Volume 24, Issue 7, July 1981
Index of content:

Motion of two superposed viscous fluids
View Description Hide DescriptionThe initial‐value problem associated with the small‐amplitude waves on the interface between two superposed viscous fluids is studied in the stable and in the unstable case (Rayleigh–Taylor instability). The results are illustrated with examples obtained by numerical and analytic inversion of the Laplace‐transformed solution for the wave amplitude. The diffusion of vorticity from the free surface into the fluids is also shown.

Stability of shear flow of stratified fluids with fine dust
View Description Hide DescriptionThe linear stability of plane parallel shear flows of a stably stratified incompressible fluid laden with uniformly distributed fine dust particles is studied. It is shown that the Miles criterion for stability, the Rayleigh–Synge criterion for instability, and Howard’s semicircle theorem can be extended under the assumptions that the mass concentration is very small and that the relaxation time of the dust particles is very much less than the time scale characterizing the basic flow. The effect of fine dust is found to increase the region of instability. In addition, a semi‐ellipse criterion for instability has been given as an improvement over Howard’s semicircle criterion.

A bubble in an axially symmetric shear flow
View Description Hide DescriptionThe shape of a three‐dimensional bubble in an axially symmetric shear flow of an inviscid incompressible fluid is calculated numerically. The shape depends upon a single dimensionless constant related to the Weber number. It is found that there is a maximum Weber number above which there is no steady solution.

Evolution of the centerline probability density function of temperature in a plane turbulent wake
View Description Hide DescriptionMeasurements are presented of the evolution of the probability density function of the temperature fluctuation along the centerline of the wake of a heated circular cylinder in the range 4⩽x/d⩽300.

Interface related velocities in turbulent plane jets
View Description Hide DescriptionThe velocities of the flow in the vicinity of the interface, of the apparent lateral motion of the interface, and of entrainment are compared for a plane two‐dimension jet.

Ponderomotive effects in collisionless plasma: A Lie transform approach
View Description Hide DescriptionA new method for the kinetic analysis of ponderomotive effects in collisionless plasma is presented. This method involves the application of the Lie‐transform perturbation technique to the Hamiltonian formulation of the Vlasov equation. Basically, a new system, in which the high frequency oscillations are absent, is found. In this system the distribution function evolves according to a ponderomotive Hamiltonian, which is the kinetic generalization of the ponderomotive potential. It is shown that the ponderomotive Hamiltonian can easily be determined from the well‐known linear susceptibility. This formalism is used to calculate several new results. Among these results are the general formula for the quasi‐static density perturbation produced by a hot magnetoplasma wave, a generalization of previous formulas for the laser‐generated quasi‐static magnetic field, and the general formula for the ponderomotive gyrofrequency shift produced by an electromagnetic wave propagating at an arbitrary angle.

Theory of mode‐conversion in weakly inhomogeneous plasma
View Description Hide DescriptionThe theory of pair‐wise coupled modes excited at a real frequency ω in a weakly nonuniform plasma (coordinate x) is developed on the basis of general local dispersion relations D(k,z) = 0 (such as from the Vlasov equation), which define a many‐valued mapping of the real axis x = Rez onto the complex plane of the wavenumber k. Mode coupling, by definition, is the analytical continuation of the branch of the mapping and only occurs at the branch points. This requires that the z plane be cut along contours C _{ b } given by D(k _{ c },C _{ b }) = 0, ∂D(k _{ c },z)/∂k = 0, where z traces a contour passing through the branch points. The coupled modes can be analyzed by expanding the dispersion relation to second order in k around the saddle points and along the lines k _{ c }(x), yielding a system of dispersion relations corresponding to unambiguous second‐order differential equations, guaranteed to have the right turning point behavior.

Collisional drift waves in a plasma with electron temperature inhomogeneity
View Description Hide DescriptionA fluid theory of collisional electrostatic drift waves in a plasma slab with magnetic shear is presented. Both electron temperature and density gradients are included. The equations are solved analytically in all relevant regions of the parameter space defined by the magnetic shear strength and the perpendicular wavelength and explicit expressions for the growth rates are given. For shear strengths appropriate for present‐day tokamak discharges the temperature gradient produces potential wells which localize the mode in the electron resistive region, well inside the ion sound turning points. Mode stability arises from a competition between the destabilizing influence of the time dependent thermal force and the stabilizing influence of electron energy dissipation. Convective energy loss is not important for shear parameters of present‐day fusion devices.

Drift modes in axisymmetric tandem mirrors
View Description Hide DescriptionThe drift mode analysis of the tandem mirror is developed for a large aspect ratio, axisymmetric model of the equilibrium. The axial drift wave eigenmodes are shown to change character as the plasma pressure varies with respect to the inverse aspect ratio. In the high beta regime the drift modes are transformed into a finite frequency convective cell and the flute‐like ion drift wave. Quasi‐linear formulas are given for the anomalous radial losses.

Finite Larmor radius theory of resistive instabilities
View Description Hide DescriptionThe hybrid‐kinetic model, previously used to investigate stability properties of collisionless high‐beta plasmas characterized by finite ion gyroradius, is generalized to include the effects of finite resistivity and gravitation (to simulate toroidal field curvature). Possible applications of the present formalism include investigations of finite ion Larmor radius effects on the stability of screw‐pinch and tokamak equilibria against ideal and resistive interchange instabilities as well as tearing modes. A simple example of such an application is carried out for the case of local slow resistive interchange modes within the context of a near‐theta‐pinch ordering.

Finite ion Larmor radius stabilization of m = 1 modes of a high‐β screw pinch
View Description Hide DescriptionIn the near ϑ‐pinch limit, a radially diffuse high‐β screw pinch may be rendered completely stable through a combination of finite ion Larmor radius effects and wall effects.

Interaction of lower hybrid wave fields with drift‐cyclotron loss‐cone instability
View Description Hide DescriptionThe dispersion relation for the drift‐cyclotron loss‐cone mode in the presence of the lower hybrid wave is calculated using both electrostatic and finite β models. It is found that lower hybrid wave fields with frequency ω_{0} can stabilize the mode if ω_{ l h }<ω_{0}<ω_{+}, or ω_{0}<ω_{−}<ω_{ l h }, where ω_{ l h } = ω_{ p i }/(1+ω^{2} _{ p e }/ω^{2} _{ c e })^{1/2}, ω_{±} = [±A+(A ^{2}+4ω^{2} _{ l h })^{1/2}]/2, and A = ω^{2} _{ l h }ε/ω_{ c i } k. If the plasma β is greater than a critical value β_{ c }, there is another stabilization region, namely, ω_{0}≳δω_{ l h }, where δ is a numerical constant. Even though the stabilization effect is small in this region, the lower hybrid wave frequency for electron heating should be in this region in order to avoid enhancing the particle loss rate.

Effects of the fast wave on the drift‐cyclotron loss‐cone mode
View Description Hide DescriptionParticle orbits in the presence of the fast wave fields are calculated. It is found that the resonant interaction of ions with the wave can be neglected at ω_{0} = 2ω_{ c i }, provided that the wave electric field strength is less than a critical value E _{ c }( = 4k _{∥} T/e). The dominant effect on the dispersion relation of the drift‐cyclotron loss‐cone mode is thus found to be due to the nonresonant effects of the wave on electrons and ions. It is found that a fast wave with ω_{0} = 2ω_{ c i } destabilizes the mode. For ω_{0}<2ω_{ c i }, fast waves can stabilize the mode for certain values of the plasma β, which is calculated numerically.

Turbulence driven tearing modes in a tokamak plasma
View Description Hide DescriptionThe influence of a random spectrum of lower hybrid waves on the evolution of tearing modes in a tokamak plasma is investigated. For rather modest levels of fluctuations, it is found that new tearing instabilities driven by turbulence are excited with significantly enhanced growth rates and comparable real frequencies. The growth rates for the new m = 1 and m = 2 modes vary as fractional powers of τ_{ R } and δ, where τ_{ R } is the turbulence induced diffusion time and δ is the ratio of Alfvénic time to the ion gyration period.

Analytic solution for a weakly rotating, collisionless z pinch
View Description Hide DescriptionThe system of time independent Vlasov and Maxwell’s equation is solved for a cylindrically symmetric z pinch including weak rotation of both electrons and ions. The solution clearly shows the effect of rotation on the density profile and azimuthal magnetic field. Limitations on the rotation frequency and radius are found by insisting on self consistency of the approximation scheme.

Kinetic effects on ballooning modes in mirror machines
View Description Hide DescriptionA general procedure for examining the influence of kinetic effects on the stability of magnetohydrodynamic ballooning modes in mirror machines is presented. In particular, the basic kinetic ballooning mode equation for a nonaxisymmetric, arbitrary beta system with anisotropic pressure is derived. Considering a long‐thin equilibrium typical of the tandem mirror, it is shown that this governing eigenmode equation reduces to a simple form independent of wave‐particle resonant effects.

Viscous plasma flow in a multiple‐mirror configuration
View Description Hide DescriptionThe steady‐state, axial plasma confinement by a multiple‐mirror device is studied in the ’’viscous fluid regime,’’ λ/l _{ m }≪1 (but not negligible), where λ is the ion‐ion mean‐free‐path and l _{ m } is the scale length of magnetic field variations. An approximate analytical solution is obtained by averaging over the rapid variations caused by the individual mirrors and compared to a numerical solution without averaging. A smooth transition is found, with increasing values of λ/l _{ m }, from sonic flow to a diffusive flow. Studies are made of the variation in density profiles and confinement times versus λ/l _{ m }, mirror ratio, relative mirror width, and number of mirror cells. The viscous fluid regime is found to have the same characteristics as the ideal multiple‐mirror regime for λ/l _{ m }→1.

Analytic equilibria with quadrupole symmetry in the paraxial limit
View Description Hide DescriptionMirror equilibria for arbitrary mirror ratio and flux‐tube eccentricity are obtained to leading order in the plasma pressure (beta expansion) in the paraxial limit (axial scale lengths long compared with radial scale lengths). The solutions are given in terms of quadratures over known functions. The theory is applied to a tandem‐mirror configuration.

Spheromak tilting instability in cylindrical geometry
View Description Hide DescriptionThe internal tilting instability in a force‐free spheromak plasma in cylindrical geometry is examined. It is found that this instability, originally found in spherical geometry, also occurs in cylindrical geometry. The analysis proceeds by first demonstrating that if no mode rational surface is present in the plasma, a necessary and sufficient condition for ideal magnetohydrodynamic instability is that there exist a solution to ∇×B_{ m } = μ_{ m }B_{ m }, where B_{ m }∼exp(i mϑ), with μ_{ m }<μ_{0}. Solutions to this equation are investigated using two approaches, by a series expansion and by a numerical solution of a modified set of linearized magnetohydrodynamic equations. The eigenvalue for the m = 1 mode satisfies μ_{1}<μ_{0} for L/a≳1.67. Since no mode rational surface exists for this elongation, an ideal magnetohydrodynamic mode, identified as the tilting mode, is unstable for these parameters. All modes with m≳1 are shown to be stable.

Anomalous toroidal field penetration in Tormac V
View Description Hide DescriptionMagnetic field penetration into a cool, collisional, magnetized plasma has been investigated in Tormac V. Magnetic probe and laser interferometer studies reveal anomalous penetration of the applied toroidal field into a plasma with an initial parallel bias toroidal field. The applied poloidal field, however, formed a well‐defined magnetic front which was effective at sweeping up particles. Strong shear in the vacuum magnetic field does not inhibit the apparent decoupling of the applied toroidal field from the applied poloidal field.