**Superluminous laser pulse in an active medium **

D.L. Fisher, T. Tajima

**Abstract**

Physical conditions are obtained to make the propagation velocity of a laser pulse and thus the phase velocity of the excited wake be at any desired value, including that equal to or greater than the speed of light. The provision of an active-plasma laser medium with an appropriately shaped pulse allows not only replenishment of laser energy loss to the wake field but also acceleration of the group velocity of photons. A stationary solitary solution in the accelerated frame is obtained from our model equations and simulations thereof for the laser, plasma, and atoms.

**Anomalous thermalization of fast ions in magnetized plasma **

K.R. Chen

**Abstract**

A novel anomalous process causing the perpendicular energy of fast ions to be thermalized and lost on average to bulk ion heating, instead of classical slowing down and bulk electron heating, is investigated with particle-in-cell simulations. More than half of the fast ions are slowed down to the thermal ion level, although some are heated to twice their birth energy. The fast ion density perturbation is large. This process is excited by a new two-gyro-stream instability and may continually occur in a burning plasma. The implications for fusion ignition and fast ion confinement are assessed.

**Nonlinear growth of strongly unstable tearing modes **

F.L. Waelbroeck

**Abstract**

Rutherford's theory of the tearing instability is extended to cases where current nonlinearities are important, such as long wavelength modes in current slabs and the m = 1 instability in tokamaks with moderately large aspect ratios. Of particular interest is the possibility that the associated magnetic islands, as a result of secondary instabilities, have a singular response to the ohmic diffusion of the current. A family of islands is used to test this possibility; it is found that the response remains bounded.

**Reduced Braginskii equations **

M. Yagi, W. Horton

**Abstract**

A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that the perpendicular component of Ohm's law be solved to ensure ·j=0 for energy conservation.

**Transport suppression by shear flow generation in multihelicity resistive-g turbulence **

H. Sugama, W. Horton

**Abstract**

Turbulent momentum transport given by the Reynolds stress is considered as a candidate for explaining the production and sustainment of the mean shear flow in the high confinement "(H)" mode. The fluctuation mechanism for the shear flow generation and transport reduction in the three-dimensional (3-D) multihelicity system is given. The profiles of the Reynolds stress, shear flow, and thermal flux in the 3-D case are compared with those in the two-dimensional (2-D) case. The Beklemishev–Horton theory for the anomalous transport which multiplies the 2-D transport by the density of distinct mode rational surfaces is found to overestimate the observed flux due to the disappearance of a subset of modes on certain rational surfaces. The mixing-length theory, in which the anomalous transport is independent of the density of mode rational surfaces, underestimates the thermal flux.

**Canonization and diagonalization of an infinite dimensional noncanonical Hamiltonian system: Linear Vlasov theory **

P.J. Morrison, B.A. Shadwick

**Abstract**

The Vlasov-Poisson equation, which is an infinite dimensional noncanonical Hamiltonian system, is linearized about a stable homogeneous equilibrium. Canonical variables for the resulting linear system are obtained. A coordinate transformation is introduced taht brings the system, which possesses a continuous spectrum, into the action-angle form where the linearized energy is diagonal.

**δ f Algorithm**

R.E. Denton, M. Kotschenreuther

**Abstract**

The δ*f* Algorithm is a low noise particle code algorithm. The perturbation of the distribution function (δ*f*) away from a large equilibrium is evolved rather than the total distribution function. "Particles" in the code are actually Lagrangian markers at which the value of the distribution function is known, The magnitude of the numerical noise is characteristic of the size of the perturbation rather than the equilibrium and scales roughly as the inverse of the number of particles, In this paper, the algorithm is described, and conserved energies are derived for linear and nonlinear sets of equations. Two different forms of the energy principle test separately adequate resolution in time and space and adequacy of the number of simulation particles. A semi-implicit time step method is described which allows violation of the Courant condition. Low noise capabilities of a linear code using the algorithm are demonstrated.

**On the interpretation of diamagnetic loop measurements for a current-carrying plasma column in a conducting chamber **

A.N. Aleynikov, B.N. Breizman, V.S. Cherkassky, B.A. Knyazev

**Abstract**

An expression is derived for the signal of a magnetic loop encircling a plasma column inside a conducting chamber with nonuniform current distribution over the plasma cross section. The plasma with radius much smaller than the length of the column is assumed to be in a quasistationary force-balanced state. The ratio of the paramagnetic component to the diamagnetic component of the signal is shown to be independent of the loop radius. Both components increase as the loop radius decreases from the chamber radius to the plasma radius. From the derived expressions, the paramagnetic component of the signal is calculated numerically for several current distributions including those of interest for the experiments. At a given total current, the paramagnetic component of the signal may vary considerably, which generally has to be taken into account in interpreting experimental data. The results of the calculations are used to process the data obtained in the experiments on the SPIN plasma device [Knyazev et al., "Characteristics of plasma produced by a linear discharge in the metal chamber of the SPIN device," J. Tech. Phys. 63, 49 (1993) (in Russian)].

**Toroidal plasma β-finite larmor radius limit in a toroidally linked mirror system **

H. Vernon Wong, H.L. Berk, V.I. Ilgisonis, V.P. Pastukhov

**Abstract**

The ballooning stability limit in one toroidal section of the recently proposed toroidally linked neutron source is investigated within the framework of the Wentzel–Kramers–Brillouin (WKB) approximation. Rotational effects induced by radial electric fields present in the equilibrium are neglected. The equilibrium pressure and density profiles are taken to be linear in the flux variable. For a reasonable set of design parameters, ideal ballooning instability limits the toroidal plasma β to less than 1%. However, the inclusion of stabilizing kinetic effects due to finite ion Larmor radius approximately doubles the predicted critical beta limit, and makes possible a choice of design parameters compatible with ballooning stability.

**A two-dimensional kinetic model of the scrape-off layer **

P.J. Catto, R.D. Hazeltine

**Abstract**

A two-dimensional (radius and poloidal angle), analytically tractable kinetic model of the ion (or energetic electron) behavior in the scrape-off layer of a limiter or divertor plasma in a tokamak is presented. The model determines the boundary conditions on the core ion density and ion temperature gradients, the power load on the limiter or divertor plates, the energy carried per particle to the walls, and the effective flux limit. The self-consistent electrostatic potential in the quasineutral scrape-off layer is determined by using the ion kinetic model of the layer along with a Maxwell–Boltzmann electron response that occurs because most electrons are reflected by the Debye sheaths (assumed to be infinitely thin) at the limiter or divertor plates.

**Kinetic and fluid modelling of plasmas: Energetic beams and tilt stability in field reversed configurations **

J. Cobb

**Abstract**

Plasma problems include features on disparate scales. One example is the dynamical coupling of the microscopic details of individual particle motion to the macroscopic plasma flow. A small energetic particle component can modify the low frequency fluid response of a plasma. This work extends fluid models to include a minority component of energetic kinetic particles and applies it to the study of the tilt mode in Field Reversed Configurations (FRC`s), an advanced fusion concept with high beta, a natural divertor, and a non-linked-to-coil geometry. A model is presented which treats the majority plasma in an extended magnetohydrodynamic (MHD) approximation (MHD plus viscous, resistive, and Hall effects) while maintaining a kinetic representation of the minority energetic component. The model is applied to simulations of FRC`s. Several different computational techniques for fluids and particles are used. Some are melded together for self-consistent fluid and kinetic simulations. Two separate fluid algorithms, the semi-implicit FLX algorithm and a Lax-Wendroff algorithm, and a kinetic algorithm (Particle-in-Cell) are used. Additional non-self-consistent simulations treat particle effects as given. The algorithms numerically solve initial value problems with starting conditions given by the output of two different equilibrium solvers. One solver uses cubic splines with direct banded matrix inversion. The other is an iterative conjugate gradient solver of a finite difference formulation. Four problems are addressed: realistic capture of injected energetic neutral beams, tilt mode stabilization of an injected neutral beam, tilt mode stabilization by equilibrium profile modification, and spin-up and viscous angular momentum transport in an FRC. Results indicate that energetic particles can change both the equilibrium and stability characteristics of FRC`s.

**Instability due to axial shear and surface impedance **

Yu.A. Tsidulko, H.L. Berk, R.H. Cohen

**Abstract**

The stability of plasma flow in the scrape-off layer of a tokamak, taking into account the surface sheath impedance and the axial shear in the E×B flow is analyzed. An interesting stability problem arises in the limit that end plates are sufficiently far apart, so that stability can be analyzed when the plasma is taken to interact with a single end plate. As parameters are varied, windows of instability are found, and it is shown that growth rates are maximized for an insulating end plate and are also quite sensitive to the ratio of the ion diamagnetic and E×B drift frequencies. Mixing-length estimates of the diffusivity are comparable to experimentally observed values.

**Kinetic resonance damping rate of the toroidal ion temperature **

J.Y. Kim, Y. Kishimoto, W. Horton, T. Tajima

**Abstract**

The linear damping rates of the toroidal ion temperature gradient (η_{i}) mode due to the toroidal resonance are calculated in the local kinetic limit. The well-known Landau contour method is generalized to treat the analytic continuation problem of the guiding center dispersion function in the toroidal resonance system where the resonance occurs from both the magnetic [del]B-curvature drift and the parallel ion transit drift. A detailed numerical analysis is presented for the dependence of the damping rate of the toroidal η_{i} mode on various parameters such as ε_{n}, k_{y}, and the trapped electron fraction. In addition, a consideration is presented on the decay problem of the ballistic response by phase mixing in the toroidal system, which is directly related to the kinetic damping problem of the normal modes by the toroidal resonance.

**Impurity transport studies in text **

W. Horton, W. Rowan

**Abstract**

The results of impurity transport experiments in the Texas Experimental Tokamak (TEXT) [Nucl. Technol./Fusion 1, 479 (1981)] are compared with the predictions of a turbulence-based transport model. In the experiments, Sc was injected into the plasma using laser ablation and the time-resolved profiles of critical Sc ionization stages were measured along with the potential fluctuation profile. The experiment was simulated using a one-dimensional (1-D) radial transport code with the standard transport flux Γ=−D(∂n/∂r)+nV. The diffusion coefficient D and convective velocity V parameters were varied until the time-dependent 1-D simulations reproduce the data. This representation for the empirical impurity transport is compared with the E×B turbulent diffusivities and mobility based on the fluctuation data and the measured radial electric field. The agreement is best with the E×B diffusivity taken in the strong turbulence regime, where D=c_{D}(φ˜/B_{T}), while the comparison with the weak turbulence diffusivity and the collisional (no fluctuations) diffusivity results in qualitative disagreements. The percolation theory diffusivity (φ˜/B_{T})^{7/10}(Δω/k^{2}_{⊥})^{3/10} is also briefly discussed.

**Continuum damping of ideal toroidal Alfven eigenmodes **

X.D. Zhang, Y.Z. Zhang, S.M. Mahajan

**Abstract**

A perturbation theory based on the two-dimensional (2-D) ballooning transform is systematically developed for ideal toroidal Alfvén eigenmodes (TAEs). A formula, similar to the Fermi golden rule for decaying systems in quantum mechanics, is derived for the continuum damping rate of the TAE; the decay (damping) rate is expressed explicitly in terms of the coupling of the TAE to the continuum spectrum. Numerical results are compared with previous calculations. It is found that in some narrow intervals of the parameter mε, the damping rate varies very rapidly. These regions correspond precisely to the root missing intervals of the numerical solution by Rosenbluth et al. [Phys. Fluids B 4, 2189 (1992)].

**Finite larmor radius flute mode theory with end loss **

I.A. Kotelnikov, H.L. Berk

**Abstract**

The theory of flute mode stability is developed for a two-energy- component plasma partially terminated by a conducting limiter. The formalism is developed as a preliminary study of the effect of end-loss in open-ended mirror machines where large Larmor radius effects are important.

**Effect of limiter end loss in finite larmor radius theory **

H.L. Berk, I.A. Kotelnikov

**Abstract**

The effect of incomplete line tying on the magnetohydrodynamic (MHD) flute mode with FLR (finite Larmor radius) effects was examined. It was shown that the combination of line tying and FLR effects can slow down MHD instability, but cannot produce complete stabilization.

**On the stability of shear-Alfven vortices **

D. Jovanovic, W. Horton

**Abstract**

Linear stability of shear-Alfvén vortices is studied analytically, using the Lyapunov method. Vortices belonging to the drift mode, which is a generalization of the standard Hasegawa–Mima vortex to the case of large parallel phase velocities, are proved to be unstable. In the case of the convective-cell mode, short perpendicular-wavelength perturbations are stable for a broad class of vortices. Eventually, instability of convective-cell vortices may occur on the perpendicular scale comparable with the vortex size, but it is followed by a simultaneous excitation of coherent structures with a better localization than the original vortex.

**Shear flow generation from the interaction of neoclassical and drift wave transport processes **

X.N. Su, P.N. Yushmanov, J.Q. Dong, W. Horton

**Abstract**

Self-consistent shear flow generation from the interaction of neoclassical and drift wave turbulence effects is investigated. The neoclassical poloidal flow damping is shown to compete with the plasma flow generation driven through the divergence of the Reynolds stress. When there is no external driving force except for the free energy released from toroidal shear flow, the turbulent fluctuations occur as a transient pulse which takes the system along an equilibrium path to a relaxed state. External torques, such as parallel neutral beam injection, are needed to maintain significant fluctuation levels. For a system driven by a fixed ion temperature gradient, although linearly the poloidal shear flow generated substantially reduces the growth rate, the simulation results show that a sequence of nonlinear pulses occurs that eventually build the fluctuations up to a level that is not significantly affected by the poloidal flow. In this new, highly nonlinear state the transport is intermittent, with high fluxes occurring through a sequence of pulses of duration 100 L_{n}/c_{s} for typical system parameters.

**Finite beta plasma equilibrium in toroidally linked mirrors **

V.I. Ilgisonis, H.L. Berk, V.P. Pastukhov

**Abstract**

The problem of finite pressure plasma equilibrium in a system with closed magnetic field lines consisting of quadrupole mirrors linked by simple toroidal cells with elliptical cross sections is analyzed. An appropriate analytical procedure is developed, that uses conformal mapping techniques, which enables one to obtain the magnetic field structure for the free boundary equilibrium problem. This method has general applicability for finding analytic solutions of the two-dimensional Dirichlet problem outside an arbitrary closed contour. Using this method, the deformations of the plasma equilibrium configuration due to finite plasma pressure in the toroidal cell are calculated analytically to the second order in a λ expansion, where λ~β/εE, β is the ratio of plasma pressure to the magnetic field pressure, ε is the inverse aspect ratio, and E is the ellipticity of the plasma cross section. The outer displacement of the plasma column is shown to depend nonlinearly on the increase of plasma pressure, and does not prevent the achievement of substantial β~10% in the toroidal cells.

**Can computer simulation predict the real behavior of turbulence? **

M.B. Isichenko

**Abstract**

Most of laws of physics have the form of ordinary or partial differential equations, amenable to various numerical approaches. If the number of absolute degrees of freedom, unconstrained by conservation laws, is three or more, the dynamics are generally chaotic. The chaos primarily manifests itself in the sensitive dependence on the initial conditions (the Lyapunov exponentiation of phase-space orbits). As any initial inaccuracy, which may be due to finite space grid, time step, or roundoff error, is exponentially enhanced by the dynamics, there is no chance, whatever the available computer power, to have quantitatively accurate predictions in terms of individual orbits, as far as many Lyapunov time scales are concerned. In terms of average quantities, however, numerical predictions can be satisfactory and correct. This is the case, when the original system and its finite-difference model, both chaotic, are characterized by (almost) the sam statistical consistency can be fulfilled. We arrive at the conclusion that there is a wide class of turbulent systems, for which this question remains completely unanswered, namely those where turbulence systems, for which this question remains completely unanswered, namely those where turbulence in non-Gaussion. Plasma turbulence in tokamaks and weather prediction are practically important examples of such systems. For these kinds of turbulence, the predictions of high-resolution computation may have nothing to do with real dynamics. The reason for this difficulty is that finite-dimensional discretizations, used to cast partial differential equations into a form understood by computer, do not generally converge to what we expect in the high-resolution limit.

**An explanation for experimental observations on harmonic cyclotron emission induced by fast ions **

K.R. Chen, W. Horton, J.W. Van Dam

**Abstract**

An explanation, supported by numerical simulations and analytical theory, is given for the harmonic cyclotron emission induced by fast ions in tokamak plasmas—in particular, for the emission observed at low harmonics in deuterium–deuterium and deuterium–tritium experiments in the Joint European Torus [e.g., Phys. Rev. Lett. 60, 33 (1988)]. It is shown that the first proton harmonic, whose field energy amplitude scales as the 0.84 power of the proton density, is one of the highest spectral peaks, whereas the first alpha harmonic is weak. The relative spectral amplitudes of different harmonics are compared. The results are consistent with the experimental observations. The simulations verify that the instabilities are caused by a weak relativistic mass effect. Simulation also shows that a nonuniform magnetic field leads to no appreciable change in the growth rate and saturation amplitude of the waves.

**Filamentation, current profiles and transport in a tokamak **

J.B. Taylor

**Abstract**

A tokamak with slightly imperfect magnetic surfaces should have a microscopically filamented current structure. If so, its equilibrium has an analog in the dynamics of interacting charged rods. Then there will be a natural current profile, analogous to thermal equilibrium of the rods (and the natural profile can be calculated by conventional statistical mechanics). This would account for the phenomenon of profile consistency or resilience in tokamaks. In addition to the natural profiles, this filamentary model also predicts an anomalous inward flux of both heat and particles in a tokamak, as well as an anomalous diffusion. These "inward-pinch'' components are related to the current gradient.

**Transport in the self-organized relaxed state of ion temperature gradient instability**

T. Tajima, Y. Kishimoto, M.J. Lebrun, M.G. Gray, J.-Y. Kim, W. Horton, V. Wong, M. Kotschenreuther

**Abstract**

We investigate the anomalous heat conduction in a tokamak plasma analytically and computationally. Our toroidal particle simulation shows a new emerging physical picture that the toroidal plasma exhibits marked properties distinct from a cylindrical plasma: (1) the development of radially extended potential streamers localized to the outside of the torus, (2) more robust ion temperature gradient instability, (3) radially constant eigenfrequency, (4) global temperature relaxation, and (5) radially increasing heat conductivity χ_{i}. These results are analyzed by linear and quasilinear kinetic theory. A relaxation theory based on the reductive perturbation theory in the quasilinear equation is developed. The theory constrains the thermal flux so that χ_{i} increases radially. The Bohm-like scaling is found in connection with the radially extended mode structure.