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Numerical solutions of magnetohydrodynamic stability of axisymmetric toroidal plasmas using cubic B-spline finite element method

Description: A nonvariational ideal MHD stability code (NOVA) has been developed. In a general flux coordinate (/psi/, theta, /zeta/) system with an arbitrary Jacobian, the NOVA code employs Fourier expansions in the generalized poloidal angle theta and generalized toroidal angle /zeta/ directions, and cubic-B spline finite elements in the radial /psi/ direction. Extensive comparisons with these variational ideal MHD codes show that the NOVA code converges faster and gives more accurate results. An extended version of NOVA is developed to integrate non-Hermitian eigenmode equations due to energetic particles. The set of non-Hermitian integro-differential eigenmode equations is numerically solved by the NOVA-K code. We have studied the problems of the stabilization of ideal MHD internal kink modes by hot particle pressure and the excitation of ''fishbone'' internal kink modes by resonating with the energetic particle magnetic drift frequency. Comparisons with analytical solutions show that the values of the critical ..beta../sub h/ from the analytical theory can be an order of magnitude different from those computed by the NOVA-K code. 24 refs., 11 figs., 1 tab.
Date: December 1, 1988
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Kinetic extensions of magnetohydrodynamic models for axisymmetric toroidal plasmas

Description: A nonvariational kinetic-MHD stability code (NOVA-K) has been developed to integrate a set of non-Hermitian integro-differential eigenmode equations due to energetic particles for axisymmetric toroidal plasmas in a general flux coordinate system with an arbitrary Jacobian. The NOVA-K code employs the Galerkin method involving Fourier expansions in the generalized poloidal angle theta and generalized toroidal angle /zeta/ directions, and cubic-B spline finite elements in the radial /Psi/ direction. Extensive comparisons with the existing variational ideal MHD codes show that the ideal MHD version of the NOVA-K code converges faster and gives more accurate results. The NOVA-K code is employed to study the effects of energetic particles on MHD-type modes: the stabilization of ideal MHD internal kink modes and the excitation of ''fishbone'' internal kink modes; and the alpha particle destabilization of toroidicity-induced Alfven eigenmodes (TAE) via transit resonances. Analytical theories are also presented to help explain the NOVA-K results. For energetic trapped particles generated by neutral beam injection (NBI) or ion cyclotron resonant heating (ICRH), a stability window for the n = 1 internal kink mode in the hot particle beta space exists even in the absence of the core ion finite Larmor radius effect. On the other hand, the trapped alpha particles are found to have negligible effects on the stability of the n = 1 internal kink mode, but the circulating alpha particles can strongly destabilize TAE modes via inverse Landau damping associated with the spatial gradient of the alpha particle pressure. 60 refs., 24 figs., 1 tab.
Date: April 1, 1989
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

High-n collisionless ballooning modes in axisymmetric toroidal plasmas

Description: A collisionless kinetic ballooning mode equation, which includes the full ion finite Larmor radius (FLR), the magnetic drift, and the trapped electron effects is derived and investigated for a large aspect ratio circular flux surface equilibrium in the frequency regime, ..omega../sub bi/, ..omega../sub ti/ < ..omega.. < ..omega../sub be/, ..omega../sub te/. The finite Larmor radius effects can reduce the growth rate, but do not stabilize the ballooning modes due to the destabilizing influence of the ion magnetic drift reonances. It is, in general, incorrect to simulate the FLR effects by employing the often used FLR modified MHD model for (k/sub theta/rho/sub i/)/sup 2/approx. greater than or equal to 0.1 and epsilon/sub n/ approx. greater than or equal to 0.1, where k/sub theta/rho/sub i/ is the ion FLR parameter and epsilon/sub n/ = L/sub n//R measures the magnetic drift frequency. The trapped electrons have a stabilizing effect due to the reduction of the destabilizing circulating electron parallel current perturbation. For typical tokamak aspect ratio, the critical ..beta.. can be improved by 40%.
Date: October 1, 1981
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Kinetic theory of collisionless ballooning modes

Description: A kinetic ballooning mode equation retaining full finite ion Larmor radius and ion magnetic drift resonance effects is derived by employing the high n ballooning mode formalism. We find that the critical ..beta.. is smaller than the ideal MHD critical ..beta.., except when eta/sub i/ = 0 (eta/sub i/ identical with dlnT/sub i//dlnN) they are identical. The finite Larmor radius effects reduce the growth rate but do not stabilize the mode. The ion magnetic drift resonance effects are destabilizing.
Date: April 1, 1981
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Theory and numerical simulations on collisionless drift instabilities in three dimensions

Description: Nonlinear behavior of the collisionless drift instabilities and the resultant anomalous plasma diffusion have been studied by means of computer simulations and analytic theory. The simulation model used is a full three dimensional electrostatic model in a cylindrical geometry in an external magnetic field. Full dynamics is employed for the ion motion while the guiding center approximations are used for the motion of electrons which allows us to use rather realistic plasma parameters in the simulations. The results of simulations indicate that a strong turbulence develops through the nonlinear interaction of the drift instabilities which results in the formation of convective cells and anomalous particle diffusion. The broad frequency spectrum resembles to those observed recently in toroidal confinement devices. Analytic theory is developed based on the mode-coupling process to explain the generation of convective cells and strong plasma turbulence along with the estimate of the resultant particle diffusion.
Date: November 1, 1977
Creator: Cheng, C.Z. & Okuda, H.
Partner: UNT Libraries Government Documents Department

Alpha particle destabilization of the toroidicity-induced Alfven eigenmodes

Description: The high frequency, low mode number toroidicity-induced Alfven eigenmodes (TAE) are shown to be driven unstable by the circulating and/or trapped {alpha}-particles through the wave-particle resonances. Satisfying the resonance condition requires that the {alpha}-particle birth speed v{sub {alpha}} {ge} v{sub A}/2{vert bar}m-nq{vert bar}, where v{sub A} is the Alfven speed, m is the poloidal model number, and n is the toroidal mode number. To destabilize the TAE modes, the inverse Landau damping associated with the {alpha}-particle pressure gradient free energy must overcome the velocity space Landau damping due to both the {alpha}-particles and the core electrons and ions. The growth rate was studied analytically with a perturbative formula derived from the quadratic dispersion relation, and numerically with the aid of the NOVA-K code. Stability criteria in terms of the {alpha}-particle beta {beta}{sub {alpha}}, {alpha}-particle pressure gradient parameter ({omega}{sub {asterisk}}/{omega}{sub A}) ({omega}{sub {asterisk}} is the {alpha}-particle diamagnetic drift frequency), and (v{sub {alpha}}/v{sub A}) parameters will be presented for TFTR, CIT, and ITER tokamaks. The volume averaged {alpha}-particle beta threshold for TAE instability also depends sensitively on the core electron and ion temperature. Typically the volume averaged {alpha}-particle beta threshold is in the order of 10{sup {minus}4}. Typical growth rates of the n=1 TAE mode can be in the order of 10{sup {minus}2}{omega}{sub A}, where {omega}{sub A}=v{sub A}/qR. Other types of global Alfven waves are stable in D-T tokamaks due to toroidal coupling effects.
Date: October 1, 1990
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Theory of drift-wave eigenmodes in toroidal plasmas

Description: The eigenmode equation describing ballooning drift waves in toroidal plasmas is analyzed using the WKB method. Two branches of eigenmodes are identified. One is slab-like and the other is a new branch induced by the finite toroidicity. The slab-like eigenmodes correspond to unbounded states and experience finite shear damping. The toroidicity-induced eigenmodes, however, correspond to local quasibounded states with negligible shear damping. Both branches of eigenmodes may exist simultaneously. Corresponding analytical theories are also presented.
Date: July 1, 1979
Creator: Chen, L. & Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Unstable universal drift eigenmodes in toroidal plasmas

Description: The eigenmode equation describing ballooning collisionless drift instabilities is analyzed both analytically and numerically. A new branch of eigenmodes, which corresponds to quasi-bound states due to the finite toroidicity, is shown to be destabilized by electron Landau damping for typical Tokamak parameters. This branch cannot be understood by the strong coupling approximation. However, the slab-like (Pearlstein-Berk type) branch is found to remain stable and experience enhanced shear damping due to finite toroidicity.
Date: August 1, 1979
Creator: Cheng, C.Z. & Chen, L.
Partner: UNT Libraries Government Documents Department

Ballooning-mode theory of trapped-electron instabilities in tokamaks

Description: Employing the ballooning-mode formalism, the two-dimensional eigenmode equation for trapped-electron instabilities in tokamaks is reduced to a one-dimensional integro-differential equation along the magnetic field lines; which is then analyzed both analytically and numerically. Dominant toroidal coupling effects are due to ion magnetic drifts which create quasi-bounded states. The trapped-electron response can be treated as perturbation and is found to destablize the quasi-bounded states.
Date: December 1, 1979
Creator: Cheng, C.Z. & Chen, L.
Partner: UNT Libraries Government Documents Department

Numerical simulation of trapped electron instabilities in toroidal geometry

Description: Dissipative trapped particle instabilities are studied by means of three-dimensional particle simulations in toroidal geometry. In the linear stage growth rate of the unstable modes, radial and poloidal ballooning mode structures observed in the simulation agree reasonably well with linear theory. In the nonlinear stage, the ballooning effect diminishes due to the generation of strong plasma turbulence resulting in the broad frequency spectrum and convective cells consistent with experimental observations.
Date: June 1, 1978
Creator: Cheng, C.Z. & Okuda, H.
Partner: UNT Libraries Government Documents Department

Nonlinear excitation of convective cells by mode coupling of drift waves

Description: It is shown that the nonlinear interaction of drift waves resonantly excites the convective cells at the wavelengths comparable to the ion gyroradius (k/sub perpendicular to rho/sub i/ approximately 1) where the drift waves are highly dispersive and the growth rates are the maximum. The results of numerical simulations are consistent with theoretical predictions. The effect of magnetic shear is also considered.
Date: May 1, 1977
Creator: Okuda, H. & Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Eigenmode analysis of compressional waves in the magnetosphere

Description: A field-aligned eigenode analysis of compressional Alfven instabilities has been performed for a two component anisotropic plasma in a dipole magnetic field. The eigenmode equations are derived from the gyrokinetic equations in the long wavelength (k rho < 1) and low frequency (..omega.. < ..omega../sub b/) limits, where rho is the hot particle gyroradius and ..omega../sub b/ is the hot particle bounce frequency. Two types of compressional instabilities are identified. One is the drift mirror mode which has an odd parity compressional magnetic component with respect to the magnetic equator. The other is the drift compressional mode with an even parity compressional magnetic component. For typical storm time plasma parameters neargeosynchronous orbit, the drift mirror mode is most unstable and the drift compressional mode is stable. The storm time compressional Pc 5 waves, observed by multiple satellites during November 14-15, 1979 (Takahashi et al., 1987), can be explained by the drift mirror instability.
Date: April 1, 1987
Creator: Cheng, C.Z. & Lin, C.S.
Partner: UNT Libraries Government Documents Department

Neoclassical diffusion of heavy impurities in a rotating tokamak plasma

Description: Particle orbits in a rotating tokamak plasma are calculated from the equation of motion in the frame that rotates with the plasma. It is found that heavy particles in a rotating plasma can drift away from magnetic surfaces significantly faster, resulting in a diffusion coefficient much larger than that for a stationary plasma. Particle simulation is carried out and the results offer a qualitative explanation for some experimental data from the Tokamak Test Reactor (TFTR). 13 refs., 2 figs.
Date: August 1, 1987
Creator: Wong, K.L. & Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

NOVA: a nonvariational code for solving MHD stability of axisymmetric toroidal plasmas

Description: A nonvariational approach for determining the ideal MHD stability of axisymmetric toroidal confinement systems is presented. The code (NOVA) employs cubic B-spline finite elements and Fourier expansion in a general flux coordinate (psi, theta, zeta) system. Better accuracy and faster convergence were obtained in comparison with the variational PEST and ERATO codes. The nonvariational approach can be extended to problems having non-Hermitian eigenmode equations where variational energy principles cannot be obtained.
Date: April 1, 1986
Creator: Cheng, C.Z. & Chance, M.S.
Partner: UNT Libraries Government Documents Department

Low-n shear Alfven spectra in axisymmetric toroidal plasmas

Description: In toroidal plasmas, the toroidal magnetic field is nonuniform over a magnetic surface and causes coupling of different poloidal harmonics. It is shown both analytically and numerically that the toroidicity not only breaks up the shear Alfven continuous spectrum, but also creates new, discrete, toroidicity-induced shear Alfven eigenmodes with frequencies inside the continuum gaps. Potential applications of the low-n toroidicity-induced shear Alfven eigenmodes on plasma heating and instabilities are addressed. 17 refs., 4 figs.
Date: November 1, 1985
Creator: Cheng, C.Z. & Chance, M.S.
Partner: UNT Libraries Government Documents Department

Propagations of drift waves in toroidal plasma systems

Description: Drift wave patterns in toroidal plasmas are studied. The dispersion relation was simplified to retain both the shear and the toroidal coupling effects. Since the dispersion relation does not depend on the toroidal angle, {phi}, the dispersion is solved in the two- dimensional space made up with minor radius and poloidal angle. The dispersion relation can be reduced into second-order, partial differential equations of a hyperbolic type. The one-dimensional convective mode analysis, which was originated in the 1960's, was extended into the two-dimensional analysis. Depending on the strength of the magnetic shear, one can obtain either the convective or the localized solutions. The results show that the plasma is expected to be unstable for large azimuthal mode number and that the plasma instability tends to be more stabilized for large mass ions. 8 refs., 3 figs., 1 tab.
Date: May 1, 1990
Creator: Yoshikawa, S. & Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Electrostatic drift wave eigenmodes in tokamaks

Description: The eigenmode equation governing both the toroidicity-induced and the ion temperature gradient driven modes of arbitrary wavelengths in tokamaks is derived by employing the ballooning mode formalism. For the toroidicity-induced mode (electron driftwave), the ion magnetic drift resonances are destabilizing and can have a significant effect when the temperature ratio, T/sub e//T/sub i/, is of order unity. The ion temperature gradient driven mode (ion drift wave), which is unstable in a shear slab geometry, is found to be further destabilized by the ion magnetic drifts, resulting in a lower critical ion temperature gradient (eta/sub i/).
Date: March 1, 1981
Creator: Cheng, C.Z. & Tsang, K.T.
Partner: UNT Libraries Government Documents Department

Analytical theory of interchange and compressional Alfven instabilities in EBT

Description: The local stability of the EBT plasma is analyzed for the long wavelength perturbations in the frequency regime, ..omega.. approx. less than or equal to ..cap omega../sub i/(..cap omega../sub i/ is ion cyclotron frequency). In addition to the low frequency interchange instability, the plasma can be unstable to the compressional Alfven wave. Contrary to the previously obtained quadratic dispersion relation in ..omega.. for the interchange mode, our dispersion relations for both types of instabilities are cubic in ..omega... New stability boundaries are found, for the hot electron interchange mode, to relate to the enhanced compressibility of the core plasma in the presence of hot electrons. The compressional Alfven instability is driven due to the coupling of hot electron magnetic drifts and diamagnetic drift with the compressional Alfven wave. The stability conditions of these two types of instabilities are opposite to each other.
Date: July 1, 1981
Creator: Cheng, C.Z. & Tsang, K.T.
Partner: UNT Libraries Government Documents Department

Physics of Substorm Growth Phase, Onset, and Dipolarization

Description: A new scenario of substorm growth phase, onset, and depolarization during expansion phase and the corresponding physical processes are presented. During the growth phase, as a result of enhanced plasma convection, the plasma pressure and its gradient are continued to be enhanced over the quiet-time values in the plasma sheet. Toward the late growth phase, a strong cross-tail current sheet is formed in the near-Earth plasma sheet region, where a local magnetic well is formed, the plasma beta can reach a local maximum with value larger than 50 and the cross-tail current density can be enhanced to over 10nA/m{sup 2} as obtained from 3D quasi-static magnetospheric equilibrium solutions for the growth phase. The most unstable kinetic ballooning instabilities (KBI) are expected to be located in the tailward side of the strong cross-tail current sheet region. The field lines in the most unstable KBI region map to the transition region between the region-1 and region-2 currents in the ionosphere, which is consistent with the observed initial brightening location of the breakup arc in the intense proton precipitation region. The KBI explains the AMPTE/CCE observations that a low-frequency instability with a wave period of 50-75 seconds is excited about 2-3 minutes prior to substorm onset and grows exponentially to a large amplitude at the onset of current disruption (or current reduction). At the current disruption onset higher frequency instabilities are excited so that the plasma and electromagnetic field fluctuations form a strong turbulent state. Plasma transport takes place due to the strong turbulence to relax the ambient plasma pressure profile so that the plasma pressure and current density are reduced and the ambient magnetic field intensity increases by more than a factor of 2 in the high-beta(sub)eq region and the field line geometry recovers from tail-like to dipole-like dipolarization.
Date: October 22, 2003
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Global mirror modes in the magnetosheath

Description: A global stability analysis of mirror modes in the magnetosheath is presented. The analysis is based upon the kinetic-MHD formulation which includes relevant kinetic effects such as Landau resonance and gradient drift effects related to inhomogeneities in the background density, temperature, pressure and its anisotropy, magnetic field, and plasma flow velocity. Pressure anisotropy provides the free energy for the global mirror mode. The local theory of mirror modes predicts purely growing modes confined in the unstable magnetosheath region; however, the nonlocal theory that includes the effects of gradients and plasma flow predicts modes with real frequencies which propagate with the flow from the magnetosheath toward the magnetopause boundary. The real frequency is on the order of a combination of the diamagnetic drift frequency and the Doppler shift frequency associated with the plasma flow. The diamagnetic drift frequency provides a wave phase velocity in the direction of the magnetopause so that wave energy accumulates against the magnetopause boundary, and the amplitude is skewed in that direction. On the other hand, plasma flow also gives rise to a real phase velocity, but the phase velocity is smaller than the flow velocity. As a result, the wave amplitude is increased in the wake of the plasma flow and piles up against the bow shock boundary.
Date: May 1, 1996
Creator: Johnson, J.R. & Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Alfven cyclotron instability and ion cyclotron emission

Description: Two-dimensional solutions of compressional Alfven eigenmodes (CAE) are studied in the cold plasma approximation. For finite inverse aspect ratio tokamak plasmas the two-dimensional eigenmode envelope is localized at the low magnetic field side with the radial and poloidal localization on the order of a/{radical}m and a/(fourth root of m), respectively, where m is the dominant poloidal mode number. Charged fusion product driven Alfven Cyclotron Instability (ACI) of the compressional Alfven eigenmodes provides the explanation for the ion cyclotron emission (ICE) spectrum observed in tokamak experiments. The ACI is excited by fast charged fusion products via Doppler shifted cyclotron wave-particle resonances. The ion cyclotron and electron Landau dampings and fast particle instability drive are calculated perturbatively for deuterium-deuterium (DD) and deuterium-tritium (DT) plasmas. Near the plasma edge at the low field side the velocity distribution function of charged fusion products is localized in both pitch angle and velocity. The poloidal localization of the eigenmode enhances the ACI growth rates by a factor of {radical}m in comparison with the previous results without poloidal envelope. The thermal ion cyclotron damping determines that only modes with eigenfrequencies at multiples of the edge cyclotron frequency of the background ions can be easily excited and form an ICE spectrum similar to the experimental observations. Theoretical understanding is given for the results of TFTR DD and DT experiments with {upsilon}{sub {alpha}0}/{upsilon}{sub A} < 1 and JET experiments with {upsilon}{sub {alpha}0}/{upsilon}{sub A} > 1.
Date: July 1995
Creator: Gorelenkov, N. N. & Cheng, C. Z.
Partner: UNT Libraries Government Documents Department

MHD Field Line Resonances and Global Modes in Three-Dimensional Magnetic Fields

Description: By assuming a general isotropic pressure distribution P = P (y,a), where y and a are three-dimensional scalar functions labeling the field lines with B = -y x -a, we have derived a set of MHD eigenmode equations for both global MHD modes and field line resonances (FLR). Past MHD theories are restricted to isotropic pressures with P = P (y only). The present formulation also allows the plasma mass density to vary along the field line. The linearized ideal-MHD equations are cast into a set of global differential equations from which the field line resonance equations of the shear Alfvin waves and slow magnetosonic modes are naturally obtained for general three-dimensional magnetic field geometries with flux surfaces. Several new terms associated with the partial derivative of P with respect to alpha are obtained. In the FLR equations, a new term is found in the shear Alfvin FLR equation due to the geodesic curvature and the pressure gradient in the poloidal flux surface. The coupling between the shear Alfvin waves and the magnetosonic waves is through the combined effects of geodesic magnetic field curvature and plasma pressure as previously derived. The properties of the FLR eigenfunctions at the resonance field lines are investigated, and the behavior of the FLR wave solutions near the FLR surface are derived. Numerical solutions of the FLR equations for three-dimensional magnetospheric fields in equilibrium with high plasma pressure will be presented in a future publication.
Date: May 30, 2002
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department

Energetic particle effects on global MHD modes

Description: The effects of energetic particles on MHD type modes are studied by analytical theories and the nonvariational kinetic-MHD stability code (NOVA-K). In particular we address the problems of (1) the stabilization of ideal MHD internal kink modes and the excitation of resonant fishbone'' internal modes and (2) the alpha particle destabilization of toroidicity-induced Alfven eigenmodes (TAE) via transit resonances. Analytical theories are presented to help explain the NOVA-K results. For energetic trapped particles generated by neutral-beam injection (NBI) or ion cyclotron resonant heating (ICRH), a stability window for the n=1 internal kink mode in the hot particle beat space exists even in the absence of core ion finite Larmor radius effect (finite {omega}{sub *i}). On the other hand, the trapped alpha particles are found to resonantly excite instability of the n=1 internal mode and can lower the critical beta threshold. The circulating alpha particles can strongly destabilize TAE modes via inverse Landau damping associated with the spatial gradient of the alpha particle pressure. 23 refs., 5 figs.
Date: January 1, 1990
Creator: Cheng, C.Z.
Partner: UNT Libraries Government Documents Department