Collisional effects on resonant particles in quasilinear theory

2020 ◽  
Vol 86 (3) ◽  
Author(s):  
Peter J. Catto
Keyword(s):  
Boundary Layer ◽  
Collision Frequency ◽  
Resonance Condition ◽  
Delta Function ◽  
New Physics ◽  
Current Drive ◽  
Careful Examination ◽  
Quasilinear Theory ◽  
Quasilinear Operator ◽  
Resonant Wave

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.

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

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Peter J. Catto
Keyword(s):  
Magnetic Field ◽  
Resonance Condition ◽  
Delta Function ◽  
Current Drive ◽  
Lower Hybrid ◽  
Quasilinear Operator ◽  
Field Form ◽  
Lower Hybrid Current Drive ◽  
Analytic Expression ◽  
Trapped Electron

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.

scholarly journals A quasilinear operator retaining magnetic drift effects in tokamak geometry

2017 ◽  
Vol 83 (6) ◽  
Author(s):  
Peter J. Catto ◽  
Jungpyo Lee ◽  
Abhay K. Ram
Keyword(s):  
Magnetic Field ◽  
Diffusion Equation ◽  
Radio Frequency ◽  
Uniform Magnetic Field ◽  
Resonance Condition ◽  
Kinetic Equations ◽  
Current Drive ◽  
Magnetized Plasma ◽  
Angle Variation ◽  
Quasilinear Operator

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.

Resonant wave-filament interactions as a loss mechanism for HHFW heating and current drive

2021 ◽  
Author(s):  
Wouter Tierens ◽  
James R Myra ◽  
Roberto Bilato ◽  
Laurent Colas
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
High Harmonic ◽  
Current Drive ◽  
Fast Wave ◽  
Power Losses ◽  
Loss Mechanism ◽  
Resonant Wave ◽  
Plasma Filaments ◽  
Analytic Expression

Abstract Perkins et al. PRL 2012 [1] reported unexpected power losses during High Harmonic Fast Wave (HHFW) heating and current drive in NSTX. Recently, Tierens et al [2] proposed that these losses may be attributable to surface waves on field-aligned plasma filaments, which carry power along the filaments, to be lost at the endpoints where the filaments intersect the limiters. In this work, we show that there is indeed a resonant loss mechanism associated with the excitation of these surface waves, and derive an analytic expression for the power lost to surface wave modes at each filament.

Direct and large-eddy simulations of internal tide generation at a near-critical slope

2011 ◽  
Vol 681 ◽  
pp. 48-79 ◽  
Author(s):  
BISHAKHDATTA GAYEN ◽  
SUTANU SARKAR
Keyword(s):  
Boundary Layer ◽  
Wave Propagation ◽  
Kinetic Energy ◽  
Internal Wave ◽  
Turbulent Kinetic Energy ◽  
Wave Energy ◽  
Slope Length ◽  
Resonant Wave ◽  
Large Eddy ◽  
Free Wave Propagation

A numerical study is performed to investigate nonlinear processes during internal wave generation by the oscillation of a background barotropic tide over a sloping bottom. The focus is on the near-critical case where the slope angle is equal to the natural internal wave propagation angle and, consequently, there is a resonant wave response that leads to an intense boundary flow. The resonant wave undergoes both convective and shear instabilities that lead to turbulence with a broad range of scales over the entire slope. A thermal bore is found during upslope flow. Spectra of the baroclinic velocity, both inside the boundary layer and in the external region with free wave propagation, exhibit discrete peaks at the fundamental tidal frequency, higher harmonics of the fundamental, subharmonics and inter-harmonics in addition to a significant continuous part. The internal wave flux and its distribution between the fundamental and harmonics is obtained. Turbulence statistics in the boundary layer including turbulent kinetic energy and dissipation rate are quantified. The slope length is varied with the smaller lengths examined by direct numerical simulation (DNS) and the larger with large-eddy simulation (LES). The peak value of the near-bottom velocity increases with the length of the critical region of the topography. The scaling law that is observed to link the near-bottom peak velocity to slope length is explained by an analytical boundary-layer solution that incorporates an empirically obtained turbulent viscosity. The slope length is also found to have a strong impact on quantities such as the wave energy flux, wave energy spectra, turbulent kinetic energy, turbulent production and turbulent dissipation.

Collisional alpha transport in a weakly non-quasisymmetric stellarator magnetic field

2019 ◽  
Vol 85 (2) ◽  
Author(s):  
Peter J. Catto
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Boundary Layers ◽  
Collision Frequency ◽  
Alpha Particles ◽  
Slowing Down ◽  
Tail Distribution ◽  
Strong Motivation ◽  
Particle Confinement ◽  
Background Ions

Alpha particle confinement is a serious concern in stellarators and provides strong motivation for optimizing magnetic field configurations. In addition to the collisionless confinement of trapped alphas in stellarators, excessive collisional transport of the trapped alpha particles must be avoided while they tangentially drift due to the magnetic gradient (the $\unicode[STIX]{x1D735}B$ drift). The combination of pitch angle scatter off the background ions and the $\unicode[STIX]{x1D735}B$ drift gives rise to two narrow boundary layers in the trapped region. The first is at the trapped–passing boundary and enables the finite trapped response to be matched to the vanishing passing response of the alphas. The second layer is a region that encompasses the somewhat more deeply trapped alphas with vanishing tangential $\unicode[STIX]{x1D735}B$ drift. Away from (and between) these boundary layers, collisions are ineffective and the alpha $\unicode[STIX]{x1D735}B$ drift simply balances the small radial drift of the trapped alphas. As this balance does not vanish as the trapped–passing boundary is approached, the first collisional boundary layer is necessary and gives rise to $\surd \unicode[STIX]{x1D708}$ transport, with $\unicode[STIX]{x1D708}$ the collision frequency. The vanishing of the tangential drift results in a separate, somewhat wider boundary layer, and significantly stronger superbanana plateau transport that is independent of collisionality. The constraint imposed by the need to avoid significant energy depletion loss in the slowing down tail distribution function sets the allowed departure of a stellarator from an optimal quasisymmetric configuration.

A Two-Parameter Method for Calculating the Two-Dimensional Incompressible Laminar Boundary Layer

1967 ◽  
Vol 71 (674) ◽  
pp. 117-123 ◽  
Author(s):  
N. Curle
Keyword(s):  
Boundary Layer ◽  
Exact Solutions ◽  
Careful Examination ◽  
Parameter Method ◽  
Boundary Layer Equations ◽  
First Approximation ◽  
Dimensional Parameters ◽  
Image Position ◽  
Momentum Integral ◽  
Two Parameter

Summary:In most one-parameter methods of calculating laminar boundary layers it is assumed that the non-dimensional parameters H=δ1/δ2, I=δ2τW/μu1 and L = 2{I−λ(H+2)}, depend only upon the pressure gradient parameter λ=u1δ22/v. In this paper it is shown theoretically that a more accurate, two-parameter representation isL=F0(λ)−μG0(λ)I2=F1(λ)−μG1(λ),where μ=λ2U1U1/(U1)2. Careful examination of the available range of exact solutions of the boundary layer equations has enabled the four functions F0, G0, F1, G1, to be tabulated, and the above functional forms agree with the exact solutions to a remarkable accuracy.In view of the fact that a reasonable first approximation to L is usually , we write,and it is then shown that the momentum integral equation becomesThis equation is easily solved by iteration, setting g=0 in the first approximation, and convergence is extremely rapid.The method is, in effect, a refinement of that due to Thwaites, which is universally accepted as one of the better of the existing calculation methods. Detailed calculations made by the present method indicate that the errors are only 5% of those given by the Thwaites method.

Reimagining full wave rf quasilinear theory in a tokamak

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Peter J. Catto ◽  
Elizabeth A. Tolman
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Collision Frequency ◽  
Velocity Space ◽  
Wave Number ◽  
Temporal Behaviour ◽  
Full Wave ◽  
Effective Collision Frequency ◽  
Space Boundary ◽  
Effective Collision

The velocity dependent resonant interaction of particles with applied radiofrequency (rf) waves during heating and current drive in the presence of pitch angle scattering collisions gives rise to narrow collisional velocity space boundary layers that dramatically enhance the role of collisions as recently shown by Catto (J. Plasma Phys., vol. 86, 2020, 815860302). The behaviour is a generalization of the narrow collisional boundary layer that forms during Landau damping as found by Johnston (Phys. Fluids, vol. 14, 1971, pp. 2719–2726) and Auerbach (Phys. Fluids, vol. 20, 1977, pp. 1836–1844). For a wave of parallel wave number ${k_{||}}$ interacting with weakly collisional plasma species of collision frequency $\nu$ and thermal speed ${v_{\textrm{th}}}$ , the effective collision frequency becomes of order $\nu {({k_{||}}{v_{th}}/\nu )^{2/3}} \gg \nu $ . The narrow boundary layers that arise because of the diffusive nature of the collisions allow a physically meaningful wave–particle interaction time to be defined that is the inverse of this effective collision frequency. The collisionality implied by the narrow boundary layer results in changes in the standard quasilinear treatment of applied rf fields in tokamaks while remaining consistent with causality. These changes occur because successive poloidal interactions with the rf are correlated in tokamak geometry and because the resonant velocity space dependent interactions are controlled by the spatial and temporal behaviour of the perturbed full wave fields rather than just the spatially local Landau and Doppler shifted cyclotron wave–particle resonance condition associated with unperturbed motion of the particles. The correlation of successive poloidal circuits of the tokamak leads to the appearance in the quasilinear operator of transit averaged resonance conditions localized in velocity space boundary layers that maintain negative definite entropy production.

Sign in / Sign up

Export Citation Format

Share Document