Christer Wahlberg

Stabilization of the internal kink mode in a tokamak by toroidal plasma rotation

The stability of the internal m=n=1 kink mode is analyzed for a tokamak with a toroidally rotating plasma, by a large aspect ratio expansion of the compressible magnetohydrodynamic equations. Assuming that the central poloidal beta is of order unity, it is found that the internal kink mode is stabilized by rotational frequencies of order Ω/ωA∼e, where ωA is the Alfven frequency and e is the inverse aspect ratio. The internal kink then turns into a stable oscillation with a Doppler-shifted frequency ∼ΩM(1−1/Γ)1/2, where Γ is the adiabatic index and M is the sonic Mach number. The stabilization comes from the centrifugal force which gives a stable density (or entropy) distribution within each magnetic surface. The parallel motion associated with the internal kink mode then behaves as the Brunt–Vaisala oscillations of a stably stratified fluid in a gravitational field. At lower rotational frequencies, Ω/ωA∼e2, the only effect of the rotation is a co-rotation of the usual (nonrotating) m=n=1 instability, wher...

Stability analysis of internal ideal modes in low-shear tokamaks

The stability of internal, ideal modes in tokamaks with low magnetic shear in the plasma core is analyzed. For equilibria with large aspect ratio, a parabolic pressure profile and a flat q profile in the core, an exact solution of the ideal magnetohydrodynamic (MHD) stability equations is found. The solution includes the eigenfunctions and the complete spectra of two distinctly different MHD phenomena: A family of fast-growing, Mercier-unstable global eigenmodes localized in a low-shear region with q 1 in the core. In the latter case the solution in addition includes one unstable eigenmode, if beta is larger than a critical value depending on the width of the low-shear region and on the q-profile in the edge region.

Importance of centrifugal effects for the internal kink mode stability in toroidally rotating tokamak plasmas

Analytical theory and two different magnetohydrodynamical stability codes are used in a study of the effects of toroidal plasma rotation on the stability of the ideal, internal kink mode in tokamak ...

The Effect of Plasma Profile Variation on the Stability of the n = 1 Internal Kink Mode in Rotating Tokamak Plasmas

The sensitivity of the stability of the ideal n = 1 internal kink mode to variations in the plasma profiles is analysed both analytically and numerically in rotating tokamak plasmas. These stability analyses have been carried out including the centrifugal effects of toroidal plasma rotation upon the equilibrium, and also inconsistently when the equilibrium is treated as static. The change in plasma stability due to rotation is partially (consistent equilibrium) or wholly (inconsistent treatment) determined by the radial profiles of the plasma density and rotation velocity. It is found that the internal kink mode stability is strongly influenced by small variations in these plasma profiles. The implications of this extreme sensitivity are discussed, with particular reference to experimental data from MAST.

Analytical stability condition for the ideal m=n=1 kink mode in a toroidal plasma with elliptic cross section

An analytical stability condition for the ideal kink mode with toroidal mode number n=1 and (dominating) poloidal mode number m=1 in a toroidal plasma with elliptic cross section in derived from a large aspect ratio expansion of the ideal magnetohydrodynamic equations. The ellipticity (e) is treated as a small parameter independent of the inverse aspect ratio (e), and the expansion is performed up to order e2e in the potential energy δW. It is found that the ellipticity has a strong, destabilizing effect on the kink mode in vertically elongated tokamaks, particularly when Δq=1−q0 is small (q0 is the safety factor at the magnetic axis), whereas an ellipticity of opposite sign (horizontal elongation) is stabilizing. By means of an additional expansion, in Δq, this effect is shown to be due to a Δq-independent term in δW proportional to the ellipticity and to the poloidal beta value (βp) at the q=1 surface. Since the leading order term in δW in the absence of ellipticity is of order Δq [Bussac et al., Phys. ...

Fast growing resistive two fluid instabilities in hybrid-like tokamak configuration

An analytic derivation of the dispersion relation for resistive instabilities in a low-shear tokamak configuration is presented. The resistive infernal mode model (Charlton et al 1989 Phys. Fluids B 1 798) is generalized to include plasma diamagnetism, subsonic equilibrium toroidal flow shear and viscosity. An estimate of the transition point between the fast S−3/13 infernal-like (S is the Lundquist number) and the slow S−3/5 tearing-like scaling is given. A novel S−3/8 scaling is found close to the ideal ion-diamagnetic magnetohydrodynamic stability boundary. New moderately fast scalings in S are also found when sheared toroidal E × B flow and viscosity are considered. An analytic treatment of the m = n = 1 quasi-interchange mode in presence of density gradients with flat temperature profiles has been made.

Effect of combined triangularity and ellipticity on the stability limit of the ideal internal kink mode in a tokamak

The sawtooth period and amplitude in tokamaks is known to depend strongly on the shape of the q = 1 surface [1]. Furthermore, this shape dependence seems to correlate, at least to some extent, with the stability limit of the ideal, internal kink mode. The strong dependence on shaping, especially on ellipticity e and triangularity δ, of the stability limit of this mode is well-known from numerical computations [2]. The strongly destabilizing effect of ellipticity alone, especially at small ∆ q = 1 − q0, has recently also been given an analytical explanation by including terms of order e 2 e, where e is the inverse aspect ratio, in a perturbation expansion of the potential energy δW of the ideal m = n = 1 mode [3]. In the case of a parabolic current profile near the axis, and for small values of ∆ q and r1, a normalized form of this contribution to δW is given by [3]

Stabilization of the Mercier modes in a tokamak by toroidal plasma rotation

The stability of localized modes (Mercier modes) in a tokamak with a toroidally rotating plasma is analyzed within the framework of compressible, ideal magnetohydrodynamics. For equilibria with large aspect ratio, poloidal beta value of order unity, and isothermal magnetic surfaces, it is found that sonic, toroidal rotation provides a strongly stabilizing effect for the Mercier modes, similar to the stabilization recently found for the internal kink mode in a rotating plasma [Wahlberg and Bondeson, Phys. Plasmas 7, 923 (2000)]. A finite oscillation frequency (Brunt–Vaisala frequency), of the order of the sound frequency, is shown to be associated with each magnetic surface. If Γ>1, where Γ is the exponent in the equation of state, the rotation transforms the Mercier instabilities to stable oscillations at the local Brunt–Vaisala frequency associated with the magnetic surface where the mode is located. If the plasma satisfies an isothermal equation of state (Γ=1), however, the stability of the Mercier mode...

Structure of the compressible MHD equations for the internal kink mode in a toroidal plasma with large aspect ratio

The equations for the ideal, internal m = n = 1 kink mode in a toroidal plasma are derived from a direct, large-aspect-ratio perturbation expansion of the compressible magnetohydrodynamic (MHD) equ ...

Electron oscillations of a collisionless plasma in a static electric field

The influence of a static electric field on electron oscillations of (i) a cold beam of electrons and (ii) a hot non-drifting plasma is investigated. The importance of the relative strength of the field is emphasized. In the weak field limit the oscillations are adiabatic, manifested by the existence of adiabatic invariants, preservation of eigenmodes and absence of reflexion. If the field is strong, however, interesting new features, such as mutual coupling of eigenmodes and wave reflexion, appear. Possible stability properties of a monotonic electrostatic transition (shock, double layer) are also discussed.