Ripple modifications to alpha transport in tokamaks

2018 ◽  
Vol 84 (5) ◽  
Author(s):  
Peter J. Catto
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Power Plants ◽  
Collision Operator ◽  
Radial Transport ◽  
Flux Surface ◽  
Slowing Down ◽  
Transport Losses ◽  
Maxwellian Distribution Function ◽  
Major Radius

Magnetic field ripple is inherent in tokamaks since the toroidal magnetic field is generated by a finite number of toroidal field coils. The field ripple results in departures from axisymmetry that cause radial transport losses of particles and heat. These ripple losses are a serious concern for alphas near their birth speed $v_{0}$ since alpha heating of the background plasma is required to make fusion reactors into economical power plants. Ripple in tokamaks gives rise to at least two alpha transport regimes of concern. As the slowing down time $\unicode[STIX]{x1D70F}_{s}$ is much larger than the time for an alpha just born to make a toroidal transit, a regime referred to as the $1/\unicode[STIX]{x1D708}\propto \unicode[STIX]{x1D70F}_{s}$ regime can be encountered, with $\unicode[STIX]{x1D708}$ the appropriate alpha collision frequency. In this regime the radial transport losses increase as $v_{0}\unicode[STIX]{x1D70F}_{s}/R$, with $R$ the major radius of the tokamak. The deleterious effect of ripple transport is mitigated by electric and magnetic drifts within the flux surface. When drift tangent to the flux surface becomes significant another ripple regime, referred to as the $\sqrt{\unicode[STIX]{x1D708}}$ regime, is encountered where a collisional boundary layer due to the drift plays a key role. We evaluate the alpha transport in both regimes, taking account of the alphas having a slowing down rather than a Maxwellian distribution function and their being collisionally scattered by a collision operator appropriate for alphas. Alpha ripple transport is found to be in the $\sqrt{\unicode[STIX]{x1D708}}$ regime where it will be a serious issue for typical tokamak reactors as it will be well above the axisymmetric neoclassical level and can be large enough to deplete the alpha slowing down distribution function unless toroidal rotation is strong.

Cyclotron instabilities of a magnetized electron plasma with anisotropic temperatures

1977 ◽  
Vol 17 (3) ◽  
pp. 453-465 ◽  
Author(s):  
C. Chiuderi ◽  
G. Einaudi ◽  
R. Giachetti
Keyword(s):  
Magnetic Field ◽  
Distribution Function ◽  
Dispersion Relation ◽  
Growth Rates ◽  
Linear Growth ◽  
Electron Plasma ◽  
Maxwellian Distribution ◽  
Analytical Treatment ◽  
Maxwellian Distribution Function ◽  
Instability Regions

The dispersion relation for an electron plasma in a magnetic field is investigated for a bi-Maxwellian distribution function. A new set of solutions for non-perpendicular propagation is found. The linear growth rates are computed and the instability regions in the (k, cos θ) plane are determined. An approximate analytical treatment of the problem is also given for certain ranges of the parameters.

A comparative study of the filamentation and Weibel instabilities and their cumulative effect. I. Non-relativistic theory

2009 ◽  
Vol 75 (1) ◽  
pp. 19-33 ◽  
Author(s):  
M. LAZAR ◽  
A. SMOLYAKOV ◽  
R. SCHLICKEISER ◽  
P. K. SHUKLA
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Distribution Function ◽  
Comparative Study ◽  
Relativistic Theory ◽  
Thermal Emission ◽  
Cumulative Effect ◽  
Fusion Plasma ◽  
Thermal Anisotropy ◽  
Maxwellian Distribution Function

AbstractA comparative study of the electromagnetic instabilities in anisotropic unmagnetized plasmas is undertaken. The instabilities considered are the filamentation and Weibel instability, and their cumulative effect. Dispersion relations are derived and the growth rates are plotted systematically for the representative cases of non-relativistic counterstreaming plasmas with isotropic or anisotropic velocity distributions functions of Maxwellian type. The pure filamentation mode is attenuated by including an isotropic Maxwellian distribution function. Moreover, it is observed that counterstreaming plasmas can be fully stabilized by including bi-Maxellian distributions with a negative thermal anisotropy. This effect is relevant for fusion plasma experiments. Otherwise, for plasma streams with a positive anisotropy the filamentation and Weibel instabilities cumulate leading to a growth rate by orders of magnitude larger than that of a simple filamentation mode. This is noticeable for the quasistatic magnetic field generated in astrophysical sources, and which is expected to saturate at higher values and explain the non-thermal emission observed.

scholarly journals Landau Modes are Eigenmodes of Stellar Systems in the Limit of Zero Collisions

2021 ◽  
Vol 923 (2) ◽  
pp. 271
Author(s):  
C. S. Ng ◽  
A. Bhattacharjee
Keyword(s):  
Distribution Function ◽  
Unstable Mode ◽  
Stellar System ◽  
Collision Operator ◽  
Thermal Fluctuations ◽  
Maxwellian Distribution ◽  
Gravitational Forces ◽  
Gravitational Systems ◽  
Maxwellian Distribution Function ◽  
Spectrum Of Eigenmodes

Abstract We consider the spectrum of eigenmodes in a stellar system dominated by gravitational forces in the limit of zero collisions. We show analytically and numerically using the Lenard–Bernstein collision operator that the Landau modes, which are not true eigenmodes in a strictly collisionless system (except for the Jeans unstable mode), become part of the true eigenmode spectrum in the limit of zero collisions. Under these conditions, the continuous spectrum of true eigenmodes in a collisionless system, also known as the Case–van Kampen modes, is eliminated. Furthermore, because the background distribution function in a weakly collisional system can exhibit significant deviations from a Maxwellian distribution function over long times, we show that the spectrum of Landau modes can change drastically even in the presence of slight deviations from a Maxwellian, primarily through the appearance of weakly damped modes that may be otherwise heavily damped for a Maxwellian distribution. Our results provide important insights for developing statistical theories to describe thermal fluctuations in a stellar system, which are currently a subject of great interest for N-body simulations as well as observations of gravitational systems.

Semi-analytical inspection of the quasi-linear absorption of lower hybrid wave in presence of -particles in tokamak reactor

2018 ◽  
Vol 84 (5) ◽  
Author(s):  
A. Cardinali ◽  
C. Castaldo ◽  
R. Ricci
Keyword(s):  
Distribution Function ◽  
Planck Equation ◽  
Current Drive ◽  
Dielectric Tensor ◽  
Lower Hybrid ◽  
Linear Diffusion ◽  
Tokamak Reactor ◽  
Slowing Down ◽  
Fokker Planck Equation ◽  
Fokker Planck

In a reactor plasma like demonstration power station (DEMO), when using the radio frequency (RF) for heating or current drive in the lower hybrid (LH) frequency range (Frankeet al.,Fusion Engng Des., vol. 96–97, 2015, p. 46; Cardinaliet al.,Plasma Phys. Control. Fusion, vol. 59, 2017, 074002), a large fraction of the ion population (the continuously born$\unicode[STIX]{x1D6FC}$-particle, and/or the neutral beam injection (NBI) injected ions) is characterized by a non-thermal distribution function. The interaction (propagation and absorption) of the LH wave must be reformulated by considering the quasi-linear approach for each species separately. The collisional slowing down of such an ion population in a background of an electron and ion plasma is balanced by a quasi-linear diffusion in velocity space due to the propagating electromagnetic wave. In this paper, both propagations are considered by including the ion distribution function, solution of the Fokker–Planck equation, which describes the collisional dynamics of the$\unicode[STIX]{x1D6FC}$-particles including the effects of frictional slowing down, energy diffusion and pitch-angle scattering. Analytical solutions of the Fokker–Planck equation for the distribution function of$\unicode[STIX]{x1D6FC}$-particles with a background of ions and electrons at steady state are included in the calculation of the dielectric tensor. In the LH frequency domain, ray tracing (including quasi-linear damping), can be analytically solved by iterating with the Fokker–Planck solution, and the interaction of the LH wave with$\unicode[STIX]{x1D6FC}$-particles, thermal ions and electrons can be accounted self-consistently and the current drive efficiency can be evaluated in this more general scenario.

scholarly journals On the size of tokamak fusion power plants

2019 ◽  
Vol 377 (2141) ◽  
pp. 20170437 ◽  
Author(s):  
Hartmut Zohm
Keyword(s):  
Power Plants ◽  
New Technology ◽  
Fusion Energy ◽  
Plasma Parameters ◽  
Fusion Power ◽  
Figures Of Merit ◽  
Superconducting Coils ◽  
Critical Technology ◽  
Alternative Approach ◽  
Major Radius

Figures of merit for future tokamak fusion power plants (FPPs) are presented. It is argued that extrapolation from present-day experiments to proposed FPPs must follow a consistent development path, demonstrating the largest required leaps in intermediate devices to allow safe extrapolation to an FPP. This concerns both plasma physics and technology. At constant plasma parameters, the figures of merit depend on both major radius R and magnetic field B . We propose to use the term ‘size’ for a combination of R and B to avoid ambiguities in scaling arguments. Two routes to FPPs are discussed: the more conventional one increasing R , based on the assumption that B is limited by present technology; and an alternative approach assuming the availability of new technology for superconducting coils, allowing higher B . It is shown that the latter will lead to more compact devices, and, assuming a criterion based on divertor impurity concentration, is in addition more favourable concerning the exhaust problem. However, in order to obtain attractive steady-state tokamak FPPs, the required plasma parameters still require considerable progress with respect to present experiments. A credible strategy to arrive at these must hence be shown for both paths. In addition, the high-field path needs a demonstration of the critical technology items early on. This article is part of a discussion meeting issue ‘Fusion energy using tokamaks: can development be accelerated?’.

scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

2018 ◽  
Vol 84 (4) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Distribution Function ◽  
Projection Operator ◽  
Linear Response ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Transport Equations ◽  
Transport Processes ◽  
Projection Operators ◽  
Covariant Representation ◽  
Classical Transport

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

Bootstrap-current computations for a stellarator-reactor plasma

1990 ◽  
Vol 44 (3) ◽  
pp. 431-453 ◽  
Author(s):  
W. D. D'Haeseleer ◽  
W. N. G. Hitchon ◽  
C. D. Beidler ◽  
J. L. Shohet
Keyword(s):  
Distribution Function ◽  
Kinetic Equation ◽  
Physical Interpretation ◽  
Collision Operator ◽  
Numerical Code ◽  
Parallel Flows ◽  
Bootstrap Current ◽  
The Difference ◽  
Rotational Transform ◽  
The Impact

Numerical results for the bootstrap current in a stellarator-reactor plasma are presented. The distribution function f is computed numerically from a kinetic equation that is averaged over the helical ripple. The parallel flows and the current are obtained as v‖ moments of f. The physics issues embedded in the code are discussed concisely, concentrating on the justification as to why the bootstrap current can be estimated from an averaged scheme. Results are presented for typical stellarator-reactor parameters. The numerical code FLOCS predicts that the momentum-restoring terms in the collision operator have no significant impact on the value of the bootstrap current (the difference being about 10%). The results obtained are related to the equilibrium flows, and a physical interpretation based on the kinetic picture is presented. Finally, an estimate for the impact of J‖ on the rotational transform is given.

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