**Summary of the international "Dawson" symposium on the physics of plasmas **

T. Tajima

**Abstract**

The Dawson Symposium was held on September 24 and 25, 1990 in honor of John Dawson's 60th birthday to reflect on various physics of plasma that he had pioneered. The international speakers touched on a wide range of subjects: magnetic fusion, laser fusion, isotope separation, computer simulation, basic plasma physics, accelerators and light sources, space physics, and international scientific collaboration. Highlighted in this article are magnetic fusion and laser fusion investigation that Dawson has been engaged in and the reviews of the present status of their development. The impact of the two-component fusion plasma idea, reactor concepts for advanced fuels, hot electron production by lasers and other nonlinear effects in laser fusion are discussed. Dawson's contributions in the allied areas are also reviewed.

**Statistical geometry of multi-scale isolines, Part II. 2D transport of passive scalar **

M.B. Isichenko, J. Kalda

**Abstract**

The propagation of a passive tracer in a vorticous incompressible fluid with a small background molecular diffusivity D_{0} is usually asymptotically diffusional with an effective diffusivity D_{eff}, which is much greater than D_{0}. The convective-diffusive transport in a two-dimensional steady flow v = ∇ψ(x,y) * Z admits an approach, based on the statistical geometry of stream-lines (= isolines of the stream function ψ(x,y)). This kind of analysis was completed previously [1,2] for a zero-mean, random flow with a single characteristic space scale λ_{0} using the percolation theory. The result D_{eff} ≈ ψ_{0}^{10/13}D_{0}^{3/13}, for ψ_{0} = <|ψ|> >> D_{0}, was expressed in terms of 2D percolation exponents. In the present paper, this approach is extended to a random multiscale flow with spectral components ψ_{λ} ∝ λ^{H}in a wide scaling range λ_{0} < λ < λ_{ m}. Using the stream-lines analysis based on the method of the virtual separation of scales, which is introduced in a companion paper [3], the expressions for the effective diffusivity D_{eff} and the corresponding mixing length for various flow spectra are obtained.

**Statistical geometry of multi-scale isolines, Part I. Fractal dimension of coastlines and number-area rule for islands **

M.B. Isichenko, J. Kalda

**Abstract**

Statistical topography involves the study of the geometrical properties of the iso-sets (contour lines or surfaces) of a random potential Ψ(r). Previous work [1,2] has addressed coastlines on a random relief Ψ(x, y) that possess a single characteristic spatial scale λ with topography belonging to the universality class of the random percolation problem. In the present paper this previous analytical approach is extended to the case of a multiscale random function with a power spectrum of scales, &Psiλ_{α}λ^{H} , in a wide range of wavelengths, λ_{0} < λ < λ_{m} . It is found that the pattern of the coastline differs significantly from that of a monoscale landscape provided that –3/4 <H < 1, with the case –3/4 <H < 0 corresponding to the long-range correlated percolation and 0 <H < 1 to the fractional Brownian relief. The expression for the fractal dimension of an individual coastline is derived, D_{h} = (10 – 3H)/7, the maximum value of which D_{h} = 7/4, corresponds to the monoscale relief. The distribution function F(a) of level lines over their size *a* is calculated: F(a)_{α}a^{–4(1-H)/7}, for λ_{0} << a << λ m . A comparison of the theoretical results with geographical data is presented.

**Explanation of Kapitza's linear law of magnetoresistance **

M.B. Isichenko, J. Kalda

**Abstract**

A theoretical interpretation of the linear dependence of magnetoresistance of metallic samples *R(B) ∝ B* on the strong external magnetic field *B* is presented. This behavior was discovered by Kapitza [1,2] and, to the best of our knowledge, has not previously been explained. It is shown that Kapitza’s law can be ascribed to the current redistribution in the sample produced by the Hall effect. As a result, the main Ohmic dissipation takes place in two types thin layer. The first type resides in the vicinity of electrodes. The second type, which is due to three-dimensional geometry of the sample, lies on a surface in the interior of the sample. It is concluded that the linear behavior *R(B)* observed in [1,2] can be directly related to the (inappropriate) two-terminal scheme of Kapitza’s magnetoresistance measurements. In this case the voltage drop is proportional to the Hall component ρ_{xy} of the microscopic resistivity tensor. The slowly varying linear magnetoresistance of simple metals is also briefly discussed. This may be attributed to the distortion introduced by potential leads.

**Dynamo effect and current drive due to magnetic fluctuations in sheared magnetic field **

K. Avinash

**Abstract**

It is shown that when the mean electromotive force in a plasma is evaluated in terms of the two-point correlation of driven magnetic field fluctuations (as would be appropriate for problems related to current drive by waves), there is no β effect; i.e., the terms associated with gradients of the mean field vanish. The relevance of this result to the problem of current drive is discussed.

**Effective plasma heat conductivity in "braided" magnetic field. Part II. Percolation limit **

M. Isichenko

**Abstract**

For pt.I see ibid., vol.33, no.7, p.795-807 (1991). This paper is devoted to the problem of anomalous transport across a magnetic field that includes a small stochastic component Δ B. The perturbation is assumed to be so strongly stretched along the background magnetic field B_{0} that the parameter R is large: R identical to b_{0}L_{0}/ Δ >> 1 (here b_{0} identical to ΔB_{⊥}/B_{0} << 1, and L_{0} is the longitudinal and Δ the transverse correlation length of the magnetic perturbation). This strong turbulence limit, which is opposite to the quasi-linear one (R<<1), has certain notable features. The principal result is that the main transport is concentrated in very thin regions, being fractal sets with the dimension df, which can range in value from 2 to 2.75, depending on the spectrum of the magnetic perturbation. These regions consist of a small fraction of magnetic lines that percolate, that is, walk from the nonperturbed magnetic flux surfaces to a distance large compared to the transverse correlation length Δ. Due to such a strong inhomogeneity of the transport distribution, as well as the long correlations, the standard transport averaging techniques fail, and one should make use of the percolation theory methods. Thus the strong turbulence regime is referred to here as the percolation limit. In comparison with the quasi-linear limit, the percolation limit has several additional intermediate regimes and the expressions for the effective heat conductivity χ_{eff} include the critical exponents of 2-D percolation theory. The estimates of χ_{eff} are obtained both in the collisional and collisionless limits, including the case of nonstationary magnetic perturbations.

**Effective plasma heat conductivity in "braided" magnetic field. Part I. Quasi-linear limit **

M. Isichenko

**Abstract**

Anomalous cross-field electron transport in a specified magnetic field with weakly-destroyed flux surfaces is discussed. Following the approach developed previously in the present paper the regimes of effective transverse plasma transport are studied systematically, both in the collisional and collisionless limits. The present analysis incorporates the nonstationary of magnetic perturbations, which was not included in the previous works. Part I of this study deals with the quasi-linear approximation, which may be written as b_{0}L_{0}/ Δ << 1, where b_{0} = ΔB_{⊥}/B_{0} represents the relative magnitude of the transverse magnetic perturbation, while L_{0} and Δ represent the longitudinal and transverse correlation lengths, respectively. It is found that some of the previously-described transport regimes cannot be considered as anomalous (in the sense that effective heat conduction χ_{eff} >> χ _{⊥}) for time-independent magnetic perturbations. However, these regimes can exist in the presence a finite-frequency magnetic flutter. A unified classification of quasi-linear regimes of anomalous transport is introduced in order to further extend the analysis to the strong magnetic turbulence limit b_{0}L_{0}/ Δ >> 1 (see Part II), which is considered as a companion paper. (Isichenko, ibid., vol.33, no.7, p.809-26 (1991)).

**Modelling of drift wave turbulence with a finite ion temperature gradient **

W. Horton, S. Hamaguchi

**Abstract**

With the use of consistent orderings in ε ≈ ρ_{s}/a and Δ approximately k_{⊥} to ρ_{s} model equations are derived for the drift instabilities from the electrostatic two-fluid equations. The electrical resistivity η included in the system allows the dynamics of both the collisional drift wave instability (η ≠ 0) and the collisionless ion temperature gradient driven instability (η = 0). The model equations also include effects of sheared velocity flows in the equilibrium plasma. The model equations used extensively in earlier nonlinear studies are obtained as appropriate limits of the model equations derived in the authors' work. The effects of electron temperature fluctuations are also discussed.

**Particle simulation of finite beta interchange modes in a sheared magnetic field **

J.H. Han

**Abstract**

Particle simulations of ideal and resistive interchange modes driven by density gradient and magnetic field curvature in finite β plasmas and sheared slab geometry have been performed. A 2-1/2D bounded magneto-inductive guiding center electron particle code, which accurately follows the perturbed magnetic field in the shear Alfvén limits has been implemented and used. To simulate effective collisions between electrons and ions in the guiding center electron plasma, an accurate algorithm for the Lorentz collision operator has been constructed and used for resistive interchange modes. With the particle code, linear and quasilinear behaviors of interchange modes in finite β plasmas have been investigated along with their electrostatic limit as a reference for both collisionless and collisional regimes. Salient features of the results are good agreement between kinetic linear theory and simulations for the linear phase of the interchange instability, and verification of the Suydam’s criterion for stability in finite β sheared slab geometry. Analytical estimates from quasilinear theory are also in agreement with the measured electrostatic potential and radial magnetic field fluctuation levels at saturation. Nonlinear features of particle simulations include mode cascadings in wave number space at saturation in electrostatic collisional plasmas and a significant amount of parallel ion temperature profile modification associated with quasilinear density profile modification in finite β collisional plasmas.

**A sufficient condition for the ideal instability of shear flow with parallel magnetic field **

X.L. Chen, P.J. Morrison

**Abstract**

A simple sufficient condition is given for the linear ideal instability of plane parallel equilibria with antisymmetric shear flow and symmetric or antisymmetric magnetic field. Application of this condition shows that plane Couette flow, which is stable in the absence of a magnetic field, can be driven unstable by a symmetric magnetic field. Also, although strong magnetic shear can stabilize shear flow with a hyperbolic tangent profile, there exists a range of magnetic shear that causes destabilization.

**Statistical dynamics of dissipative drift wave turbulence **

F.Y. Gang, P.H. Diamond, J.A. Crotinger, A.E. Koniges

**Abstract**

The statistical dynamics of a two-field model of dissipative drift wave turbulence is investigated using the EDQNM (eddy damped quasinormal Markovian) closure method [J. Fluid Mech. 41, 363 (1970)]. The analyses include studies of statistical closure equations, derivation of an H theorem, and its application to formulation of selective decay hypotheses for turbulent relaxation process. The results show that the dynamics of the two-field model is fundamentally different from that of the familiar, one-field Hasegawa–Mima model [Phys. Fluids 21, 87 (1978)]. In particular, density fluctuations nonlinearly couple to small scales, as does enstrophy. This transfer process is nonlinearly regulated by the dynamics of the density–vorticity cross correlation. Since density perturbations are not simply related to potential perturbations, as is vorticity, their transfer rate is greater. As a result, turbulent relaxation processes exhibit both dynamic alignment of density and vorticity and coherent vortex formation.

**A kinetic theory of trapped electron driven drift wave turbulence in a sheared magnetic field **

F.Y. Gang, P.H. Diamond, M.N. Rosenbluth

**Abstract**

A kinetic theory of collisionless and dissipative trapped electron driven drift wave turbulence in a sheared magnetic field is presented. Weak turbulence theory is employed to calculate the nonlinear electron and ion responses and to derive a wave kinetic equation that determines the nonlinear evolution of trapped electron mode turbulence. Saturated fluctuation spectrum is calculated using the condition of nonlinear saturation. The turbulent transport coefficients are in turn calculated using saturated fluctuation spectrum. Due to the disparity in the three different radial scale lengths of the slab-like eigenmode: Δ (trapped electron layer width), x_{t} (turning point width) and x_{i} (Landau damping point), Δ < x_{t} < x_{i}, we find that ion Compton scattering rather than trapped electron Compton scattering is the dominant nonlinear saturation mechanism. Ion Compton scattering transfers wave energy from short to long wavelengths where the wave energy is shear damped. As a consequence, a saturated fluctuation spectrum |φ|^{2}(k_{θ}) ≈ k_{θ}^{-α} (α = 2 and 3 for the dissipative and collisionless regime, respectively) occurs for k_{θρs < 1 and is heavily damped for k θρs > 1. The predicted fluctuation level and transport coefficients are well below the mixing length" estimate. This is due to the contribution of radial wavenumbers xt-1 < kr ≤ ρi-1 to the nonlinear couplings, the effect of radial localization of trapped electron response to a layer of width, Δ, and the weak turbulence factor ⟨(γel)/(ω→κ)⟩→k < 1, which enters the saturation level.}

**Nonlinear saturation of ideal interchange modes in a sheared magnetic field **

A.D. Beklemishev

**Abstract**

Pressure-driven ideal modes cannot completely interchange flux tubes of a sheared magnetic field; instead, they saturate, forming new helical equilibria. These equilibria are studied both analytically and numerically with reduced MHD equations in a flux conserving Lagrangian representation. For unstable localized modes, the structure of the nonlinear layer generated around the resonant flux surface depends on the value of Mercier parameter D _{M}. Its width is found to be proportional to the position of the inflection point on the linear eigenfunction. Perturbed surfaces in equilibrium always have folds, i.e., areas where the direction of the original reduced magnetic field is reserved. But only far from the instability threshold does the internal structure of the nonlinear layer resemble bubble formation. The appearance of sheet currents and island-like structures along the resonant flux surface may be of interest for the description of forced reconnection in models with finite resistivity. Analytic results are found to be in agreement with 2-D numerical simulations. The case of ballooning instability is included by representing nonlocal driving terms through the matching parameter (Δ'), which defines the outer boundary conditions for the interchange layer.

**Drift wave vortices in inhomogeneous plasmas **

X.N. Su, W. Horton, P.J. Morrison

**Abstract**

The effects of density and temperature gradients on drift wave vortex dynamics are studied using a fully nonlinear model with the Boltzmann density distribution. The equation based on the full Boltzmann relation, in the short wavelength (~ρ_{s}) region, possesses no localized monopole solution, while in the longer wavelength [~(ρ_{s}r_{n})^{1/2}] region the density profile governs the existence of monopolelike solutions. In the longer wavelength regime, however, the results of analysis show that due to the inhomogeneity of the plasma the monopoles cannot be localized sufficiently to avoid coupling to propagating drift waves. Thus, the monopole drift wave vortex is a long-lived coherent structure, but it is not precisely a stationary structure since the coupling results in a "flapping" tail. The flapping tail causes energy of the vortex to leak out, but the effect of the temperature gradient-induced nonlinearity is to reduce the leaking of this energy.

**On the origin of cosmological magnetic fields **

T. Tajima, K. Shibata, S. Cable, R.M. Kulsrud

**Abstract**

It is shown that a plasma with temperature T sustains fluctuations of electromagnetic fields and particle density even if it is assumed to be in a thermal equilibrium. The level of fluctuations in the plasma for a given wavelength and frequency of electromagnetic fields is rigorously computed by the fluctuation-dissipation theorem. A large zero frequency peak of electromagnetic fluctuations is discovered. We show that the energy contained in this peak is complementary to the energy lost by the plasma cutoff effect. The level of the zero frequency magnetic fields is computed as ⟨B^{2}⟩^{0}/8π = 1/2π^{3}T(ω_{p}/c)^{3}, where T and ω_{p} are the temperature and plasma frequency. This is the theoretical minimum magnetic field strength, as no turbulence is assumed. The size of the fluctuations is λ ≈ (c/ω_{p})(η/ω)^{1/2}, where η and ω are the collision frequency and the frequency of magnetic fields oscillations. These results are not in contradiction with the conventional black-body radiation spectra but its extension, and as such, do not contradict the observed lack of structure in the cosmic microwave background. The level of these is computed B_{λ} = 9.4 X 10^{-7} (a/a_{0})^{-1/2} (λ/1 cm)^{-3/2} Gauss, where a and a_{0} are the scale factors at time t and at present and λ the wavelength of magnetic fluctuations. Our computer particle simulation shows the support of the theory and in fact exhibits a peaking of the magnetic energy spectrum at zero frequency. The level of magnetic fields is significant at the early radiation epoch of the Universe. Implications of these magnetic fields in the early Universe are discussed.

**Effects of compressibility, diamagnetic drift, and thermal conduction on resistive ballooning modes in the 2nd stability regime **

J-Y. Kim, D-I. Choi, J.W. Van Dam, W. Horton

**Abstract**

The stabilizing effects of various terms such as compressibility, diamagnetic drift, and parallel thermal conduction are investigated on the type of resistive ballooning modes whose driving force comes from the resistive region, which are known to be unstable in the high-beta second stability regime when analyzed in the incompressible limit. It is found that compressibility gives a significant stabilizing effect mainly through the perpendicular magnetic compression, which suggests the possibility of a second stable window for these resistive ballooning modes. The diamagnetic drift terms slightly reduce the growth rate in the incompressible limit, but, with finite compressibility, lead to fairly strong stabilization. The compression due to ion polarization, which becomes significant at large diamagnetic drift, contributes to this stabilization. On the other hand, parallel thermal conduction and perpendicular magnetic compression, which enter through the equation for temperature evolution, are shown to have a negligible effect on the stability of these modes.

**Numerical study of compressible magnetoconvection with an open transitional boundary **

H. Hanami, T. Tajima

**Abstract**

Nonlinear evolution of magnetoconvection is studied by computerized simulation in a system with a dynamical open boundary between the convection region and corona of the sun. A model was studied in which the fluid is subject to the vertical gravitation, magnetohydrodynamics (MHD), and high stratification, through an MHD code with the MacCormack-Donner cell hybrid scheme in order to well represent convective phenomena. Initially the vertical fluid flux penetrates from the convectively unstable zone at the bottom into the upper diffuse atmosphere. As the instability develops, the magnetic fields are twisted by the convection motion and the folding magnetic fields is observed. When the magnetic pressure is comparable to the thermal pressure in the upper layer of convective zone, strong flux expulsion from the convective cell interior toward the cell boundary appears. Under appropriate conditions the simulation exhibits no shock formation incurred by the fluid convected to the photosphere, in contrast to earlier works with box boundaries. The magnetic field patterns observed are those of concentrated magnetic flux tubes, accumulation of dynamo flux near the bottom boundary, pinched flux near the downdraft region, and the surface movement of magnetic flux toward the downdraft region. Many of these computationally observed features are reminiscent of solar observations of the fluid and magnetic structures of their motions.

**Resistive tearing mode instability with shear flow and viscosity **

L. Ofman, X.L. Chen, P.J. Morrison, R.S. Steinolfson

**Abstract**

The linear theory of the resistive tearing mode instability in slab geometry, has been recently extended by introducing the effect of equilibrium shear flow and viscosity [Phys. Fluids 29, 2563 (1986); Phys. Fluids B 1, 2224 (1989); ibid. 2, 495 (1990); ibid. 2, 2575 (1990)]. In the present analysis, numerical solutions of the time-dependent resistive equations are generalized to this problem and growth rate scaling is obtained. The results of the computations are compared to previous work, and the computed growth rate scalings agree with analytical predictions. Namely, the "constant-Ψ" growth rate scales as S^{-1/2} and the "nonconstant-Ψ" growth rate scales as S^{-1/3}, where S is the magnetic Reynolds number. The Furth–Killeen–Rosenbluth (FKR) scaling of S^{-3/5} is reproduced for small values of shear flow. The presence of flow introduces a new peak in the eigenfunction, which is outside of the peak that occurs in the case without flow. The introduction of viscosity and small shear alters the growth rate scaling to S^{-2/3}(S_{v}/S)^{1/6} where S_{v} is the ratio of the viscous time to the Alfvén time. When the shear flow is large, the growth rate behaves in a more complex way, and Kelvin–Helmholtz instability effects are present.

**Rigorous upper bound for turbulent electromotive force in reversed-field pinches **

C.B. Kim, J.A. Krommes

**Abstract**

An upper bound is determined for the turbulently generated axial electromotive force in reversed-field pinches, constrained solely by energy conservation in the approximation of incompressible magnetohydrodynamics. The field reversal is predicted and comparisons are made with the minimum-energy state with the invariant magnetic helicity.

**Natural current profiles in a tokamak **

J.B. Taylor

**Abstract**

In this paper I show how one may arrive at a universal, or natural, family of Tokamak profiles using only accepted physical principles. These particular profiles are similar to ones proposed previously on the basis of ad hoc variational principles and the point of the present paper is to provide a justification for them. However in addition, the present work provides an interesting view of Tokamak fluctuations and leads to a new result -- a relationship between the inward particle pinch velocity, the diffusion coefficient and the current profile. The basic Tokamak model is described in this paper. Then an analogy is developed between Tokamak profiles and the equilibrium of a realisable dynamical system. Then the equations governing the natural Tokamak profiles are derived by applying standard statistical mechanics to this analog. The profiles themselves are calculated and some other results of the theory are described.

**Propagation of magnetoacoustic waves in the solar atmosphere with random inhomogeneities of density and magnetic fields **

M. Ryutova, M. Kaisig, T. Tajima

**Abstract**

Effects of strong and random inhomogeneities of the magnetic fields, plasma density, and temperature in the solar atmosphere on the properties of magnetoacoustic waves of arbitrary amplitudes are studied. The procedure which allows one to obtain the averaged equation containing the nonlinearity of a wave, dispersion properties of a system, and dissipative effects is described. It is shown that depending on the statistical properties of the medium, different scenarios of wave propagation arise: in the predominance of dissipative effects the primary wave is damped away in the linear stage and the efficiency of heating due to inhomogeneities is much greater than that in homogeneous medium. Depending on the interplay of nonlinear and dispersion effects, the process of heating can be afforded through the formation of shocks or through the storing of energy in a system of solitons which are later damped away. Our computer simulation supports and extends the above theoretical investigations. In particular the enhanced dissipation of waves due to the strong and random inhomogeneities is observed and this is more pronounced for shorter waves.

**Numerical simulation of ion temperature gradient driven modes in the presence of ion-ion collisions **

X.Q. Xu, M.N. Rosenbluth

**Abstract**

Ion temperature gradient driven modes in the presence of ion-ion collisions in a toroidal geometry with trapped ions have been studied by using a 1 2/2 d linearized gyro-kinetic particle simulation code in the electrostatic limit. The purpose of the investigation is to try to understand the physics of flat density discharges, in order to test the marginal stability hypothesis. Results giving threshold conditions of L_{Ti}/R_{0}, an upper bound on k_{χ}, and linear growth rates and mode frequencies over all wavelengths for the collisionless ion temperature gradient driven modes are obtained. The behavior of ion temperature gradient driven instabilities in the transition from slab to toroidal geometry, with trapped ions, is shown. A Monte Carlo scheme for the inclusion of ion-ion collisions, in which ions can undergo Coulomb collisional dynamical friction, velocity space diffusion and random walk of guiding centers, has been constructed. The effects of ion-ion collisions on the long wave length limit of the ion modes is discussed.

**Tearing modes in tokamaks with lower hybrid current drive **

X.Q. Xu, M.N. Rosenbluth

**Abstract**

n this paper, the effect of current drive on the tearing modes in the semi-collisional regime is analyzed using the drift-kinetic equation. A collisional operator is developed to model electron parallel conductivity. For the pure tearing modes the linear and quasilinear growth rates in the Rutherford regimes have been found to have roughly the same forms with a modified resistivity as without current drive. One interesting result is the prediction of a new instability. This instability, driven by the current gradient inside the tearing mode layer, is possibly related to MHD behavior observed in these experiments.

**Unified theory of ballooning instabilities and temperature gradient driven trapped ion modes **

X.Q. Xu, M.N. Rosenbluth

**Abstract**

A unified theory of temperature gradient-driven trapped ion modes and ballooning instabilities is developed using kinetic theory in banana regimes. All known results such as electrostatic and purely magnetic trapped particle modes and ideal magnetohydrodynamic ballooning modes (or shear Alfven waves) are readily derived from the present single general dispersion relation. Several new results from ion--ion collision, finite beta stabilization of ion temperature gradient-driven trapped particle modes, and trapped particle modification of ballooning modes are derived and discussed. The interrelationship between these modes is established.

**The effect of trapped ions and current drive on tokamak microinstabilities: theory and simulations **

X.Q. Xu

**Abstract**

In this work, a range of low-frequency microinstabilities in tokamaks and the related anomalous cross field transports have been investigated analytically and numerically. A unified theory of temperature gradient driven trapped ion modes and ballooning instabilities is developed using kinetic theory in banana regimes. All known results, such as electrostatic and purely magnetic trapped particle modes and ideal MHD ballooning modes (or shear Alfven waves) are readily derived from our single general dispersion relation. Several new results from ion-ion collision and trapped particle modification of ballooning modes are derived and discussed and the interrelationship between those modes is established. Ion temperature gradient driven modes in the presence of ion-ion collisions in a toroidal geometry with trapped ions have been studied by using a 1⋅2/2⋅d linearized gyrokinetic particle simulation code in the electrostatic limit. The purpose of the investigation is to try to understand the physics of flat density discharges, in order to test the marginal stability hypothesis. Results giving threshold conditions of L_{Ti}/R_{0}, an upper bound on k_{χ}, and linear growth rates and mode frequencies over all wavelengths for the collisionless ion temperature gradient driven modes are obtained. The behavior of ion temperature gradient driven instabilities in the transition from slab to toroidal geometry, with trapped ions, is shown. A Monte-Carlo scheme for the inclusion of ion-ion collisions, in which ions can undergo Coulomb collisional dynamical friction, velocity space diffusion and random walk of guiding centers, has been constructed. The effects of ion-ion collisions on the long wave length limit of the ion modes is discussed. Finally, the effect of current drive on the tearing modes in the semicollisional regime is analyzed using the drift-kinetic equation. A collisional operator is developed to model the electron parallel conductivity. For the pure tearing modes the linear and quasilinear growth rates in the Rutherford regimes have been found to have roughly the same forms with a modified resistivity as without current drive. One interesting result is the prediction of a new instability. This instability, driven by the current gradient inside the tearing mode layer, is possibly related to MHD behavior observed in these experiments.

**Transition from toroidal to slab temperature gradient driven modes **

J.Y. Kim, W. Horton

**Abstract**

In the local approximation the electrostatic dispersion relation for the (ion or electron) temperature gradient driven modes is solved retaining the full guiding-center resonance condition ω=*k*_{||}*v*_{||} + **k** ⋅ **v**_{D} (*v*^{2}_{⊥}, *v*^{2}_{||}). Increase of the slab-to-toroidal ratio parameter *k*_{||}*v*_{T}/*k*_{⊥}*v*_{D} is shown to directly increase the threshold value of the temperature-to-density gradient parameter η_{c}. Therefore, the local anomalous heat flux has a strong dependence on the safety factor *q* when *k*_{||} ~ 1/*qR*.

**Vlasov-Maxwell equilibria of solar coronal loops **

V. Krishan, T.D. Sreedharan, S.M. Mahajan

**Abstract**

A Vlasov-Maxwell description of the ubiquitous solar coronal structures is presented. It is found that an equilibrium plasma configuration can live with spatial gradients in density, temperature, current and drift speeds of the charged particles. Any stability study must be carried out over this inhomogeneous equilibrium state. In addition, the Vlasov description admits the investigation of kinetic processes like heating and radiation and unlike a fluid description, it does not require an equation of state to determine the individual variations of temperature and density.

**Resonant MHD modes with toroidal coupling Part II. ballooning-twisting modes **

J.W. Connor, R.J. Hastie, J.B. Taylor

**Abstract**

This is Part II of a study of resonant perturbations, such as resistive tearing and ballooning modes, in a torus. These are described by marginal ideal magnetohydrodynamic (MHD) equations in the regions between resonant surfaces; matching across these surfaces provides the dispersion relation. Part I (Phys. Fluids B 3, 1532 (1991)) described how all the necessary information from the ideal MHD calculations could be represented by a so-called E matrix. The calculation of this E matrix for tearing modes (even parity in perturbed magnetic field) in a large-aspect-ratio torus was also described. There the toroidal modes comprise coupled cylinder tearing modes and the E matrix is a generalization of the familiar Δ′ quantity in a cylinder. In the present paper, resistive ballooning, or twisting modes, which have odd parity in perturbed magnetic field, are discussed. Unlike the tearing modes, these odd-parity modes are intrinsically toroidal and are not directly related to the odd-parity modes in a cylinder. This is evident from the analysis of the high- n limit in ballooning space, where the twisting mode exhibits a singular transition at large aspect ratio when the interchange effect is small (as in a tokamak). Analysis of the high- n limit in coordinate space, rather than ballooning space, clarifies this singular behavior. It also yields a prescription for treating low- n twisting modes and a method for calculating an E matrix for resistive ballooning modes in a large-aspect-ratio tokamak in the limit the interchange term vanishes. The elements of this matrix are given in terms of cylindrical tearing mode solutions.

**Nonlinear tearing mode; Rutherford regime and global characteristics **

Y.Z. Zhang, R. Denton, S.M. Mahajan, C. Jiayu, A.J. Wootton

**Abstract**

Experiments on the TEXT tokamak show that the temporal evolution of a single helicity tearing mode is dominated by the Rutherford regime. Further analysis of the Mirnov signal and sawtooth activity combined with numerical simulation emphasizes the global characteristics of the tearing mode.