Lower hybrid current drive in a tokamak for correlated passes through resonance

Standard quasilinear descriptions are based on the constant magnetic field form of the quasilinear operator so improperly treat the trapped electron modifications associated with tokamak geometry. Moreover, successive poloidal transits of the Landau resonance during lower hybrid current drive in a tokamak are well correlated, and these geometrical details must be properly retained to account for the presence of trapped electrons that do not contribute to the driven current. The recently derived quasilinear operator in tokamak geometry accounts for these features and finds that the quasilinear diffusivity is proportional to a delta function with a transit or bounce averaged argument (rather than a local Landau resonance condition). The new quasilinear operator is combined with the Cordey (Nucl. Fusion, vol. 16, 1976, pp. 499–507) eigenfunctions to properly derive a rather simple and compact analytic expression for the trapped electron modifications to the driven lower hybrid current and the efficiency of the current drive.

Characterization in terms of K-functionals of quasilinear operators of weak types $\left( \Lambda_{\varphi_0, 1}, \Lambda_{\psi_0, \infty} \right)$, $\left( \Lambda_{\varphi_1, 1}, \Lambda_{\psi_1, 1} \right)$

We obtain necessary and sufficient condition in terms of K-functionals of pairs of Lorentz spaces for quasilinear operator to act boundedly from pair of Lorentz spaces $\left( \Lambda_{\varphi_1 1}(\mathbb{R}^n), \Lambda_{\varphi_2 1}(\mathbb{R}^n) \right)$ to pair of Lorentz spaces $\left( \Lambda_{\psi_1 \infty}(\mathbb{R}^n), \Lambda_{\psi_2 \infty}(\mathbb{R}^n) \right)$.

Collisional effects on resonant particles in quasilinear theory

A careful examination of the effects of collisions on resonant wave–particle interactions leads to an alternate interpretation and deeper understanding of the quasilinear operator originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388) for collisionless, magnetized plasmas, and widely used to model radio frequency heating and current drive. The resonant and nearly resonant particles are particularly sensitive to collisions that scatter them out of and into resonance, as for Landau damping as shown by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). As a result, the resonant particle–wave interactions occur in the centre of a narrow collisional boundary when the collision frequency $\unicode[STIX]{x1D708}$ is very small compared to the wave frequency $\unicode[STIX]{x1D714}$ . The diffusive nature of the pitch angle scattering combined with the wave–particle resonance condition enhances the collision frequency by $(\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708})^{2/3}\gg 1$ , resulting in an effective resonant particle collisional interaction time of $\unicode[STIX]{x1D70F}_{\text{int}}\sim (\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714})^{2/3}/\unicode[STIX]{x1D708}\ll 1/\unicode[STIX]{x1D708}$ . A collisional boundary layer analysis generalizes the standard quasilinear operator to a form that is fully consistent with Kennel–Englemann, but allows replacing the delta function appearing in the diffusivity with a simple integral (having the appropriate delta function limit) retaining the new physics associated with the narrow boundary layer, while preserving the entropy production principle. The limitations of the collisional boundary layer treatment are also estimated, and indicate that substantial departures from Maxwellian are not permitted.

A quasilinear operator retaining magnetic drift effects in tokamak geometry

The interaction of radio frequency waves with charged particles in a magnetized plasma is usually described by the quasilinear operator that was originally formulated by Kennel & Engelmann (Phys. Fluids, vol. 9, 1966, pp. 2377–2388). In their formulation the plasma is assumed to be homogenous and embedded in a uniform magnetic field. In tokamak plasmas the Kennel–Engelmann operator does not capture the magnetic drifts of the particles that are inherent to the non-uniform magnetic field. To overcome this deficiency a combined drift and gyrokinetic derivation is employed to derive the quasilinear operator for radio frequency heating and current drive in a tokamak with magnetic drifts retained. The derivation requires retaining the magnetic moment to higher order in both the unperturbed and perturbed kinetic equations. The formal prescription for determining the perturbed distribution function then follows a novel procedure in which two non-resonant terms must be evaluated explicitly. The systematic analysis leads to a diffusion equation that is compact and completely expressed in terms of the drift kinetic variables. The equation is not transit averaged, and satisfies the entropy principle, while retaining the full poloidal angle variation without resorting to Fourier decomposition. As the diffusion equation is in physical variables, it can be implemented in any computational code. In the Kennel–Engelmann formalism, the wave–particle resonant delta function is either for the Landau resonance or the Doppler shifted cyclotron resonance. In the combined gyro and drift kinetic approach, a term related to the magnetic drift modifies the resonance condition.