S.P. Hirshman

Neoclassical transport of impurities in tokamak plasmas

Tokamak plasmas are inherently comprised of multiple ion species. This is due to wall-bred impurities and, in future reactors, will result from fusion-born alpha particles. Relatively small densities nI, of highly charged non-hydrogenic impurities can strongly influence plasma transport properties whenever . The determination of the complete neoclassical Onsager matrix for a toroidally confined multispecies plasma, which provides the linear relation between the surface averaged radial fluxes and the thermodynamic forces (i.e. gradients of density and temperature, and the parallel electric field), is reviewed. A closed set of one-dimensional moment equations is presented for the time evolution of thermodynamic and magnetic field quantities which results from collisional transport of the plasma and two-dimensional motion of the magnetic flux surface geometry. The effects of neutral-beam injection on the equilibrium and transport properties of a toroidal plasma are consistently included.

Three-dimensional free boundary calculations using a spectral Green's function method

Abstract The plasma energy W p = ∫ ω p ( 1 2 B 2 +p d V is minimized over a toroidal domain ω p using an inverse representation for the cylindrical coordinates R = ΣR mn ( S ) cos( mθ − nζ ) and Z = ΣZ mn ( s ) sin( mθ − nζ ), where ( s , θ, ζ) are radial, poloidal and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when ω p is varied to make the total pressure 1 2 B 2 + p continuous at the plasma surface Σ p and when the vacuum magnetic field B v satisfies the Neumann condition B v ·d Σ p = 0 . The vacuum field is decomposed as B v = B 0 + ∇φ , where B 0 is the field arising from plasma currents and external coils and φ is an single-valued potential necessary to satisfy B v d Σ p = 0 when p ≠ 0. A Greens function method is used to obtain an integral equation over Σ p for the scalar magnetic potential φ = Σφ mn sin( mθ − nζ ). A linear matrix equation is solved for φ mn to determine 1 2 B 2 v on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to l = 2 stellarator equilibria are presented.

Bootstrap current and neoclassical transport in tokamaks of arbitrary collisionality and aspect ratio

A multi-species fluid model is described for the steady state parallel and radial force balance equations in axisymmetric tokamak plasmas. The bootstrap current, electrical resistivity, and particle and heat fluxes are evaluated in terms of the rotation velocities and friction and viscosity coefficients. A recent formulation of the neoclassical plasma viscosity for arbitrary shape and aspect ratio (including the unity aspect ratio limit), arbitrary collisionality, and orbit squeezing from strong radial electric fields is used to illustrate features of the model. The bootstrap current for the very low aspect ratio National Spherical Torus Experiment [J. Spitzer et al., Fusion Technol. 30, 1337 (1996)] is compared with other models; the largest differences occur near the plasma edge from treatment of the collisional contributions. The effects of orbit squeezing on bootstrap current, thermal and particle transport, and poloidal rotation are illustrated for an enhanced reverse shear plasma in the Tokamak Fusi...

Finite‐aspect‐ratio effects on the bootstrap current in tokamaks

The bootstrap current in tokamaks is determined in the low collision frequency regime for arbitrary values of the aspect ratio and effect charge. The resulting expression should be useful for a quantitative analysis of currents in tokamaks.

Variational bounds for transport coefficients in three‐dimensional toroidal plasmas

A variational principle is developed for the linearized drift‐kinetic, Fokker–Planck equation, from which both upper and lower bounds for neoclassical transport coefficients can be calculated for plasmas in three‐dimensional toroidal confinement geometries. These bounds converge monotonically with the increasing phase‐space dimensionality of the assumed trial function. This property may be used to identify those portions of phase space that make dominant contributions to the transport process. A computer code based on this principle has been developed that uses Fourier–Legendre expansions for the poloidal, toroidal, and pitch‐angle dependences of the distribution function. Numerical calculations of transport coefficients for a plasma in the TJ‐II flexible heliac [Nucl. Fusion 28, 157 (1988)] are used to demonstrate the application of this procedure.

Preconditioned descent algorithm for rapid calculations of magnetohydrodynamic equilibria

Abstract Conjugate gradient descent algorithms have been used in several magnetohydrodynamic (MHD) equilibrium codes to find numerical minima of the MHD energy and thus to locate local stable equilibria. Numerical convergence studies with the spectral equilibrium code VMEC (variational moments equilibrium code) have shown that the number of descent iterations required to obtain a fixed level of convergence grows linearly with the number of radial mesh points. This undesirable mesh dependence is due to the quadratic dependence on the radial mesh spacing of the condition number for the linearized discrete MHD equations. By use of a preconditioning matrix to coalesce the eigenvalues of the linearized MHD forces around unity, it is possible to reduce the condition number substantially and thereby nearly eliminate the mesh size dependence of the convergence rate of the descent algorithm. An invertible, positive-definite tridiagonal preconditioning matrix is derived from the force equations used in VMEC, and the improvement in temporal convergence is demonstrated for several three-dimensional equilibria.

MOMCON: A spectral code for obtaining three-dimensional magnetohydrodynamic equilibria

Abstract A new code, MOMCON (spectral moments code with constraints), is described that computes three-dimensional ideal magnetohydrodynamic (MHD) equilibria in a fixed toroidal domain using a Fourier expansion for the inverse coordinates ( R, Z ) representing nested magnetic surfaces. A set of nonlinear coupled ordinary differential equations for the spectral coefficients of ( R, Z ) is solved using an accelerated steepest descent method. A stream function, λ, is introduced to improve the mode convergence properties of the Fourier series for R and Z . The convergence rate of the R - Z spectra is optimized on each flux surface by solving nonlinear constraint equations relating the m ≥ 2 spectral coefficients of R and Z .

Neoclassical transport fluxes in the plateau regime in nonaxisymmetric toroidal plasmas

The neoclassical fluxes in the plateau regime in an arbitrary large aspect ratio nonaxisymmetric toroidal system are calculated using fluid equations. The results are evaluated explicitly for a stellarator and for a rippled tokamak. For typical stellarator parameters, the helical magnetic field contribution to the particle and heat fluxes is comparable to the toroidicity contribution, and therefore, cannot be neglected. In addition, the helical magnetic field contribution to the parallel viscosity is capable of reversing the directions of the bootstrap current and Ware pinch flux in the plateau regime. In contrast, the neoclassical particle and heat fluxes in a rippled tokamak are comparable to those in an axisymmetric tokamak after the radial electric field and parallel flow velocity are determined (without external sources).

Physics issues of compact drift optimized stellarators

Physics issues are discussed for compact stellarator configurations which achieve good confinement by the fact that the magnetic field modulus |B| in magnetic co-ordinates is dominated by poloidally symmetric components. Two distinct configuration types are considered: (1) those which achieve their drift optimization and rotational transform at low β and low bootstrap current by appropriate plasma shaping; and (2) those which have a greater reliance on plasma β and bootstrap currents for supplying the transform and obtaining quasi-poloidal symmetry. Stability analysis of the latter group of devices against ballooning, kink and vertical displacement modes has indicated that stable β values on the order of 15% are possible. The first class of devices is being considered for a low β near term experiment that could explore some of the confinement features of the high β configurations.

V3FIT: a code for three-dimensional equilibrium reconstruction

The V3FIT code for performing equilibrium reconstruction in three-dimensional plasmas is described. It is a modular code that has the potential to be coupled with a variety of equilibrium solvers to compute the externally measured response to an arbitrary internal state of the plasma. Singular-value decomposition is used to identify the dominant components of the plasma state that can be accurately determined by the reconstruction process and to guide the minimization of the χ2 variance-normalized mismatch between the measured and computed signals. Comparison of a tokamak plasma equilibrium computed by V3FIT and by the axisymmetric equilibrium reconstruction code EFIT is presented. V3FIT is used to reconstruct an axisymmetric DIII-D equilibrium using experimentally observed magnetic diagnostic signals. Three-dimensional reconstructions of stellarator plasma equilibria in the CTH device show the code behaves as expected in the presence of experimental noise, appropriately ignores near-singular directions in parameter space and robustly reconstructs equilibria starting from substantially different initial parameter values.