%0 Book Section %1 statphys23_0123 %A Sonnino, G.S. %A Peeters, P.H.P. %A Zonca, F.Z. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K elementary equilibrium non physics plasmas processes statistical statphys23 thermodynamics topic-3 transport %T A Heuristic Kinetic Model for Transport Coefficients of Tokamak Plasmas %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=123 %X The main goal of neoclassical transport theory is to study the behaviour of a plasma in the presence of an inhomogeneous and curved magnetic field. One of the most important results of the neoclassical theory is that the global geometry of the magnetic field has a very strong influence on the transport processes. The neoclassical theory is able to derive the transport coefficients for ions and electrons when the plasma is magnetically confined in tokamak reactors. In spite of its elegant and coherent formulation, the theoretical predictions of the neoclassical theory are in strong disagreement with experience. The experimental ion heat flux measured in tokamak plasmas is roughly in agreement with the neoclassical theory. However, the electron particle flux and the electronic heat flux are about $1010^2$ greater than the values computed by the neoclassical theory. The difference between the experimental and the neoclassical flux is referred to as the anomalous flux. For many physicists, the origin of this discrepancy is mainly attributed to turbulence phenomena existing in tokamak plasmas. Fluctuations in plasmas can become unstable and therefore amplified. According to this interpretation, fluctuations will successively interact in a nonlinear way leading the plasma to a state, which is far away from equilibrium. In this condition, the transport properties are supposed to change significantly. In this paper we develop a heuristic kinetic model able to interpret the theoretical predictions of the thermodynamic field theory (TFT) applied to transport processes in tokamak plasmas. Through our model, we shall roughly estimate the transport coefficients. We shall see that, consistently with intuitive expectations, the electric diffusion coefficient and the electrical thermal coefficient behave as the inverse of the electronic collision time, $\tau_e$, while the electrical conductivity is proportional to $\tau_e$. Since, the collision time decreases as the intensity of the thermodynamic forces increase, our model predicts that the electrical conductivity decreases as the intensity of the electric field increases, while the values of the electrical thermal and diffusion coefficients increase as the intensity of the thermodynamic forces increases. These results are in line with numerical simulations of a Lorentz gas subject to a strong electric field. Successively, we describe two simple effects generated by the gradient of the radial electric drift : the banana orbit deformation effect (the squeezing orbit effect) and the banana orbit rotation effect. The last effect is expected to be visible when the up-down symmetry of the Tokamak plasma is broken. These two mechanisms can enhance the value of the radial diffusion coefficient of the plasma as well as the magnitude of the electron bootstrap current. Summarizing, an increment of the value of the thermodynamic forces reduces the value of the electron collision time and, at the same time, induces, through the electric drift, a deformation and/or a rotation of the guiding centres orbits of the trapped particles. Another effect, which may affect the deformation of the banana orbit of the particles and, therefore, the value of the diffusion coefficient, is the toroidal geometry of the tokamak. The final part of our work is specifically devoted to the analysis of this effect. We shall prove that this effect may significantly enhance the value of the electron radial diffusion coefficient and, without involving turbulence phenomena, we can estimate that the value of this coefficient may exceed the value estimated by the Berk, Galeev, Sagdeev and Balescu theory (BGSB) by a factor, which may be of order $10$. The value of the ion diffusion coefficient remains, however, substantially unaltered.

@incollection{statphys23_0123, abstract = {The main goal of neoclassical transport theory is to study the behaviour of a plasma in the presence of an inhomogeneous and curved magnetic field. One of the most important results of the neoclassical theory is that the global geometry of the magnetic field has a very strong influence on the transport processes. The neoclassical theory is able to derive the transport coefficients for ions and electrons when the plasma is magnetically confined in tokamak reactors. In spite of its elegant and coherent formulation, the theoretical predictions of the neoclassical theory are in strong disagreement with experience. The experimental ion heat flux measured in tokamak plasmas is roughly in agreement with the neoclassical theory. However, the electron particle flux and the electronic heat flux are about $10\div 10^2$ greater than the values computed by the neoclassical theory. The difference between the experimental and the neoclassical flux is referred to as the {\it anomalous flux}. For many physicists, the origin of this discrepancy is mainly attributed to turbulence phenomena existing in tokamak plasmas. Fluctuations in plasmas can become unstable and therefore amplified. According to this interpretation, fluctuations will successively interact in a nonlinear way leading the plasma to a state, which is far away from equilibrium. In this condition, the transport properties are supposed to change significantly. In this paper we develop a heuristic kinetic model able to interpret the theoretical predictions of the thermodynamic field theory (TFT) applied to transport processes in tokamak plasmas. Through our model, we shall roughly estimate the transport coefficients. We shall see that, consistently with intuitive expectations, the electric diffusion coefficient and the electrical thermal coefficient behave as the inverse of the electronic collision time, $\tau_e$, while the electrical conductivity is proportional to $\tau_e$. Since, the collision time decreases as the intensity of the thermodynamic forces increase, our model predicts that the electrical conductivity {\it decreases} as the intensity of the electric field increases, while the values of the electrical thermal and diffusion coefficients {\it increase} as the intensity of the thermodynamic forces increases. These results are in line with numerical simulations of a Lorentz gas subject to a strong electric field. Successively, we describe two simple effects generated by the gradient of the radial electric drift : the banana orbit deformation effect (the squeezing orbit effect) and the banana orbit rotation effect. The last effect is expected to be visible when the {\it up-down} symmetry of the Tokamak plasma is broken. These two mechanisms can enhance the value of the radial diffusion coefficient of the plasma as well as the magnitude of the electron bootstrap current. Summarizing, an increment of the value of the thermodynamic forces reduces the value of the electron collision time and, at the same time, induces, through the electric drift, a deformation and/or a rotation of the guiding centres orbits of the trapped particles. Another effect, which may affect the deformation of the banana orbit of the particles and, therefore, the value of the diffusion coefficient, is the toroidal geometry of the tokamak. The final part of our work is specifically devoted to the analysis of this effect. We shall prove that this effect may significantly enhance the value of the electron radial diffusion coefficient and, without involving turbulence phenomena, we can estimate that the value of this coefficient may exceed the value estimated by the Berk, Galeev, Sagdeev and Balescu theory (BGSB) by a factor, which may be of order $10$. The value of the ion diffusion coefficient remains, however, substantially unaltered.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sonnino, G.S. and Peeters, P.H.P. and Zonca, F.Z.}, biburl = {https://www.bibsonomy.org/bibtex/27c49a572e73182af4df4f8e60af51d17/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {6bd0461570c03408fd273103ab39fde2}, intrahash = {7c49a572e73182af4df4f8e60af51d17}, keywords = {elementary equilibrium non physics plasmas processes statistical statphys23 thermodynamics topic-3 transport}, month = {9-13 July}, timestamp = {2007-06-20T10:16:12.000+0200}, title = {A Heuristic Kinetic Model for Transport Coefficients of Tokamak Plasmas}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=123}, year = 2007 }