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.

Theory of energy flux and polarization changes of a radio wave with two magnetoionic components undergoing self demodulation in the ionosphere

1974 ◽  
Vol 341 (1626) ◽  
pp. 361-381 ◽  
Keyword(s):  
Electron Temperature ◽  
Radio Wave ◽  
Phase Difference ◽  
Incident Wave ◽  
Unit Volume ◽  
Collision Frequency ◽  
Lower Ionosphere ◽  
Effective Collision Frequency ◽  
Composite Wave ◽  
Effective Collision

The absorption of a powerful plane radio wave vertically incident on the lower ionosphere is studied. If it contains the two magnetoionic components with roughly equal amplitudes, the power absorbed per unit volume can be either greater or less than the sum of the powers for the separate components, depending on their phase difference. This is determined by the polarization of the incident wave, and the heights where the absorption is a maximum can be changed by changing this polarization. The power absorbed causes an increase in the electron temperature and thence in the effective collision frequency. This is studied first for an unmodulated wave. If the wave is amplitude modulated, the increase of collision frequency varies periodically in the modulation cycle. This results in self demodulation which is different for the two magnetoionic components because of their different rates of absorption. The result is that the polarization of the composite wave varies periodically over the modulation cycle.

THE NONLINEAR INTERACTION OF A RADIO-FREQUENCY WAVE WITH AN INHOMOGENEOUS PLASMA SLAB: I. KINETICS OF HIGH-TEMPERATURE AIR IN THE PRESENCE OF AN ELECTROMAGNETIC FIELD

1965 ◽  
Vol 43 (11) ◽  
pp. 2021-2035 ◽  
Author(s):  
Robert J. Papa ◽  
Carl T. Case
Keyword(s):  
High Temperature ◽  
Radio Frequency ◽  
Electron Temperature ◽  
Collision Frequency ◽  
Inhomogeneous Plasma ◽  
Frequency Wave ◽  
Effective Collision Frequency ◽  
Plasma Slab ◽  
Radio Frequency Wave ◽  
Effective Collision

A radio-frequency wave is normally incident upon an inhomogeneous plasma slab. The plasma slab is composed of partially ionized high-temperature air corresponding to the characteristics of the plasma sheath surrounding hypersonic reentry vehicles. The isotropic part of the electron velocity distribution function is Maxwellian because of electron–electron collisions. The electromagnetic wave is intense enough to heat selectively the electron gas, altering the various electron production and loss processes. The high-frequency limit is considered, and expressions are obtained for the electron number density and effective collision frequency as a function of electron temperature. The effective collision frequency takes into account the effects of electron–neutral and electron–ion collisions for momentum transfer. From an energy balance equation, the electron temperature is found to be a function of both the frequency and field strength of the wave. The electron temperature is found also to exhibit an instability that gives rise to a hysteresis effect.

Theoretical Analysis of Oscillating and Streaming Fields Between Two Parallel Beams and Numerical Investigation of the Cooling Efficiency

2003 ◽  
Author(s):  
Qun Wan ◽  
A. V. Kuznetsov
Keyword(s):  
Boundary Layer ◽  
Nusselt Number ◽  
Boundary Layers ◽  
Standing Wave ◽  
Critical Value ◽  
Core Region ◽  
Wave Number ◽  
Wave Form ◽  
Channel Height ◽  
The Core

The main purpose of this paper is to investigate the oscillating and streaming flow fields and the heat transfer efficiency across a channel between two long parallel beams, one of which is stationary and the other oscillating in standing wave form. The oscillating amplitude is assumed much smaller than the channel height. When the Reynolds number, which is defined by the oscillating frequency and the standing wave number, is much greater than unity, boundary layer structures are found near both beams, which are separated by a core region in the center of the channel. The oscillating fields within the core region and both boundary layers are obtained analytically. Based on the oscillating fields, the streaming fields within both boundary layers are also analytically obtained. Further investigation of boundary layer streaming fields shows that the streaming velocities approach constant values at the edges of the boundary layers and provide slip velocities for the streaming field in the core region. The core region streaming velocity field is numerically obtained by solving the mass and momentum conservation equations in their stream function–vorticity form. The temperature field is also computed for two cases: both beams are kept at constant but different temperatures (case A) or the oscillating beam is kept at a constant temperature and the stationary beam is prescribed a constant heat flux (case B). Cases of different channel heights are computed and a critical height is found. When the channel height is smaller than the critical value, for each half standing wavelength distance along the beams, two symmetric eddies are observed, which occupy the whole channel. In this case, the Nusselt number increases with the increase of the channel height. After the critical value, two layers of asymmetric eddies are observed near the oscillating beam and the Nusselt number decreases and approaches unity with the increase of the gap size. The abrupt change of the streaming field and the Nusselt number as the channel height goes through its critical value may be due to the bifurcation caused by instability of the vortex structure in the fluid layer.

On the generation of spatially growing waves in a boundary layer

1965 ◽  
Vol 22 (3) ◽  
pp. 433-441 ◽  
Author(s):  
M. Gaster
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Wave Number ◽  
Wave System ◽  
Velocity Fluctuations ◽  
Forced Wave ◽  
General Terms ◽  
Complex Wave ◽  
Complex Wave Number

The solution is obtained in general terms for the velocity fluctuations generated in a laminar boundary layer by an oscillating disturbance on the boundry wall. The form of excitation is chosen to represent a vibrating ribbon of the type used by Schubauer to force disturbance in boundary layers. The forced wave system generated by the ribbon is shown to be a spatially growing one, which is described far downstream by an eigenmode of the system which has a complex wave-number.

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.

THE NONLINEAR INTERACTION OF A RADIO-FREQUENCY WAVE WITH AN INHOMOGENEOUS PLASMA SLAB: II. NONLINEAR REFLECTION AND TRANSMISSION COEFFICIENTS OF AN INHOMOGENEOUS PLASMA SLAB

1965 ◽  
Vol 43 (11) ◽  
pp. 2036-2044 ◽  
Author(s):  
Robert J. Papa ◽  
Carl T. Case
Keyword(s):  
Collision Frequency ◽  
Inhomogeneous Plasma ◽  
Critical Value ◽  
Effective Dielectric Constant ◽  
Reflection And Transmission ◽  
Electric Field Amplitude ◽  
Transmission Coefficients ◽  
Effective Collision Frequency ◽  
Plasma Slab ◽  
Effective Collision

When a radio-frequency plane wave is incident upon an inhomogeneous, lossy plasma slab, part of the electromagnetic energy is reflected, part is absorbed by the plasma medium, and part is transmitted. At sufficiently high power levels (about 1 watt cm2 at X-band frequencies), the equation of state of the plasma is altered, which produces a change in the effective dielectric constant of the medium. In the high-frequency limit, the effective dielectric constant of the medium is expressed as a function of the electron density and an effective collision frequency. The electron density and effective collision frequency at each point in the medium are functions of the electron temperature. In Part I, the electron temperature (Te) at each point (z) in the slab has been expressed in terms of the local value of E2/ω2, where E = electric field amplitude and ω = signal frequency, i.e., Te = Te(z, E2/ω2). Using the predetermined functional dependence of Te on z and E2/ω2, the electromagnetic field distribution and the net reflection and transmission coefficients of the plasma slab are computed by employing the Runge–Kutta technique to integrate Maxwell's equations numerically step by step. For each value of ω, relatively small changes in the reflection and transmission coefficient are produced as [Formula: see text] is increased up to a critical value of [Formula: see text], where Einc = incident electric field amplitude. For values of [Formula: see text] greater than this critical value, there is a sharp increase in the reflection coefficient and a sharp drop in the transmission coefficient.

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