Inverse Aspect Ratio Research Articles

Transport scaling in the collisionless-detrapping regime in stellarators

Stellarator transport scalings with electric field, geometry, and collision frequency in the reactor-relevant collisionless-detrapping regime are determined from numerical solutions of the drift kinetic equation. A new geometrical scaling, proportional to ε3/2t rather than εtε1/2h, is found, where εt is the inverse aspect ratio and εh is the helical ripple. With the new scaling, no reduction in energy confinement time is associated with large helical ripple, which provides design flexibility. Integral expressions for the particle and heat fluxes that are useful for transport simulations are given.

Kinetic equilibrium and stability properties of high-current betatrons

Kinetic stability properties of an intense relativistic electron ring in modified and conventional betatron configurations are investigated using the linearized Vlasov–Maxwell equations. Included is the important influence of intense equilibrium self-fields. It is assumed that the ring is thin, and that ν/γb≪1, where ν is Budker’s parameter and γbmc2 is the characteristic electron energy. The stability analysis is carried out for eigenfrequency ω close to harmonics of the cyclotron frequency ωcz in the vertical betatron field. Also included in the analysis is the influence of transverse electromagnetic effects and surface-wave perturbations. Dispersion relations for longitudinal perturbations are obtained, where it is assumed that the ring is located inside a perfectly conducting toroidal shell. There are several noteworthy points. First, transverse electromagnetic effects can completely stabilize the negative-mass instability for sufficiently high-current rings when betatron focusing forces exceed defocusing self-field forces (ω2cz>ω2pe/γ2b). Second, for ω2cz>ω2pe/γ2b with no charge neutralization or stabilizing spread in canonical angular momentum ( f=0 and Δ=0), surface-wave instabilities can be completely stabilized at a sufficiently low transverse beam temperature. Third, for ω2cz<ω2pe/γ2b, sufficiently low transverse temperature together with surface effects combine to drive a radial kink instability. Finally, the dispersion relation is analyzed numerically for parameters representative of the Naval Research Laboratory’s modified betatron (National Technical Information Service Document No. AD-A108359/1) and the Los Alamos National Laboratory’s Liner Driven Ring Accelerator (National Technical Information Service Document No. DE84007994) and Phermex Injected Conventional Betatron (National Technical Information Service Document No. DE86002441/XAB). Detailed stability results are presented for the projected operating regimes of these devices, including the effects of canonical angular momentum spread, inverse aspect ratio, slowly varying accelerating fields, location of the conducting wall, and radial elongation of the minor ring cross section.

Effect of toroidicity during lower-hybrid mode conversion

The effect of toroidicity during lower-hybrid mode conversion is examined by treating the wave propagation in an inhomogeneous medium as an eigenvalue problem for ω2 (m, n), m, n poloidal and toroidal wavenumbers. Since the fre-quency regime near ω = ω2LH is an accumulation point for the eigenvalue spectrum, the degenerate perturbation technique must be applied. The toroidal eigenmodes are constructed by a zeroth-order superposition of monochromatic solutions with different poloidal dependence m; thus they generically exhibit a wide spectrum in k‖ for given fixed ω2 even for small inverse aspect ratio є. When the average 〈k‖〉 is in the neighbourhood of kmin, the minimum wave-number for accessibility of the mode conversion regime, it is possible that excitation of toroidal modes rather than geometrie optics may determine the wave coupling to the plasma. Our results are not changed significantly by a small amount of dissipation. The level of density fluctuations in modem tokamaks, on the other hand, may cause enough k‖ scattering to mask the toroidicity effects. Nevertheless, it is shown that a wide k‖ spectrum excited by a monochromatic pump will persist even with vanishing fluctuation level.

Stability analysis for the Big Dee upgrade of the Doublet III tokamak

Ideal magnetohydrodynamic stability analysis has been carried out for configurations expected in the Big Dee tokamak, an upgrade of the Doublet III tokamak into a non-circular cross-section device which began operation early in 1986. The results of this analysis support theoretical predictions as follows: Since the maximum value of beta stable to ballooning and Mercier modes, which we denote βc, increases with inverse aspect ratio, elongation and triangularity, the Big Dee is particularly suited to obtain high values of βc and there exist high βc Big Dee equilibria for large variations in all relevant plasma parameters. The beta limits for the Big Dee are consistent with established theory as summarized in present scaling laws. High beta Big Dee equilibria are continuously accessible when approached through changes in all relevant input parameters and are structurally stable with respect to variations of input plasma parameters. Big Dee beta limits have a smooth dependence on plasma parameters such as βp and elongation. These calculations indicate that in the actual running of the device the Big Dee high beta equilibria should be smoothly accessible. Theory predicts that the limiting plasma parameters, such as beta, total plasma current and plasma pressure, which can be obtained within the operating limits of the Big Dee are reactor relevant. Thus the Big Dee should be able to use its favourable ideal MHD scaling and controlled plasma shaping to attain reactor relevant parameters in a moderate sized device.

Fully toroidal ion temperature gradient driven drift modes

A simple quadratic dispersion relation is derived for electrostatic ion temperature gradient driven modes without expansion in the inverse aspect ratio. It is also shown that these modes experience the local curvature only on the outside of the magnetic surface.

Poloidal electric field generated by electron cyclotron heating in a collisional tokamak plasma

The poloidal electric field generated by electron heating in a collision dominated tokamak plasma is calculated by the adjoint method. The physical mechanism is elucidated as due to a combination of the Doppler shifted cyclotron resonance and magnetic field inhomogeneity. For uniform rf field on a flux surface, the normalized electrostatic potential is found to be much smaller than the inverse aspect ratio for realistic power levels.

Tokamak plasma equilibrium with non-uniform current distribution

An equilibrium solution correct to the second order of the inverse aspect ratio is obtained for the axisymmetric Tokamak with circular cross section. The toroidal current density distribution profile is non-uniform. The equilibrium parameters are computed. The values of the displacement of the magnetic axis xm and the limiting value of the poloidal beta beta P,max are determined. The optimum toroidal current density profile is found. The second order correction increases the value of beta P,max and decreases the value of xm. The shape of the conducting shell surrounding the plasma is estimated.

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.

Ion cyclotron electromagnetic wave structure in Elmo bumpy torus

The problem of ion cyclotron wave structure in EBT has been analyzed using the smallness of the inverse aspect ratio ε = a/R0 and of the inverse cavity number ε' = 1/N. The procedure is to expand in these two parameters, reducing the complete toroidal problem to a system of equations to be solved in sequence. To second order in ε and ε', this system contains two ordinary differential equations of second order and one partial differential equation with periodic coefficients in a magnetically adapted system of co-ordinates. The smallness of the mean bumpiness parameter εB reduces the problem to a single second-order differential equation which, for a parabolic density profile, is a Whittaker equation. The EM wave structure corresponds to a simple mode with only a few wavelengths across the plasma radius, permitting multi-harmonic ion cyclotron heating with interesting efficiency, as observed in experiments.

Nonlinear theory of large-mode-number ballooning modes in fully toroidal geometry

A nonlinear model equation describing ballooning modes for β of the order of the inverse aspect ratio and including magnetic shear is derived. For the special case of circular concentric flux surfaces, we obtain an approximate dispersion relation, including finite Larmor radius effects, and show the possibility of explosive instability in the nonlinear regime. The results are shown to remain valid in the presence of a low-density hot-ion constituent.

Unsteady lifting-line theory as a singular perturbation problem

Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of James (1975), and points out a limitation in that of Van Holten (1975, 1976, 1977).

Reduced equations for finite beta tearing modes in tokamaks

The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have ∇ ⋅ B=0 satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio ε of the torus and for β∼ε or smaller. This is demonstrated by deriving a reduced set of MHD equations that are correct to fifth order in ε. These equations contain the exact equilibrium relation and, as such, can be used to find three-dimensional stellarator equilibria. In addition, if a subsidiary ordering in η, the resistivity, is made, the equations of Glasser, Greene, and Johnson [Phys. Fluids 8, 875 (1967); 19, 567 (1967)] are recovered. This set of reduced equations has been coded by extending the initial value code hilo [Phys. Fluids 26, 3066 (1983)]. Results obtained for both ideal and resistive linear stability from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. Good agreement is shown for both zero and finite-beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter.

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.

Tokamak beta limit

The beta limit for stability against MHD modes is calculated for a large aspect ratio tokamak. The results are found to be similar to those calculated numerically for small aspect ratio. For ideal modes, the beta limit is found to be proportional to current, having the form (28ϵ/qa)% (or 5.6 I (MA)/a(m) B(T)), with a maximum value of 14ϵ% at qa = 2, ϵ being the inverse aspect ratio. A limit of 7ϵ% is found when tearing-mode stability is also required.

A differential equation for the ambipolar electric field in a multiple-helicity torsatron

In a torsatron, the ambipolar electric field is obtained by equating the ion and electron fluxes. This formulation, when solved simultaneously with the density and temperature rate equations, gives continuous electric fields but may lead to the radial derivative of the electric field being discontinuous. This unphysical situation arises from the neglect of the finite orbit deviation from the flux surfaces. If the fluxes are calculated to higher order in inverse aspect ratio to include the finite orbit deviation, then a second-order differential equation is obtained that will give a continuous first derivative for the electric field.

Second Order Effect of a Toroidal Equilibrium With Anisotropic Pressure

A class of equilibria for a high beta Tokamak with highly anisotropic pressure and circular plasma cross section is solved to second order in the inverse aspect ratio. The effect of anisotropy on the limiting values of the poloidal beta βP is slight. The second order correction increases the limiting values of βP.

A FIR grating polychromator using conical diffraction as a diagnostic tool for ECE plasma measurements

A new 6(12) grating polychromator based on the principle of conical diffraction has been designed for the millimetre wavelength range. The application will be the study of electron cyclotron emission spectra of large tokamaks. The conical diffraction principle has two relevant advantages, namely a wavelength-independent efficiency and a very large wavelength range, accessible by one grating, 85–425 GHz. The experimental dispersion and imaging characteristics are shown. The resolution of the instrument is shown to depend linearly on λ, which combined with the typical 1 R dependence of the tokamak magnetic field gives almost constant spatial resolution in the line of sight. The spectral range of the instrument (Δλ = 0.33 mm independent of grating angle) results in a spatial coverage of the plasma in second harmonic radiation of Δd d = 0.03 B 0ϵ , where d is the diameter of the plasma, B 0 the magnetic field on axis and ϵ the inverse aspect ratio, R a . The radiation is detected by cryogenically-cooled InSb, with ≈ 1 μs temporal resolution. The theoretical efficiency is given.

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.

Development of a differential equation for the ambipolar electric field in a bumpy torus in the low collision frequency limit

In a bumpy torus the radial electric field is determined by the constraint of ambipolarity. If both species are dominated by nonresonant diffusion, then the ambipolarity relationship as it is usually formulated is algebraic in the electric field. The solutions of this algebraic equation may have a discontinous first derivative of the electric field. The unphysical situation arises from the neglect of the finite orbit deviation from the pressure surface. If the fluxes are calculated to higher order in the inverse aspect ratio, a second-order differential equation for the electric field is obtained that will give a continuous first derivative for the electric field.

Neoclassical transport associated with collisionless detrapping in a bumpy torus

In the low-collisionality nonresonant regime in a bumpy torus the transitional particles can make a large contribution to neoclassical transport. This contribution can be moderated by the toroidally induced radial drift which causes transitional particles to detrap and retrap in the mirror sectors. This effect leads to diffusion coefficients which are linear in the collision frequency and scale with the inverse aspect ratio instead of the more usual square of the inverse aspect ratio.