Geodesic acoustic modes and zonal flows in toroidally rotating tokamak plasmas

a b s t r a c tThe continuous spectrum of analytical toroidally rotating magnetically confined plasma equilibria is investigated analytically and numerically. In the presence of purely toroidal flow, the ideal magnetohydrodynamic equations leave the freedom to specify which thermodynamic quantity is constant on the magnetic surfaces. Introducing a general parametrization of this quantity, analytical equilibrium solutions are derived that still posses this freedom. These equilibria and their spectral properties are shown to be ideally suited for testing numerical equilibrium and stability codes including toroidal rotation. Analytical expressions are derived for the low-frequency continuous Alfvén spectrum. These expressions still allow one to choose which quantity is constant on the magnetic surfaces of the equilibrium, thereby generalizing previous results. The centrifugal convective effect is shown to modify the lowest Alfvén continuum branch to a buoyancy frequency, or Brunt-Väisälä frequency. A comparison with numerical results for the case that the specific entropy, the temperature, or the density is constant on the magnetic surfaces yields excellent agreement, showing the usefulness of the derived expressions for the validation of numerical codes.

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“…[44][45][46][47] for isothermal flux surfaces and the recent result of Ref. [48] for magnetic surfaces of constant density.…”

Section: Introductionmentioning

confidence: 51%

“…1) for magnetic surfaces of constant density the recent result of Refs. [48] for axisymmetric modes is recovered. The present result generalizes these particular results to arbitrary c f .…”

Section: A Torusmentioning

confidence: 90%

The Brunt–Väisälä frequency of rotating tokamak plasmas

Haverkort

Blank

Koren

2012

Journal of Computational Physics

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“…This particular situation was considered by Lahkin et al [5] for the case of purely toroidal rotation and isochoric magnetic surfaces. In their work the perturbed energy equation does not contain heat flux vector terms either, even though this is not an adiabatic regime.…”

Section: Perturbed Equilibriummentioning

confidence: 99%

Heat flux effects on the dispersion relation for geodesic modes in rotating plasmas

Duarte

Sgalla

Elfimov

et al. 2015

J. Phys.: Conf. Ser.

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Abstract. The MHD theory of the effect of toroidal and poloidal rotations on the dynamics of Zonal Flows -ZFs and Geodesic Acoustic Modes -GAMs in axisymmetric magnetic confinement configurations is revisited. The MHD model has an arbitrariness regarding the energy conservation equation and previous works on the effect of rotation on ZFs and GAMs adopted an adiabatic law, or other simplifying assumptions, to treat this problem. However, in fusion grade plasmas, the heat transport along the magnetic field lines is rather fast and, therefore, a somewhat more appropriate model is to assume isothermal flux surfaces. This implies to take into account the heat transport equation in the model and, in the presence of rotation, this leads to an increase in the degree of the dispersion relation for these modes, giving rise to a low-frequency third branch of these modes. This has been previously obtained by Elfimov, Galvão and Sgalla [1] employing a model of circular flux surfaces from the outset. In this paper, the theoretical development is generalized by using flux coordinates, following the method of Ilgisonis et al [2]. This allows a better assessment of the applicability of the results and to investigate the relevance of the low frequency mode in non-circular tokamaks. Specific results for the TCABR tokamak are presented. IntroductionGeodesic acoustic modes (GAMs) are plasma eigenmodes characterized by a perturbed electric field constrained to oscillate only in the radial direction (i.e., m = n = 0, where m and n are the poloidal and toroidal mode numbers, respectively) and by a perturbed density that varies with the first order poloidal harmonic, m = ±1. GAMs, whose existence was predicted in the 1960s [1], are stable oscillations over magnetic surfaces. Their generation mechanism is based on the effect of the (radial) perturbed electric field, constant and perperdicular to the magnetic surfaces, which induces a poloidal flow over the surface due to the drift velocity, E × B. In a toroidal magnetic surface, the geodesic curvature of the field lines gives rise to a restoring force against that flux, thus inducing an oscillatory motion. GAMs are the branch of nonzero frequency of the same dispersion relation associated with zonal flows (ZFs), which are zero-frequency modes.The nonlinear process behind the existence of the GAMs could be described as follows. Density and temperature gradients trigger drift micro-instabilities, which give rise to a turbulent spectrum of fluctuations with large convective cells that cause the anomalous transport. After certain thresholds, the correlation between the same fluctuations triggers GAMs and ZFs by means of the Reynolds stress tensor and other parametric mechanisms. Sheared flows, associated with these modes, tend to tear the convective cells, thus reducing the anomalous transport.

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“…It was just such a degeneration, which allowed us to find analytically the unstable zonal flows in Ref. 6. A large piece of the Comment is devoted to the proposition to go out of use in MHD the conventional sonic Mach number used in II and in a lot of papers by the other authors (see, e.g., Refs.…”

mentioning

confidence: 99%

“…A large piece of the Comment is devoted to the proposition to go out of use in MHD the conventional sonic Mach number used in II and in a lot of papers by the other authors (see, e.g., Refs. [4][5][6][7]. Although this subject is rather general and has no specific relation to the commented paper II, there is a sense to note the following.…”

mentioning

confidence: 99%

Response to “Comment on ‘Continuum modes in rotating plasmas: General equations and continuous spectra for large aspect ratio tokamaks’ ” [Phys. Plasmas 19, 064701 (2012)]

Lakhin

Ilgisonis

2012

Physics of Plasmas

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The equations for the continuous spectra derived in our paper [V. P. Lakhin and V. I. Ilgisonis, Phys. Plasmas 18, 092103 (2011)] can be reduced to the matrix form used by Goedbloed et al. [Phys. Plasmas 11, 28 (2004)]. It is shown that the assumptions made in our paper provide the elliptic flow regime and guarantee the existence of plasma equilibrium with nested magnetic surfaces of circular cross-section. The new results on magnetohydrodynamic instabilities of such tokamak equilibria obtained in our paper but absent in the paper by Goedbloed et al. are emphasized.

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Geodesic acoustic modes and zonal flows in toroidally rotating tokamak plasmas

Cited by 25 publications

References 15 publications

The Brunt–Väisälä frequency of rotating tokamak plasmas

The Brunt–Väisälä frequency of rotating tokamak plasmas

Heat flux effects on the dispersion relation for geodesic modes in rotating plasmas

Response to “Comment on ‘Continuum modes in rotating plasmas: General equations and continuous spectra for large aspect ratio tokamaks’ ” [Phys. Plasmas 19, 064701 (2012)]

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