Assuming only small gyromotion periods and Larmor radii compared to any other time and length scales, and retaining the lowest significant order in δ = ρ i /L 1, the general expression of the ion gyroviscous stress tensor is presented. This expression covers both the "fast dynamics" (or "magnetohydrodynamic") ordering, where the time derivative and ion gyroviscous stress are first order in δ relative to the ion gyrofrequency and scalar pressure respectively, and the "slow dynamics" (or "drift")ordering, where the time derivative and ion gyroviscous stress are respectively second order in δ. This general stress tensor applies to arbitrary collisionality and does not require the distribution function to be close to a Maxwellian. Its exact divergence (gyroviscous force) is written in closed vector form, allowing for arbitrary magnetic geometry, parallel gradients and flow velocities. Considering in particular the contribution from the velocity gradient (rate of strain) term, the final form of the momentum conservation equation after the "gyroviscous cancellation" and the "effective renormalization of the perpendicular pressure by the parallel vorticity" is precisely established.

At finite ratios of the kinetic plasma pressure to the magnetic pressure, the magnetic confinement configurations of axisymmetric plasma columns tend to acquire characteristics that hinder the onset of instabilities driven by the combined effects of magnetic curvature and pressure gradient. A simple analytical dispersion relation that contains the main physical factors affecting an important class of these modes is given.PACS numbers: 52.30.+rThe purpose of this Letter is to discuss the maximum value of the ratio /3 of the plasma kinetic pressure to the pressure associated with the confining magnetic field that can be reached without inducing loss of confinement. This ratio is important in order to assess the main characteristics, such as transport properties and rates of radiation emission, of a given magnetic confinement configuration, as well as the type of fusion reactor that can be developed out of it.Here we limit most of our attention to the ideal magnetohydrodynamic (MHD) approximation and notice that when /3 increases toward finite values, instabilities driven by the combined effects of the magnetic field curvature and the pressure gradient can be expected to develop. 1 An essential feature of the ideal MHD model is the strong interaction between the developing plasma instability and its magnetic confinement configuration. As /3 is increased, the configuration of the plasma isobaric surfaces also changes, and, since the plasma motion is tied to that of the magnetic lines, this will have a dual effect 2 : Not only will it increase the instability driving pressure gradient, but it also will enhance the stabilizing magnetic tension in that region of unfavorable magnetic curvature where the relevant modes develop.In the earliest stability analysis of these modes, the considered equilibrium models neglected the modification of the confinement configuration that takes place as j3 evolves and becomes finite. In particular, only linear terms in the plasma ,pressure gradient were retained in the relevant normal mode equation, and rather pessimistic upper limits on j3 for stable configurations were obtained. However, as we shall show in this Letter, important nonlinear terms must also be retained; as was first demonstrated for simplified equilibrium configurations, these terms can lead to production of a "second stability region/ This circumstance is illustrated by the following dispersion relation that we have derived from a consistent description of the ideal MHD equilibrium configuration in the vicinity of the magnetic axis:• or 3/^AQG 2 \ 2 / 32 -£ • (1)Here we have employed familiar notations except for the dimensionless parameters s{ip) = dlxiq{Wdlnr{$), Gti>)=-8irR 0 q 0 B 0 ml r(mp(Wdip that measure the magnetic shear and the plasma pressure gradient. For G 2 < 1.2s, corresponding to the first stability region in the (G,s) plane, the pressure gradient is not strong enough to overcome the stable shear-Alfvgn oscillations. For G 2 > 6.4s the tendency of the plasma to expand is h...

An analytical theory of ideal-MHD ballooning modes that can be excited in finite-β equilibria is carried out on model configurations which include the effects of the increase of the poloidal field toward the outer edge of the plasma column and the dependence of the rate of magnetic shear on the poloidal angle. The relevant growth rates and eigensolutions are, in fact, significantly different from those derived on the basis of ‘low-β’ model configurations that omit one or both of the effects mentioned above, and provide different indications for the expected interaction between ideal-MHD and kinetic modes. For each value of the shear parameter ŝ, the normalized growth rate Γ becomes real at a critical value of the dimensionless pressure gradient parameter G. When the latter is increased at constant ŝ, Γ is found to increase only up to a saturation point, after which it decreases and tends to vanish at a second critical value of G.