Axisymmetric Toroidal Plasma Research Articles

Numerical studies of toroidal resistive magnetohydrodynamic instabilities

A time-stepping code has been constructed to study the dominant resistive magnetohydrodynamic (MHD) instability of an axisymmetric toroidal plasma. The model used is based on the linearized, incompressible MHD equations with constant density and includes the toroidal ideal model if the resistivity is taken to be zero. The equations are solved fully implicitly using a coordinate system for which one set of coordinate surfaces coincides with a set of surfaces of constant poloidal flux. This is crucial for the accurate representation of modes for which the perturbed quantities vary rapidly near surfaces with rational values of the safety factor. The code is checked by comparison with an exactly soluble model, cylindrical resistive MHD codes and a toroidal ideal MHD code (ERATO). Results are presented showing the effect of resistivity on the unstable internal modes near nq 0 = 1 for an INTOR-like numerically generated equilibrium.

Toroidal plasma equilibria in an external magnetic multipole field

The equilibrium of an axisymmetric toroidal plasma, with almost circular cross-section, situated in an external magnetic multipole field is considered. The external field is created by a number of circular current carrying conductors symmetrically arranged outside of the plasma (Extrap configuration). The equilibrium field is expanded to first orders in the inverse aspect ratio and in a small parameter describing the non-circularity of the plasma cross-section. The current density is taken on the form j= kappa 2 psi +C. With this choice, both 'peaked' ( kappa 2>0) and 'skin-type' ( kappa 2<0) profiles can be described. The results obtained indicate that, in the range of parameters where equilibria with non-elongated magnetic surfaces ('square-shaped' separatrix) can be found, in the case of four external conductors, there is a high sensitivity of the plasma location (toroidal radius) to the precise values of the currents in the external conductors.

Saturated tearing modes in toroidal geometry

A quasilinear method is developed for determining saturated tearing mode magnetic island widths in axisymmetric toroidal plasmas with arbitrary cross-sectional shape, aspect ratio, or plasma pressure (beta). The method is applied to compute magnetic island widths in force-free toroidal plasmas with aspect ratio as low as 2.0 and elongation between 1.0 and 2.0. It is found that current suppression within the magnetic island strongly increases the saturated width while current peaking reduces width. The effects of current profile, geometry, and harmonic mixing are also studied.

External inductance of an axisymmetric plasma

The external inductance of an axisymmetric toroidal plasma with an arbitrary aspect ratio and cross section is obtained using a Green’s function method. By varying an equivalent skin current density over the plasma surface, while keeping the total toroidal current fixed, the plasma boundary is made to coincide with a magnetic surface. Numerical computations of the self-inductance and mutual inductance as functions of aspect ratio and elongation are fitted to simple analytic formulas. The effect of distributed plasma current on the volt-seconds required to reach a prescribed net current is considered.

Theory of the rippling instability in toroidal devices

The theory of the rippling instability is developed for axisymmetric toroidal plasmas including ion viscosity and parallel electron heat conduction, but assuming that the growth rate is small compared to the wave angular frequency omega 0=(1+1.71 eta e) omega e*. omega e* is the electron diamagnetic frequency and eta e identical to (dlnTe/dr)/(dlnN/dr). Three (in)stability regions must be considered, viz. (ci/$R omega 0)2<0.12 where toroidal effects are negligible and the plasma is stable; 0.12<(ci/R omega 0)2<0.26 q2 where the magnetic curvature destabilizes the mode; and 0.26 q2<(ci/R omega 0)2 where the plasma is stable again. (The numerical values are given for eta e= eta i=2 and Te=Ti, not heat conduction or viscosity.) Parallel electron heat conduction is stabilizing; ion viscosity broadens or narrows the instability domain, according to the value of the parameter omega i*0,i/Pi. Under certain conditions, an important up-down asymmetry of the density fluctuation spectrum may arise.

A matrix method for resistive MHD stability analysis of axisymmetric toroidal plasma

A matrix method to solve a resistive MHD stability problem has been developed. The equations are reduced to an eigenvalue problem of block tridiagonal matrices. The inverse iteration method is employed as a solution method of the eigenvalue problem.

Viscous effects in a collisional tokamak plasma with strong rotation

The full viscosity tensor for an axisymmetric toroidal plasma in the collisional regime (with strong rotation) is calculated, including gyroviscosity and O(ε) poloidal variations over the flux surface. It is shown that the resulting viscous force is of sufficient magnitude to account for the radial transfer of toroidal momentum that must be inferred in order to explain the rotation measurements in tokamak experiments. The consequences of a viscous force of this form and magnitude on particle transport and on the evolution of toroidal and poloidal rotation velocities are discussed.

Finite Larmor radius equations for waves in a Tokamak plasma near ion cyclotron harmonics

Starting from the linearized Vlasov equation, the authors derive the complete finite Larmor radius equations for electromagnetic waves propagating in an axisymmetric toroidal plasma in the ion cyclotron frequency domain.

Resistive ballooning modes in an axisymmetric toroidal plasma with long mean free path

Tokamak devices normally operate at such high temperatures that the resistive fluid description is inappropriate. In particular, the collision frequency may be low enough for trapped particles to exist. However, on account of the high conductivity of such plasmas, one can identify two separate scale lengths when discussing resistive ballooning modes. By describing plasma motion on one of these, the connection length, in terms of kinetic theory the dynamics of trapped particles can be incorporated. On the resistive scale length, this leads to a description in terms of modified fluid equations in which trapped particle effects appear. The resulting equations are analyzed and the presence of trapped particles is found to modify the stability properties qualitatively.

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.

The Effect of Trapped Electrons on ECRH Current Drive in a Weakly Relativistic Plasma

The wave-induced current by electron cyclotron resonance heating (ECRH) is investigated in an axisymmetric toroidal plasma. The plasma is assumed to be so weakly relativistic that the only relativistic correction to the resonance condition is considered. The effect of trapped electrons are also calculated to the order of \(\sqrt{\varepsilon}\), where e is the inverse aspect ratio. The trapped electrons and the relativistic resonance condition are shown to have important effect on the efficiency of current drive by ECRH.

Mode conversion near ion-ion hybrid and IC harmonic resonances in Tokamaks

The authors propose a model for the investigation of mode conversion near two ion hybrid and cyclotron harmonic resonances in axisymmetric toroidal plasmas. The wave differential equations are derived from Maxwell-Vlasov equations using this model and the assumption of weak dispersion, and are shown to coincide with equations recently obtained in the literature for a slab model including a (B0. Del )B0 term. At the same time the limits of applicability of these equations in a Tokamak are clarified. The wave equation is solved explicitly in two cases of physical interest: near an isolated two-ion hybrid resonance, where the fast wave couples to the cold torsional Alfven wave, leading mainly to electron heating; and near the first harmonic resonance of a single species plasma, where it couples to an ion Bernstein wave. These solutions clarify some aspects of the physics of these resonances, and can be useful for applications in numerical ICR heating codes.

The Effect of Trapped Electrons on the Beam-Induced Current. II

The toroidal effect on the beam-induced current for an axisymmetric toroidal plasma is investigated by means of an expansion in \(\sqrt{\varepsilon }\) ( ε : inverse aspect ratio). An expression for the induced current is obtained up to the order of ε . This current is numerically calculated when the beam ions are monoenergetic and the beam velocity is much smaller than the electron thermal velecity. By use of results, an approximate expression for the induced current is derived as a function of ε and effective charge \bar Z of the plasma.

Continuous Double Adiabatic Spectrum in Toroidal Plasmas

The continuous spectrum of an anisotropic and axisymmetric toroidal plasma is investigated using the double adiabatic theory. The continuum is given by an eigenvalue problem of a fourth order system of ordinary differential equations. In contrast to the magnetohydrodynamic continuum the double adiabatic continuum may become unstable. The stability depends upon the parallel and perpendicular pressure distributions along the field lines. In absence of a toroidal magnetic field, the fourth order system decouples into two second order differential equations for which specific stability criteria are derived.

Axisymmetric toroidal equilibria with plasma flow

A general equilibrium equation for an axisymmetric toroidal plasma having both toroidal and poloidal components of the magnetic field and plasma flow is derived. The possibility of a numerical treatment is discussed and polynomial solutions are obtained for several limiting cases. These solutions show that with flow rotation, the pressure and magnetic surfaces are no longer concentric, and that there is a significant displacement between these surfaces. A new result of hydrodynamic interest is the equilibrium equation for a compressible fluid.

A plasma resistive diffusion model

A self-consistent study of the slow resistive evolution of an axisymmetric toroidal plasma gives rise to a set of transport equations involving one space variable which require input from the solution of a generalized differential equation obtained from the time-differentiated Grad-Shafranov equation. An iterative scheme is presented for the numerical solution of this generalized differential equation which overcomes the problems of the non-standard boundary conditions. As an illustration this method is used to compute the instantaneous diffusion velocity of a class of model toroidal equilibria. A more detailed study is presented of the time evolution of this model in the cylindrical limit in order to illustrate techniques which can be used in a more complete toroidal simulation.

The Effect of Trapped Electrons on the Wave-Induced Current

The toroidal effect on the beam-induced current for an axisymmetric toroidal plasma is investigated by means of an expansion in \(\sqrt{\varepsilon }\) ( e : inverse aspect ratio). An expression for the induced current is obtained up to the order of e . This current is numerically calculated when the beam ions are monoenergetic and the beam velocity is much smaller than the electron thermal velecity. By use of results, an approximate expression for the induced current is derived as a function of e and effective charge \bar Z of the plasma.

RF current drive in a toroidal plasma in the banana regime

The use of travelling waves for the steady-state current drive in an axisymmetric toroidal plasma in the banana regime is studied. The treatment is based on a quasi-linear equation for the electron distribution function, averaged over the period of particle motion along the small azimuth of the torus. It is shown that the trapped electrons do not absorb the energy of the monochromatic (in frequency) RF field and thus only the circulating electrons contribute to the driving current and the absorbed RF power. The current and the absorbed power are calculated by using the electron distribution function obtained for the case of a narrow wave packet, both the toroidial magnetic field and the distortion of the electron distribution with respect to transverse velocities being taken into account. The important role of the marginally circulating electrons is revealed. It is pointed out that the toroidal satellite resonances can affect the RF current drive by spreading and splitting the region of wave-particle interaction.

Finite-n ballooning mode theory for axisymmetric toroidal plasmas

Ballooning mode theory for finite toroidal mode number n is derived for shearfree and sheared axisymmetric toroidal plasmas. The resulting finite-n theory is applicable for both compressible and incompressible perturbations. The inclusion of finite compressibility changes the ballooning mode eigenfunctions and eigenvalues, but not the form of the finite-n correction.

High-n collisionless ballooning modes in axisymmetric toroidal plasmas

A collisionless kinetic ballooning-mode equation, which includes the full ion finite-Larmor-radius (FLR), the magnetic-drift, and the trapped-electron effects, is derived and investigated for a large-aspect-ratio, circular-flux-surface equilibrium in the frequency regime, ωbi, ωti<ω<ωbe, ωte. The finite-Larmor-radius effects can reduce the growth rate, but do not stabilize the ballooning modes due to the destabilizing influence of the ion-magnetic-drift resonances. It is, in general, incorrect to simulate the FLR effects by employing the often used FLR-modified MHD model for (kθρi)2⪆0.1 and ϵn⪆0.1, where kθρi is the ion FLR parameter and ϵn = Ln/R measures the magnetic-drift frequency. The trapped electrons have a stabilizing effect due to the reduction of the destabilizing circulating-electron parallel-current perturbation. For a typical tokamak aspect ratio, the critical β can be improved by 40%.