# A simple model for collisional drift waves

12 Dec 1997-Canadian Journal of Physics (NRC Research Press Ottawa, Canada)-Vol. 75, Iss: 12, pp 891-906

TL;DR: A simple model for drift waves in high-density, low-temperature, magnetized plasmas is presented in this paper, where the nonadiabatic response of the electrons is obtained perturbatively.

Abstract: A simple model for drift waves in high-density, low-temperature, magnetized plasmas is presented. The nonadiabatic response of the electrons is obtained perturbatively. A solution to the two-fluid equation describing the electron temperature is presented. Curvature effects due to the magnetic field inhomogeneity are retained. R´ esum´ e :N ous presentons un modele simple des ondes de deplacement dans un plasma de haute densite et de basse temperature. Une approche perturbative permet d'obtenir la reponse non-adiabatique des electrons. Nous proposons une solution de l'equation a deux fluides qui decrit la temperature electronique et tenons compte des effets de courbure dus a l'inhomogeneite du champ magnetique. (Traduit par la r ´

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01 Jan 1996

TL;DR: In this article, three-dimensional computations of turbulence arising from the nonlinear collisional drift wave equations are carried out, where the flux-surface-based coordinate system is aligned with the magnetic field, and the geometry is that of an actual model tokamak with arbitrary poloidal cross section.

Abstract: Three-dimensional computations of turbulence arising from the nonlinear collisional drift wave equations are carried out. The flux-surface-based coordinate system is aligned with the magnetic field, and the geometry is that of an actual model tokamak with arbitrary poloidal cross section. The physical periodicity constraint is rigorously respected. The results show that the dominant process arising from this system is the three-dimensional version of the collisional drift wave nonlinear instability, in which fluctuation free energy transfer among parallel wavelengths plays an enhanced role. Poloidal asymmetry in the fluctuations and associated transport are found to result primarily from the poloidal variation in the ion polarization drift and not the more traditional ballooning (magnetic curvature) effects. Magnetic curvature is found to be very important only in the case of reversed magnetic shear: with it, reversing the shear causes a drop in the thermal energy flux by a factor of three. The contrast with concurrent work on ballooning is suggested to result from the latter's neglect of the electron temperature dynamics. As in previous results of two-dimensional slab computations, the electron temperature gradient is the principal free energy source. The turbulence appears to be non-local over the radial range of the 4 cm covered by the computations; the non-locality is a form of weighted averaging of the free energy sources and sinks by the turbulence, and is sufficient to explain the rise in relative amplitude with increasing radius since the absolute amplitude is relatively constant. Initial tests with an isothermal ion pressure suggest that the ion dynamics could make up the quantitative difference between these results and the experimental observations, once the ion temperature is properly incorporated.

82 citations

01 Nov 1983

TL;DR: In this article, an energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, ρ, π, σ, ω, φ, υ, τ, ϵ, ϳ, ς, ψ, ϩ, ϸ, ϴ, Ϡ, ϖ, ϓ, ό, ϐ, Ϻ, ϔ

Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

81 citations

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TL;DR: In this paper, a 3-field model for collisional drift waves, in the ballooning representation, for a low-pressure stellarator plasma is presented, which includes the effects of a finite radial mode number (θk).

Abstract: A 3-field model for collisional drift waves, in the ballooning representation, for a low-pressure stellarator plasma is presented. The 3-field model, which includes the effects of a finite radial mode number (θk) , is solved as an initial-value problem along the magnetic field line. It is shown that for a stellarator with low global magnetic shear, θk= 0 corresponds to the fastest linear growth rate. The effects of the magnetic field structure for the tokamak and stellarator configurations are discussed in a comparative way. PACS Nos.: 52.35Kt, 52.30Jb, 52.35Ra

3 citations

01 Jan 1972

TL;DR: In this article, an extensive investigation of the dependence of drift wave characteristics upon parallel wavelength in collisional potassium Q-machine plasmas is presented, which includes the effects of parallel resistivity and thermal conductivity for electrons and collisional viscosity, and motion parallel to the magnetic field for ions.

Abstract: An extensive investigation of the dependence of drift wave characteristics upon parallel wavelength in collisional potassium Q‐machine plasmas is presented. Results are compared with a two‐fluid slab‐model theory which includes the effects of parallel resistivity and thermal conductivity for electrons and Larmor radius, collisional viscosity, and motion parallel to the magnetic field for ions. The general stability characteristics of the drift wave are confirmed: stability at low magnetic field (B/k⊥≲0.4 kG·cm) due to ion viscosity, stability at both long‐ and short‐parallel wavelength (λ‖ ≳ 200 and λ‖≲100 cm, respectively) due to the combined effects of ion viscosity and parallel electron fluid expansion and stability of individual modes at high magnetic field (B/k⊥ ≈ 1 kG·cm, λ‖ ≈ 100 cm) due to ion parallel motion. Quantitative agreement between experiment and theory is demonstrated for the detailed parametric dependences of instability onset, complex frequency, and oscillation amplitude on parallel wa...

##### References

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TL;DR: In this article, an energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, ρ, π, σ, ω, φ, υ, τ, ϵ, ϳ, ς, ψ, ϩ, ϸ, ϴ, Ϡ, ϖ, ϓ, ό, ϐ, Ϻ, ϔ

Abstract: An energy principle is used to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J×B−∇p=0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x=x(ρ, θ, ζ). Here, θ are ζ are poloidal and toroidal flux coordinate angles, respectively, and p=p(ρ) labels a magnetic surface. Ordinary differential equations in ρ are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x. A steepest‐descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive‐definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter λ is introduced to ensure the rapid convergence of the Fourier series for x, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self‐consistent value for λ.

750 citations

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TL;DR: A review of measurements of microscopic fluctuations and theories of turbulence and anomalous transport for tokamaks is given in this article, and some comparisons between theory and experiment are presented.

Abstract: A review of measurements of microscopic fluctuations and theories of turbulence and anomalous transport for tokamaks is given, and some comparisons between theory and experiment are presented. The results of the measurements indicate that all tokamaks have rather similar, broadband, incoherent microscopic fluctuations. Such fluctuations have been measured in the density, potential, electric field, and magnetic field. In the edge regions of three tokamaks, the particle transport caused by the turbulent electric field fluctuations has been measured directly. Although tokamak microturbulence has been studied extensively, neither its source nor its role in anomalous energy transport is yet understood. The incoherent, turbulent nature of the fluctuations has made it difficult to understand them theoretically. Recently, however, significant theoretical progress has been made in several areas including non-linear models of drift wave turbulence and transport, models of anomalous electron thermal conduction by stochastic magnetic field fluctuations, and non-linear models of localized resistive-MHD instabilities.

726 citations

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TL;DR: In this article, the self-consistent classical plasma equilibrium with diffusion was studied in a toroidal geometry having a sheared magnetic field and it was found that the pressure gradient is zero unless the Fourier component of 1/B2, which resonates with that surface, vanishes.

Abstract: The self‐consistent classical plasma equilibrium with diffusion is studied in a toroidal geometry having a sheared magnetic field. Near each rational surface it is found that the pressure gradient is zero unless the Fourier component of 1/B2, which resonates with that surface, vanishes. Despite the resonances, the overall plasma confinement is, in practice, only slightly modified by the rational surfaces.

373 citations

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01 Jan 1991

336 citations

### "A simple model for collisional drif..." refers background in this paper

...Using (36) the covariant basis vectors ev Co@Cv, e Co@C , and e Co@C are easily computed, followed by the contravariant basis vectors Qn @ e M 4 lmnelfem for +l> m> n, @ +v> > ,, where lmn is the usual Levi–Civita symbol for permutations [30]....

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TL;DR: In this paper, a review of the literature in the area of tokamak microinstability theory is presented, focusing on low-frequency electrostatic drift-type modes.

Abstract: Significant investigations in the area of tokamak microinstability theory are reviewed. Special attention is focused on low-frequency electrostatic drift-type modes, which are generally believed to be the dominant tokamak microinstabilities under normal operating conditions. The basic linear formalism including electromagnetic (finite-beta) modifications is presented along with a general survey of the numerous papers investigating specific linear and non-linear effects on these modes. Estimates of the associated anomalous transport and confinement times are discussed, and a summary of relevant experimental results is given. Studies of the non-electrostatic and high-frequency instabilities associated with the presence of high-energy ions from neutral-beam injection (or with the presence of alpha-particles from fusion reactions) are also surveyed.

309 citations