Jeffrey P. Freidberg

“One size fits all” analytic solutions to the Grad–Shafranov equation

An extended analytic solution to the Grad–Shafranov equation using Solov’ev profiles is presented. The solution describes standard tokamaks, spherical tokamaks, spheromaks, and field reversed configurations. It allows arbitrary aspect ratio, elongation, and triangularity as well as a plasma surface that can be smooth or possess a double or single null divertor X-point. The solution can also be used to evaluate the equilibrium beta limit in a tokamak and spherical tokamak in which a separatrix moves onto the inner surface of the plasma.

Radial discontinuities in tokamak magnetohydrodynamic equilibria with poloidal flow

It is shown that transonic poloidal flow leads to ideal magnetohydrodynamic tokamak equilibria with radial discontinuities in the density, pressure, and flow velocity profiles. Transonic profiles are defined as having flow velocities ranging from subsonic to supersonic with respect to the poloidal sound speed (csBp/B). The jump of the equilibrium quantities occurs approximately at the sonic surface and its magnitude is of order e1/2 (e is the inverse aspect ratio). Because of the large velocity shear at the sonic surface, transonic profiles may improve energy confinement as suggested by current understanding of tokamak plasma turbulence suppression.

A family of analytic equilibrium solutions for the Grad–Shafranov equation

A family of exact solutions to the Grad–Shafranov equation, similar to those described by Atanasiu et al. [C. V. Atanasiu, S. Gunter, K. Lackner, and I. G. Miron, Phys. Plasmas 11, 3510 (2004)], is presented. The solution allows for finite plasma aspect ratio, elongation and triangularity, while only requiring the evaluation of a small number of well-known hypergeometric functions. Plasma current, pressure, and pressure gradients are set to zero at the plasma edge. Realistic equilibria for standard and spherical tokamaks are presented.

A fast, high-order solver for the Grad-Shafranov equation

Abstract We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

A general formulation of magnetohydrodynamic stability including flow and a resistive wall

A general formulation is presented for determining the ideal magnetohydrodynamic stability of an axisymmetric toroidal magnetic configuration including the effects of an arbitrary equilibrium flow velocity and a resistive wall. The system is inherently non-self-adjoint with the eigenvalue appearing in both the equations and the boundary conditions. Even so, after substantial analysis we show that the stability problem can be recast in the form of a standard eigenvalue problem, ωA⋅z=B⋅z, which is highly desirable for numerical computation.

Universal scaling laws for quench and thermal hydraulic quenchback in CICC coils

A set of universal scaling relations is presented describing the propagation of quench in CICC (cable-in-conduit superconducting magnet) magnets. Four distinct types of behavior are possible depending upon the length of the coil and the magnitude of the quench induced pressure rise. The boundaries separating these regions can be simply expressed in terms of L/sub q/ the initial quench length, and J the stabilizer current density, the two parameters likely to vary during standard operation. The phenomenon of thermal hydraulic quenchback (THQB) is also considered. It is shown that the conditions for the onset of THQB can also be cast as a set of universal scaling relations and easily superimposed on the quench diagram.<<ETX>>

Stabilization of the resistive wall mode by flowing metal walls

The effect of flowing metal walls on the resistive wall instabilities is analyzed for a general cylindrically symmetric diffusive pinch configuration. Two types of liquid metal flow are analyzed: a uniform flow which is poloidally symmetric, and a two-stream flow consisting of two opposite streams splitting at the top and merging at the bottom. It is found in both configurations that when the liquid wall flow velocity exceeds a critical value, the resistive wall mode is stabilized. However, for the two-stream flow the critical velocity is several times smaller than that for the uniform flow. Still in a realistic experiment one needs a flow velocity of a few tens m/s to stabilize the resistive wall mode.

Designing a tokamak fusion reactor—How does plasma physics fit in?

This paper attempts to bridge the gap between tokamak reactor design and plasma physics. The analysis demonstrates that the overall design of a tokamak fusion reactor is determined almost entirely by the constraints imposed by nuclear physics and fusion engineering. Virtually, no plasma physics is required to determine the main design parameters of a reactor: a, R0, B0, Ti, Te, p, n, τE, I. The one exception is the value of the toroidal current I, which depends upon a combination of engineering and plasma physics. This exception, however, ultimately has a major impact on the feasibility of an attractive tokamak reactor. The analysis shows that the engineering/nuclear physics design makes demands on the plasma physics that must be satisfied in order to generate power. These demands are substituted into the well-known operational constraints arising in tokamak physics: the Troyon limit, Greenwald limit, kink stability limit, and bootstrap fraction limit. Unfortunately, a tokamak reactor designed on the basi...

Magnetohydrodynamic stability comparison theorems revisited

Magnetohydrodynamic (MHD) stability comparison theorems are presented for several different plasma models, each one corresponding to a different level of collisionality: a collisional fluid model (ideal MHD), a collisionless kinetic model (kinetic MHD), and two intermediate collisionality hybrid models (Vlasov-fluid and kinetic MHD-fluid). Of particular interest is the re-examination of the often quoted statement that ideal MHD makes the most conservative predictions with respect to stability boundaries for ideal modes. Some of the models have already been investigated in the literature and we clarify and generalize these results. Other models are essentially new and for them we derive new comparison theorems. Three main conclusions can be drawn: (1) it is crucial to distinguish between ergodic and closed field line systems; (2) in the case of ergodic systems, ideal MHD does indeed make conservative predictions compared to the other models; (3) in closed line systems undergoing perturbations that maintain...

Finite shift stabilization of ballooning modes in a high‐beta tokamak

The widely used s vs α stability diagram for ballooning modes is based on the assumption of circular flux surfaces with small shifts. (Here s=rq’/q and α=−R0q2β’.) This analysis is extended to include finite shifts. There again results a universal stability diagram, although in this case it is parametrized by three quantities s, α, and σ’, the differential shift between flux surfaces. The analysis shows that the inclusion of a finite shift leads to a significant further stabilization of the ballooning modes over that from the standard case.