**Continuum damping of low-n toroidicity-induced shear Alfven eigenmodes **

H.L. Berk, J.W. Van Dam, Z. Guo, D.M. Lindberg

**Abstract**

The effect of resonant continuum damping is investigated for the low-mode-number, toroidicity-induced, global shear Alfven eigenmodes, which can be self-excited by energetic circulating alpha particles in an ignited tokamak plasma. Resonant interaction with the shear Alfven continuum is possible for these eigenmodes, especially near the plasma periphery, leading to significant dissipation, which is typically larger than direct bulk plasma dissipation rates. Two perturbation methods are developed for obtaining the Alfven resonance damping rate from the ideal fluid zeroth-order shear Alfven eigenvalue and eigenfunction. In both methods the real part of the frequency is estimated to zeroth order, and the imaginary part, which includes the damping rate, is then obtained by perturbation theory. One method, which is applicable when the eigenfunction is nearly real, can readily be incorporated into general magnetohydrodynamic (MHD) codes. In the second method, the zeroth-order eigenfunctions may be complex; however, the application of this method to general MHD codes needs more detailed development. Also, an analytical estimate is found for the next-order real frequency shift of the fluid global Alfven mode. Analytical and numerical studies of this continuum damping effect indicate that it can substantially reduce the alpha particle-induced growth rate. Thus, either it is possible to prevent instability or, if unstable, to use the Alfven resonance damping to estimate the saturation amplitude level predicted from quasilinear theory.

**Numerical linear and nonlinear investigation of vertical slot convection **

A.H. McAllister

**Abstract**

A stressed hydrodynamical system undergoes a series of changes in its behavior as the control parameter (proportional to the stress) is increased. We take the specific case of a slab of fluid bounded by rigid vertical walls horizontally, but of infinite extent in height and depth, that is subject to a vertical gravitational field and a horizontal temperature gradient imposed by maintaining different uniform temperatures on the side walls. This is called the vertical slot convection (VSC) problem. This system exhibits a one dimensional buoyancy-driven convective flow at any finite temperature difference. This flow, in turn, becomes unstable as the dimensionless control parameter, the Grashof number, is increased. We derive 2-D vorticity/stream function equations in the Boussinesq approximation (eliminating pressure waves, but allowing buoyancy forces). We linearize these and perform a thorough linear stability analysis, obtaining accurate eigenvalues, neutral stability curves, and eigenfunctions for the VSC problem and several related and limiting cases, including the Benard problem. This demonstrates the role of each term in the equations. We constructed a fully nonlinear pseudo-spectral semi-implicit fluid simulation code using a Fourier-Chebychev decomposition and working at a Prandtl number of 7.5 (water), we use the linear eigenfunctions at the linearly critical wave number, α = 1.383, as perturbations to the 1-D basic shear flow. Above the critical Grashof number, G_ c, the system becomes unstable to the onset of secondary vortices. The Landau expansion theory is applied and checked in the near supercritical regime. At higher Grashof numbers a second bifurcation (Hopf) leading to oscillatory behaviors is found. Above G_ h the oscillations become anharmonic and evolve into nonlinear thermal relaxation oscillations which are analyzed in detail. These show similarities to the well-known 'sawtooth' oscillations found in tokamak discharge. After a probable third bifurcation we find an apparently chaotic attractor that resembles the well-known Lorenz attractor. Preliminary results are presented to support this interpretation. Finally we examine extensions to other parameter regimes and consider future extensions and applications to plasma physics in solar and astrophysical problems.

**Nonlinear dynamic models displaying temporal features of sawtooth oscillations **

A. Thyagaraja, F.A. Haas

**Abstract**

There is at present no completely satisfactory theory of sawteeth in tokamaks. Although the simulations of Denton et all (Phys. Rev. Letts. 56, 2477, 1986) and Aydemir et al (Phys. Fluids B1, 744, 1989) capture some of the periodicity properties, they do not show the partial reconnection currently observed in experiments. It would seem useful to construct the simplest possible nonlinear dynamical model to illustrate some of the temporal features of sawteeth. To this end, we introduce the variables *y(t)* and *x(t)*, which represent the internal energy and the magnetic turbulence amplitude respectively…

**Nonlinear dynamics of the relativistic standard map **

Y. Nomura, Y.H. Ichikawa, W. Horton

**Abstract**

The acceleration and heating of charged particles by electromagnetic fields has been extensively investigated by the standard map. The question arises as to how the relativistic effects change the dynamical behavior described with the classical standard map. The relativistic standard map is a two-parameter (K,β=ω/kc) family of dynamical systems reduced to the standard map when β→0. For β≠0 the relativistic mass increase suppresses the onset of stochasticity. It is shown that the speed of light limits the rate of advance of the phase in the relativistic standard map and introduces Kolmogorov-Arnold-Moser surfaces persisting in the high-momentum region. An intricate structure in the higher-order periodic orbits and chaotic orbits is analyzed using the symmetry properties of the relativistic standard map. An interchange of the stability of the periodic orbits is observed and explained by the local linear stability of the periodic orbits.

**Exact solutions for a system of nonlinear plasma fluid equations **

M.G. Prahovic, R.D. Hazeltine, P.J. Morrison

**Abstract**

A method is presented for constructing exact solutions to a system of nonlinear plasma fluid equations that combines the physics of reduced magnetohydrodynamics and the electrostatic drift-wave description of the Charney-Hasegawa-Mima equation. The system has nonlinearities that take the form of Poisson brackets involving the fluid field variables. The method relies on modifying a class of simple equilibrium solutions, but no approximations are made. A distinguishing feature is that the original nonlinear problem is reduced to the solution of two linear partial differential equations, one fourth-order and the other first-order. The first-order equation has Hamiltonian characteristics and is easily integrated, supplying information about the general structure of solutions.

**Transport profiles induced by radially localized modes in a tokamak **

A.D. Beklemishev, W. Horton

**Abstract**

A new approach to the calculation of turbulent transport coefficients for radially localized modes is presented. The theory takes into account the nonuniformity of the distribution of rational (resonant) magnetic surfaces in minor radius. This distribution function is proportional to the density of available states of excitation. The resulting density of states correction qualitatively changes the radial profile of the transport coefficients, as compared to the usual local diffusivity formulas. The correction factor calculated for the η_{i} -mode turbulence increases toward the plasma edge and allows a better qualitative agreement of the radial χ_{i} profile with experimental data than the conventional approach.

**Turbulence modifications in Ohm's law and tokamak transport phenomenology **

A. Thyagaraja, F.A. Haas

**Hamiltonian chaos and transport in quasi-geostrophic flows **

D. del Castillo-Negrete, P.J. Morrison

**Abstract**

Chaotic advective transport in quasigeostrophic flows is studied. Of particular interest is to compare theory with recent rotating tank experiments. Ideas regarding chaotic advection are briefly reviewed. A derivation of the quasigeostrophic equation relevant to the tank experiments is given, from which a model for the stream function is extracted and compared to experimental data. Linear theory is shown to predict correctly the onset of observed sinuous Rossby waves. A model stream function, composed of a zonal flow equilibrium with linear eigenfunctions, is used to study chaotic transport. Upon applying the Chirikov overlap criterion to the model it is seen, in agreement with experiments, that banded chaos, i.e., regions of chaos bounded by invariant surfaces, is to be expected. It is also shown that global chaos, and hence transport across the zonal flow, is inconsistent with linear theory and in general requires resonances with phase velocities near the peak velocity of the zonal flow equilibrium. Speculations regarding the consistency of chaos and conservation of potential vorticity are made.

**Magnetic interchange instability of accretion disks **

M. Kaisig, T. Tajima, R.V.E. Lovelace

**Abstract**

The nonlinear evolution of the magnetic interchange or buoyancy instability of a differentially rotating disk threaded by an ordered vertical magnetic field is investigated. A 2D ideal fluid in the equatorial plane of a central mass in the corotating frame of reference is considered as a model for the disk. If the rotation rate of the disk is Keplerian, the disk is found to be stable. If the vertical magnetic field is sufficiently strong, and the field strength decreases with distance from the central object, and thus the rotation of the disk deviates from Keplerian, if is found that an instability develops. The magnetic flux and disk matter expand outward in certain ranges of azimuth, while disk matter with less magnetic flux moves inward over the remaining range of azimuth, showing a characteristic development of an interchange instability.

**Steady-state dynamo and current drive in a non-uniform bounded plasma **

R.R. Mett, J.B. Taylor

**Abstract**

Current drive due to helicity injection and the dynamo effect are examined in an inhomogeneous bounded plasma. Averaged over a magnetic surface, there is, in general, no dynamo effect independent of resistivity---contrary to the results found previously for an unbounded plasma. The dynamo field is calculated explicitly for an incompressible viscoresistive fluid in the plane-slab model. In accord with the authors` general conclusion, outside the Alfven resonant layer it is proportional to the resistivity. Within the resonant layer there is a contribution which is increased, relative to its value outside the layer, by a factor (ω*a*^{2}/(η+ν)), where ω is the wave frequency, *a* is the plasma radius, η is the magnetic diffusivity, and ν is the kinematic viscosity. However, this contribution vanishes when integrated across the layer. The average field in the layer is resonance enhanced by a factor (ω*a*^{2}/(η+ν))^{2/3} and is proportional to the shear in the magnetic field and the cube root of the gradient of the Alfven speed. These results are interpreted in terms of helicity balance, and reconciled with the infinite medium calculations.

**Analytical calculation of neutral transport and its effect on ions **

M. Calvin, R.D. Hazeltine, P.M. Valanju, E.R. Solano

**Abstract**

The authors present an analytical calculation of the neutral particle distribution and its effects on ion heat and momentum transport in three-dimensional plasmas with arbitrary temperature and density profiles. A general variational principle, taking advantage of the simplicity of the charge exchange (CX) operator, is derived to solve self-consistently the problem of neutral-plasma interaction. To facilitate an extremal solution, the short CX mean-free-path (λ_{x}) ordering is used. Furthermore, a non-variational, analytical solution providing a full set of transport coefficients is derived by making the realistic assumption that the product of the CX cross-section with relative velocity is constant. The effects of neutrals on plasma energy loss and rotation appear in simple, sensible forms. It is found that neutral viscosity dominates ion viscosity everywhere, and in the edge region by a large factor.

**Double tearing instability with shear flow **

L. Ofman

**Abstract**

The linear evolution of the double tearing mode with parallel to the magnetic field equilibrium shear flow and viscosity is investigated numerically. Numerically obtained growth rates are found to agree with the solutions of the double tearing dispersion relation in the parameter range of validity. Solutions of the incompressible, time-dependent, linearized, viscoresistive magnetohydrodynamic equations for the double tearing mode with parallel flow are found for wide relevant parameter ranges. Large (weakly coupled) and small (strongly coupled) rational surface separation y_{s} are investigated. The magnetic Reynolds number S is varied up to 10^{8}, and ambient flow velocities up to 0.57 of the Alfvén speed V_{A} far from the tearing layer are considered. The normalized wave number α is 0.05 (long wavelength) and 0.5 (short wavelength). Spatial variations of the perturbed magnetic field Ψ and flow W indicate the "nonconstant-Ψ" effects for small y_{s}. Shear flow decouples the rational surfaces, reduces the growth rate, and transforms the instability to the standard tearing mode. Overstable modes are found from the solutions of the dispersion relation and in the numerical computations, and their frequencies are not affected by the value of viscosity. The temporal oscillations of the solutions increase with the flow at the resonant surfaces at a rate slower than that of the Doppler shift. For viscous Reynolds number S_{v} comparable to or larger than the magnetic Reynolds number a stabilizing effect was found, and in the presence of large flow the real growth rate γ_{R} scaling approaches the standard tearing mode scaling γ_{R}~S_{ν}^{1/6}.

**Inertia effects on the rigid displacement approximation of tokamak plasma vertical motion **

R. Khayrutdinov, E.A. Azimov, R. Carrera

**Abstract**

A widely used method of plasma stability analysis uses the Rigid Displacement Model (RDM) of plasma behavior. In the RDM it is assumed that the plasma displacement is small and usually plasma inertia effects are neglected. In addition, it is considered that no changes in plasma shape, plasma current, and plasma current profile take place throughout the plasma motion. The massless-filament approximation (instantaneous force-balance) accurately reproduces the unstable root of the passive stabilization problem. Then, on the basis that the instantaneous force-balance approximation is correct in the passive stabilization analysis, the massless approximation is utilized also in the study of the plasma vertical stabilization by active feedback. It is shown here that the RDM (without mass effects included) does not provide correct stability results for a tokamak configuration (plasma column, passive conductors, and feedback control coils). Therefore, it is concluded that inertia effects have to be retained in the RDM system of equations. It is shown analytically and numerically that stability diagrams with and without plasma-mass corrections differ significantly. When inertia effects are included the stability region is more restricted than obtained in the massless approximation.

**General analysis of magnetic loop positioning for plasma control in ignition tokamaks **

R. Khayrutdinov, E.A. Azimov, R. Carrera

**Abstract**

The control of the plasma vertical position in tokamak configurations and the positioning of magnetic, pick-up loops are considered. The equilibrium problem for a plasma with free-boundary is solved using the inverse-variable technique. Circuit equations for eddy currents in the vacuum vessel, eddy currents in structures, and currents in active coils coupled with the plasma equilibrium and transport equations are solved. The influence of the location of pick-up coils on control of the plasma vertical position is examined. It is shown that there are geometrical arrangements of the magnetic loops such that the plasma vertical position cannot be controlled using a conventional control law (this is regardless of the resistivity of the passive conductors and the gain value in the control system). An explanation of this phenomenon is given. A new control law is proposed such that plasma control is possible with general positioning of the magnetic loops. Our conclusions should be important for the operation of ignition tokamaks, where elongated, high-current plasmas have to be stabilized with magnetic loop positioning subject to severe constraints.

**Analysis of tokamak plasma vertical stability by rigid displacement and resistive MHD methods **

R. Khayrutdinov, E.A. Azimov, R. Carrera

**Abstract**

Stability and control of the vertical position of tokamak plasmas are studied using the “rigid displacement” and the resistive MHD models. In the rigid displacement model the plasma current and plasma shape are assumed not to change with time. The plasma, the vacuum vessel, and the active conductors are represented by a set of rigid, axisymmetric filaments. In the resistive MHD model no limitations on the time variation of the plasma shape and current are imposed. The circuit equations for the eddy currents in the vacuum vessel and the passive coils, and the currents induced in active conductors are solved simultaneously with the equilibrium and transport equations for the plasma. The plasma response to a step-control signal is studied. A detailed comparison between the two simulation methods is presented. It is shown that the resistive MHD model predicts a more stable plasma than the rigid displacement model. The computational time advantage of the rigid displacement analysis over conventional resistive MHD calculations is almost outweighed by the numerical algorithm used in the integration scheme of the DINA code employed here.

**Stability of annular equilibrium of energetic large orbit ion beam **

H.V. Wong, H.L. Berk, R.V. Lovelace, N. Rostoker

**Abstract**

The low-frequency stability of a long thin annular layer of energetic ions in a background plasma with finite axial and zero azimuthal magnetic field is studied analytically. It is found that although the equilibrium is susceptible to the kink instability, low mode number perturbations can be stabilized in the limit of *N _{i}*/

**IFS numerical laboratory tokamak **

M.J. Lebrun, T. Tajima

**Abstract**

A numerical laboratory of a tokamak plasma is being developed. This consists of the backbone (the overall manager in terms of the MPPL programming language), and the modularized components that can be plugged in or out for a particular run and their hierarchical arrangement. The components include various metrics for overall geometry, various dynamics, field calculations, and diagnoses.

**Effects of sheared flows on ion temperature gradient driven turbulent transport **

S. Hamaguchi, W. Horton

**Abstract**

Previous studies of the ion-temperature-gradient (ITG) driven turbulence are expanded to include the effect of sheared E×B flows in sheared magnetic fields. The radial eigenmodes are shown to substantially change character by shifting the modes off the rational surface. The new mode structure and growth rate directly affects the transport of both thermal energy and momentum in the sheared flows. The growth rate first increases with small shear flow and then decreases. The theoretical correlation of the shear flow with the thermal transport is important with respect to the transitions observed in tokamaks of a low (L mode) to a high (H mode) thermal confinement state as a function of the poloidal rotation velocity in the shear flow layer. The three-dimensional nonlinear simulations show that the anomalous ion thermal diffusivity is reduced significantly when dv_{E}/dx ≈ 2(c_{s}/L_{s}) ((1+η_{i})T_{i}/T_{e})^{1/2}. This condition is thought to be satisfied in a boundary layer in tokamaks with shear flow.

**On broken ballooning symmetry **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

A 2-D ballooning transform is devised to investigate ballooning symmetry breaking effects. It is found that there are stringent limitations on the use of 1-D eigenmode equations to describe plasma stability.

**Stability of drift-wave modons in the presence of temperature gradients **

D. Jovanovic, W. Horton

**Abstract**

In the homogeneous Hasegawa-Mima equation the dipole vortex or modon solution is well known to be robustly stable from both analytic and numerical studies. In the inhomogeneous plasma where T_{ε} ≠ 0 the corresponding vortex has an external structure extending into the high temperature region. Lyapunov stability method is used to determine the stability properties of these extended vortex structures. Negative value of the Lyapunov functional, as a measure of the instability, is shown to be bounded by (R/L_{τ})^{2} where R is the radius of the core of vortex and L_{τ} is scale length of the temperature gradient.

**Poisson bracket for Vlasov equation on a symplectic leaf **

H. Ye, P.J. Morrison, J.D. Crawford

**Abstract**

The degeneracy in the Lie-Poisson bracket, associated with the Hamiltonian structure of the Vlasov equation, is removed by restriction to a given symplectic leaf. The restricted equation of motion in terms of a generating function and it manifestly preserves the Casimir constraints of the system. A nondegenerate Poisson bracket in terms of the generating function is presented.

**The cyclone-anticyclone asymmetry in rotating shallow water **

S.V. Bazdenkov

**Abstract**

The cyclone-anticyclone asymmetry; i.e., the predominant generation of anticyclones in rotating shallow water, is considered from the viewpoint of flow relaxation toward vortices with minimal energy and fixed enstrophy (“selective decay” process). Three invariants of the set of equations for rotating shallow water are taken into account: total energy, enstrophy, and “mass.” A nonlinear second order differential equation is obtained that describes the relaxed flows. It is shown that the anticyclone-like solution corresponds to a minimal energy vale, in comparison with the cyclone-like solution for the same generalized enstrophy and “mass.”

**Linear studies M=1 modes in high temperature with a four field model **

A.Y. Aydemir

**Abstract**

The *m*=1 mode in high temperature plasmas is examined using a simple four-field model of tokamak dynamics derived by Hazeltine *et* *al*. (Phys. Fluids 30, 3204 (1987)). It is shown that, despite its simplicity, the model reproduces with remarkable accuracy results obtained with more sophisticated kinetic treatments in various collisionality regimes. The effects of parallel compressibility on the *m*=1 mode in the collisional, semicollisional, and collisionless regimes are also discussed. Coupling to the ion sound waves is found to be weakly destabilizing in the collisional regime, and stabilizing in the semicollisional and collisionless regimes.

**Scaling laws of stochastic ExB plasma transport **

M.B. Isichenko, W. Horton

**Abstract**

We discuss two distinct regimes of cross-field test particle diffusion due to electrostatic turbulence. In the *quasi-linear regime*, which takes place for small amplitude ∅_{0} of the potential perturbation, the local turbulent diffusion coefficient scales as D_{t} ∝ ∅^{2}_{0}. For the turbulence amplitude approaching or exceeding the mixing length level, the **E x B** drift of particles becomes sensitive to the topology of contours of constant potential ∅(r,t) due to the longer correlations of drift orbits. In this regime, the percolation theory is used to describe the long flights of particles along the critical level contours. For a single-scale, random distribution of the electric potential, *the percolation scaling* for the diffusivity D_{Τ} ∝ ∅_{0}^{7/10} is derived and applied to edge tokamak transport.

**Toroidal plasma reactor with a low external magnetic field **

A.D. Beklemishev, V.A. Gordin, R.R. Khayrutdinov, V.I. Petviashvili, T. Tajima

**Abstract**

The Lyapunov conditions for plasma stability are shown to be met in a toroidal pinch configuration with the safety factor q < 0.5, decreasing from the centre to the periphery without field reversal. This magnetic configuration is capable or containing high pressure plasma with only a small external toroidal magnetic field. Stable configurations are round with average beta near 15% and with the magnetic field associated mainly with the plasma current. The beta value calculated with the external magnetic field can be > 100%. Fast charged particles produced by fusion reactions are asymmetrically confined by the poloidal magnetic field (owing to the lack or a strong toroidal field). They thus generate a current in the non-central part or the plasma volume, which reinforces the poloidal field.

**Temperature anisotropy effect on the toroidal ion temperature gradient mode **

J.Y. Kim, W. Horton, S. Migliuolo, D.I. Choi, B. Coppi

**Abstract**

Using the local kinetic and fluid approaches, the effects of the anisotropies in the ion temperature and in the ion temperature gradient on the toroidal ion temperature gradient driven mode are investigated. A study of the effect of the increasing ion temperature relative to the fixed electron temperature in the peaked density limit is also made. Comparisons are given between the new toroidal results and the previous slab results.

**Relaxed states with plasma flow **

K. Avinash, J.B. Taylor

**Abstract**

n the theory of relaxation, a plasma reaches a state of minimum energy subject to constant magnetic helicity. In this state, the plasma velocity is zero. Several authors have attempted to extend the theory, by introducing a number of different helicity invariants, so as to obtain relaxed states with plasma flow. It is shown here that these generalized invariants are special cases of two basic self-helicities: one for electrons and one for ions. The validity of the generalized invariants is discussed and contrasted with that of original magnetic helicity.

**Studies of plasma equilibrium and transport in a tokamak fusion device with the inverse-variable technique **

R.R. Khayrutdinov, V.E. Lukash

**Abstract**

We describe an accurate and efficient model for studying the evolution of tokamak plasmas. The equilibrium problem for a plasma with a free boundary is solved using the inverse variable technique. The one-dimensional (averaged on magnetic surfaces) system of transport equations are solved together with the circuit equations for the vacuum vessel and the passive and active coils. As an example of the application of this method. We simulate the discharge in the T-3M tokamak as it transiently evolves to a separatrix configuration.

**Fractal orbits and passive transport in scaling turbulence **

M.B. Isichenko

**Abstract**

The passive advection n time-dependent 2D and 3D flows is considered. The velocity field v(r,*t*) is assumed to have power spectra of both scales and frequencies in respectively wide inertial ranges. *The quasi-linear limit* of passive transport is introduced and studied analytically using the method of *the virtual separation of scales*. The fractal dimension of particle trajectories and the propagation rates of an impurity are calculated.