Decompressive (cooling rarefaction) shock in optically thin radiative plasma

D. Kh. Morozov and M. Pekker


It is shown that the decompressive shock,i.e. a sho k where the pressure behind the front is smaller than the pressure ahead of it,is possible in a radiative plasma. This is in contrast to the situation in classic gas dynamics. An example of a steady-state decompressive shock wave for a simple, but realistic model for radiative losses, is presented. It is shown that it satisfies the Landau stability criteria.


Effects of Liquid Metal Walls on Equilibrium and Stability in Tokamaks

A. Y. Aydemir


Liquid metal wall concept has received a great deal of attention recently because of its perceived advantages in addressing high heat flux, magnetohydrodynamic (MHD) stability, and other related issues in advanced confinement schemes. Despite its inherent and clear benefits, this concept also poses potentially serious problems from the MHD equilibrium and stability point of view. In particular, a liquid metal flow strong enough to stabilize resistive wall modes can also have adverse affects on the MHD equilibrium through its interaction with the background poloidal and toroidal fields. These deleterious effects can be large enough to mitigate any expected stabilization of the unstable MHD modes.


Low Frequency Stability of Geotail Plasma

H. Vernon Wong, W. Horton, J.W. Van Dam, and C. Crabtree


The release of stored energy in the magnetosphere during magnetic storms may be triggered by plasma instabilities. We investigate the local stability of a simple but representative model of the flux surfaces of the Earth's magnetosphere in the MHD and drift frequency regimes. Magnetospheric flux surfaces at 6-10 Earth radii, with plasma beta ~ 5, are stable to MHD ballooning modes unless kv xp < 2/5 where xp is the plasma gradient scale length and kv the vacuum field line curvature at the equatorial plane. Drift modes may also be unstable unless h ~ 2/3, where h is the density gradient scale length divided by the temperature gradient scale length.


Comment on "Improved boundary layer analysis of forced magnetic reconnection due to a boundary perturbation [Phys. Plasmas 7, 875 (2000)]"

A. Bhattacharjee, R. Fitzpatrick, and Xiaogang Wang


The problem of forced magnetic reconnection in a resistive plasma is of fundamental interest for numerous laboratory and space plasma applications. Asymptotic solutions of a model of forced reconnection, originally proposed by J.B. Taylor, have been discussed in the literature. In a recent paper, Ishizawa and Tokuda (hereafter, IT) claim that previous analyses of the Taylor problem are wrong and provide an "improved boundary layer analysis" that demonstrates a new reconnection process. Unfortunately, as we show below, IT's analysis is not an improvement on previous analyses. Instead, it introduces new errors in an old problem that has been solved by boundary-layer analyses and tested by numerical simulations.


Relativistic Magnetohydrodynamicsď

R. D. Hazeltine and S. M. Mahajan


The lowest-order description of a magnetized plasma is given by magnetohydrodynamics (MHD). However, many astrophysical plasmas (as well as some laboratory plasmas) are relativistic, in that either the thermal speed or the fluid flow speed approaches the speed of light, and conventional MHD is not consistent with special relativity. Beginning with exact laws of motion, this work derives a generalization of MHD that is Lorentz covariant and therefore applicable to relativistic plasma. The resulting closed set of fluid equations is then seen to reduce to conventional MHD in the nonrelativistic limit.


Collisionless kinetic-fluid closure and its application to the three-mode ion temperature gradient driven system

H. Sugama, T.-H. Watanabe, W. Horton


A novel closure model is presented to give a set of fluid equations which describe a collisionless kinetic system. In order to take account of the time reversal symmetry of the collisionless kinetic equation, the new closure model relates the parallel heat flux to the temperature and the parallel flow in terms of the real-valued coefficients in the unstable wavenumber space. Effects of the closure model on turbulence saturation and anomalous transport are investigated based on kinetic and fluid entropy balances. When the new closure model, called NCM (for nondissipative closure model), is applied to the three-mode ion temperature gradient (ITG) driven system, the fluid system of equations reproduces the exact nonlinear kinetic solution found by Watanabe, Sugama, and Sato [Phys. Plasmas 7, 984 (2000)]. Oscillatory behaviors and initial amplitude dependence of other numerical kinetic solutions of the three-mode ITG problem can also be accurately described by the fluid system.


An Alternative Derivation of Particle Distribution with Effects of Orbit Squeezing

K.C. Shaing and R.D. Hazeltine


An alternative derivation of the particle distribution function with the effects of orbit squeezing is presented. It is based on the utilization of an approximated constant of motion. The result is identical to the one derived previously by solving the linear drift kinetic equation [Phys. Fluids B 4, 2547 (1992)].


Stability Properties of High-Beta Geotail Flux Tubes

W. Horton, H.V. Wong, J.W. Van Dam, and C. Crabtree


Kinetic theory is used to investigate the stability of ballooning-interchange modes in the high pressure geotail plasma. A variational form of the stability problem is used to compare new kinetic stability results with MHD, Fast-MHD, and Kruskal-Oberman stability results. Two types of drift modes are analyzed. A kinetic ion pressure gradient drift wave with a frequency given by the ion diamagnetic drift frequency w*pi, and a very low-frequency mode |w| << w*pi, wDi that is often called a convective cell or the trapped particle mode. In the high-pressure geotail plasma a general procedure for solving the stability problem in a 1/b expansion for the minimizing dB|| is carried out to derive an integral-differential equation for the kinetically valid displacement field xy for a flux tube. The plasma energy released by these modes is estimated in the nonlinear state. The role of these instabilities in substorm dynamics is assessed in the substorm scenarios described in Maynard etal (1996).


Control of tearing modes in toroidal fusion experiments using "designer" error-fields

Richard Fitzpatrick and Enrico Rossi


It is demonstrated that, under certain circumstances, a magnetic island chain due to a saturated tearing instability in a toroidal magnetic fusion device, can lock to a special class of externally generated magnetic perturbation in a stabilizing phase. The theoretical apparatus needed to design such perturbations is outlined. These special perturbations« --- which are termed "designer" error-fields --- could be used to control the amplitudes of tearing modes in toroidal magnetic fusion experiments without the need for fast phase modulation. This type of control would be far more feasible in a reactor environment than conventional active feedback control via external magnetic perturbations.


Conceptual design of an active feedback system for the control of the resistive shell mode in tokamaks

Richard Fitzpatrick


A quadratic dispersion relation is derived which governs the feedback-modified stability of the resistive shell mode in a large-aspect ratio, low-beta tokamak plasma. The effectiveness of a given feedback scheme is determined by a single parameter, alpha_0, which measures the coupling of different poloidal harmonics due to the non-sinusoidal nature of the feedback currents. Feedback fails when this parameter becomes either too positive or too negative. Feedback schemes can be classified into three groups, depending on the relative values of the poloidal mode number, m_0, of the intrinsically unstable resistive shell mode, and the number, M, of feedback coils in the poloidal direction. Group I corresponds to M ú 2 m0 but different from m0; Group II corresponds to M=m0; finally, Group III corresponds to M>2 m0. The optimal Group I feedback scheme is characterized by extremely narrow detector loops placed as close to the plasma - i.e., well inside the resistive shell. Of course, such a scheme would be somewhat impractical. The optimal Group II feedback scheme is characterized by large, non-overlapping detector loops, and moderately large, non-overlapping feedback coils. Such a scheme is 100% effective (i.e., it makes the resistive shell appear super-conducting) when the detector loops are located just outside the shell. Unfortunately, the scheme only works efficiently for resistive shell modes possessing one particular poloidal mode number. The optimal Group III feedback scheme is characterized by slightly overlapping detector loops, and strongly overlapping feedback coils. Such a scheme is 100% effective when the detector loops are located just outside the shell. In addition, the scheme is 100% effective when the detector loops are located just outside the shell. In addition, the scheme works efficiently for resistive shell modes with a range of different poloidal mode numbers.


The Proper Homogeneous Lorentz Transformation Operator e L =exp[-w . S - x . K]: Where is It Going, What's the Twist

H.L. Berk, K. Chaicherdsakul and T. Udagawa


A discussion of the formal Lorentz transformation eL = exp[- w . S - x . K] is given where eL transforms coordinates of an observer O to those of an observer O'.  Two methods of evaluation are presented. The first is based on a dynamical analog. It is shown that the transformation can be evaluated from the set of equations that are identical to the set of equations that determine the 4-velocity of a charged particle in response to a combined spatially uniform and temporally constant electric field E and magnetic field B, where E is parallel to x and B is anti-parallel to w, and E/B = x/w. The principal difference in the two problems is that in the dynamics problem, the initial conditions for the 4-velocity u must satisfy the constraint, u . u = 1, whereas the inner product of the coordinates acted on by eL can have any real value. In order to evaluate eL, one can then apply the simplifying techniques of transforming to the frame where E is parallel or anti-parallel to B, whereupon the transformation eL in this special frame is trivially evaluated. Then we transform back to the original frame. We determine the b and the rotation W that results from a successive boost and rotation that the operator eL produces. A second method is based on a direct summation of the power series of the matrix elements of eL. The summation is facilitated by observing that the operators, J+= K + i S and J-= K - i S commute with each other, and can be represented in terms of the Pauli spin matrices. Indeed, we can reduce the Lorentz transformation to the product of spinor operators to give a compact way to compute the elements of the Lorentz operator eL.


Potato, Banana, Local and Non-Local Transport

K. C. Shaing


The relationship between potato and banana transport theories is addressed using the solution of the drift kinetic equation. It is shown that they are two limits of a complete theory. The potato theory is the psi -> 0 limit and the banana theory is the psi -> oo limit. Here, psi is the poloidal flux function. These local transport theories are valid even in the steep gradient situations because the real orbit width is usually smaller than the gradient scale length when the appropriate orbit squeezing effects are taken into account.« The characteristic feature of a non-local theory is also discussed. It is shown that the equilibrium distribution in such a theory must be non-Maxwellian and non-expandable.


A Theory for Toroidal Momentum Pinch and Flow Reversal in Tokamaks

K. C. Shaing


It is demonstrated that besides the well-known toroidal momentum diffusion flux there is also a pinch-like flux in the fluctuation-induced toroidal viscosity. A toroidal flow profile is determined up to a constant, e.g.,the value of the flow at the magnetic axis, by balancing these two fluxes. The remaining residual toroidal viscosity determines the value of the flow at the axis. It is illustrated that the direction of the flow at the axis can change after plasma confinement is improved. The theory is applied to explain the toroidal flow reversal in tokamak experiments.


Influential Tokamak Datasets from L-mode International Database

B. Hu and W. Horton


The empirical tokamak energy confinement time $\tau_E$ is expressed in terms of the tokamak parameters in the form of power law scaling formulas. The international database, collected through the ITER project, yields the standard measure called the ITERL97-P confinement law~[Kaye et al., Nucl. Fusion 37, 1303 (1997).] for baseline tokamak performance. We rederive the L-mode scaling formula from ITERL database and perform? a statistical analysis for the prediction to larger tokamaks. A series of confinement times is generated by repeatedly performing the linear regression fit on the dataset dropping one machine each time. In this way, we can observe the influence of each particular machine on the empirical formula and the extrapolations made for the proposed IGNITOR and ITER-FEAT tokamak machines. The resulting spread in the predicted values of $\tau_E$ are 35% and 40% respectively for the two proposed machines IGNITOR and ITER-FEAT, and the influential machines defined as those which produce the largest changes when dropped from the database are discussed. Future tokamaks for burning plasma experiments are compact and of high-density. In order to better predict their performance, a subset of the ITERL database, defined as the FUSP subdatabase, is used for linear regression and projection of the Lawson product for IGNITOR and ITER-FEAT design parameters.


Hamiltonian Description of Shear Flow

N. J Balmforth and P. J. Morrison


Euler's equation linearized about a shear flow equilibrium is solved by means of a novel invertible integral transform that is a generalization of the Hilbert transform. The integral transform provides a means for describing the dynamics of the continuous spectrum that is well-known to occur in this system. The results are interpreted in the context of Hamiltonian systems theory, where it is shown that the integral transform defines a canonical transformation to action-angle variables. A means for attaching Krein signature to a continuum eigenmode is given.


Formation and Primary Heating of the Solar Corona - Theory and Simulation

S.M. Mahajan, R. Miklaszewski K.I. Nikol'skaya, and N.L. Shatashvili


An integrated magneto-fluid model, that accords full treatment to the velocity? fields associated with the directed plasma motion, is developed to investigate the dynamics of coronal structures. It is suggested that the interaction of the fluid and the magnetic aspects of plasma may be a crucial element in creating so much diversity in the solar atmosphere. It is shown that the structures which comprise the solar corona can be created by particle (plasma) flows observed near the Sun's surface --- the primary heating of these structures is caused by the viscous dissipation of the flow kinetic energy.


Chaos and the Limits of Predictability for the Solar-Wind Driven Magnetosphere-Ionosphere System

Wendell Horton, R. S. Weigel and J. Clint Sprott


The solar wind driven magnetosphere-ionosphere exhibits a variety of dynamical states including low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionosphere known broadly as substorms, and states of continuous strong unloading identified as magnetic storms [J.P. Smith, J.-L. Thiffeault, W. Horton, J. Geophys. Res. 105, 12983 (2000)]. The WINDMI model is a six-dimensional substorm model that captures through a set of ordinary differential equations the energy flow through the solar wind-magnetosphere-ionosphere system. There are six major energy components of the system with conservation of energy and charge described by the coupling coefficients. We show when the d=6 model can be reduced to the minimal d=3 model for deterministic chaos. The reduced model is of the class of chaotic equations given in Sprott [J.C. Sprott, Am. J. Phys. 68, 758 (2000)]. The bifurcation diagram remains similar and the limited prediction time is preserved (~3hr). Determining all three Lyapunov exponents for the d = 3 model also allows us to determine the dimension of the chaotic attractor for the system.


Single-Pass Ion Cyclotron Resonance Absorption

Boris N. Breizman and Alexey V. Arefiev


The ion response to the rf-field during single-pass ion-cyclotron resonance heating (ICRH) can be essentially nonlinear. This paper presents a self-consistent theory of the rf-wave propagation and ion motion through the resonance. An important ingredient of the problem is the ion flow along the magnetic field. The flow velocity limits the time the ions spend at the resonance, which in turn limits the ion energy gain. A feature that makes the problem nonlinear is that the flow accelerates under the effect of the grad B force and rf-pressure. This acceleration can produce a steep decrease in the plasma density at the resonance, resulting in partial reflection of the incident wave.


WINDMI Optimization and Performance Validation

R.S. Weigel, W. Horton, and I. Doxas


n optimization study of the prediction performance for the substorm model WINDMI is presented.? The model is based on the Earth's magnetospheric dynamics and provides a low-order description of the nightside energy loading and unloading that takes place during the substorm process.? Previous studies of this model on isolated substorms have indicated that it can be a good predictor of solar wind driven substorm activity as measured by fluctuations in the AL index for selected substorms.? Because the model is based on a set of VB_s driven nonlinear ordinary differential equations which can exhibit bifurcation and catastrophe-like behavior, an optimization of the model using conventional minimization techniques over a large data set does not work well.? For such systems the genetic algorithm method of optimization is more efficient at exploring the parameter space.? We present the results of a genetic algorithm optimization of WINDMI using the Blanchard--McPherron and the Bargatze data set and test statistically alternative forms of the model which include the effects of ionospheric conductivity enhancements and region 2 coupling.? A key result from the large scale computations used to search for a uniform convergence of the prediction over the 117 substorm database, is the finding that there are three distinct types of VB_s-AL wave forms characterizing the substorms in the Blanchard--McPherron database.? Two types are given by the internally triggered WINDMI model and the third type requires an external trigger such as the northward turning of the IMF model of Lyons (1995).


Vortex Solitons - Mass, Energy, and Angular Momentum Bunching in Relativistic Electron-Positron Plasmas

T. Tatsuno, V. Berezhiani, and S.M. Mahajan


It is shown that the interaction of large amplitude electromagnetic waves with a hot electron-positron (e-p) plasma (a principal constituent of the universe in the MeV epoch) leads to a bunching of mass, energy, and angular momentum in stable, long-lived structures. ?Electromagnetism in the MeV epoch, then, could provide a possible route for seeding the observed large-scale structure of the universe.


The role of polarization current in magnetic island evolution

J. W. Connor, F. L. Waelbroeck and H. R. Wilson


The polarization current plays an important role in the evolution of magnetic islands with a width comparable to the characteristic ion orbit width. Understanding the evolution of such small magnetic islands is important for two reasons: (1) to investigate the threshold mechanisms for growth of large-scale islands (e.g. neoclassical tearing modes), and (2) to describe the drive mechanisms for small scale magnetic turbulence and consequent transport. This paper presents a two-fluid, cold ion, collisional analysis of the role of the polarization current in magnetic island evolution in slab geometry. It focuses on the role played by the conjunction of parallel electron dynamics and perpendicular transport (particle diffusion and viscosity) in determining the island rotation frequency and the distribution of the polarization current within the island.


Nonlinear dynamics of feedback modulated magnetic islands in toroidal plasmas

Richard Fitzpatrick and Francois L. Waelbroeck


We present a comprehensive analysis of the dynamics of a helical magnetic island chain, embedded in a toroidal plasma, in the presence of an externally imposed, rotating, magnetic perturbation of the same helicity. Our calculations are carried out in the large aspect-ratio, zero-beta, resistive magnetohydrodynamical (MHD) limit, and incorporate a realistic treatment of plasma viscosity. We find three regimes of operation, depending on the modulation frequency (i.e., the difference in rotation frequency between the island chain and the external perturbation). For slowly modulated islands, the perturbed velocity profile extends across the whole plasma. For strongly modulated islands, the perturbed velocity profile is localized around the island chain, but remains much wider than the chain. Finally, for very strongly modulated islands, the perturbed velocity profile collapses to a boundary layer on the island separatrix, plus a residual profile which extends a few island widths beyond the separatrix. We obtain analytic expressions for the perturbed velocity profile, the island equation of motion, and the island width evolution equation in each of these three regimes. We find that the ion polarization correction to the island width evolution equation, which has previously been reported to be stabilizing, is, in fact, destabilizing in all three regimes.


Nonlinear dynamics of dynamo modes in reversed field pinches

Richard Fitzpatrick and Edmund P. Yu


The nonlinear dynamics of a dynamo mode in a reversed field pinch under the action of the braking torque due to eddy currents excited in a resistive vacuum vessel and the locking torque due to an external error-field is investigated. A simple set of phase evolution equations for the mode is derived. These equations represent an important extension of the well-known equations of Zohm, et al. [H. Zohm, A. Kallenbach, H. Bruhns, G. Fussmann, and O. Kluber, Europhys. Lett. 11,, 745 (1990).], so as to include a self-consistent calculation of the radial extent of the region of the plasma which co-rotates with the mode; the width of this region being determined by plasma viscosity. A comprehensive theory of the influence of a resistive vacuum vessel on error-field locking and unlocking thresholds is developed. Under certain circumstances, a resistive vacuum vessel is found to strongly catalyze locked mode formation.


Effect of a resistive vacuum vessel on dynamo mode rotation in reversed field pinches

R. Fitzpatrick, S.C. Guo, D.J. Den Hartog, and C.C. Hegna


Abstract Locked ( i.e., non-rotating) dynamo modes give rise to a serious edge loading problem during the operation of high current reversed field pinches. Rotating dynamo modes generally have a far more benign effect. A simple analytic model is developed in order to investigate the slowing down effect of electromagnetic torques due to eddy currents excited in the vacuum vessel on the rotation of dynamo modes in both the Madison Symmetric Torus (MST) [Fusion Technology 19, 131 (1991)] and the Reversed Field Experiment (RFX) [Fusion Engineering and Design 25, 335 (1995)]. This model strongly suggests that vacuum vessel eddy currents are the primary cause of the observed lack of mode rotation in RFX. The eddy currents in MST are found to be too weak to cause a similar problem. The crucial difference between RFX and MST is the presence of a thin, highly resistive vacuum vessel in the former device. The MST vacuum vessel is thick and highly conducting. Various locked mode alleviation methods are discussed.


Feedback stabilization of resistive shell modes in a reversed field pinch

Richard Fitzpatrick and Edmund P. Yu


A reactor relevant reversed field pinch (RFP) must be capable of operating successfully when surrounded by a close-fitting resistive shell whose L/R time is much shorter than the pulse length. Resonant modes are largely unaffected by the resistivity of the shell, provided that the plasma rotation is maintained against the breaking effect of non-axisymmetric eddy currents induced in the shell. This may require an auxiliary momentum source, such as a neutral beam injector. Non-resonant modes are largely unaffected by plasma rotation, and are expected to manifest themselves as non-rotating resistive shell modes growing on the L/R time of the shell. A general RFP equilibrium is subject to many simultaneously unstable resistive shell modes. The only viable control mechanism for resistive shell modes in an RFP reactor is active feedback. It is demonstrated that an N-fold toroidally symmetric arrangement of feedback coils, combined with a strictly linear feedback algorithm, is capable of simultaneously stabilizing all intrinsically unstable resistive shell modes over a wide range of different RFP equilibria. The number of coils in the toroidal direction N must be greater than, or equal to, the range of toroidal mode numbers of the unstable resistive shell modes. However, this range is largely determined by the aspect-ratio of the device. The optimum coil configuration corresponds to one in which each feedback coil overlaps its immediate neighbours in the toroidal direction. The critical current which must be driven around each feedback coils is, at most, a few percent of the equilibrium toroidal plasma current. The feedback scheme is robust to small deviations from pure N-fold toroidal symmetry or a pure linear response of the feedback circuits.


Formation and locking of the ``slinky mode'' in reversed field pinches

Richard Fitzpatrick


The formation and breakup of the ``slinky mode'' in an RFP is investigated using analytic techniques previously employed to examine mode locking phenomena in tokamaks. The slinky mode is a toroidally localized, coherent interference pattern in the magnetic field which co-rotates with the plasma at the reversal surface. This mode forms, as a result of the nonlinear coupling of multiple m=1 core tearing modes, via a bifurcation which is similar to that by which toroidally coupled tearing modes lock together in a tokamak. The slinky mode breaks up via a second bifurcation which is similar to that by which toroidally coupled tearing modes in a tokamak unlock. However, the typical m=1 mode amplitude below which slinky breakup is triggered is much smaller than that above which slinky formation occurs. Analytic expressions for the slinky formation and breakup thresholds are obtained in all regimes of physical interest. The locking of the slinky mode to a static error-field is also investigated analytically. Either the error-field arrests the rotation of the plasma at the reversal surface before the formation of the slinky mode, so that the mode subsequently forms as a non-rotating mode, or the slinky mode forms as a rotating mode and subsequently locks to the error-field. Analytic expressions for the locking and unlocking thresholds are obtained in all regimes of physical interest. The problems associated with a locked slinky mode can be alleviated by canceling out the accidentally produced error-field responsible for locking the slinky mode, using a deliberately created ``control'' error-field. Alternatively, the locking angle of the slinky mode can be swept toroidally by rotating the control field.


Bifurcated states of a rotating tokamak plasma in the presence of a static error-field

Richard Fitzpatrick


The bifurcated states of a rotating tokamak plasma in the presence of a static, resonant, error-field are found to be strongly analogous to the bifurcated states of a conventional induction motor. The two plasma states are the ``unreconnected'' state, in which the plasma rotates and error-field driven magnetic reconnection is suppressed, and the ``fully reconnected'' state, in which the plasma rotation at the rational surface is arrested and driven magnetic reconnection proceeds without hindrance. The response regime of a rotating tokamak plasma in the vicinity of the rational surface to a static, resonant, error-field is determined by three parameters: the normalized plasma viscosity, P, the normalized plasma rotation, Q_0, and the normalized plasma resistivity, R. There are eleven distinguishable response regimes. The extents of these regimes in P-Q_0-R space are calculated. Furthermore, an expression for critical error-field amplitude required to trigger a bifurcation from the ``unreconnected'' to the ``fully reconnected'' state is obtained in each regime. The appropriate response regime for low density, ohmically heated, tokamak plasmas is found to be the nonlinear constant-psi regime for small tokamaks and the linear constant-psi regime for large tokamaks. The critical error-field amplitude required to trigger error-field driven magnetic reconnection in such plasmas is found to be a rapidly decreasing function of machine size, indicating that particular care may be needed to be taken to reduce resonant error-fields in a reactor-sized tokamak.


Optimal design of feedback coils for the control of external modes in tokamaks

R. Fitzpatrick and E.P. Yu


A formalism is developed for optimizing the design of feedback coils placed around a tokamak plasma in order to control the resistive shell mode. It is found that feedback schemes for controlling the resistive shell mode fail whenever the distortion of the mode structure by the currents flowing around the feedback coils becomes too strong, in which case the mode escapes through the gaps between the coils, or through the centres of the coils. The main aim of the optimization process is to reduce this distortion by minimizing the coupling of different Fourier harmonics due to the feedback currents. It is possible to define a quantity alpha_0 which parameterizes the strength of the mode coupling. Feedback fails for alpha_0 > 1. The optimization procedure consists of minimizing alpha_0 subject to practical constraints. If there are very many evenly spaced feedback coils surrounding the plasma in the poloidal direction then the optimization can be performed analytically. Otherwise, the optimization must be performed numerically. The optimal configuration is to have many, large, overlapping coils in the poloidal direction.


The effect of a partial resistive shell on the magnetohydrodynamical stability of tokamak plasmas

Richard Fitzpatrick


A comprehensive theory is developed to determine the effect of a partial resistive shell on the growth-rate of the external kink mode in a low-beta, large aspect-ratio, circular flux-surface tokamak. In most cases, it is possible to replace a partial shell by a complete ``effective shell'' of somewhat larger radius. In fact, the radius of the effective shell can be used to parameterize the ability of a partial shell to moderate the growth of the external kink mode. It is necessary to draw a distinction between ``resonant shells,'' for which the eddy currents excited in the shell are able to flow in unidirectional continuous loops around the plasma, and ``nonresonant shells,'' for which this is not possible. As a general rule, resonant shells perform better than similar nonresonant shells. The theory is used to derive some general rules regarding the design of incomplete passive stabilizing shells. The theory is also employed to determine the effectiveness of two realistic feedback stabilization schemes for the resistive shell mode, both of which only require a relatively small n umber of independent feedback controlled conductors external to the plasma.


Feedback stabilization of the resistive shell mode in a tokamak fusion reactor

Richard Fitzpatrick


Stabilization of the ``resistive shell mode'' is vital to the success of the ``advanced tokamak'' concept. The most promising reactor relevant approach is to apply external feedback using, for instance, the previously proposed ``fake rotating shell'' scheme [R. Fitzpatrick & T.H. Jensen, Phys. Plasmas 3, 2641 (1996)]. This scheme, like other simple feedback schemes, only works if the feedback controlled conductors are located inside the ``critical radius'' at which a perfectly conducting shell is just able to stabilize the ideal external kink mode. In general, this is not possible in a reactor, since engineering constraints demand that any feedback controlled conductors be placed outside the neutron shielding blanket (i.e., relatively far from the edge of the plasma). It is demonstrated that the fake rotating shell feedback scheme can be modified so that it works even when the feedback controlled conductors are located well beyond the critical radius. The gain, bandwidth, current, and total power requirements of such a feedback system for a reactor sized plasm a are estimated to be less than 100, a few Hz, a fews tens of kA, and a few MW, respectively. These requirements could easily be met using existing technology. It is concluded that feedback stabilization of the resistive shell mode is possible in a tokamak fusion reactor.


Gyrokinetic study of the ion temperature gradient instability in the vicinity of fux surfaces with reversed magnetic shear

J. Q. Dong, W. Horton, and Y. Kishimoto


The Ion temperature gradient (ITG) driven instability is investigated in the vicinity of a flux surface where the magnetic shear reverses. The generic properties of the profile of the magnetic shear are taken into account with gyrokinetic stability theory in the local sheared slab geometry integral equation. The stability analysis shows that there are four distinct unstable ITG branches with significantly different eigenvalues and mode structures existing simultaneously in the vicinity of the minimum q layer. The variation of eigenmode structures with magnetic shear is investigated in detail. Mixing length estimation of the induced plasma transport is performed. Detailed numerical results are presented and general correlations with simulations and experiments are noted.


Study of Photonic Crystal Based Waveguide and Channel Drop Filter and Localization of Light in Photonic Crystal

Wei Jiang


The major part of thesis addressed theoretical problems arising in Photonic Crystal based novel devices, like waveguides and channel drop filters. A perturbation theory based on band theory is proposed for 2-dimensional (2-D) waveguides and is verified in 1-D waveguides. This theory shows excellent agreement with the exact theory in 1-D case for enough large band gap. This theory also predicts that the propagating constant of the modes may not be zero at the lower edge of the band gap, no matter for 1-D or 2-D case. This is verified by the exact theory in 1-D case. In 2 -D case, where an exact theory is not possible, it is opposed to the numerical results o f another group of researchers but supported by experimental data. We proposed a full group theory approach to channel drop filters. In light of this theory, we show the difficulty lying generate high order flat-top can be easily overcome by enhancing the symmetry of the system, instead of simply increasing the number of cavities. As an example, we calculate the 3-fold symmetry case. We find that the localization of light in Photonic Crystal undergoes a trilogy by simulation. This unexpected phenomenon is analyzed by introducing a novel measure for localization problems: the C number. Finally, we understand the trilogy is due to the competition of two localization mechanisms with different ranges of coherence. We show a localization picture where Anderson model is insufficient to explain. theory based on band theory is proposed for 2-dimensional (2-D) waveguides and is verified in 1-D waveguides. This theory shows excellent agreement with the exact theory in 1-D case for enough large band gap. This theory also predicts that the propagating constant of the modes may not be zero at the lower edge of the band gap, no matter for 1-D or 2-D case. This is verified by the exact theory in 1-D case. In 2 -D case, where an exact theory is not possible, it is opposed to the numerical results o f another group of researchers but supported by experimental data. We proposed a full group theory approach to channel drop filters. In light of this theory, we show the difficulty lying generate high order flat-top can be easily overcome by enhancing the symmetry of the system, instead of simply increasing the number of cavities. As an example, we calculate the 3-fold symmetry case. We find that the localization of light in Photonic Crystal undergoes a trilogy by simulation. This unexpected phenomenon is analyzed by introducing a novel measure for localization problems: the C number. Finally, we understand the trilogy is due to the competition of two localization mechanisms with different ranges of coherence. We show a localization picture where Anderson model is insufficient to explain.


Diffusion in Chaotic Systems (Dissertation)

Ilyia Osipenkov


Diffusion in shear and reverse-shear chaotic systems is analyzed. First, a map model for transition to chaos is created. Diffusion in the Symmetric Cubic Map at various stages of chaos is examined. The number of particles, which have not diffused, (N) is found to be well described by the function N = A/(B-Bcrit)C, where A, C and Bcrit depend on the map parameters and initial conditions. Diffusion in a two-wave system, described by differential equations, was also analyzed. A comparison between diffusion in systems with constant, shear and reverse-shear profiles is made.


Modeling of Plasma Rotation in Radio-Frequency-Wave-Heated Tokamak Plasmas

K. C. Shaing


It is shown by considering the general properties of solution of the kinetic equation that plasma rotation in Radio-Frequency (RF) wave heated tokamak plasmas can be modeled by the RF-modified plasma viscous force. This approach is qualitatively different from a model that uses a radial current to model the RF effects on plasma rotation.


Predictive transport simulations of internal transport barriers using the Multi-Mode Model

P. Zhu, G. Bateman, A. H. Kritz, and W. Horton


The formation of internal transport barriers observed in both JET and DIII-D are reproduced in predictive transport simulations. These simulations are carried out for two JET optimized shear discharges and two DIII-D negative central shear discharges using the Multi-Mode model in the time-dependent 1-1/2-D}BALDUR transport code [C.E.Singer et al., Comput. Phys. Commun. 49, 275 (1988)]. The Weiland model is used for drift modes in the Multi-Mode model in combination with either Hahm-Burrell or Hamaguchi-Horton flow shear stabilization mechanisms, where the radial electric field is inferred from the measured toroidal velocity profile and the poloidal velocity profile computed using neoclassical theory. The transport barriers are apparent in both the ion temperature and thermal diffusivity profiles of the simulations. The timing and location of the internal transport barriers in the simulations and experimental data for the DIII-D cases are in good agreement, though some differences remain for the JET discharges. The formations of internal transport barriers are interpreted as resulting from a combination of E x B flow shear and weak magnetic shear mechanisms.


Vertical Displacement Events in Shaped Tokamaks

A. Y. Aydemir


Computational studies of vertical displacement events (VDE's)in shaped tokamaks are presented. The calculations are performed with our nonlinear, 3D resistive MHD code, CTD, which can efficiently treat resistive walls and free boundary displacements in moderately-shaped geometries. This work has a number of related goals: First,the mechanisms for generation of halo currents, the paths taken by them in the plasma and surrounding conductors, and their relative magnitude, are elucidated. Second, coupling between an n =0 vertical instability and an n =1 external kink mode are examined to offer a possible explanation for the nonuniformities observed in the poloidal halo currents during VDE's,and the forces generated by them on plasma-facing components. Finally, effects of a rotating liquid metal wall on the equilibrium and n =0 stability of elongated plasmas are briefly discussed.


Sheared-Flow Generalization of the Harris Sheet

S. M. Mahajan and R. D. Hazeltine


A novel, exact class of solutions to the Vlasov--Maxwell system, with self--generated magnetic fields and nonuniform plasma flows, are constructed. It is shown that a gyrotropic distribution function (independent of gyrophase) does not allow equilibrium shear flow; introduction of agyrotropy is essential for the maintenance of spatially nonuniform velocity fields. The new self--consistent sheared--flow solutions include the shearless Harris Sheet [E.G. Harris, Nuovo Cimento 23, 117 (1962)] solution as a special case. These equilibria are likely to be relevant to a variety of astrophysical flows (most natural flows are sheared) and to a better understanding of the laboratory phenomena observed, for example, in the device MRX (Magnetic Reconnection Experiment, M. Yamada, H. Ji, S. Hsu, T. Carter, R. Kulsrud, N. Bretz, F. Jodes, Y. Ono, and F. Perkins, Phys. Plasma 4, 1936 (1997)) designed to study magnetic reconnection.


Transport Barrier Dynamics

W. Horton and P. Zhu


The properties of the internal transport barriers are developed using theory and radial transport simulations that evolve local turbulent energy density with the temperature profiles. Standard ITG models for the nonlinear radial fluxes driven by drift wave turbulence and stabilized by flow shear are implemented in a new high resolution multiple space--time transport code. A dimensionless parameterization of the input power is introduced and shown to characterize the bifurcation to an internal transport barrier. Examples of the interaction and feedback loops of the turbulence with the transport profiles are given for transport barriers as in TFTR and JT-60U. For JT-60U the high performance discharge E 27969, which reached an equivalent QDT of unity, is modeled with an appropriate set of turbulent thermal, angular momentum and particle diffusivities. The bifurcation analysis suggests a scaling law for the critical power for the onset of internal transport barriers.


Synergism between Liquid Metal Walls,Tokamak Physics Performance,and Reactor Attractiveness

M. Kotschenreuther


Liquid metal walls appear capable of allowing stable tokamak operation with increased elongation under reactor conditions. Code results from models indicate that the magnitude of improvement can be large with up to a factor of three improvement in stable beta (from 5-7% to 20-22%) at aspect ratio 4 and 3, respectively. This would enable a reduction in the size of a 1~GW advanced tokamak reactor from 5.5 m to 3.15 m. Flowing walls can stabilize resistive wall modes. Stability benefits allow operation in parameters which enable higher confinement with less extrapolation


Analysis and Prediction of a Laser Beam Position Time Series by Neural Networks

Frank Breitling


This thesis describes the time series analysis and prediction of a laser beam position. In addition to the theoretical discussion of the predictability, an actual prediction is realized by the application of artificial neural networks. It is shown that the standard deviation of spatial fluctuations can be reduced (achieved was a reduction of the standard deviation) by 20% beyond that of a feedback controlled system. This is an experimental demonstration of predictability and shows a feasible way of realization by the methods of neural prediction.


Global Drift Wave Map Test Particle Simulations

J.-M. Kwon, W. Horton, P. Zhu, P. J. Morrison, and D.-I. Choi


Global drift wave map equations that allow the integration of particle orbits on long time scales are implemented to describe transport. Ensembles of test particles are tracked to simulate the low-confinement mode/reversed shear/enhanced reversed shear plasmas in the Tokamak Fusion Test Reactor (TFTR) tokamak and the Optimized Shear plasma in the Joint European Torus (JET) tokamak. The simulations incorporate a radial electric field, Er , obtained from a neoclassical calculation [Zhu et al., Phys. Plasmas 6, 2503 (1999)] and a model for drift wave fluctuations that takes into account change in the mode structure due to Er [Taylor et al., Plasma Phys. Controlled Fusion 38, 1999 (1996)]. Steady state particle density profiles along with two different measures of transport, the diffusion coefficient based on a running time average of the particle displacement and that calculated from the mean exit time, are obtained. For either weak or reversed magnetic shear and highly sheared Er, particle transport barriers are observed to be established. In the presence of such a transport barrier, it is shown that there is, in general, a difference between the two measures of transport. The difference is explained by a simple model of the transport barrier.


Parallel Current in Anisotropic High-Beta Extremely Low Aspect-Ratio Tokamak Plasmas

K. C. Shaing


Parallel current in an anisotropic high-beta extremely low aspect ratio tokamak plasma is calculated from the linearized drift kinetic equation. It is found that it depends on the radial gradients of the parallel plasma pressure and the magnetic field strength. It can be expressed as the sum of the bootstrap current and the Pfirsch--Schl\"uter current.


A Collisionless Self-Organizing Model for the H-mode Boundary Layer

S. M. Mahajan and Z. Yoshida


It is shown that in a collisionless two-fluid model, a combination of the Hall term and fluid vorticity can lead to the formation of a self-organized singular layer which displays the essential observational features of the thin shear-layer associated with the H-mode tokamak discharges: the layer width is of the order of a poloidal gyro-radius, the poloidal velocity of the order of poloidal Mach number unity, and an electrostatic potential (yielding a negative electric field) of the order of the edge plasma temperature.


Bifurcation and Relaxation of Radial Electric Field in Enhanced Reversed Shear Tokamak Plasmas

K. C. Shaing, A. Y. Aydemir, R. Hiwatari, W. A. Houlberg, Y. Ogawa, and M. C. Zarnstorff


It is shown that toroidal magnetic field ripple induced thermal and fast ion loss can drive the radial electric field to bifurcate over the local maximum of the parallel (or poloidal) viscosity. The subsequent plasma profile evolution reduces the hot particle density and relaxes the radial electric field. This behavior is consistent with the experimental observations in Enhanced Reversed Shear mode in tokamaks.


Attaining Neoclassical Transport in Ignited Tokamaks

M. Kotschenteuther, W. Dorland, Q. P. Liu, M. C. Zarnstorff, R. L. Miller, and Y. R. Lin-Liu


The requirements to stabilize microinstabilities with velocity shear in tokamaks are examined for both aspect ratio $A=3$ and $A=1.4$. A comprehensive linear gyrokinetic code is used to compute growth rates in realistic numerical equilibria. Growth rates for $A=3$ and $A=1.4$ are generally similar for electron drift modes and ion temperature gradient (ITG) modes. Velocity shear is stronger at low aspect ratio; however, low $A$ profiles have a stronger microtearing mode which is more difficult to shear stabilize. Nonetheless, low aspect ratio devices are predicted to have good confinement and may ignite at very small size. Profiles are presented which may allow high $\beta$, MHD-stable operation at $A=1.4$ with high confinement. The possibility of using a controlled application of non-turbulent loss processes to control profiles to prevent turbulent transport and to maximize $\beta$ is suggested.


Computing Casimir Invariants from Pffafian Systems

Tom Yudichak and Philip J. Morrison


We describe a method for computing Casimir invariants that is applicable to both finite and infinite-dimensional Poisson brackets. We apply the method to various finite and infinite-dimensional examples, including a Poisson bracket embodying both finite and infinite-dimensional structure.


Radially Localized Helicon Modes in Nonuniform Plasma

Boris N. Breizman and Alexey V. Arefiev


Radial density gradient in a cylindrical plasma column forms a potential well for nonaxisymmetric helicon modes (m not equal to 0). This paper presents an analytic description of such modes in the limit of small parallel wavenumbers. The corresponding mode equation indicates the possibility of efficient resonant absorption of rf power in helicon discharges at unusually low frequencies.


Magnetic Field Lines, Hamiltonian Dynamics, and Nontwist Systems

P. J. Morrison


Magnetic field lines typically do not behave as described in the symmetri- cal situations treated in conventional physics textbooks. Instead, they behave in a chaotic manner; in fact, magnetic field lines are trajectories of Hamiltonian systems. Consequently the quest for fusion energy has interwoven, for 50 years, the study of magnetic field configurations and Hamiltonian systems theory. The manner in which invariant tori break-up in symplectic twist maps, maps that embody one and a half degree-of-freedom Hamiltonian systems in general and describe magnetic field lines in tokamaks in particular, will be reviewed, including symmetry methods for finding periodic orbits and Greene ăs residue criterion. In nontwist maps, which describe e.g. reverse shear tokamaks and zonal flows in geophysical fluid dynamics, a new theory is required for describing tori break-up. The new theory is discussed and comments about renormalization are made.