We investigate percolation in binary and ternary mixtures of patchy colloidal particles theoretically and using Monte Carlo simulations. Each particle has three identical patches, with distinct species having different types of patch. Theoretically we assume tree-like clusters and calculate the bonding probabilities using Wertheim's first-order perturbation theory for association. For ternary mixtures we find up to eight fundamentally different percolated states. The states differ in terms of the species and pairs of species that have percolated. The strongest gel is a trigel or tricontinuous gel, in which each of the three species has percolated. The weakest gel is a mixed gel in which all of the particles have percolated, but none of the species percolates by itself. The competition between entropy of mixing and internal energy of bonding determines the stability of each state. Theoretical and simulation results are in very good agreement. The only significant difference is the temperature at the percolation threshold, which is overestimated by the theory due to the absence of closed loops in the theoretical description.
Large scale structures in the E × B shearing rate, known as staircases, are shown to form in nonlinear gyro-kinetic turbulence simulations with kinetic electrons. However, in many cases, a small scale structure in the shearing rate is observed that appears to prevent the formation of staircases. The small scale structures are interpreted to be linked to the self-interaction of turbulent modes connected with the double periodic boundary conditions on the torus. The self-interaction is a newly discovered mechanism for zonal flow generation and is shown to scale proportional to the normalized Larmor radius. The mechanism is also affected by magnetic shear, being weaker at larger values.
The dependence of the heat flux on the temperature gradient length in collisionless ion temperature gradient turbulence has recently been revisited. It has been found that the heat flux is discontinuous at a finite heat flux threshold larger than the (Dimits) interpolated threshold. In this paper, the influence of collisions on the heat flux close to the threshold is investigated. It is found that up to relatively high collision frequencies, relevant to the modern day experiments, a discontinuous behaviour of the heat flux as a function of the gradient length persists. Collisions, however, do lead to a reduction in the gradient length at which the discontinuity is observed. Below the finite heat flux threshold, a state of low turbulence with a vanishing small heat flux persists, which can drive the zonal flow against the collisional dissipation. This state is characterised by the fully developed staircases in the radial ExB shearing profile. Increasing the collision frequency at a fixed gradient length leads to the loss of the fully developed staircase structure with the ExB shearing profile having the form of a sawtooth that allows for avalanche formation and a finite heat flux. At very high collision frequencies or gradient lengths well above the threshold the staircase structure is lost. The simulations indicate that the long wave length zonal flow saturates through a mechanism that directly involves the turbulence intensity.
This paper investigates the so-called tertiary instabilities driven by the zonal flow in gyro-kinetic tokamak core turbulence. The Kelvin Helmholtz instability is first considered within a 2D fluid model and a threshold in the zonal flow wave vector kZF>kZF,c for instability is found. This critical scale is related to the breaking of the rotational symmetry by flux-surfaces, which is incorporated into the modified adiabatic electron response. The stability of undamped Rosenbluth-Hinton zonal flows is then investigated in gyro-kinetic simulations. Absolute instability, in the sense that the threshold zonal flow amplitude tends towards zero, is found above a zonal flow wave vector kZF,cρi≈1.3 (ρi is the ion thermal Larmor radius), which is comparable to the 2D fluid results. Large scale zonal flows with kZF<kZF,c are unstable for sufficiently large amplitudes with increasing trend for an increasing radial scale. However, the critical E × B-shearing rate associated with the stability boundary ωE×B,c exceeds typical values connected to the pure flow state at marginal stability by more than an order of magnitude, which therefore lies deeply in the stable parameter region. Furthermore, the impact of zonal temperature perturbations on the tertiary instability is examined. Although temperature perturbations favor instability, the realistic values of gradient-driven gyro-kinetic simulations still lie deeply in the stable parameter regime. Therefore, the relevance of the tertiary instability as a saturation mechanism to the zonal flow amplitude is questioned, as most of the zonal flow intensity is concentrated in modes satisfying kZF≪kZF,c as well as ωE×B≪ωE×B,c.
This paper investigates the influence of turbulent dynamics on the neo-classical equilibrium in a tokamak, with an emphasis on the turbulence driven stationary electric current. The neo-classical solution is evaluated using the Hirschmann-Sigmar formalism, in which the turbulent dynamics enter as a forcing term. The latter forcing terms are evaluated through time averages of gyro-kinetic turbulence simulations and are linked with the velocity non-linearity in the gyro-kinetic equation. The time averaged turbulent forcing terms connected with the velocity non-linearity provide a non-negligible current drive, despite being a correction of second order in the normalized Larmor radius. For ITG turbulence, the force exerted due to the heat flux balance is the dominant contribution to the current. The parallel fluctuations of electron density/temperature and the electrostatic potential drive the majority of the current, which is in magnitude comparable to the bootstrap current in the kinetic cyclone base case and increases the total current by a few percent in cases with an experimentally relevant heat flux. An up-down symmetry breaking mechanism is required for turbulent current drive, which is provided in this study by a background rotation or rotation gradient. Consequently, the current is nearly linear in the plasma rotation or its gradient.
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