Stability Of Ballooning Modes Research Articles

Ballooning mode stability of elongated high-beta tokamaks

The variational principle derived by Choe and Freidberg [Phys. Fluids 29, 1766 (1986)] and used to estimate the geometry of high-beta tokamak equilibria is extended to include elongation. Ballooning mode stability is then investigated, illustrating the influence of elongation on local and global stability.

Finite pressure ballooning mode stability in toroidal equilibriums

In this paper the effect of finite pressure on the ballooning instability in toroidal magnetohydrodynamics (MHD) equilibria of steep boundary stellarators and tokamaks is examined. Ballooning modes tend to arise near the place where the local shear vanishes and the normal curvature (the curvature component perpendicular to the flux surface, pointing away from the magnetic axis) is negative. It is shown how the pressure gradient determines the position of the shearless points, and demonstrated in detail how this effect explains the existence of second stability in tokamaks. For large aspect ratio circular cross-section tokamaks the second stability condition is found to scale as α=const S1.25. Stellarators are inherently more stable because of the negative vacuum shear, which at moderate pressure gradients allows the zero shear point to localize on the inner side of the flux surface. However, at high pressure gradients the Pfirsch–Schlüter current produces a positive mean shear when the total toroidal current on a flux surface is zero. This causes the zero shear point to localize on the outer edge near the vertical extremes of the flux surface. This effect, together with helical contributions to the helical curvature, allows for ballooning instability to arise. At higher pressure gradients, with zero net toroidal current, an unstable ballooning mode which localizes to within a helical period always arises where the normal curvature is unfavorable.

Beta limit and localized ballooning mode stability in Doublet III

The ideal localized ballooning mode stability criterion has been analysed using equilibria obtained in Doublet III. Near the experimental beta limit, portions of the flux surfaces could have developed the pressure gradient set by this stability criterion. At low q (≤3.2), this is not sufficient to affect the trend in the development of the plasma internal inductance, , and total βt. At high q (≥3.5), both and βt achieve stationary values. This correlates well with the hypothesis that a large portion of the plasma reaches the ballooning mode limit for more than one energy confinement time.

Resistive ballooning modes driven by anomalous transport effects

Using Braginskii’s two fluid equations, the stability of resistive ballooning modes is examined in the presence of parallel thermal conduction, anomalous electron viscosity, and radial thermal conductivity. A generalized set of coupled second-order differential equations in φ and ψ is derived in ballooning space and is solved to obtain analytical solutions in two interesting frequency regimes, SCs/ qR ≪ ‖ω‖ ≪ Cs/ qR and ‖ω‖ ≫ Cs/qR. It is shown that the anomalous thermal transport term excites the new m=1 resistive ballooning mode (‖ω‖ ≫ Cs/qR) with a large growth rate. The excitation of the m=2 type (or Δ′ driven) mode, on the other hand, is found to be strongly influenced by both anomalous electron viscosity and radial thermal conduction. Finally, the additional effect of parallel electron thermal conduction is shown to give new resistive ballooning modes with significantly large growth rates varying as fractional powers of anomalous electron viscosity and classical thermal conductivity.

KINETIC THEORY ON BALLOONING-MODE IN COLLISIONAL PLASMAS

We have studied the kinetic effects on ballooning-mode more completely from the gyro-kinetic theory. Here we have consideres the effects of the following factors, i. e., collision, hot-ion components such as neutral beam injection or fusion α-particles, particle diamagnetitc drift, ion gyro-radius ,as well as trapped particles. It has been shown numerically that both gyro-radius of ion and trapped particles have stabilizing effects on ballooning-mode, whereas the collision term has the destabilizing effect besides, the kinetic theory reproduces the results given in Ideal MHD theory, i.e. the shear can improve the stability of ballooning-mode.

The stability of resistive ballooning modes in a high temperature plasma

It has been suggested that the combined effects of electron pressure gradient in the parallel component of Ohm's law and finite parallel electron thermal conduction can stabilise the resistive ballooning modes in a high temperature plasma. The authors have analysed this problem and conclude that an unstable ballooning mode persists. This contrasts with the situation for tearing modes which are indeed stabilised by these effects.

Resistive ballooning modes in high-beta tokamaks

Resistive ballooning mode stability is examined using one-fluid, two-fluid, and kinetic theories to determine the domain of validity of the resistive-fluid theories in the high-beta Doublet III tokamak. In general, it is found that the requirements for the validity of the resistive-fluid theories in Doublet III are only satisfied on the cold, outermost edge of the plasma with temperatures T ≤ 25 eV. In the domain of validity, the two-fluid and kinetic theories are qualitatively the same and indicate that the dominant effect is parallel resistivity with small corrections from perpendicular resistivity, fluid compression, thermal conduction, and ion collisional viscosity. The generic role of resistive ballooning modes in tokamaks and the implications for anomalous transport are also discussed.

Two fluid equations for resistive ballooning modes in axisymmetric toroidal plasmas

The equations governing linear stability of resistive ballooning modes are obtained from the full set of two-fluid equations, with diamagnetic, viscous and thermal conduction effects retained. A cold ion model in which the full electron physics is retained is then developed further and the averaged equations valid in the resistive regime of ballooning space are derived.

Kinetic theory of resistive ballooning modes

A linear and nonlinear kinetic theory of resistive ballooning modes that includes diamagnetic drifts and finite Larmor radius effects is presented. The linear stability of resistive ballooning modes is examined analytically and numerically. A renormalized resistive ballooning equation is derived, and the saturation level of the instabilities is analytically calculated. Finally, a calculation of the electron thermal conductivity for the large ω* regime is presented.

Stabilization of Resistive Ballooning Modes in High-Temperature Plasmas

The linear stability of resistive ballooning modes in a high-temperature tokamak plasma is examined, with account taken of the effect of temperature gradients and finite parallel electron thermal conductivity. It is found that for moderate values of ..beta.., where ..beta.. is the ratio of the thermal to magnetic pressure, resistive ballooning modes may be completely stabilized.

Safety factor profile optimization of high-n ideal ballooning mode stability of Tokamaks

Conditions of improved stability to high-n ideal MHD modes are obtained with equilibria generated by flattening and raising the safety factor (q) profile. Flat q profiles shorten the connection length for high-n ballooning modes and diminish the destabilizing impact of the geodesic curvature. High values of q stabilize the residual interchange modes.

Kinetic effects on ballooning modes in mirror machines

A general procedure for examining the influence of kinetic effects on the stability of magnetohydrodynamic ballooning modes in mirror machines is presented. In particular, the basic kinetic ballooning mode equation for a nonaxisymmetric, arbitrary beta system with anisotropic pressure is derived. Considering a long-thin equilibrium typical of the tandem mirror, it is shown that this governing eigenmode equation reduces to a simple form independent of wave-particle resonant effects.

Neutral beam effects on Tokamak ballooning mode stability

An equation is derived that gives a necessary, as well as sufficient, stability criterion for the guiding center plasma to instabilities of large toroidal mode number. This equation is applied to D-shaped scalar pressure and beam-induced tensor pressure Tokamak equilibria of aspect ratio 3. The tensor and scalar pressure equilibria generated have comparable pressure profiles. The tensor pressure equilibria are valid in the limit that the beam particle injection energy EO>>Ec and (Zeff-1)<<1, where Ec is the critical energy. The scalar pressure equilibria are valid when EO<or=Ec. Therefore, the effects of neutral beam injection on the stability of a Tokamak are bracketed. It is found that quasi-perpendicular injection induces a displacement of the rho perpendicular to contours with respect to the flux surfaces away from the destabilizing curvature region, which enhances the stability of tensor pressure equilibria over comparable scalar pressure equilibria. Also found is a difference between the necessary and sufficient stability limits when the most unstable modes concentrate in regions of relatively high pressure. Both stability coalesce when ballooning modes, which appear near the edge of the plasma where the pressure is low, dominate.