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First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear, because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the existence of modes, driven by the ion temperature gradient and the parallel velocity gradient, which grow transiently. Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. A parametric model is constructed which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in the reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical parallel-velocity-gradient-driven turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence (which cannot be modelled by the quasi-linear methods used in current transport codes) and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows. [ABSTRACT FROM AUTHOR]

The characteristics of near-unity-β equilibria are investigated with two codes. CUBE is a multigrid Grad–Shafranov solver [Gourdain et al., J. Comput. Phys. 216, 275 (2006)], and Ophidian was written to compute solutions using analytic unity-β equilibria [Cowley et al., Phys. Fluids B 3, 2066 (1991)]. Results from each method are qualitatively and quantitatively compared across a spectrum of mutually relevant parameters. These comparisons corroborate the theoretical results and provide benchmarks for high-resolution numerical results available from CUBE. Both tools facilitate the exploration of the properties of high-β equilibria, such as a highly diamagnetic plasma and its ramifications for stability and transport. [ABSTRACT FROM AUTHOR]

A new analytically and numerically manageable model collision operator is developed specifically for turbulence simulations. The like-particle collision operator includes both pitch-angle scattering and energy diffusion and satisfies the physical constraints required for collision operators: it conserves particles, momentum, and energy, obeys Boltzmann’s H-theorem (collisions cannot decrease entropy), vanishes on a Maxwellian, and efficiently dissipates small-scale structure in the velocity space. The process of transforming this collision operator into the gyroaveraged form for use in gyrokinetic simulations is detailed. The gyroaveraged model operator is shown to have more suitable behavior at small scales in phase space than previously suggested models. Model operators for electron-ion and ion-electron collisions are also presented. [ABSTRACT FROM AUTHOR]

Current sheets formed in magnetic reconnection events are found to be unstable to high-wavenumber perturbations. The instability is very fast: its maximum growth rate scales as S1/4vA/LCS, where LCS is the length of the sheet, vA the Alfvén speed, and S the Lundquist number. As a result, a chain of plasmoids (secondary islands) is formed, whose number scales as S3/8. [ABSTRACT FROM AUTHOR]

Observations of galaxy clusters show that the intracluster medium (ICM) is likely to be turbulent and is certainly magnetized. The properties of this magnetized turbulence are determined both by fundamental nonlinear magnetohydrodynamic interactions and by the plasma physics of the ICM, which has very low collisionality. Cluster plasma threaded by weak magnetic fields is subject to firehose and mirror instabilities. These saturate and produce fluctuations at the ion gyroscale, which can scatter particles, increasing the effective collision rate and, therefore, the effective Reynolds number of the ICM. A simple way to model this effect is proposed. The model yields a self-accelerating fluctuation dynamo whereby the field grows explosively fast, reaching the observed, dynamically important, field strength in a fraction of the cluster lifetime independent of the exact strength of the seed field. It is suggested that the saturated state of the cluster turbulence is a combination of the conventional isotropic magnetohydrodynamic turbulence, characterized by folded, direction-reversing magnetic fields and an Alfvén-wave cascade at collisionless scales. An argument is proposed to constrain the reversal scale of the folded field. The picture that emerges appears to be in qualitative agreement with observations of magnetic fields in clusters. [ABSTRACT FROM AUTHOR]

One of the outstanding challenges of solar physics is understanding the mechanism for heating the solar corona, a low-β, low-resistivity plasma driven slowly by the shuffling of "footpoints" (ends of flux tubes) anchored in the photosphere. The corona's low resistivity means that enormous current densities must be present for Ohmic dissipation to be effective. Two ways that this can occur in a slowly evolving, "quasiequilibrium" medium are by the formation of current sheets or by the excitation of very low-frequency Alfvén resonances. Both processes lead to singular current densities in the ideal limit, and both may result from the excitation of very low- (or zero-) frequency shear Alfvén resonances. Thus, understanding the conditions under which zero-frequency Alfvén resonances can occur may ultimately be important in understanding coronal heating. This paper examines shear Alfvén resonances in line-tied magnetic geometries, with special attention being paid to geometries with discontinuities in the field line mapping as candidates for having zero-frequency Alfvén resonances. Two such geometries have been examined in detail: a two-dimensional X-point and a three-dimensional magnetic null. Qualitative differences in the two spectra are observed. The X-point geometry has Alfvén resonances of vanishing frequency, whereas the magnetic null geometry has a finite cutoff in resonant frequency for all field lines in the system. The Alfvén continuum for a null depends on its geometry, however, and much lower (though still nonzero) resonant frequencies are found when the flux surfaces are moderately eccentric. Consequently, a subset of magnetic nulls in complex, line tied geometries may have flux surfaces that can be driven resonantly by a driver of much lower frequency than for similar media without nulls. To the extent that nulls are generic features of complex magnetic geometries {and there is some reason to... [ABSTRACT FROM AUTHOR]

Stochastic magnetic fields in a weakly collisional plasma enhance the collisionality of electrons in the medium compared to the classical Spitzer values. The reduction in the thermal conductivity may play an important role in the evolution of galaxy clusters. This work extends prior work by examining the parallel propagation of electrons in plasma media where the magnetic field amplitude sampled along a field line is a stochastic function of the distance. New physics is exhibited when many large-amplitude magnetic mirrors are present; the parallel transport becomes subdiffusive rather than diffusive in this limit. The transition between diffusion and subdiffusion can be obtained from properties of the asymptotic solution to the kinetic equation in the vicinity of a solitary magnetic mirror. This transition has been computed as a function of the power law exponent in the distribution of mirror maxima, and the asymptotic scaling of the mean-squared electron displacement with time has been obtained in these systems as a function of this exponent. Monte Carlo comparisons are in agreement with these results. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The theory of beat wave excitation in a slightly inhomogeneous plasma is presented. While the theory is general it applies directly to the experiments performed in Alaska by the ionospheric heating facilities HIPAS [High Power Auroral Simulation, Radio Sci. 25, 1269 (1990)] and HAARP [High Frequency Active Auroral Research Project, Geophys. Res. Lett. 25, 257 (1998)]. A ray tracing (WKB) formalism appropriate for computations is developed. The computational implementation of this formalism and extensive results will be presented in a follow up to this paper. Also the beat wave excitation of upper-hybrid waves is investigated analytically. The complicated trajectories of these waves in the plasma are described. When this beat wave reaches the plasma wave resonance it can, in the HIPAS-HAARP experiments, attain nonlinear amplitude. The electrostatic upper-hybrid waves are trapped around the density maximum of the ionosphere. This trapping is investigated in detail. Beat wave pumping of the trapped modes is possible using HAARP and HIPAS or with split beams from HAARP. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Through the principal use of the reduced magnetohydrodynamic version of the finite aspect ratio code [L. A. Charlton et al., J. Comput. Phys. 86, 270 (1990)], an m/n=1/1 resistive kink mode was poloidally rotated with the accompanying rotational shear. It was observed that the growth rate of this unstable mode can either decrease or increase as the applied equilibrium rotation is increased to near poloidal sonic speeds. Shear in the poloidal rotation profile is stabilizing, but only if the destabilizing effects of bulk rotation can be overcome. Therefore, the mode's stability was sensitive to the location of the rotation's peak relative to the eigenmode's spatial extent. The destabilizing effects of bulk rotation are apparently a rotationally enhanced beta, and the stabilizing effects appear to be caused by exceeding a critical rotational shear spatially averaged over the eigenmode structure. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

High-beta (β>ε/q2) large-aspect-ratio (ε<1) tokamak equilibria are shown to be always stable to ideal magnetohydrodynamics (MHD) modes that are localized about a flux surface. Both the ballooning and interchange modes are shown to be stable. This work uses the analytic high-beta large-aspect-ratio tokamak equilibria developed by Cowley et al. [Phys. Fluids B 3, 2069 (1991)], which are valid for arbitrary pressure and safety factor profiles. The stability results make no assumption about these profiles nor the shape of the boundary. [ABSTRACT FROM AUTHOR]

The ion-temperature-gradient-driven instability is considered in this paper. Physical pictures are presented to clarify the nature of the instability. The saturation of a single eddy is modeled by a simple nonlinear equation. It is shown that eddies that are elongated in the direction of the temperature gradient are the most unstable and have the highest saturation amplitudes. In a sheared magnetic field, such elongated eddies twist with the field lines. This structure is shown to be an alternative to the usual Fourier mode picture in which the mode is localized around the surface where k|| =0. These elongated twisting eddies, which are an integral part of the ‘‘ballooning mode’’ structure, could survive in a torus. The elongated eddies are shown to be unstable to secondary instabilities that are driven by the large gradients in the long eddy. It is argued that the ‘‘mixing length’’ is affected by this nonlinear process, and is unlikely to be a linear eigenmode width. [ABSTRACT FROM AUTHOR]

An analytic solution of the high-beta (∈&βmacr;[sub p] ∼ βq² / ∈ » 1) equilibrium of a large aspect ratio tokamak is presented. Two arbitrary flux functions, the pressure profile p(ψ) and the safety factor profile q(ψ), specify the equilibrium. The solution splits into two asymptotic regions: the core region where ψ is a function of the major radius alone and a narrow boundary layer region adjoining the conducting wall. The solutions in the two regions are asymptotically matched to each other. For monotonic pressure profiles, the Shafranov shift is equal to the minor radius. For beta much bigger than 1, the solution contains a region (in place of the magnetic axis) of zero magnetic field and constant pressure. At high beta the quantity β[sub I] which is essentially proportional to the pressure over the total current squared, is largely independent of pressure. The important ramifications of limited β[sub I] for high-beta reactors are discussed. Generalizations to shaped cross sections and hollow pressure profiles are outlined. The problem of equilibrium reconstruction in the high-beta regime is also considered. [ABSTRACT FROM AUTHOR]

Toroidal calculations on m = 2 magnetic feedback are presented. These calculations are performed using a coupled Δ' formulation and show that the feedback can have a pronounced effect on the sideband (m = 1 and 3) content. [ABSTRACT FROM AUTHOR]

The separation of the cylindrical tearing mode stability problem into a resistive resonant layer calculation and an external marginal ideal magnetohydrodynamic (MHD) calculation (Δ’ calculation) is generalized to axisymmetric toroidal geometry. The general structure of this separation is analyzed and the marginal ideal MHD information (the toroidal generalization of Δ’) required to discuss stability is isolated. This can then, in principle, be combined with relevant resonant layer calculations to determine tearing mode growth rates in realistic situations. Two examples are given: the first is an analytic treatment of toroidally coupled (m=1, n=1) and (m=2, n=1) tearing modes in a large aspect ratio torus; the second, a numerical treatment of the toroidal coupling of three tearing modes through finite pressure effects in a large aspect ratio torus. In addition, the use of a coupling integral approach for determining the stability of coupled tearing modes is discussed. Finally, the possibility of using initial value resistive MHD codes in realistic toroidal geometry to determine the necessary information from the ideal MHD marginal solution is discussed. [ABSTRACT FROM AUTHOR]

This paper examines the stability of tearing modes in a sheared slab when the width of the tearing layer is much smaller than the ion Larmor radius. The ion response is nonlocal, and the quasineutrality retains its full integral form. An expansion procedure is introduced to solve the quasineutrality equation in powers of the width of the tearing layer over the ion Larmor radius. The expansion procedure is applied to the collisionless and semicollisional tearing modes. It is found that first-order terms in the expansion are strongly stabilizing. The physics of the mode and of the stabilization are discussed. Tearing modes are observed in experiments even though the slab theory predicts stability. It is proposed that these modes grow from an equilibrium with islands at the rational surfaces. If the equilibrium islands are wider than the ion Larmor radius, the mode is unstable when δ' is positive. [ABSTRACT FROM AUTHOR]

The usual kinetic description of a plasma is extended to include variables to describe the nuclear spin. The distribution function, over phase space and the new spin variables, provides a sufficient description of a spin-polarized plasma. The evolution for the distribution function is given. The equations derived are used to calculate depolarization caused by four processes: inhomogeneous fields, collisions, collisions in inhomogeneous fields, and waves. It is found that depolarization by field inhomogeneity on scales large compared with the gyroradius is totally negligible. The same is true for collisional depolarization. Collisions in inhomogeneous fields yield a depolarization rate of order 10-3 sec-1 for deuterons and a negligible rate for tritons in a typical fusion reactor design. This is still sufficiently small on reactor time scales. However, small-amplitude magnetic fluctuations (of order 1 G) resonant with the spin precession frequency can lead to significant depolarization (depolarizes triton in 10 sec and deuteron in 100 sec). [ABSTRACT FROM AUTHOR]

The scope and detail of physics contained in computational models for fluid (density, momentum, energy) transport in toroidal plasmas have steadily increased during the past two decades. There has been considerable success in the development and verification of models for sources and sinks of particles, energy, momentum, and magnetic flux. Transport codes have collectively become very useful tools in interpreting experimental data and in providing guidance for new experiments. However, a more thorough understanding of the fundamental transport processes of magnetically confined plasmas and development of improved computational models are needed to enhance the predictive capabilities of transport codes. It is argued that fluid transport modeling by itself cannot lead to a complete understanding of transport—there must be a very strong collaboration among theory, experiment, and modeling on both the fluid and kinetic levels. [ABSTRACT FROM AUTHOR]

Using a simple model for the layer of the tearing mode, it is demonstrated that toroidally coupled tearing modes with two rational surfaces are most unstable when the ω*’s of the electrons at the rational surfaces are equal. The onset of instability may then occur because of the tuning of ω* rather than the passage of Δ’-like quantities through zero. This mechanism for the onset of instability is sharp since the resonance is narrow. The effect of toroidal rotation is also discussed. [ABSTRACT FROM AUTHOR]

A numerical study is carried out, based on a simple toroidal tokamak equilibrium, to demonstrate the radial re-distribution of the electromagnetic torque density, as a result of a rotating resistive plasma (linear) response to a static resonant magnetic perturbation field. The computed electromagnetic torque peaks at several radial locations even in the presence of a single rational surface, due to resonances between the rotating response, in the plasma frame, and both Alfvén and sound continuum waves. These peaks tend to merge together to form a rather global torque distribution, when the plasma resistivity is large. The continuum resonance induced net electromagnetic torque remains finite even in the limit of an ideal plasma. [ABSTRACT FROM AUTHOR]

Within the single fluid theory for a toroidal, resistive plasma, the favorable average curvature effect [Glasser et al., Phys. Fluids 18, 875 (1975)], which is responsible for the strong stabilization of the classical tearing mode at finite pressure, can also introduce a strong screening effect to the externally applied resonant magnetic field. Contrary to conventional understanding, this screening, occurring at slow plasma rotation, is enhanced when decreasing the plasma flow speed. The plasma rotation frequency, below which this screening effect is observed, depends on the plasma pressure and resistivity. For the simple toroidal case considered here, the toroidal rotation frequency has to be below ∼10-5ωA, with ωA being the Alfvén frequency. In addition, the same curvature effect leads to enhanced toroidal coupling of poloidal Fourier harmonics inside the resistive layer, as well as reversing the sign of the electromagnetic torque at slow plasma flow. [ABSTRACT FROM AUTHOR]

Theoretically, high-β (β∼1) tokamaks offer a large fusion power efficiency advantage over low-β devices. However, if high-β tokamaks are inherently unstable then such devices will never be realized. In particular, kink modes are thought to be the most serious obstacle to high-β operations. High-β tokamaks are characterized by a very large Shafranov shift with a thin "boundary layer" on the outboard side of the device and a large "core" region of vertical flux surfaces comprising most of the central volume. In this paper, the energy principle is used to compute the magnetohydrodynamic internal kink stability of such devices in the large aspect ratio limit with a low toroidal mode number. A class of internal kink mode similar to the usual low-β kink is present; the stability against these modes is computed. A set of parameters describing a kink stable high-β equilibrium is given. Stability is shown to be dependent on the shape of the plasma boundary. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]