompressible viscous fluids with periodic boundaries, and (3) compressible resistive viscous MHD fusion plasmas. It is shown that the final analytical solutions for the self-similar, slowest decay phase with the smallest eigenvalue for the two applications (1) and (2) agree very well respectively with simulations reported in [4] and with new simulation data as well as simulations with friction free boundaries reported in [5]. Specifically, the correlation coefficient of the analytic relation for (2) with the simulation results is 0.999, being so close to unity and much better than that for the sinh-Poisson solution of [6]. The Euler-Lagrange equations for the application (3) are shown to include both the Taylor state and the Beltrami flow as the limiting case of uniform resistivity, uniform viscousity, and quasi-steady zero-pressure plasma.

References

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