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Journal ArticleDOI

High- n ideal and resistive shear Alfvén waves in tokamaks

15 Apr 1985-Annals of Physics (Academic Press)-Vol. 161, Iss: 1, pp 21-47
TL;DR: In this paper, the ideal and resistive MHD equations for the shear Alfven waves are studied in a low-β toroidal model by employing the high-n ballooning formalism.
About: This article is published in Annals of Physics.The article was published on 1985-04-15 and is currently open access. It has received 450 citations till now. The article focuses on the topics: Resistive touchscreen.

Summary (2 min read)

I. INTRODUCTION

  • The properties of the shear Alfven waves in a magnetically confined plasma have been extensively studied due to their important implications in plasma heating and instabilities.
  • The authors will systematically examine the shear Alfven waves, the toroidal coupling effects on the shear Alfven spectrum, and the resistivity dependence of the shear Alfven modes in a low-g, large aspect ratio, axisymmetric toroidal plasma.
  • The authors will show that the nonuniform toroidal field not only breaks up the shear Alfven continuum, hut also creates new discrete marginally stable modes in the continuum gaps.

II. SHEAR ALFVEN WAVE EQUATIONS

  • The authors model is based on the linearized MHD fluid equations in a nonuniform medium.
  • These equations are solved for perturbations subject to certain boundary conditions in $ space and to the initial conditions of the perturbations.
  • G where the contour integration path C is parallel to the real axis of the complex u-plane and is above all singularities of t(x",<u) due to the causality condition.
  • The Green's function G has branch cuts and poles.
  • By ignoring effects associated with the fast magnetosonic waves and the slow sound waves, the authors will also neglect the v $ terra in the artiabatic pressure law.'.

IV, BREAKUP OF THE CONTINUOUS SPECTRUM AND THE TOROIDAL SHEAR ALFVEN WAVES IN A TOROIDAL SVSTEM

  • It has been extensively studied [12] [13] [14] [15] that unstable ballooning modes are introduced by the local bad curvature effects when ou >.
  • The authors numerical results also show that the shear Alfven-Landau modes are insensitive to the toroidal effects.
  • The authors find that the coupling of neighboring poloidal harmonics results in the breakup of the continuous spectrum into small continuum bands with the gaps located at the intersections of the shear Alfven frequencies of different poloidal harmonics.
  • Since the curvature (finite a ) has little effect on the discrete Alfven waves and the continuum gaps, for simplicity the authors will set a_ = 0 in the following discussion.
  • Numerical eigenfrequenoies from solving Eq. ( 23) are consistent with the analytical solutions of Eqs. ( 53) and ( 54.

V. RESISTIVE HHD EIGEHMODES

  • When the plasma resistivity v is introduced, its effects on the discrete ideal modes (the shear Alfven-Landau modes, the toroidal shear Alfven modes, and the ideal ballooning modes) can be considered as perturbations.
  • In addition to these ideal modes, the plasma resistivity introduces four branches of resistive modes, the resistive entropy modes, the tearing modes, the resistive periodic shear Alfven modes, and the resistive ballooning nodes.
  • Connor et al. [18] had studied this eigenmode branch in an infinite sheared slab in the Fourier transformed space.
  • R^»rameters have only minor effects on its solutions.
  • Matching the resistive solution, Bj. ( 65), to the ideal solution, Eq. ( 59), the authors have the eigenvalue condition EQUATION ) and the eigenfrequencies of the resistive entropy mode with even parity is given by EQUATION ) Nbte that Kj. ( 67) is identical with Eq. ( 56) obtained by Cbnnor _et _a^. [18] .

V-2. RESISTIVE PERIODIC SHEAR ALFVEN WAVES

  • The authors have shown that in Sec. IV that the periodic nature of the toroidal magnetic field can break up the ideal slab continuous spectrum into bands of continuous spectra separated by the gaps.
  • The authors see that these numerical results are consistent with their analytical solutions H}s. ( 94) and ( 96), except for the m = 0 even parity eigenmode.
  • When ou > ap C » the in = 0 even parity resistive ballooning mode transforms into the ideal ballooning mode, which is destabilized by, and localized in, the local bad curvature region.
  • Iqnoring ion sound effects, the previous studies [24, 25] obtained only the leading order contribution in CL for the even parity modes.

VI. SUMMARY AND DISCUSSIONS

  • The authors have extensively studied shear Alfven waves for a low-g, large aspect ratio toroidal plasma using both the ideal and the f\ resistive MHD models, which can be described by a single second order differential equation, Eq. ( 24), in terms of the extended poloidal coordinate 9 by employing the high-n WKB-ballooning mode formalism.
  • By employing the method of asymptotic matching, the authors have shown that only the even parity eigenmodes exist inside the lowest continuum gaps and the corresponding eigenfrequencies vary from the lower endpoints of the gaps when s = 0 to the upper limits of the gaps when s + ».
  • For the toroidal shear Alfven waves the eigenfrequency approaches the ideal limit frequency linearly with v along tha -90° line in the complex w-plane.
  • The authors results may explain many features of the numerical results of Glasser et al. [22] , who integrated the high-n resistive MHD ballooning mode equation (a fourth order differential equation) by using numerically generated axisymmetric toroidal equilibria.
  • The authors numerical calculations have found that the toroidal shear Alfven mode may account for these oscillations.

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Citations
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Journal ArticleDOI
TL;DR: A review of the progress accomplished since the redaction of the first ITER Physics Basis (1999 Nucl Fusion 39 2137-664) in the field of energetic ion physics and its possible impact on burning plasma regimes is presented in this paper.
Abstract: This chapter reviews the progress accomplished since the redaction of the first ITER Physics Basis (1999 Nucl Fusion 39 2137-664) in the field of energetic ion physics and its possible impact on burning plasma regimes New schemes to create energetic ions simulating the fusion-produced alphas are introduced, accessing experimental conditions of direct relevance for burning plasmas, in terms of the Alfvenic Mach number and of the normalised pressure gradient of the energetic ions, though orbit characteristics and size cannot always match those of ITER Based on the experimental and theoretical knowledge of the effects of the toroidal magnetic field ripple on direct fast ion losses, ferritic inserts in ITER are expected to provide a significant reduction of ripple alpha losses in reversed shear configurations The nonlinear fast ion interaction with kink and tearing modes is qualitatively understood, but quantitative predictions are missing, particularly for the stabilisation of sawteeth by fast particles that can trigger neoclassical tearing modes A large database on the linear stability properties of the modes interacting with energetic ions, such as the Alfven eigenmode has been constructed Comparisons between theoretical predictions and experimental measurements of mode structures and drive/damping rates approach a satisfactory degree of consistency, though systematic measurements and theory comparisons of damping and drive of intermediate and high mode numbers, the most relevant for ITER, still need to be performed The nonlinear behaviour of Alfven eigenmodes close to marginal stability is well characterized theoretically and experimentally, which gives the opportunity to extract some information on the particle phase space distribution from the measured instability spectral features Much less data exists for strongly unstable scenarios, characterised by nonlinear dynamical processes leading to energetic ion redistribution and losses, and identified in nonlinear numerical simulations of Alfven eigenmodes and energetic particle modes Comparisons with theoretical and numerical analyses are needed to assess the potential implications of these regimes on burning plasma scenarios, including in the presence of a large number of modes simultaneously driven unstable by the fast ions

519 citations

Journal ArticleDOI
TL;DR: In this article, it was shown both analytically and numerically that the toroidicity not only breaks up the shear Alfven continuous spectrum, but also creates new discrete toroidic induced shear eigenmodes with frequencies inside the continuum gaps.
Abstract: In toroidal plasmas, the toroidal magnetic field is nonuniform over a magnetic surface and causes coupling of different poloidal harmonics. It is shown both analytically and numerically that the toroidicity not only breaks up the shear Alfven continuous spectrum, but also creates new, discrete, toroidicity‐induced shear Alfven eigenmodes with frequencies inside the continuum gaps. Potential applications of the low‐n toroidicity‐induced shear Alfven eigenmodes on plasma heating and instabilities are addressed.

509 citations

Journal ArticleDOI
TL;DR: Alfven wave instability in toroidally confined plasmas is studied in this paper, where the authors identify three types of Alfven wave instabilities: frequency crossings of counterpropagating waves, extremum of the continuous spectrum, and reversed shear Alfven eigenmode.
Abstract: Superthermal energetic particles (EP) often drive shear Alfven waves unstable in magnetically confined plasmas. These instabilities constitute a fascinating nonlinear system where fluid and kinetic nonlinearities can appear on an equal footing. In addition to basic science, Alfven instabilities are of practical importance, as the expulsion of energetic particles can damage the walls of a confinement device. Because of rapid dispersion, shear Alfven waves that are part of the continuous spectrum are rarely destabilized. However, because the index of refraction is periodic in toroidally confined plasmas, gaps appear in the continuous spectrum. At spatial locations where the radial group velocity vanishes, weakly damped discrete modes appear in these gaps. These eigenmodes are of two types. One type is associated with frequency crossings of counterpropagating waves; the toroidal Alfven eigenmode is a prominent example. The second type is associated with an extremum of the continuous spectrum; the reversed shear Alfven eigenmode is an example of this type. In addition to these normal modes of the background plasma, when the energetic particle pressure is very large, energetic particle modes that adopt the frequency of the energetic particle population occur. Alfven instabilities of all three types occur in every toroidal magnetic confinement device with an intense energetic particle population. The energetic particles are most conveniently described by their constants of motion. Resonances occur between the orbital frequencies of the energetic particles and the wave phase velocity. If the wave resonance with the energetic particle population occurs where the gradient with respect to a constant of motion is inverted, the particles transfer energy to the wave, promoting instability. In a tokamak, the spatial gradient drive associated with inversion of the toroidal canonical angular momentum Pζ is most important. Once a mode is driven unstable, a wide variety of nonlinear dynamics is observed, ranging from steady modes that gradually saturate, to bursting behavior reminiscent of relaxation oscillations, to rapid frequency chirping. An analogy to the classic one-dimensional problem of electrostatic plasma waves explains much of this phenomenology. EP transport can be convective, as when the wave scatters the particle across a topological boundary into a loss cone, or diffusive, which occurs when islands overlap in the orbital phase space. Despite a solid qualitative understanding of possible transport mechanisms, quantitative calculations using measured mode amplitudes currently underestimate the observed fast-ion transport. Experimentally, detailed identification of nonlinear mechanisms is in its infancy. Beyond validation of theoretical models, the future of the field lies in the development of control tools. These may exploit EP instabilities for beneficial purposes, such as favorably modifying the current profile, or use modest amounts of power to govern the nonlinear dynamics in order to avoid catastrophic bursts.

431 citations

Journal ArticleDOI
TL;DR: In this article, the interactions of these energetic particles with linear and nonlinear Alfve'n waves generated in the magnetized plasma are reviewed, and the interaction of the alpha particles produced in the nuclear reactions is discussed.
Abstract: In magnetic fusion reactors relying on the burning of deuterium and tritium, sufficient confinement of the alpha particles produced in the nuclear reactions is crucial to sustaining the burning plasma. In this article the interactions of these energetic particles with linear and nonlinear Alfve'n waves generated in the magnetized plasma are reviewed.

379 citations


Cites background from "High- n ideal and resistive shear A..."

  • ...…ingredients in these analyses were nonuniform equilibrium profiles of EP sources, of SAW continuous spectrum (Chen, White, and Rosenbluth, 1984; Cheng, Chen, and Chance, 1985; Chen, 1988, 1994), the corresponding continuum damping by phase mixing (Grad, 1969), the specific equilibrium…...

    [...]

  • ...An important theoretical result was that discrete Alfvén eigenmodes (AEs), such as toroidal AEs (TAEs), can exist essentially free of continuum damping in the frequency gaps of the SAW continuous spectrum (Cheng, Chen, and Chance, 1985)....

    [...]

Journal ArticleDOI
TL;DR: The physics of magnetically confined plasmas has had much of its development as part of the program to develop fusion energy and is an important element in the study of space and astrophysical Plasmas as mentioned in this paper.
Abstract: The physics of magnetically confined plasmas has had much of its development as part of the program to develop fusion energy and is an important element in the study of space and astrophysical plasmas. Closely related areas of physics include Hamiltonian dynamics, kinetic theory, and fluid turbulence. A number of topics in physics have been developed primarily through research on magnetically confined plasmas. The physics that underlies the magnetic confinement of plasmas is reviewed here to make it more accessible to those beginning research on plasma confinement and for interested physicists.

349 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity, and the effect of a small layer curvature is simulated by a gravitational field.
Abstract: The stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity. The effect of a small layer curvature is simulated by a gravitational field. In an incompressible fluid, there can be three basic types of ``resistive'' instability: a long‐wave ``tearing'' mode, corresponding to breakup of the layer along current‐flow lines; a short‐wave ``rippling'' mode, due to the flow of current across the resistivity gradients of the layer; and a low‐g gravitational interchange mode that grows in spite of finite magnetic shear. The time scale is set by the resistive diffusion time τR and the hydromagnetic transit time τH of the layer. For large S = τR/τH, the growth rate of the ``tearing'' and ``rippling'' modes is of order τR−3/5τH−2/5, and that of the gravitational mode is of order τR−1/3τH−2/3. As S → ∞, the gravitational effect dominates and may be used to stabilize the two nongravitational modes. If the zero‐order configuration is in equilibrium, the...

2,314 citations

Journal ArticleDOI
TL;DR: In this article, the Mercier criterion for the stability of the ideal magnetohydrodynamic interchange mode is derived, the generalization of the earlier stability criterion for resistive interchange mode was obtained, and a relation between the two was noted.
Abstract: Previous work by Johnson and Greene on resistive instabilities is extended to finite‐pressure configurations. The Mercier criterion for the stability of the ideal magnetohydrodynamic interchange mode is rederived, the generalization of the earlier stability criterion for the resistive interchange mode is obtained, and a relation between the two is noted. Conditions for tearing mode instability are recovered with the growth rate scaling with the resistivity in a more complicated manner than η3/5. Nyquist techniques are used to show that favorable average curvature can convert the tearing mode into an overstable mode and can often stabilize it.

543 citations

Journal ArticleDOI
TL;DR: In this paper, a wave equation that shows a coupling between a surface wave and a shear Alfven wave is derived, and a theory of long-period magnetic pulsations (Pc 3 to Pc 5) is presented as an initial value problem to explain impulse-excited pulsations.
Abstract: A theory of long-period magnetic pulsations (Pc 3 to Pc 5) is presented as an initial value problem to explain impulse-excited pulsations. By using a one-dimensional model a wave equation that shows a coupling between a surface wave and a shear Alfven wave is derived. By solving this equation on the basis of initial value approach we conclude that there is a continuous spectrum with damping proportional to inverse power of time and that there are weakly damped discrete eigenmodes (surface eigenmodes) due to sharp variations in the plasma parameters. The frequency ωr and the damping rate γ of the surface eigenmode are given approximately by ωr = k∥[(BI² + BII²)/μ0(ρI + ρII)]1/2 and γ/ωr = |k⊥/▽(ln υA²)| respectively, where υA(=B/(μ0ρ)1/2) is the Alfen speed, k∥ and k⊥ are wave numbers parallel and perpendicular to the magnetic field, and subscripts I and II refer to quantities associated with each side of the surface. The result is used to explain recent observations of plasmapause-associated magnetic pulsations as well as magnetic pulsations excited by sudden commencements and sudden impulses near the magnetopause.

338 citations

Journal ArticleDOI
TL;DR: In this article, the authors examined the dispersion relation for plasmas with non-uniform profiles and compared the results with those of a sharp boundary model, showing that the frequency of the waves is a complex quantity having a real and imaginary part.
Abstract: It is well known, [1–6], that the linearized equations of motion of ideal MHD possess a continuous spectrum which leads to damping of propagating waves through phase mixing. We show how this arises by examining the dispersion relation for plasmas with non-uniform profiles and comparing the results with those of a sharp boundary model. In this paper the special case of the non-uniform sheet-pinch is examined in order to present the mathematical details as clearly as possible. It is shown that as a result of the non-uniformity the frequency of the waves is a complex quantity having a real and imaginary part. The corresponding eigenfunctions and their mathematical pathology are discussed.

282 citations

Journal ArticleDOI
TL;DR: In this article, a systematic procedure for studying the influence of kinetic effects on the stability of MHD ballooning modes is presented, which includes effects due to finite gyroradius, trapped particles, and wave-particle resonances.
Abstract: A systematic procedure for studying the influence of kinetic effects on the stability of MHD ballooning modes is presented. The ballooning mode formalism, which is particularly effective for analysing high-mode-number perturbations of a plasma in toroidal systems, is used to solve the Vlasov-Maxwell equations for modes with perpendicular wavelengths on the scale of the ion gyroradius. The local stability on each flux surface is determined by the solution of three coupled integro-differential equations which include effects due to finite gyroradius, trapped particles, and wave-particle resonances. More tractable forms of these equations are then obtained in the low (ω < ωbi, ωti) and intermediate- (ωbi, ωti < ω < ωbe, ωte) frequency regimes with ωbj and ωtj being the average bounce and transit frequencies of each species. After further simplifying approximations, the kinetic results here are shown to be reducible to the MHD-ballooning-mode equations in the analogous limits, ω ωs where ωs = cs/Lc, with cs being the acoustic speed and Lc the connection length.

225 citations