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Stabilization of the axisymmetric instability in the PDX tokamak

Description: Passive stabilization, associated with the mutual inductances of the field shaping coils with the plasma and with each other, can stabilize the axisymmetric magnetohydrodynamic instability in the Princeton Poloidal Divertor Experiment device that was observed in earlier numerical studies.
Date: December 1, 1977
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Ideal magnetohydrodynamic stability of the spheromak configuration

Description: Results are presented of a parametric study of the ideal magnetohydrodynamic stability properties of the spheromak, or compact torus, configuration. In the absence of a nearby conducting wall, the spheromak is always unstable to at least one current driven mode. With a conducting wall at the surface, the spheromak can be unstable to current driven modes if the current is too peaked, i.e., q/sub o/(R/a) less than or equal to 2/3, or if the shear is too low at the origin. The Mercier criterion sets an upper limit on the pressure gradient everywhere, but configurations that are everywhere Mercier stable can be unstable to pressure driven low-n modes. Stable toroidal configurations exist with a spherical wall separated by half a minor radius, and with ..beta../sub theta/ = 30%.
Date: January 19, 1982
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Multiple time scale methods in tokamak magnetohydrodynamics

Description: Several methods are discussed for integrating the magnetohydrodynamic (MHD) equations in tokamak systems on other than the fastest time scale. The dynamical grid method for simulating ideal MHD instabilities utilizes a natural nonorthogonal time-dependent coordinate transformation based on the magnetic field lines. The coordinate transformation is chosen to be free of the fast time scale motion itself, and to yield a relatively simple scalar equation for the total pressure, P = p + B/sup 2//2..mu../sub 0/, which can be integrated implicitly to average over the fast time scale oscillations. Two methods are described for the resistive time scale. The zero-mass method uses a reduced set of two-fluid transport equations obtained by expanding in the inverse magnetic Reynolds number, and in the small ratio of perpendicular to parallel mobilities and thermal conductivities. The momentum equation becomes a constraint equation that forces the pressure and magnetic fields and currents to remain in force balance equilibrium as they evolve. The large mass method artificially scales up the ion mass and viscosity, thereby reducing the severe time scale disparity between wavelike and diffusionlike phenomena, but not changing the resistive time scale behavior. Other methods addressing the intermediate time scales are discussed.
Date: January 1, 1984
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Two-dimensional modeling of the formation of spheromak configurations

Description: A reduced set of two-dimensional MHD equations have been derived describing the axisymmetric time evolution of a MHD stable plasma evolving slowly due to resistive diffusion and changing boundary conditions. The equations are restricted to low ..beta.. but allow changing topology. They are integrated in time to demonstrate a possible spheromak formation method. External circuit equations are integrated simultaneously with the plasma equations to determine the electromagnetic boundary conditions self consistently. The effects of a finite conductivity vacuum chamber are included.
Date: September 1, 1980
Creator: Jardin, S.C. & Park, W.
Partner: UNT Libraries Government Documents Department

Nonlinear saturation of non-resonant internal instabilities in a straight spheromak

Description: An initial value numerical solution of the time dependent nonlinear ideal magnetohydrodynamic equations demonstrates that spheromak equilibria which are linearly unstable to nonresonant helical internal perturbations saturate at low amplitude without developing singularities. These instabilities thus represent the transition from an axisymmetric to a non-axisymmetric equilibrium state, caused by a peaking of the current density.
Date: April 1, 1982
Creator: Park, W. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Feedback stabilization of the axisymmetric instability of a deformable tokamak plasma

Description: We analyze the magnetohydrodynamic (MHD) stability of the axisymmetric system consisting of a free boundary, non-circular cross-section tokamak plasma, finite resistivity passive conductors, and an active feedback system with magnetic flux pickup loops, a proportional amplifier with gain G, and current carrying poloidal field coils. Numerical simulation of a system that is unstable with G = 0 shows that for some placements of the pickup loops, the system will remain unstable for all values of G, while for other placements of the loops, the system will be stable for G > G/sub crit/. This behavior is explained by analysis using an extended energy principle, and it is shown to result from the deformability of the plasma cross section. 9 refs., 5 figs.
Date: September 1, 1987
Creator: Pomphrey, N. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Nonlinear evolution of the resistive interchange mode in the cylindrical spheromak

Description: Results are presented of a study of various aspects of the single helicity nonlinear development of the resistive interchange mode in the cylindrical spheromak. A formulation of the helically symmetric resistive MHD equations that partially separates the ideal MHD characteristics is developed. Mode saturation can occur due to the quasilinear flattening of the pressure profile in the vicinity of the mode rational surface. However, this saturation process is defeated when the plasma overheats and in regions of the plasma where the shear is low. Finite fluid compression has significant, and optimistic, consequences on the long-term nonlinear behavior of this mode. For a tearing mode stable cylindrical spheromak equilibrium configuration with an axial beta value of 6%, complete overlap of the m = 1 islands occurs in about 3% of the resistive skin time for a magnetic Reynold's number of S = 10/sup 5/. For typical parameters of the S-1 device at Princeton, this time corresponds to nearly one-millisecond.
Date: February 1, 1984
Creator: DeLucia, J. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Stabilizing windings for the tilting and shifting modes in an inductively formed spheromak

Description: It is shown that a set of 4 passive conducting coils, each with the figure 8 topology, will stabilize a spheromak to the rigid body toroidal mode number n = 1 tilting and shifting modes. The inclusion of these coils does not interfere with the inductive formation process.
Date: August 1, 1981
Creator: Jardin, S.C. & Christensen, U.
Partner: UNT Libraries Government Documents Department

Dynamic modeling of transport and positional control of tokamaks

Description: We describe here a numerical model of a free boundary axisymmetric tokamak plasma and its associated control systems. The plasma is modeled with a hybrid method using two-dimensional velocity and flux functions with surface-averaged MHD equations describing the evolution of the adiabatic invariants. Equations are solved for the external circuits and for the effects of eddy currents in nearby conductors. The method is verified by application to several test problems and used to simulate the formation of a bean-shaped plasma in the PBX experiment.
Date: October 1, 1985
Creator: Jardin, S.C.; Pomphrey, N. & DeLucia, J.
Partner: UNT Libraries Government Documents Department

Amplification of S-1 Spheromak current by an inductive current transformer

Description: We attempt to predict the consequences of adding an inductive current transformer (OH Transformer) to the present S-1 Spheromak experiment. Axisymmetric modeling with only classical dissipation shows an increase of toroidal current and a shrinking and hollowing of the current channel, conserving toroidal flux. These unstable profiles will undergo helical reconnection, conserving helicity K = ..integral.. A-vector x B-vector d tau while increasing the toroidal flux and decreasing the poloidal flux so that the plasma relaxes toward the Taylor state. This flux rearrangement is modeled by a new current viscosity term in the mean-field Ohm's law which conserves helicity and dissipates energy.
Date: November 1, 1985
Creator: Jardin, S.C.; Janos, A. & Yamada, M.
Partner: UNT Libraries Government Documents Department

Feedback stabilization of axisymmetric modes in tokamaks

Description: Noncircular tokamak plasmas can be unstable to ideal MHD axisymmetric instabilities. Passive conductors with finite resistivity will at best slow down these instabilities to the resistive (L/R) time of the conductors. An active feedback system far from the plasma which responds on this resistive time can stabilize the system provided its mutual inductance with the passive coils is small enough.
Date: January 1, 1982
Creator: Jardin, S.C. & Larrabee, D.A.
Partner: UNT Libraries Government Documents Department

The spheromak as a prototype for ultra-high-field superconducting magnets

Description: In view of current progress in the development of superconductor materials, the ultimate high-field limit of superconducting magnets is likely to be set by mechanical stress problems. Maximum field strength should be attainable by means of approximately force-free magnet windings having favorable ''MHD'' stability properties (so that small winding errors will not grow). Since a low-beta finite-flux-hole spheromak configuration qualifies as a suitable prototype, the theoretical and experimental spheromak research effort of the past decade has served to create a substantial technical basis for the design of ultra-high-field superconducting coils. 11 refs.
Date: August 1, 1987
Creator: Furth, H.P. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Stability of elongated cross-section tokamaks to axisymmetric even poloidal mode number deformations

Description: A recent paper by Nakayama, Sato and Matsuoka suggests that elliptical cross section tokamaks with aspect ratio R/a = 3.2 and with elongation kappa = 2.6 are unstable to a splitting (m = 2, n = 0) instability for plasma ..beta.. > 5%, and that kappa /> =/ 4.0 plasmas are unstable to splitting for ..beta.. /> =/ 1%. We have tried to reproduce these results using the MHD evolution code TSC, but find these configurations to be stable, not even near a stability boundary. Even a kappa = 3.7 plasma with ..beta.. = 23.0% is stable to the splitting mode. However, the addition of pinching coils at the waist will cause the plasma to split if the current in these coils exceeds a critical value I/sub c/ which decreases with increasing ..beta... 8 refs., 11 figs., 1 tab.
Date: June 1, 1989
Creator: Weiner, R.; Jardin, S.C. & Pomphrey, N.
Partner: UNT Libraries Government Documents Department

Two-dimensional transport of tokamak plasmas. [Neoclassical transport theory]

Description: A reduced set of two-fluid transport equations is obtained from the conservation equations describing the time evolution of the differential particle number, entropy, and magnetic fluxes in an axisymmetric toroidal plasma with nested magnetic surfaces. Expanding in the small ratio of perpendicular to parallel mobilities and thermal conductivities yields as solubility constraints one-dimensional equations for the surface-averaged thermodynamic variables and magnetic fluxes. Since Ohm's law E + U x B = R' accounts for any nonideal effects, only determines the particle flow relative to the diffusing magnetic surfaces, it is necessary to solve a single two-dimensional generalized differential equation, (delta/delta t) nabla psi x (nabla p - J x B) = O, to find the absolute velocity of a magnetic surface enclosing a fixed toroidal flux. A time-dependent coordinate transformation is used to describe the diffusion of plasma quantities through magnetic surfaces of changing shape.
Date: October 1, 1978
Creator: Hirshman, S.P. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Modeling of the sawtooth instability in tokamaks using a current viscosity term

Description: We propose a new method for modeling the sawtooth instability and other MHD activity in axisymmetric tokamak transport simulations. A hyper-resistivity (or current viscosity) term is included in the mean field Ohm's law to describe the effects of the three-dimensional fluctuating fields on the evolution of the inverse transform, q, characterizing the mean fields. This term has the effect of flattening the current profile, while dissipating energy and conserving helicity. A fully implicit MHD transport and 2-D toroidal equilibrium code has been developed to calculate the evolution in time of the q-profile and the current profile using this new term. The results of this code are compared to the Kadomtsev reconnection model in the circular cylindrical limit. 17 refs., 8 figs.
Date: August 1, 1988
Creator: Ward, D.J. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Tilt and shift mode stability in a spheromak with a flux core

Description: The stability of spheromak equilibria with a flux core, or reversal coil, is studied by means of an ideal MHD code. Results depend critically upon whether the flux hole region (the current free area just inside the separatrix) is treated as perfectly conducting plasma or as vacuum. This indicates that the tilt and shift modes persist as resistive instabilities if they are stable in ideal MHD. Specifically, for nonoptimally shaped equilibria, the flux core must nearly touch the current channel if the flux hole is vacuum, whereas the core may be slightly outside the separatrix if the flux hole has conducting plasma. A larger margin exists for optimally shaped equilibria.
Date: July 1, 1984
Creator: Finn, J.M. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Princeton spectral equilibrium code: PSEC

Description: A fast computer code has been developed to calculate free-boundary solutions to the plasma equilibrium equation that are consistent with the currents in external coils and conductors. The free-boundary formulation is based on the minimization of a mean-square error epsilon while the fixed-boundary solution is based on a variational principle and spectral representation of the coordinates x(psi,theta) and z(psi,theta). Specific calculations using the Columbia University Torus II, the Poloidal Divertor Experiment (PDX), and the Tokamak Fusion Test Reactor (TFTR) geometries are performed.
Date: March 1, 1984
Creator: Ling, K.M. & Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Stabilizing windings for tilting and shifting modes

Description: This invention provides simple, inexpensive, independent and passive, conducting loops for stabilizing a plasma ring having externally produced equilibrium fields on opposite sides of the plasma ring and internal plasma currents that interact to tilt and/or shift the plasma ring relative to the externally produced equilibrium field so as to produce unstable tilting and/or shifting modes in the plasma ring. More particularly this invention provides first and second passive conducting loops for containing first and second induced currents in first and second directions corresponding to the amplitude and directions of the unstable tilting and/or shifting modes in the plasma ring. To this end, the induced currents provide additional magnetic fields for producing restoring forces and/or restoring torques for counteracting the tilting and/or shifting modes when the conducting loops are held fixed in stationary positions relative to the externally produced equilibrium fields on opposite sides of the plasma ring.
Date: February 26, 1982
Creator: Jardin, S.C. & Christensen, U.R.
Partner: UNT Libraries Government Documents Department

Time-Scales for Non-Inductive Current Buildup in Low-Aspect-Ratio Toroidal Geometry

Description: The fundamental differences between inductive and non-inductive current buildup are clarified and the associated time-scales and other implications are discussed. A simulation is presented whereby the plasma current in a low-aspect-ratio torus is increased primarily by the self-generated bootstrap current with only 10% coming from external current drive. The maximum obtainable plasma current by this process is shown to scale with the toroidal field strength. The basic physics setting the time-scales can be obtained from a 1D analysis. Comparisons are made between the timescales found here and those reported in the experimental literature.
Date: November 1, 1999
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

A Triangular Finite Element with First-derivative Continuity Applied to Fusion MHD Applications

Description: We describe properties of the reduced quintic triangular finite element. The expansion used in the element will represent a complete quartic polynomial in two dimensions, and thus the error will be of order h5 if the solution is sufficiently smooth. The quintic terms are constrained to enforce C1 continuity across element boundaries, allowing their use with partial differential equations involving derivatives up to fourth order. There are only three unknowns per node in the global problem, which leads to lower rank matrices when compared with other high-order methods with similar accuracy but lower order continuity. The integrations to form the matrix elements are all done in closed form, even for the nonlinear terms. The element is shown to be well suited for elliptic problems, anisotropic diffusion, the Grad-Shafranov-Schlter equation, and the time-dependent MHD or extended MHD equations. The element is also well suited for 3D calculations when the third (angular) dimension is represented as a Fourier series.
Date: January 30, 2004
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Plasma Science Contribution to the SCaLeS Report

Description: In June of 2003, about 250 computational scientists and mathematicians being funded by the DOE Office of Science met in Arlington, VA, to attend a 2-day workshop on the Science Case for Large-scale Simulation (SCaLeS). This document was the output of the Plasma Science Section of that workshop. The conclusion is that exciting and important progress can be made in the field of Plasma Science if computer power continues to grow and algorithmic development continues to occur at the rate that it has in the past. Full simulations of burning plasma experiments could be possible in the 5-10 year time frame if an aggressive growth program is launched in this area.
Date: October 9, 2003
Creator: Jardin, S.C.
Partner: UNT Libraries Government Documents Department

Resistive stability of the cylindrical spheromak

Description: The growth rates for resistive instabilities in a straight circular cylinder with spheromak profiles are computed by using two complementary methods. The first method employs boundary layer analysis and asymptotic matching, most valid for values of the magnetic Reynolds number S greater than or equal to 10/sup 5/. The second method solved the full linearized resistive MHD equations as an initial value problem, utilizing zone packing around the mode rational surface. Resolution requirements limit this to S less than or equal to 10/sup 7/. The results from these two methods agree to better than 1 in 10/sup 3/ in the overlap region 10/sup 7/ greater than or equal to S greater than or equal to 10/sup 5/. A scan of parameter space reveals that for parabolic q-profiles, the least unstable configurations have q/sub 0/ R/a approx. 0.67. The Hall term in Ohm's Law is easily incorporated into both methods. Recalculating the resistive MHD growth rates in the presence of this term shows that the resistive interchange mode is completely stabilized for a large enough value of the ion cyclotron time.
Date: November 1, 1983
Creator: DeLucia, J.; Jardin, S.C. & Glasser, A.H.
Partner: UNT Libraries Government Documents Department

Tearing-mode stability of a forming Spheromak plasma

Description: The results of numerical calculations of ..delta..' for a class of equilibria typical of those encountered during the early formation stage of the S1 Spheromak are presented. The equilibrium plasma is assumed to be cylindrically symmetric and pressureless. It encloses a current carrying perfect conductor (flux core) and is surrounded by a vacuum with zero longitudinal field. Stability boundaries in the space formed by the equilibrium parameters are mapped. The plasma is tearing mode stable provided B/sub z//B/sub theta/ at the flux core is below a certain critical value which depends on the equilibrium parameters. For typical equilibria, this critical value is 0.65.
Date: October 1, 1981
Creator: Heidbrink, W.W.; Jardin, S.C. & Chance, M.S.
Partner: UNT Libraries Government Documents Department