(Color online) (a) Ion perturbed density profiles across the island O-point for , at different island rotation frequencies (b) Potential profile across the island O-point for at different island rotation frequencies. The adiabatic response at the island separatrix is dominant. However, other minor non-adiabatic contributions which do not exhibit a linear dependence on the island rotation could explain the small deviations of the curves presented from a perfectly linear scaling with ω.
(Color online) Perturbed distribution in arbitrary units (averaged on s, i.e., along a magnetic field line) as a function of the velocity space coordinates at the island separatrix in correspondence with the O-point (i.e., , ξ = 0) for ions and electrons. Here, ω = −0.02 and . The triangle separates the trapped domain (inside) from the passing one (outside). It is apparent that the ion response is well inside the trapped region of the phase space, while the electrons are mainly outside it.
(Color online) Comparison between the radial profile of the radial electric field at the island O-point between the analytical formula Eq. (8) and the numerical simulations for ω = 0.04 in arbitrary units.
(Color online) Value of the “peak” of the radial electric field at the island separatrix (see Fig. 3) normalized to the corresponding electric field associated to φ 0 at ω = −0.02 and for various Te /Ti . The negative sign indicates a smooth increase of the Er radial profile towards the value inside the island (cf. Fig. 3), without any peak and any abrupt change of sign.
(Color online) Value of the “peak” of the radial electric field at the island separatrix (see Fig. 3) normalized to the corresponding electric field associated to φ 0 at ω = −0.02 and Te /Ti = 2 for various island widths (normalized to the thermal ion Larmor radius). The smoothing at the separatrix is less effective when the island is smaller. With the parameters employed in our simulations, the ion banana width amounts to ρb = 3.44ρL .
(Color online) Peak of the perpendicular ion and electron flow associated to the classical polarization current (defined as ,21 with angular brackets denoting gyroaverage) at ξ = −π/2 (i.e., between O- and X-point, cf. Fig. 7) with imposed electrostatic potential φ 0 (circles and stars) and with self-consistent electrostatic potential (triangles and crosses) for various frequencies and . Note the large difference between the self-consistent cases and the imposed φ 0 cases even for this small island size. Note also the quadratic dependence on ω for the ions. All runs have been performed with the same spatial resolution.
(Color online) Perpendicular profile in arbitrary units of the s-averaged ion and electron neoclassical flow (where j labels the species and which is expected to be dominated by the neoclassical polarization current when φ 0 is employed, see Ref. 11) at ξ = −π/2 (i.e., between the O-Point and the X-point) for various values of the island rotation frequency.
(Color online) Radial profile of the total normalized rotation frequency ωTOT for electrons (red triangles) and ions (blue circles) for islands rotating in the ion diamagnetic direction (ω = −0.02) showing that the sum of the electric and diamagnetic contributions inside the island matches the island rotation frequency (green stars) for different island widths w. The rotation frequency ωE (magenta crosses) is represented as well. In the considered simulations, the value of the logarithmic gradients (R/Ln = R/LTi = R/LTe = 1.5) has been kept low in order to exclude turbulent effects, while Te /Ti = 2. Note that, in the simulations, the total width of the computational domain amounts to ≈160ρL , but here, for clearness, only the island region has been represented.
(Color online) The same as in Fig. 8 for islands rotating in the electron diamagnetic direction (ω = 0.02). In contrary to Fig. 8, the adiabatic response opposes the flattening, and therefore, the electrostatic potential develops in order to let electrons rotate with the island. For small widths, ions are not able to follow the island.
(Color online) Normalized ion density gradient at the island O-point for an island with for various values of the island rotation frequency. The vertical dashed-dotted line identifies the ion diamagnetic frequency, while the dashed one the electron diamagnetic frequency. The normalized equilibrium density gradient has been set to .
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