Abstract.Long range frequency chirping of Bernstein-Greene-Kruskal modes, whose existence is determined by the fast particles, is investigated in cases where these particles do not move freely and their motion is bounded to restricted orbits. An equilibrium oscillating potential, which creates different orbit topologies of energetic particles, is included into the bump-on-tail instability problem of a plasma wave. With respect to fast particles dynamics, the extended model captures the range of particles motion (trapped/passing) with energy and thus represents a more realistic 1D picture of the long range sweeping events observed for weakly damped modes, e.g. global Alfven eigenmodes, in tokamaks. The Poisson equation is solved numerically along with bounce averaging the Vlasov equation in the adiabatic regime. We demonstrate that the shape and the saturation amplitude of the nonlinear mode structure depends not only on the amount of deviation from the initial eigenfrequency but also on the initial energy of the resonant electrons in the equilibrium potential. Similarly, the results reveal that the resonant electrons following different equilibrium orbits in the electrostatic potential lead to different rates of frequency evolution. As compared to the previous model [Breizman B.N. 2010 Nucl. Fusion 50 084014], it is shown that the frequency sweeps with lower rates. The additional physics included in the model enables a more complete 1D description of the range of phenomena observed in experiments.Hard nonlinear evolution in the presence of particle orbits 2

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We extend previous analytical calculations of 2D high-β equilibria in order-unity aspect ratio tokamaks with toroidal flow to include pressure anisotropy, assuming guiding-center theory for a bi-Maxwellian plasma and the ideal MHD Ohm's law. Equilibrium solutions are obtained in the core region (which fills most of the plasma volume) and the boundary layer. We find that pressure anisotropy with p∥>p⊥ (p∥<p⊥) reduces (enhances) the plasma diamagnetism relative to the isotropic case whenever an equilibrium solution exists. Sufficiently fast toroidal flows (Ω>Ωmin) were previously found to suppress the field-free region (diamagnetic hole) that exists in static isotropic high-β equilibria. We find that all equilibrium solutions with pressure anisotropy suppress the diamagnetic hole. For the static case with a volume-averaged toroidal beta of 70%, plasmas with max(p∥/p⊥)>α1=1.07 have equilibrium solutions. We find that α1 decreases with increasing toroidal flow speed, and above the flow threshold Ωmin we find α1=1, so that all p∥>p⊥ plasmas have equilibrium solutions. On the other hand, for p∥<p⊥ there are no equilibrium solutions below Ωmin. Above Ωmin (where there is no diamagnetic hole in the isotropic case), equilibrium solutions exist for α2<min(p∥/p⊥)<1, where α2 decreases from unity with increasing flow speed. The boundary layer width increases and the Shafranov shift decreases for p∥>p⊥, while the converse is true for p∥<p⊥.

Impact of pressure anisotropy on magnetic configuration and stability View the table of contents for this issue, or go to the journal homepage for more 2016 Nucl. Fusion 56 112017

We report on observations of bursty mode activity during early neutral beam heating in KSTAR plasmas, before current flat top while the q profile is still evolving. The magnitude of the activity increases with early beam heating, and reduces with the addition of resonant magnetic perturbation magnetic field coils. A mode analysis yields a toroidal mode number of n=2. The mode is observed to be downward chirping in frequency, and exists for the duration of the slowing-down of the beam. Motional Stark effect constrained equilibrium reconstructions are available at adjacent time slices: we have rescaled the total current to the measured value to obtain the q profile during the mode activity. From this we have computed the mode spectrum and identified a number of candidate gap modes. Wave-particle simulations with plausible distribution functions are computed, which demonstrate that the lowest frequency mode satisfies the condition for wave drive 1 * w w > , where * w is the fast ion diamagnetic drift frequency. An interesting finding is the change from exponential growth of the mode above n n 0.6 f i0 » , whereby the mode continues to nonlinearly grow at a reduced rate over a period of 100 wave periods up to final saturated amplitude. We believe that this may be because the two spatial resonances at s=0.4 and s=0.8 overlap for sufficiently high fast ion density, and so the phasespace volume and fast ion density available to drive the mode increases.

Three-wave interactions in plasmas are described, in the framework of kinetic theory, by the quadratic response tensor (QRT). The cold-plasma QRT is a common approximation for interactions between three fast waves. Here, the first-order thermal correction (FOTC) to the cold-plasma QRT is derived for interactions between three fast waves in a warm unmagnetized collisionless plasma, whose particles have an arbitrary isotropic distribution function. The FOTC to the cold-plasma QRT is shown to depend on the second moment of the distribution function, the phase speeds of the waves, and the interaction geometry. Previous calculations of the rate for second harmonic plasma emission (via Langmuir-wave coalescence) assume the cold-plasma QRT. The FOTC to the cold-plasma QRT is used here to calculate the FOTC to the second harmonic emission rate, and its importance is assessed in various physical situations. The FOTC significantly increases the rate when the ratio of the Langmuir phase speed to the electron thermal speed is less than about 3.

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The quadratic longitudinal response function describes the second-order nonlinear response of a plasma to electrostatic wave fields. An explicit expression for this function in the weak-turbulence regime requires the evaluation of velocity-space integrals involving the velocity distribution function and various resonant denominators. Previous calculations of the quadratic longitudinal response function were performed by approximating the resonant denominators to facilitate the integration. Here, we evaluate these integrals exactly for a non-relativistic collisionless unmagnetized isotropic Maxwellian plasma in terms of generalized plasma dispersion functions, and correct certain aspects of expressions previously derived for these functions. We show that in the appropriate limits the exact expression reduces to the approximate form used for interactions between two fast waves and one slow wave, such as the electrostatic decay of Langmuir waves into Langmuir waves and ion sound waves, and the scattering of Langmuir waves off thermal ions.