On the effective Liapunov exponent in the tokamak context
Résumé
Liapunov exponent provides a quantitative measure of the degree of stochasticity for a trajectory in the theory of both Hamiltonian and dissipative dynamical systems. In the context of fusion plasmas, the Liapunov exponent is usually associated with the regime of highly developed magnetic or collisional stochasticity (transport in the highly stochastic magnetic field, ergodic divertor problem, etc.). To treat general case with growing stochasticity, the extension of the standard concept is done introducing the effective Liapunov exponent, $L_e(t)$, as the averaged Liapunov exponent over different trajectories/magnetic field lines. The numerical analysis includes the observation of the Kolmogorov entropy, the number of the trajectories/ magnetic field lines with positive Liapunov exponent and statistics of the Liapunov exponents. Three models relevant for tokamak physics are considered: tokamap, revtokamap and the model of test electron radial diffusion in the magnetic field with irregularities. It is shown that the effective Liapunov exponent is in the correspondence with the degree of stochasticity. Additionally, its behaviour is independent on the origin of stochasticity (deterministic or statistical stochasticity). The dynamical behaviour of the effective Liapunov exponent can be associated with the type of test particle transport in different magnetic stochasticities: strange transport in the magnetic field regions with the mixture of the regular and irregular structures, uniform mixing in the highly irregular regions and standard diffusion in the case with highly frequent collisions and regular or partially destroyed magnetic field regions.
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