Inverse Aspect Ratio Research Articles

Base Excitation of Rigid Bodies. II: Periodic Slide‐Rock Response

An approximate closed‐form solution is developed for a single‐mode, steady‐state slide‐rock response resulting from a harmonic ground acceleration. The approximate solution is developed using the method of slowly varying parameters and is valid for a rectangular block undergoing small angles of rotation at the frequency of base excitation. Impacts with the foundation are assumed to be perfectly plastic and frictional impulses are included. Periodic solutions are found to exist in general only for relatively high amplitudes of ground acceleration and friction less than the inverse aspect ratio of the block. The rock component of the response is sensitive to changes in aspect ratio and friction and insensitive to changes in ground acceleration. The slide component of response is approximately equal to the amplitude of ground displacement and is insensitive to changes in friction and aspect ratio. Results compare favorably to those obtained by numerical integration. The accuracy of the approximate solution i...

Relaxation rate of poloidal rotation in the banana regime

The time-dependent parallel viscosity (B. Del Pi a) is calculated in an axisymmetric toroidal plasma with arbitrary aspect ratio. By using this parallel viscosity, the relaxation rate nu p of the poloidal rotation in Tokamak plasmas is obtained in the banana regime. The analytical fit for this relaxation rate is also computed. In large-aspect-ratio Tokamaks, this relaxation rate nu p is found to be of the order nu ii square root epsilon and the previously-derived result nu p approximately nu ii by Shaing et al. (1989) is shown to be incorrect, where nu ii and epsilon are the ion-ion collision frequency and the inverse aspect ratio.

Brillouin limit for non-neutral plasma in inhomogeneous magnetic fields

The effect of magnetic field inhomogeneity upon the maximum density of non-neutral plasma that can be confined in hydrostatic equilibrium under the combined influence of electrostatic and magnetostatic fields is explored in two cases of azimuthally symmetric geometries: first, a plasma in an arbitrary vacuum poloidal magnetic field, B=Br(r,z)r̂+Bz(r,z)ẑ, and second, a toroidal plasma in a toroidal vacuum magnetic field. For the poloidal field case, it is demonstrated that the Brillouin limit on number density can be achieved only with a homogeneous axial magnetic field. For the toroidal field case, it is demonstrated that although the Brillouin limit does not provide a limitation on the local number density confined, it nevertheless does provide an upper limit on the total amount of non-neutral plasma confined in equilibrium. It is also demonstrated that when viscosity is present in the toroidal field case, its effect to first order in the inverse aspect ratio can be opposed by a readjustment of only the external electrostatic potential.

Theory of the m=1 kink mode in toroidal plasma

The energy principle of ideal magnetohydrodynamics (MHD) is used to study the ideal MHD stability of the m=1 internal kink mode in a toroidal plasma. The equilibrium configurations that are considered allow for a broad region where the safety factor q is close to unity. This region may extend to the magnetic axis, or may be a singular layer. The minimization of the energy functional yields an implicit equation for the growth rate that can be solved by simple numerical means. The examples that are treated numerically retain the essential features of experimentally expected q profiles. The growth rate depends very sensitively on the q profile close to unity and increases with the width of the q≊1 layer. The highest values are of the order of the inverse aspect ratio ε divided by the poloidal Alfvén time. Nonmonotonic profiles with q>1 on axis are more unstable than monotonic q profiles with q<1. In the latter case, the mode tends to be localized in the q≊1 region instead of in the center.

Axisymmetric electromagnetic eigenmodes of plasma-filled toroidal resonators

The axisymmetric electromagnetic eigenmodes of a toroidal resonator of a circular cross-section filled with a homogeneous cold magnetized plasma are analyzed. The static magnetic field of the model considered has only a toroidal component, falling off inversely as the major radius. Since the basic equations are not separable in toroidal coordinates, they are solved by a perturbation method. Eigenfrequencies and eigenfunctions are calculated up to the second order in the inverse aspect ratio.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers

We derive the phase diffusion and mean drift equations for the Oberbeck–Boussinesq equations in large-aspect-ratio containers. We are able to recover all the long-wave instability boundaries (Eckhaus, zigzag, skew-varicose) of straight parallel rolls found previously by Busse and his colleagues. Moreover, the development of the skew-varicose instability can be followed and it becomes clear how the mean drift field conspires to enhance the necking of phase contours necessary for the production of dislocation pairs. We can calculate the wavenumber selected by curved patterns and find very close agreement with the dominant wavenumbers observed by Heutmaker &amp; Gollub at Prandtl number 2.5, and by Steinberg, Ahlers &amp; Cannell at Prandtl number 6.1. We find a new instability, the focus instability, which causes circular target patterns to destabilize and which, at sufficiently large Rayleigh numbers, may play a major role in the onset of time dependence. Further, we predict the values of the Rayleigh number at which the time-dependent but spatially ordered patterns will become spatially disordered. The key difficulty in obtaining these equations is the fact that the phase diffusion equation appears as a solvability condition at order ε (the inverse aspect ratio) whereas the mean drift equation is the solvability condition at order ε2. Therefore, we had to use extremely robust inversion methods to solve the singular equations at order ε and the techniques we use should prove to be invaluable in a wide range of similar situations. Finally, we discuss the introduction of the amplitude as an active order parameter near pattern defects, such as dislocations and foci.

Natural elongation and triangularity of tokamak equilibria

Quasianalytic formulas are calculated for the elongation κ and triangularity δ of the plasma surface of a free-boundary tokamak equilibrium. The final results give κ and δ as functions of five quantities: the inverse aspect ratio ε, the poloidal beta βp, the internal inductance li, and the quadrupole and hexapole moments of the externally applied field. The agreement with numerically computed equilibria is found to be quite good when A≥3, κ≤1.5, and δ≤0.2 and when the plasma is limited by the vacuum vessel wall and not diverted by the presence of a separatrix on the plasma surface.

The mechanism of sawtooth precursors and sawtooth relaxations

It is shown that certain specific properties are requested from the current density profile in order to arrive at a coherent theoretical explanation of the sawtooth phenomenon in tokamaks: (1) the shear ŝ must be small at the q=1 surface (hence the current density profile has a plateau or shoulder in the neighborhood of q=1); (2) J′ varies abruptly not far away from q=1 (in particular, the outer edge of the plateau is close to q=1). Indeed, when these conditions are satisfied, the contribution to the jump of the derivative of the magnetohydrodynamic (MHD) solution ([Ψ′]r1) from those terms that are proportional to ε2 (ε is the inverse aspect ratio) is not negligible. It is, moreover, opposite in sign to the usual contribution ([Ψ(0)′]r1∝ŝ1) so that the inverse of the logarithmic jump 1/Δ′∝Ψr1/[Ψ′]r1 can now take on either positive values (ŝ1&amp;gt;ŝ1,c), or negative ones (ŝ1&amp;lt;ŝ1,c), or can ‘‘diverge’’ (1/Δ′→±∞ for ŝ1→ŝ1,c). The growth rate reaches very large values when 1/Δ′→−∞ (i.e., [Ψ′]r1→0−) and its rapid acceleration observed shortly before the crash can now be explained: The ontology of the sawtooth rise and crash requires indeed that ŝ1 always be smaller than ŝ1,c (hence, 1/Δ′&amp;lt;0 opposite to conventional wisdom!) and approach this value from below at the onset of the collapse. Current density [Plasma Physics and Controlled Nuclear Fusion Research 1986 (IAEA, Vienna, 1987), Vol. 1, p. 263] as well as density [Controlled Fusion and Plasma Physics 1989 (EPS, Geneva, 1989), Vol. 13 B, II-565] plateaus and shoulders have been observed in TEXTOR discharges. This and other comparisons with experiment, including the acceleration rate of the perturbation, are discussed. Analysis of the particle trajectories in the perturbed magnetic structure yields an amplitude criterion for the onset of collapse that agrees well with observations on TEXTOR. The ontology of the crash phase requires an ambipolar electric field to build up inside the q=1 surface so that only suprathermal electrons are lost. This enables the calculation of ratios of the density, current density, and electron temperature crashes; these are found to agree with the measured ratios. Theory finally suggests that the sawtooth limitation of the central temperature is not unavoidable.

Eigenmodes for electromagnetic waves propagating in a toroidal cavity

A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Toroidal ion temperature gradient driven modes with dissipation and trapped electron effects

An eigenvalue equation is derived for ion temperature gradient driven modes in a toroidal geometry without any expansion in inverse aspect ratio and including the effect of dissipative trapped electrons and electron-ion collisional dissipation. In the electrostatic limit a simple quadratic dispersion relation is obtained. An upshift of the density scale length stability threshold and a downshift of the ηi stability threshold due to electron trapping is demonstrated.

Toroidal effects on current driven modes in tokamaks

The low beta stability of a toroidal circular cross-section tokamak is re-analysed. Although the structure of the unstable modes can be considerably modified by toroidal coupling, the overall stability picture is well represented by the ‘straight tokamak model’, with the major exception that the internal kink, although stabilized by toroidal effects, still has a free-boundary counterpart – the toroidal kink mode. The toroidal kink is unstable for q0 < qc ≤ 1 and is destabilized by toroidal coupling, having a growth rate γ that scales with inverse aspect ratio a/R as γ – (a/R)2[qs], where [qs] is the integer part of the edge safety factor qs.

Comments on ‘‘A fully toroidal fluid analysis of the magnetohydrodynamic ballooning mode branch in tokamaks’’ [Phys. Fluids 3 1, 359 (1988)

It is shown that, in the compressible limit considered by Andersson and Weiland, the magnetohydrodynamic ballooning mode disappears if εn (the pressure gradient inverse aspect ratio)≳0.14 irrespective of plasma pressure.

Ideal stability of cylindrical plasma in the presence of mass flow

The ideal stability of cylindrical plasma with mass flows is investigated using the guiding center plasma (GCP) model of Grad [Proceedings of the Symposium on Electromagnetic and Fluid Dynamics of Gaseous Plasmas (Polytechnic Inst. of Brooklyn, New York, 1961), p. 37]. For rotating plasmas, the kinetic treatment of the parallel motion in GCP gives significantly different results from the fluid models, where the pressures are obtained from equations of state. In particular, GCP removes the resonance with slow magnetoacoustic waves and the loss of stability that occurs in magnetohydrodynamics (MHD) for near-sonic flows. Because of the strong kinetic damping of the sound waves in an isothermal plasma, the slow waves have little influence on plasma stability in GCP at low beta. In the large aspect ratio, low-beta tokamak ordering, Alfvénic flows are needed to change the ideal GCP stability significantly. At lowest order in the inverse aspect ratio, flow can be favorable or unfavorable for stability of local modes depending on the profiles, but external kinks are always destabilized by flow if the velocity vanishes at the edge. For high-beta, reversed field pinch equilibria, numerical computations show that flow can be stabilizing for local modes, but external modes are destabilized by flow. In three dimensions, the MHD equilibrium problem becomes hyperbolic for arbitrarily small flows across the magnetic field, whereas the GCP equilibrium equation remains elliptic for sub-Alfvénic flows.

Pressure gradient-driven modes in finite beta toroidal plasmas

When the ion temperature gradient is finite, the ideal kinetic theory plasma destabilizes the MHD mode below the critical beta of MHD theory. For a temperature gradient eta i>2/3 and inverse aspect ratio epsilon (Ti)<0.35 the electrostatic toroidal eta i-mode is unstable. The coupling between the two modes is investigated by solving the fourth-order system describing the shear Alfven-drift wave and the ion acoustic wave in the finite beta Tokamak. The ion kinetic velocity integral including the gyroradius effects and the ion magnetic drift resonances are used to obtain gamma k(k,q, beta , eta i,s, epsilon n,q, tau ) and gamma k/(kx2) for the modes. The study emphasizes the beta - and q-dependence of the transport associated with gamma k/(kx2).

The stretching of a thin viscous inclusion and the drawing of glass sheets

A thin, two-dimensional inclusion of a highly viscous fluid is deformed by an external Stokes flow—a suggested mechanism by which the Earth’s oceanic crust is entrained and thinned by the mantle. The method of matched asymptotic expansions is applied with the small expansion parameter being the inverse aspect ratio of the inclusion. The kinematic condition and continuity of shear and normal stresses lead to boundary conditions for the biharmonic equation governing the outer flow in the two cases of stretching by a pure shear and by a simple shear flow. The evolution of the inclusion can then, in principle, be determined. The equations governing the drawing of glass sheets may be derived as a limiting case, and any asymmetry in the centerline of the sheet is deformed over a short time scale before stretching occurs. Several examples are computed, showing how the straightening of the sheet can be slowed down by constraints on the gradients at the two ends. It is shown that an inclusion with smooth initial data cannot break up in a finite time.

Localized resistive instabilities in general toroidal configurations, with applications to INTOR equilibria

A useful method for studying localized resistive instabilities in toroidal equilibria at and near the ideal stability limit is presented. In particular, a valuable relation is given explicitly which allows direct calculation of the resistive growth rate at the ideal ballooning limit in terms of line-averaged equilibrium quantities. The method, which is valid not only for axisymmetric equilibria but also for toroidal, nonsymmetric 3-D configurations, is derived without making any of the usual restricting assumptions such as small inverse aspect ratio, low pressure etc. The actual evaluation of stability criteria and dispersion relations is illustrated by studying the stability of typical INTOR equilibria with respect to localized ideal and resistive modes. Configurations which are stable with respect to ideal ballooning modes (and are therefore also Mercier stable) are shown to be unstable with respect to resistive ballooning instabilities, and the corresponding growth rates are calculated. Considerable growth rates ( gamma -1 approximately 1 ms) are found at values of ( beta ) that are 80-90% of the critical value. Below these values of ( beta ), the growth rates decrease very rapidly. A subject of more general interest is also treated, namely the situation which appears when the Mercier exponent sM exceeds the value 1/2, and a new dispersion relation for a model equation is derived.

ECRH current drive in Tokamak plasmas

The current drive by electron cyclotron resonance heating (ECRH) is investigated in a Tokamak with arbitrary aspect ratio. The r.f. power is assumed to be weak and the wave-induced current is calculated by using an adjoint equation to the steady-state linearized Fokker-Planck equation with a quasi-linear diffusion term. The trapped electrons and the modification of electron cyclotron resonance condition by the relativistic mass increase are shown to have significant effects on the efficiency of this current drive. The efficiency strongly depends on the values of the parallel velocity u0 of resonant electrons, the inverse aspect ratio in , the poloidal angle theta 0 of the absorption point, and the relativistic parameter S which represents the strength of the relativistic correction to the resonance condition. It is also pointed out that the electron-electron collisions enhance the trapping effects compared with those of the Lorentz gas model.

On tokamak equilibrium

Large-aspect-ratio tokamak equilibrium is studied, including effects, such as ellipticity of the flux surfaces, of second order in the inverse aspect ratio. To facilitate the interpretation of experimental data, the analysis uses simple coordinates that are readily evaluated, rather than co-ordinates based on the exact flux surfaces. An explicit formula for the safety factor, including second-order terms, is presented, along with a brief discussion of rotational effects. The analysis assumes near-circularity of the outermost closed surface and low plasma pressure.

Experimental investigation of parallel-walled and divergent-walled open thermosyphons

Natural convection experiments are performed for plane-walled, slot-like open thermosyphons having either parallel principal walls or divergent principal walls. The fluid in the thermosyphon and its surroundings is water ( Pr ≅ 5). When the two principal walls are heated and maintained at the same uniform temperature which exceeds that of the surroundings, unstable operation is encountered. The instability is believed due to inherent obstructions to the transport of cool fluid from the surroundings to the thermosyphon, such transport being needed to make up for the hot fluid discharged from the thermosyphon to the surroundings. Steady-state Nusselt number results are obtained for the one-sided-heated thermosyphon. For the parallel-walled case, a universal Nusselt number correlation is achieved as the 0.25 power of the product of the inverse aspect ratio and the Rayleigh number. This correlation yields heat transfer coefficients that are independent of the interwall spacing. Furthermore, aside from a 7% difference in a multiplicative constant, the correlation is identical to that for natural convection in a one-sided-heated vertical channel open at both the top and bottom. Divergence of the principal walls of the one-sidedheated thermosyphon has only a small effect on the Nusselt number, with departures of no more than 5% from those for the parallel-walled case.

Spheromak tilt stabilization with an energetic particle component

The tilt and shift stability of a spheromak plasma in the presence of a large orbit ion-ring component is determined by evaluating the magnetic interaction of the plasma with the ion ring and the external coils. Neither the ion-ring toroidal current nor the ring inverse aspect ratio is required to be small. The influence of conducting walls is considered to be negligible, as is appropriate for compression experiments. The plasma tilt and shift stability has been evaluated for a comprehensive set of numerically generated, self-consistent, hybrid spheromak/ion-ring equilibria. Stable equilibria are found over a considerable range of the parameters. The tilt stabilization of the plasma typically requires that more than about 25% of the total toroidal current be carried by the ion ring.