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A model is developed to simulate micro-scale turbulence driven ZFs, and their impact on the MHD tearing and kink modes is examined. The model is based on a stochastic representation of the micro-scale ZFs with a given Alfv\'{e}n Mach number, M_S. Two approaches were explored: i) passive stochastic model where the ZFs amplitudes are independent of the MHD mode amplitude, and ii) the semi-stochastic model where the amplitudes of the ZFs have a dependence on the amplitude of the MHD mode itself. The results show that the stochastic ZFs can significantly stabilise the (2,1) and (1,1) MHD modes even at very low kinematic viscosity, Pr, where the mode is linearly unstable. Our results, therefore indicate a possible mechanism for stabilisation of the MHD modes via small-scale perturbations in poloidal flow, simulating the turbulence driven ZFs., Comment: 8 pages, submitted to Physics of Plasmas August 2019

A global heat flux model based on a fractional derivative of plasma pressure is proposed for the heat transport in fusion plasmas. The degree of the fractional derivative of the heat flux, $\alpha$, is defined through the power balance analysis of the steady state. The model was used to obtain the experimental values of $\alpha$ for a large database of the JET Carbon-wall as well as ITER Like-wall plasmas. The findings show that the average fractional degree of the heat flux over the database for electrons is $\alpha \sim 0.8$, suggesting a global scaling between the net heating and the pressure profile in the JET plasmas. The model is expected to provide an accurate and a simple description of heat transport that can be used in transport studies of fusion plasmas., Comment: 6 pages submitted to Physical Review Research

The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where L\'{e}vy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable L\'{e}vy distribution as solutions to the FFP equation. The~statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The~transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data., Comment: 20 pages 7 figures

A novel possibility of self-organized behaviour of stochastically driven oscillators is presented. It is shown that synchronization by L\'evy stable processes is significantly more efficient than that by oscillators with Gaussian statistics. The impact of outlier events from the tail of the distribution function was examined by artificially introducing a few additional oscillators with very strong coupling strengths and it is found that remarkably even one such rare and extreme event may govern the long term behaviour of the coupled system. In addition to the multiplicative noise component, we have investigated the impact of an external additive L\'evy distributed noise component on the synchronisation properties of the oscillators., Comment: Accepted in J. Phys. A: Math. Theor. (2017)

Full orbit dynamics of charged particles in a $3$-dimensional helical magnetic field in the presence of $\alpha$-stable L\'evy electrostatic fluctuations and linear friction modeling collisional Coulomb drag is studied via Monte Carlo numerical simulations. The L\'evy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space resulting from intermittent electrostatic turbulence. The probability distribution functions of energy, particle displacements, and Larmor radii are computed and showed to exhibit a transition from exponential decay, in the case of Gaussian fluctuations, to power law decay in the case of L\'evy fluctuations. The absolute value of the power law decay exponents are linearly proportional to the L\'evy index $\alpha$. The observed anomalous non-Gaussian statistics of the particles' Larmor radii (resulting from outlier transport events) indicate that, when electrostatic turbulent fluctuations exhibit non-Gaussian L\'evy statistics, gyro-averaging and guiding centre approximations might face limitations and full particle orbit effects should be taken into account., Comment: 5 pages, 5 figures. Accepted as a letter in Physics of Plasmas

In plasma turbulence theory, due to the complexity of the system with many non-linearly interacting waves, the dynamics of the phases is often disregarded and the so-called random-phase approximation (RPA) is used assuming the existence of a Chirikov-like criterion for the onset of wave stochasticity. The dynamical amplitudes are represented as complex numbers, $\psi = \psi_r + i\psi_i = ae^{i\theta}$, with the amplitudes slowly varying whereas the phases are rapidly varying and, in particular, distributed uniformly over the interval $[0;2\pi)$. However, one could expect that the phase dynamics can play a role in the self-organisation and the formation of coherent structures. In the same manner it is also expected that the RPA falls short to take coherent interaction between phases into account. In this work therefore, we studied the role of phase dynamics and the coupling of phases between different modes on the characteristic time evolution of the turbulent. We assume a simple turbulent system where the so-called stochastic oscillator model can be employed. The idea of interpreting turbulence by stochastic oscillators. The stochastic oscillator models can be derived from radical simplifications of the nonlinear terms in the Navier-Stokes or Gyro-Kinetic equations. In this particular case we adopt the basic equation for the stochastic oscillator model with passive advection and random forcing from Ref., Comment: Proceedings of the 43rd EPS Conference on Plasma Physics, July 4-8, Leuven, Belgium 2016

Collective synchronous motion of the phases is introduced in a model for the stochastic passive advection-diffusion of a scalar with external forcing. The model for the phase coupling dynamics follows the well known Kuramoto model paradigm of limit-cycle oscillators. The natural frequencies in the Kuramoto model are assumed to obey a given scale dependence through a dispersion relation of the drift-wave form $-\beta\frac{k}{1+k^2}$, where $\beta$ is a constant representing the typical strength of the gradient. The present aim is to study the importance of collective phase dynamics on the characteristic time evolution of the fluctuation energy and the formation of coherent structures. Our results show that the assumption of a fully stochastic phase state of turbulence is more relevant for high values of $\beta$, where we find that the energy spectrum follows a $k^{-7/2}$ scaling. Whereas for lower $\beta$ there is a significant difference between a-synchronised and synchronised phase states, and one could expect the formation of coherent modulations in the latter case., Comment: Accepted for publication in Physics of Plasmas

A predator-prey model of dual populations with stochastic oscillators is presented. A linear cross-coupling between the two populations is introduced following the coupling between the motions of a Wilberforce pendulum in two dimensions: one in the longitudinal and the other in torsional plain. Within each population a Kuramoto type competition between the phases is assumed. Thus, the synchronisation state of the whole system is controlled by these two types of competitions. The results of the numerical simulations show that by adding the linear cross-coupling interactions predator-prey oscillations between the two populations appear which results in self-regulation of the system by a transfer of synchrony between the two populations. The model represents several important features of the dynamical interplay between the drift wave and zonal flow turbulence in magnetically confined plasmas, and a novel interpretation of the coupled dynamics of drift wave-zonal flow turbulence using synchronisation of stochastic oscillator is discussed., Comment: Accepted in Phys. Rev. E

In this paper we present a study of anomalous diffusion using a Fokker-Planck description with fractional velocity derivatives. The distribution functions are found using numerical means for varying degree of fractionality observing the transition from a Gaussian distribution to a L\'evy distribution. The statistical properties of the distribution functions are assessed by a generalized expectation measure and entropy in terms of Tsallis statistical mechanics. We find that the ratio of the generalized entropy and expectation is increasing with decreasing fractionality towards the well known so-called sub-diffusive domain, indicating a self-organising behavior., Comment: 22 pages, 14 figures

A theory of non-local linear ion-temperature-gradient (ITG) drift modes while retaining non-adiabatic electrons is presented, extending the previous work [S. Moradi, et al {\em Phys. Plasmas} {\bf 18}, 062106 (2011)]. A dispersion relation is derived to quantify the effects of the fractional velocity operator in the Fokker-Planck equation modified by temperature gradients and non-adiabatic electrons on the real frequency and growth rate. Solving the dispersion relation, it is shown here that as the plasma becomes more turbulent, it deviates from a Maxwellian distribution and becomes L\'{e}vy distributed. The resulting L\'{e}vy distribution of the plasma may thus significantly alter the transport. The relative effect of the fractional derivative is larger on the real frequency than on the growth rate of the ITG mode., Comment: 14pages 1 Figure submitted to Phys. Rev. E

Turbulence characteristics of electron cyclotron (EC) and ohmic heated (OH) discharges has been analyzed by electrostatic gyrokinetic simulations with GYRO [J. Candy, R.E. Waltz, Journal of Computational Physics 186, 545-581 (2003)] aiming to find insights into the effect of auxiliary heating on the transport. Trapped electron modes are found to be unstable in both OH and the EC heated scenarios. In the OH case the main drive is from the density gradient and in the EC case from the electron temperature gradient. The growth rates and particle fluxes exhibit qualitatively different scaling with the electron-to-ion temperature ratios in the two cases. This is mainly due to the fact that the dominant drives and the collisionalities are different. The inward flow velocity of impurities and the impurity diffusion coefficient decreases when applying EC heating, which leads to lower impurity peaking, consistently with experimental observations., Comment: Submitted to Physics of Plasmas