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Volume 4, Issue 7, July 1992

Poloidal beta scaling for a bootstrapped tokamak
View Description Hide DescriptionA new scaling of the poloidal plasma beta with the bootstrap fraction is derived for a steady‐state tokamak. In this derivation, the seed current profile is specified, but no assumptions are made regarding the functional form of the total current profile. The scaling, which holds for a wide class of density and temperature profiles, exhibits a quadratic dependence of the poloidal beta on the bootstrap fraction. The effects of the ionic charge and unequal electron and ion temperatures are considered.

The structure of ideal magnetohydrodynamics with incompressible steady flow
View Description Hide DescriptionThe equations of ideal magnetohydrodynamics (MHD) with steady incompressible mass flow are known to be integrable to (1) a scalar elliptic partial differential equation (PDE) for a flux function in the case of helically symmetric systems; and (2) a system of two coupled scalar PDE’s in the case of arbitrary nontoroidal geometries. In the present paper it is shown that these integrated equations do not have the same symmetry properties as the original MHD equations; hence are not general enough. The appropriate generalizations are given, and their structure is analyzed. As a result a classification of stationary states of ideal MHD with flow in terms of isomorphisms to simpler MHD systems is obtained. Under certain mild conditions the involved differential operators can be mapped smoothly to considerably simpler ones; however, the mapping is not necessarily analytic. This transformation technique allows the construction of complicated MHD solutions with flow out of simpler MHD solutions. Some explicit examples are presented.

Equilibrium of non‐neutral plasma in toroidal geometry with applied electric field
View Description Hide DescriptionThe equilibrium and stability of non‐neutral plasma in toroidal geometry with applied electric field is studied. The calculations are done in a large aspect ratio, thin ring approximation, using a circuit model. Toroidal corrections to the l=1 (l is the poloidal mode number) mode are obtained. These corrections reduce the l=1 cylindrical diocotron frequency. It is shown that under certain conditions, the external electric field can stabilize dissipative l=1 diocotron oscillations.

Electron–ion hybrid instabilities driven by velocity shear in a magnetized plasma
View Description Hide DescriptionThe stability of a magnetized plasma is investigated in which a sheared electron flow channel is present. The flow’s peak velocity and shear scale length are denoted by V and L, respectively. If the velocity channel is perpendicular to the confining magnetic field and L≤ ρ_{ i } (ρ_{ i } is the ion Larmor radius) an electrostatic instability develops whose frequency is on the order of the lower hybrid frequency. For V/(Ω_{ eL }) ≳ 0.02 (Ω_{ e } denotes the electron cyclotron frequency), the peak growth rate is on the order of the lower hybrid frequency when k _{∥} = 0 (in here, k _{∥} is the wave number along the magnetic field). For V/(Ω_{ eL }) ≳ 0.1 and k _{∥} = 0, the spectrum peaks when k _{ yL } ∼ 1, where k _{ y } is the wave number in the direction of the flow velocity. For this mode it is shown that (i) a net cross‐field current is not required for the onset of instability and (ii) the growth rate is not reduced by a velocity profile with no net flow (spatially averaged). Hence we conclude that velocity shear is the only source of free energy. Further, it is shown that density gradients do not stabilize this mode. It follows that the mode presented in this work cannot be identified with the well‐known modified two‐stream instability. If the velocity channel is parallel to the confining magnetic field and the plasma is weakly magnetized, an instability driven by velocity shear is shown to exist, provided that V/(ω_{ pe } L) ≳ 0.32, where ω_{ pe } is the electron plasma frequency. It is shown that a net plasma current is not required in order for this instability to be excited.

Sensitivity of the kink instability to the pressure profile
View Description Hide DescriptionThe stability of the n=1 ideal kink in DIII‐D‐like [Fusion Technol. 8, 441 (1985)] configurations is found to depend critically on the details of the current density and pressure profiles. The maximum stable normalized β, β_{ N }=β/(I/aB) (I in MA, a in meters, and B _{0} in tesla), is found to increase dramatically as the peak in the pressure gradient is shifted from the central region toward the plasma boundary. Further, for peaked pressure profiles, the β limit is insensitive to the internal inductance, l _{ i }, whereas for broad pressure profiles, the β limit depends more strongly on l _{ i } and can increase with l _{ i } almost up to the point where the q profile becomes hollow. With broad pressure profiles and a conducting wall at 1.5 minor plasma radii, kink‐stable β_{ N } values of 5.8−significantly above both the Troyon limit and the ballooning limit−have been found.

Self‐consistent theory for ion gyroresonance
View Description Hide DescriptionFor describing ion gyroresonance processes, a complete set of self‐consistent Vlasov–Maxwell equations is derived by systematically transforming a self‐consistent action principle from particle coordinates to guiding‐center/oscillation‐center coordinates. They include the oscillation‐center Vlasov equation; the equations for wave–particle resonant interactions; and Maxwell equations for the background electromagnetic fields. These equations satisfy local conservation laws for energy, momentum, and angular momentum, constructed by using the Noether algorithm. A heuristic interpretation based on the theory of linear mode conversion in ray phase space is also presented for ion gyroresonance processes, which suggests a method for obtaining analytic solutions in general geometry.

Resistive drift wave and interchange turbulence in a cylindrical plasma with magnetic and velocity shear
View Description Hide DescriptionBy solving numerically the two‐field nonlinear model equations for the potential and density fluctuations in the presence of a density gradient, magnetic curvature with shear, and poloidal velocity shear, turbulent spectra and diffusion rates are studied. When the adiabatic parameter (Ω_{ e }/ν_{ e })(ρ_{ s } ^{2}/R ^{2})/(κρ_{ s }) is small, a trend of a dual cascade, normal cascade of the density fluctuations, and an inverse cascade of the potential fluctuations, is seen in the wave number spectra, producing a large particle flux proportional to ν_{ e } ^{1/3}. Here R is the major radius, ρ_{ s } is the ion Larmor radius at the electron temperature, Ω_{ e } is the electron cyclotron frequency, ν_{ e } is the electron ion collision frequency, and κ is an inverse scale length of the background density gradient. Parallel wave numbers are represented by 1/R. For large (Ω_{ e }/ν_{ e })(ρ_{ s } ^{2}/R ^{2})/(κρ_{ s }), the electrons become adiabatic, with a significantly reduced particle flux proportional to ν_{ e }. In the presence of an externally imposed radial electric field with a negative (positive) polarity, E _{ r }<0 (E _{ r }≳0), the poloidal velocity shear in the E× B drift motion suppresses (enhances) the fluctuation level in the growth phase; however, these effects practically disappear in the saturated state. The radial electric field produced by the inverse cascade due to the convective nonlinearity is also studied; however, its effect on the particle transport is mild.

Generalized gyroviscous force and its effect on the momentum balance equation
View Description Hide DescriptionThe perturbed nonlinear gyroviscous force ( , where is the gyroviscous stress tensor) is calculated through order δ^{2} [δ∼(1/Ω)(∂/∂t), k _{⊥}ρ∼δ^{1/2}, where Ω is the particle gyrofrequency, k _{⊥}ρ denotes the finite Larmor radius effect] by solving the stress tensor evolution equation. The result shows that mostly cancels the diamagnetic convective term (where V _{*} is a generalized total diamagnetic flow) in the momentum balance equation. This formula generalizes the usual, widely used ‘‘gyroviscous cancellation’’ [in which the cancels mostly the total time derivative of the diamagnetic flow ( )] to include temperature variations and a perturbed stress tensor drift. It is proved that when temperature variations are neglected, the new formula reduces to the conventional form. When the temperature variations are considered, the new formula is simpler and more rigorous than the conventional one. A new polarization drift formula deduced from the new gyroviscous force is also derived.

Analytic expressions for mode conversion in a plasma at the peak of a parabolic density profile
View Description Hide DescriptionFor mode conversion in an unmagnetized plasma with a parabolic density profile of scale length L, analytic expressions, in terms of parabolic cylinder functions, for the energy flux coefficients (reflection, transmission, and mode conversion) and the fields for both the ‘‘direct’’ problem (incident electromagnetic wave converting to a Langmuir wave) and the ‘‘inverse’’ problem (incident Langmuir wave converting to an electromagnetic wave) are derived for the case where the incident wave frequency ω matches the electron plasma frequency ω_{ p } at the peak of the density profile. The mode conversion coefficient for the direct problem is equal in magnitude to that of the inverse problem, and the corresponding reflection and transmission coefficients satisfy energy conservation. In contrast to the linear profile problem, the conversion efficiency depends explicitly on the value of the collision frequency (in the cold, collisional limit) or electron temperature (in the warm, collisionless limit), but a transformation of parameters relates the results for these two limits.

The effect of collisions in maintaining a non‐Maxwellian plasma distribution in a spherically convergent ion focus
View Description Hide DescriptionIt is shown that, for a moderately long time scale, two‐body collisions can actually help in maintaining a non‐Maxwellian ion velocity distribution in a plasma. This effect occurs when the ions have orbits large enough to transit spatial regions with different physical parameters in a single collision time. The effect is applied to ions confined in a spherically convergent ion focus (SCIF), where collisions at the edge of a spherical potential well help maintain convergent non‐Maxwellian flow in the rest of the device.

Forced reconnection and current sheet formation in Taylor’s model
View Description Hide DescriptionThe problem of forced reconnection in Taylor’s model, first investigated by Hahm and Kulsrud [Phys. Fluids 28, 2412 (1985)], is revisited. After the linear phases A, B, and C, described by Hahm and Kulsrud, the plasma enters a nonlinear phase W with a current sheet, described by Waelbroeck [Phys. Fluids B 1, 2380 (1989)]. After the W phase, the plasma passes into the Rutherford regime. The reconnected flux at the separatrix increases monotonically from zero to its asymptotic value in the Rutherford regime. Analytical expressions for the reconnected flux and the island width are given.

Interferometry on the compact toroid formation experiments at Phillips Laboratory
View Description Hide DescriptionInterferometric measurements of the line‐averaged free electron density of plasma compact toroids as part of preliminary toroid formation experiments are presented. These experiments are in preparation for compact toroid acceleration experiments to be performed at Phillips Laboratory’s High Energy Plasma Division. The primary instrument used was a Michelson (double pass) interferometer operating with a HeNe laser at a wavelength of 633 nm. Density measurements of hydrogen and argon plasmas are correlated with magnetic field and spectroscopic measurements to characterize the toroid mass, velocity, and impurity distribution.

Continuum damping of low‐n toroidicity‐induced shear Alfvén eigenmodes
View Description Hide DescriptionThe effect of resonant continuum damping is investigated for the low‐mode‐number, toroidicity‐induced, global shear Alfvén eigenmodes, which can be self‐excited by energetic circulating alpha particles in an ignited tokamak plasma. Resonant interaction with the shear Alfvén 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 Alfvén resonance damping rate from the ideal fluid zeroth‐order shear Alfvén 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 Alfvén 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 Alfvén resonance damping to estimate the saturation amplitude level predicted from quasilinear theory.

Energy extraction from intense electron beams in hot plasmas by electron cyclotron waves
View Description Hide DescriptionA hot weakly relativistic plasma in a magnetic field is considered in the slab description with an electron distribution which is the sum of an isotropic Maxwellian function plus a second shifted Maxwellian function describing an electron population drifting along the external magnetic field. The interaction of such a beam–plasma system with ordinary and extraordinary electron cyclotron waves propagating in a direction perpendicular to the external magnetic field and to the beam is studied. The physical system is analyzed in the geometrical optics limit and also by means of a wave‐dynamical treatment. A resonance layer due to the beam appears where the condition ω_{ ce }=γω is verified, where γ=(1−v _{0} ^{2}/c ^{2})^{−1/2} and v _{0} is the beam drift velocity. If the electron beam has a sufficiently high density, an ordinary wave coming from the high‐field side can be converted into an extraordinary mode precisely at ω_{ ce }=γω, propagating then up to the upper‐hybrid layer where electron Bernstein waves are generated. When the beam density exceeds a definite threshold value, a multiple conversion process also directly involving the Bernstein waves takes place at ω_{ ce }=γω. The excitation of the Bernstein modes enhances the amplification of the injected waves. For definite values of the relevant parameters of the problem, the injected electromagnetic waves are strongly amplified, the electrostatic field associated to the Bernstein mode bunching the electrons of the beam in a direction perpendicular to that along which the beam is flowing.

Stability of resistive and ideal ballooning modes in the Texas Experimental Tokamak and DIII‐D
View Description Hide DescriptionThe stability of drift resistive ballooning modes is examined using the reduced Braginskii equations, which include electron temperature and magnetic fluctuations. The Texas Experimental Tokamak (TEXT) [Phys. Fluids B 2, 2879 (1990)] edge plasma is found to be unstable for a broad range of mode numbers. For low mode numbers (m<70), the plasma is unstable to the drift resistive ballooning mode, which has a growth rate that scales linearly with the resistivity (γ∼η). As the mode number increases, a transistion is found to the resistive ballooning mode with the usual scaling of γ∼η^{1/3}. A similar analysis is made for parameters from DIII‐D [Phys. Fluids B 2, 1405 (1990)] for both the L mode and the H mode. It is found that the L mode is unstable to the resistive mode, but β ≊ β_{ I }/6, where β_{ I } is the critical β for the ideal instability. The H mode is weakly unstable to the resistive mode and is only about a factor of 2 below the β threshold for the more robustly unstable ideal mode.

Interaction of bootstrap‐current‐driven magnetic islands
View Description Hide DescriptionThe formation and interaction of fluctuating neoclassical pressure‐gradient‐driven magnetic islands is examined. The interaction of magnetic islands produces a stochastic region around the separatrices of the islands. This interaction causes the island pressure profile to be broadened, reducing the island bootstrap current and drive for the magnetic island. A model is presented that describes the magnetic topology as a bath of interacting magnetic islands with low to medium poloidal mode number (m≂3–30). The islands grow by the bootstrap current effect and damp due to the interaction of the magnetic islands. The effect of this sporadic growth and decay of the islands (‘‘magnetic bubbling’’) is not normally addressed in theories of plasma transport due to magnetic fluctuations. The nature of the transport differs from statistical approaches to magnetic turbulence since the transfer of particles and heat occur as discrete, abrupt events, not as a random walk process. This model suggests that tokamak experiments have relatively short‐lived, coherent, long‐wavelength magnetic oscillations present in the steep pressure‐gradient regions of the plasma.

Toroidal kinetic η_{ i }‐mode study in high‐temperature plasmas
View Description Hide DescriptionA new kinetic integral equation for the study of the ion‐temperature‐gradient‐driven mode in toroidal geometry is developed that includes the ion toroidal (curvature and magnetic gradient) drift motion ω_{ D }, the mode coupling from finite k _{∥} due to the toroidal feature of the sheared magnetic configuration. The integral equation allows the stability study for arbitrary k _{∥} v _{ i }/(ω − ω_{ D }) and k _{⊥} ρ_{ i }. A systematic parameter study is carried out for the low β circular flux surface equilibrium. Possible correlations between the unstable mode characteristics and some experimental results such as the fluctuation spectrum and the anomalous ion thermal transport measurements are discussed.

The effect of the radial electric field on the L–H transitions in tokamaks
View Description Hide DescriptionIt is well known that usual assumptions of neoclassical theory become invalid if very large gradients occur at the plasma edge. Therefore neoclassical theory of plasma rotation in tokamaks is revisited in order to account for anomalous transport driven by a turbulence. It is shown that this model yields both steep and gradual profiles for the poloidal rotation velocity at the edge corresponding to the H and L regimes of confinement, respectively. Results and conclusions are focused on experiments employing the biased electrode technique. Regimes with fast poloidal rotation in excess of poloidal sound speed are considered with the emphasis on relaxation.

Resistive toroidal stability of internal kink modes in circular and shaped tokamaks
View Description Hide DescriptionThe linear resistive magnetohydrodynamical stability of the n=1 internal kink mode in tokamaks is studied numerically. The stabilizing influence of small aspect ratio [Holmes et al., Phys. Fluids B 1, 788 (1989)] is confirmed, but it is found that shaping of the cross section influences the internal kink mode significantly. For finite pressure and small resistivity, curvature effects at the q=1 surface make the stability sensitively dependent on shape, and ellipticity is destabilizing. Only a very restricted set of finite pressure equilibria is completely stable for q _{0} < 1. A typical result is that the resistive kink mode is slowed down by toroidal effects to a weak resistive tearing/interchange mode. It is suggested that weak resistive instabilities are stabilized during the ramp phase of the sawteeth by effects not included in linear resistive magnetohydrodynamics. Possible mechanisms for triggering a sawtooth crash are discussed.

The role of edge current density on kink mode stability and its implication for magnetohydrodynamic activity associated with edge localized modes
View Description Hide DescriptionThe stability of a plasma equilibrium to the external kink mode is shown to be dependent on the details of the current density near the plasma edge and the value of the safety factor, q, at the edge. The buildup of the current density near the plasma edge is shown to decrease the shear in the safety‐factor profile and lead to destabilization of the kink mode. Tokamak plasmas in the high confinement mode of operation have high current density near the edge and are known to exhibit activity referred to as edge localized modes. An important feature of this activity is intermittent bursts of magnetohydrodynamic (MHD) activity near the plasma edge. A model is proposed for the MHD activity in terms of the external kink mode. The role of the plasma geometry and equilibrium profiles is discussed.