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Theory of Perturbed Equilibria for Solving the Grad-Shafranov Equation

Description: The theory of perturbed magnetohydrodynamic equilibria is presented for different formulations of the tokamak equilibrium problem. For numerical codes, it gives an explicit Newton scheme for solving the Grad-Shafranov equation subject to different constraints. The problem of stability of axisymmetric modes is shown to be a particular case of the equilibrium perturbation theory.
Date: July 1, 1999
Creator: Pletzer, A. & Zakharov, L.E.
Partner: UNT Libraries Government Documents Department

A tokamak equilibrium with arbitrary aspect ratio

Description: A general analytical solution of the Grad-Shafranov equation is presented. It allows the simulation of plasmas with elongation and triangularity, with an independent choice of pressure and plasma current. A numerical computed fit to families of such solutions allows the direct computation of the poloidal flux function {Psi}(R,Z) from a parametric description of the plasma given by aspect ratio A, elongation {kappa}, triangularity {delta}, size (i.e. major radius), plasma beta poloidal {beta}{sub pol}, and plasma current I{sub p}.
Date: July 13, 1995
Creator: Zheng, S.B.; Wootton, A.J. & Solano, E.R.
Partner: UNT Libraries Government Documents Department

Analytic, High-beta Solutions of the Helical Grad-Shafranov Equation

Description: We present analytic, high-beta ({beta} {approx} O(1)), helical equilibrium solutions for a class of helical axis configurations having large helical aspect ratio, with the helix assumed to be tightly wound. The solutions develop a narrow boundary layer of strongly compressed flux, similar to that previously found in high beta tokamak equilibrium solutions. The boundary layer is associated with a strong localized current which prevents the equilibrium from having zero net current.
Date: May 19, 2004
Creator: Smith, D.R. & Reiman, A.H.
Partner: UNT Libraries Government Documents Department

Upgrade for the NSTX Control Computer

Description: The National Spherical Torus Experiment (NSTX) is a proof of scientific principle experiment as a magnetic fusion containment device. A primary goal of NSTX operations is control of the plasma current, position and shape in real time for a wide range of plasma pressure and current density profiles. In order to employ the best calculation of the plasma current, position and shape, it is planned to implement the equilibrium analysis code, EFIT, in real-time, RTEFIT. EFIT inverts the Grad-Shafranov equation and performs a least squares fit to the magnetics data. RTEFIT is also capable of providing the plasma current profile and the plasma pressure profile from analysis of diagnostic data. The calculation time for RTEFTI using the present NSTX control computer system is comparable to the expected energy confinement time on NSTX and is thus slower than desired. A computer upgrade based upon 604e processors will permit the RTEFIT calculation loop to complete in about 3 ms. The presence of the passive plates further complicates the control algorithm to be used in conjunction with RTEFIT. The planned approach is to measure the eddy currents in the passive plates and to use the transient response of the coils to minimize the total shell current effect.
Date: June 1, 1999
Creator: Mueller, D.; Gates, D.A. & Ferron, J.R.
Partner: UNT Libraries Government Documents Department

Control System Development Plan for the National Spherical Torus Experiment

Description: The National Spherical Torus Experiment (NSTX) has as one of its primary goals the demonstration of the attractiveness of the spherical torus concept as a fusion power plant. Central to this goal is the achievement of high plasma {beta} ( = 2{micro}{sub 0} <p>/B{sup 2} a measure of the efficiency of a magnetic plasma confinement system). It has been demonstrated both theoretically and experimentally that the maximum achievable {beta} is a strong function of both local and global plasma parameters. It is therefore important to optimize control of the plasma. To this end a phased development plan for digital plasma control on NSTX is presented. The relative level of sophistication of the control system software and hardware will be increased according to the demands of the experimental program in a three phase plan. During Day 0 (first plasma), a simple coil current control algorithm will initiate plasma operations. During the second phase (Day 1) of plasma operations the control system will continue to use the preprogrammed algorithm to initiate plasma breakdown but will then change over to a rudimentary plasma control scheme based on linear combinations of measured plasma fields and fluxes. The third phase of NSTX plasma control system development will utilize the rtEFIT code, first used on DIII-D, to determine, in real-time, the full plasma equilibrium by inverting the Grad-Shafranov equation. The details of the development plan, including a description of the proposed hardware will be presented.
Date: June 1, 1999
Creator: Neumeyer, C.; Mueller, D.; Gates, D.A. & Ferron, J.R.
Partner: UNT Libraries Government Documents Department

High-mode-number ballooning modes in a heliotron/torsatron system: 1, Local magnetic shear

Description: The characteristics of the local magnetic shear, a quantity associated with high-mode-number ballooning mode stability, are considered in heliotron/torsatron devices that have a large Shafranov shift. The local magnetic shear is shown to vanish even in the stellarator-like region in which the global magnetic shear is positive. The reason for this is that the degree of the local compression of the poloidal magnetic field on the outer side of the torus, which maintains the toroidal force balance, is reduced in the stellarator-like region of global magnetic shear because the global rotational transform in heliotron/torsatron systems is a radially increasing function. This vanishing of the local magnetic shear is a universal property in heliotron/torsatron systems with a large Shafranov shift since it results from toroidal force balance in the stellarator-like global shear regime that is inherent to such systems.
Date: May 1, 1996
Creator: Nakajima, N.
Partner: UNT Libraries Government Documents Department

NSTX Diagnostics for Fusion Plasma Science Studies

Description: This paper will discuss how plasma science issues are addressed by the diagnostics for the National Spherical Torus Experiment (NSTX), the newest large-scale machine in the magnetic confinement fusion (MCF) program. The development of new schemes for plasma confinement involves the interplay of experimental results and theoretical interpretations. A fundamental requirement, for example, is a determination of the equilibria for these configurations. For MCF, this is well established in the solutions of the Grad-Shafranov equation. While it is simple to state its basis in the balance between the kinetic and magnetic pressures, what they are as functions of space and time are often not easy to obtain. Quantities like the plasma pressure and current density are not directly measurable. They are derived from data that are themselves complex products of more basic parameters. The same difficulties apply to the understanding of plasma instabilities. Not only are the needs for spatial and temporal resolution more stringent, but the wave parameters which characterize the instabilities are difficult to resolve. We will show how solutions to the problems of diagnostic design on NSTX, and the physics insight the data analysis provides, benefits both NSTX and the broader scientific community.
Date: July 5, 2001
Creator: Kaita, R.; Johnson, D.; Roquemore, L.; Bitter, M.; Levinton, F.; Paoletti, F. et al.
Partner: UNT Libraries Government Documents Department

ELLIPT2D: A Flexible Finite Element Code Written Python

Description: The use of the Python scripting language for scientific applications and in particular to solve partial differential equations is explored. It is shown that Python's rich data structure and object-oriented features can be exploited to write programs that are not only significantly more concise than their counter parts written in Fortran, C or C++, but are also numerically efficient. To illustrate this, a two-dimensional finite element code (ELLIPT2D) has been written. ELLIPT2D provides a flexible and easy-to-use framework for solving a large class of second-order elliptic problems. The program allows for structured or unstructured meshes. All functions defining the elliptic operator are user supplied and so are the boundary conditions, which can be of Dirichlet, Neumann or Robbins type. ELLIPT2D makes extensive use of dictionaries (hash tables) as a way to represent sparse matrices.Other key features of the Python language that have been widely used include: operator over loading, error handling, array slicing, and the Tkinter module for building graphical use interfaces. As an example of the utility of ELLIPT2D, a nonlinear solution of the Grad-Shafranov equation is computed using a Newton iterative scheme. A second application focuses on a solution of the toroidal Laplace equation coupled to a magnetohydrodynamic stability code, a problem arising in the context of magnetic fusion research.
Date: March 22, 2001
Creator: Pletzer, A. & Mollis, J.C.
Partner: UNT Libraries Government Documents Department

High-mode-number ballooning modes in a heliotron/torsatron system: 2, Stability

Description: In heliotron/torsantron systems that have a large Shafranov shift, the local magnetic shear is found to have no stabilizing effect on high-mode-number ballooning modes at the outer side of the torus, even in the region where the global shear is stellarator-like in nature. The disappearance of this stabilization, in combination with the compression of the flux surfaces at the outer side of the torus, leads at relatively low values of the plasma pressure to significant modifications of the stabilizing effect due to magnetic field-line bending on high-mode-number ballooning modes-specifically, that the field-line bending stabilization can be remarkably suppressed or enhanced. In an equilibrium that is slightly Mercier-unstable or completely Mercier-stable due to peaked pressure profiles, such as those used in standaxd stability calculations or observed in experiments on the Compact Helical System, high-mode-number ballooning modes are destabilized due to these modified stability effects, with their eigenfunctions highly localized along the field line. Highly localized mode structures such as these cause the ballooning mode eigenvalues {omega} {sup 2} to have a strong field line dependence through the strong dependence of the local magnetic curvature, such that the level surfaces of {omega} {sup 2} ({psi}, {theta} {sub k}, {alpha}), (<0) become spheroids in ({theta} {sub k}, {alpha}) space, where {psi} labels flux surfaces and {theta} {sub k} is the radial wavenumber. Because the spheroidal level surfaces for unstable eigenvalues are surrounded by level surfaces for stable eigenvalues of high-mode-number toroidal Alfven eigenmodes, those high-mode-number ballooning modes never lead to low-mode-number modes. In configuration space, these high- mode-number modes are localized in a single toroidal pitch of the helical coils, and hence they may experience substantial stabilization due to finite Larmor radius effects.
Date: May 1, 1996
Creator: Nakajima, N.
Partner: UNT Libraries Government Documents Department

An analytic determination of beta poloidal and internal inductance in an elongated tokamak from magnetic probe measurements

Description: Analytic calculations of the magnetic fields available to magnetic diagnostics are performed for tokamaks with circular and elliptical cross sections. The explicit dependence of the magnetic fields on the poloidal beta and internal inductances is sought. For tokamaks with circular cross sections, Shafranov's results are reproduced and extended. To first order in the inverse aspect ratio expansion of the magnetic fields, only a specific combination of beta poloidal and internal inductance is found to be measurable. To second order in the expansion, the measurements of beta poloidal and the internal inductance are demonstrated to be separable but excessively sensitive to experimental error. For tokamaks with elliptical cross sections, magnetic measurements are found to determine beta poloidal and the internal inductance separately. A second harmonic component of the zeroth order field in combination with the dc harmonic of the zeroth order field specifies the internal inductance. The internal inductance in hand, measurement of the first order, first harmonic component of the magnetic field then determined beta poloidal. The degeneracy implicit in Shafranov's result (i.e. that only a combination of beta poloidal and internal inductance is measurable for a circular plasma cross section) reasserts itself as the elliptic results are collapsed to their circular limits.
Date: February 1, 1992
Creator: Sorci, J.M.
Partner: UNT Libraries Government Documents Department

An analytic solution of high. beta. equilibrium in a large aspect ratio tokamak

Description: An analytic solution of the high {beta} ({epsilon}{bar {beta}}{sub p} {approximately} {beta}q{sup 2}/{epsilon} {much gt} 1) equilibrium of a large aspect ratio tokamak is presented. Two arbitrary flux functions, the pressure profile p({psi}) and the safety factor profile q({psi}), specify the equilibrium. The solution splits into two asymptotic regions: the core region where {psi} is a function of the major radius alone and a narrow boundary layer region adjoining the conducting wall. The solutions in the two regions are asymptotically matched to each other. For monotonic pressure profiles, the Shafranov shift is equal to the minor radius. For {beta} much bigger than one, the solution contains a region (in place of the magnetic axis) of zero magnetic field and constant pressure. At high {beta} the quantity {beta}{sub I}, which is essentially proportional to the pressure over the total current squared, is largely independent of pressure. We discuss the important ramifications of limited {beta}{sub I} for high {beta} reactors. Generalizations to shaped cross sections and hollow pressure profiles are outlined. We also consider the problem of equilibrium reconstruction in the high {beta} regime. 8 refs., 7 figs.
Date: March 1, 1991
Creator: Cowley, S.C.; Kaw, P.K.; Kelly, R.S. & Kulsrud, R.M.
Partner: UNT Libraries Government Documents Department

Tokamak fusion reactor start-up simulation

Description: A simulation code TSEC (Time-dependent Spectral Equilibrium Code) has been developed to model the axisymmetric evolution of a tokamak on the resistive (L/R) time scale of the external coils, conductors, or shell. The electromagnetic interaction between the plasma and the external circuit is taken into account in a self-consistent manner. TSEC is Lagrangian and utilizes magnetic flux coordinates with spectral decomposition in the angle variable theta. The plasma is modeled as a finite-size, zero-inertia, finite-pressure fluid which adjusts its position and shape to remain in free-boundary equilibrium consistent with the currents in the external circuits. At the heart of TSEC is a fast method of calculating the self-consistent free-boundary plasma equilibrium at each time step which is based on the minimization of a certain mean-square error. 3 refs., 6 figs., 3 tabs.
Date: February 1, 1986
Creator: Ling, K.M.; Jardin, S.C. & Perkins, F.W.
Partner: UNT Libraries Government Documents Department

Current drive, anticurrent drive, and balanced injection

Description: In lower hybrid (LH) discharges, the number of suprathermal electrons is limited by the upper bound on the current density from the q = 1 condition, which is caused by the onset of the m = 1 MHD instability. The stored energy of suprathermal electrons, measured in terms of a poloidal beta, scales with plasma current as I/sub p//sup -1/. Potentially, these bounds represent very restrictive conditions for heating in larger machines. Consequently, it seems necessary to perform experiments where the electrons are driven in both directions, parallel and antiparallel to the magnetic field, i.e., bidirectional scenarios like anticurrent drive or balanced injection. Data from PLT relevant to these ideas are discussed. 6 refs., 4 figs.
Date: August 1, 1987
Creator: von Goeler, S.; Stevens, J.; Beiersdorfer, P.; Bell, R.; Bernabei, S.; Bitter, M. et al.
Partner: UNT Libraries Government Documents Department