Volume 28, Issue 11, November 1985
Index of content:

Viscous damping of the oscillations of a rotating simple drop
View Description Hide DescriptionThe viscous damping of the capillary oscillations of a simple drop in rotation in air is evaluated in the small viscosity limit.

Stabilization of interchange modes in mirror plasmas by resonant coupling to ion‐cyclotron sidebands
View Description Hide DescriptionBased on a nonlinear, local mode–mode coupling relation, the resonant interaction of flute‐interchange modes with radio frequency (rf ) wave fields in the ion‐cyclotron range of frequencies (ICRF) is studied. It is shown that interchange modes can be stabilized because of resonant coupling to electrostatic ion‐cyclotron sidebands. Stabilization takes place if the frequency of the rf field is somewhat higher than the local ion‐cyclotron frequency, as observed in contemporary experiments on mirror devices.

Low Reynolds number flow over cavities
View Description Hide DescriptionLow Reynolds number flow over rectangular cavities is analyzed. The problem is posed to simulate towing tank experiments of Taneda [J. Phys. Soc. Jpn. 4 6, 1935 (1979)]. A very good agreement exists between numerical and experimental results. The dividing streamline separates from the cavity side wall below the upper corner. The separation point moves toward this corner when the aspect ratio decreases. Flow structure inside the cavity changes considerably with the aspect ratio. Only corner vortices exist in a cavity of aspect ratio W/h=4.0. A decrease of the aspect ratio leads to the enlargement and eventual merger of these vortices. The merger begins with the formation of a stagnation point separating two vortex centers inside the cavity. These vortex centers become progressively weaker and merge to form a single central vortex in a cavity of aspect ratio W/h=2.0. Further decrease of the aspect ratio results in the enlargement of the new corner vortices and their eventual merger. This process begins in a cavity of aspect ratio between 0.6 and 0.575 and is finished in a cavity of aspect ratio 0.5, where two central vortices have been identified. The further decrease of the aspect ratio leads to the sequential repetition of this process and creation of additional central vortices.

Is sedimentation container‐shape dependent?
View Description Hide DescriptionThe question is addressed as to the dependence of sedimentation of a homogeneous suspension on the shape of the container. It is demonstrated, by comparing calculations for spherical and plane geometries, that shape‐dependent contributions to the sedimentation velocity remain in the limit of infinitely distant container walls. Upon transformation from the laboratory reference frame to a local frame of reference that moves with the average volume velocity, this shape‐dependence is found to disappear.

Effects of viscosity variation on the stability of film flow down heated or cooled inclined surfaces: Long‐wavelength analysis
View Description Hide DescriptionThe linear stability of liquid film flow, when the viscosity depends exponentially on temperature, is investigated here for the case of heating or cooling from below by means of an analysis valid for long waves. It is shown that cooling stabilizes the flow, while heating destabilizes it. For the case of cooling, a cutoff Prandtl number exists, above which the flow is stable with respect to long waves.

Thermoconvective waves in a binary mixture
View Description Hide DescriptionThe propagation of undamped transverse thermal waves in a layer of binary mixture is investigated. It is shown that when the thermal diffusivity K _{ T } is equal to the mass diffusivity K _{ S }, such waves do not exist. But when K _{ T }>K _{ S }, these waves may exist provided the layer is heated from below (such that thermohaline instability does not occur) and the concentration of the solute in the mixture decreases vertically upward. The analysis also reveals the interesting result that when K _{ T }<K _{ S }, undamped waves can propagate even if the layer is heated from above provided the concentration increases vertically upward. It is further shown that as in a homogeneous fluid, weakly damped low‐frequency waves can propagate and the regions in the parameter space where such propagation is possible are delineated. High‐frequency waves are shown to be always strongly damped.

An infrared method for viscous fluids. I. Point vortices and vortex sheets in two dimensions
View Description Hide DescriptionA method of extracting the behavior of velocity/vorticity fields caused by the low‐frequency, or infrared (IR), portion of nonlinear interactions in arbitrary spatial dimensions is transcribed from the Schwinger/Fradkin representation of Green’s functions in quantum field theory to problems of viscous Navier–Stokes fluids. The general IR formalism is developed and applied to certain simple, two‐dimensional situations involving point vortices and vortex sheets. As an illustration point vortices inserted into a viscous fluid are shown to perform a finite number of revolutions about each other before they dissipate and disappear.

Statistical analysis of the anisotropy of scalar diffusion in turbulent shear flows
View Description Hide DescriptionAnisotropy of the diffusion of a passive scalar is studied statistically by means of the direct‐interaction approximation incorporated with the multiscale expansion method. The scalar flux generated by turbulent shear flows is calculated up to the second order of a scale parameter. As a result, a relation between the anisotropy of scalar diffusion and the inhomogeneity of turbulent flows is shown.

Dynamics of a gas mixture in an extended kinetic theory
View Description Hide DescriptionThe dynamics of a mixture of N rarefied gases is studied by analyzing the set of nonlinear continuity equations for the number densities. Removal and regeneration effects are also taken into account. Various physical situations are considered, corresponding to N=2 and N=3. The relevant diagrams (trajectories in the phase space and curves of the single densities as functions of time) are reported and briefly discussed on physical grounds.

The paradox of the inverted temperature profiles between an evaporating and a condensing surface
View Description Hide DescriptionThe effect of internal degrees of freedom and of a class of boundary conditions on inverted temperature profiles between evaporating and condensing parallel plates is considered. It is found that such inverted profiles are present also in the case of polyatomic species and for most (but not all) modifications of the boundary conditions.

Collisional dynamics of a strongly magnetized pure electron plasma
View Description Hide DescriptionFor a pure electron plasma in a sufficiently strong magnetic field, there is a many‐electron adiabatic invariant which constrains the collisional dynamics. For the case of a uniform magnetic field, the adiabatic invariant is the total kinetic energy associated with the electron velocity components that are perpendicular to the magnetic field (i.e., ∑_{ j } m v ^{2} _{ j⊥} /2). Were the adiabatic invariant an exact constant of the motion, no exchange of energy would be possible between the parallel and the perpendicular degrees of freedom, and the plasma could acquire and maintain two different temperatures, T _{∥} and T _{⊥}. However, an adiabatic invariant is not strictly conserved. In the present case, each collision produces an exponentially small exchange of energy between the parallel and the perpendicular degrees of freedom, and these act cumulatively in such a way that T _{∥} and T _{⊥} relax to a common value. This paper provides a calculation of the equipartition rate.

Interaction between matter and radiation at very high temperatures: I. Compton scattering
View Description Hide DescriptionThe interaction between the two components of a relativistic binary gas is investigated. Starting from the relativistic Boltzmann equation, general expressions for the rate of energy exchange, the relaxation time, and the coefficient of bulk viscosity are obtained. As a particular example, the formalism is applied to the study of a gas at very high temperatures, T<∼6×10^{9} °K, assuming that the energy exchange between matter and radiation is caused by Compton scattering; relativistic effects are fully taken into account.

Renormalized geometric optics description of mode conversion in weakly inhomogeneous plasmas
View Description Hide DescriptionConventional mode conversion theory in inhomogeneous plasmas starts with a local dispersion relation, which serves as a base for constructing a differential equation describing the waves in nearly degenerate plasma regions, where the mode conversion can take place. It will be shown, however, that the usual geometric optics perturbation scheme, which in the zero order leads to the aforementioned dispersion relation, predicts either rapid variations of the local wave vector and amplitude of the wave, or large first‐order corrections to the amplitude in nearly degenerate plasma regions. A novel, general, renormalized perturbation scheme will be suggested in order to remove this singular behavior. The new method is formulated in terms of the conventional, general plasma dielectric tensor and yields two coupled, energy‐conserving differential equations describing the mode conversion. Simple asymptotic solutions of these equations exist if the mode coupling is localized and weak. The method is applied to the classical problem of transformation of the extraordinary mode propagating in a cold magnetized plasma at small angles to the magnetic field. Generalization of the method to the case of an unreduced, multicomponent wave propagation problem is discussed.

Characteristics and structure of an axially symmetric plasma echo
View Description Hide DescriptionAn axially symmetrical spatial plasma echo is obtained theoretically by using the Vlasov–Poisson equation system. Several features of the echo are studied by making use of this echo solution. Common linear relations between exciter grid separation l and the position of echo‐amplitude maximum l _{ e } are shown, and an influence of l on the amplitude is also obtained. The radial extent of the echo and the effects of the magnetic field and the plasma temperature on the echo are discussed. Most importantly, the existence of the fine structure of the echo, such as the appearance of a side lobe (besides the main lobe) in the amplitude curve and the presence of a phase gap on the interferometric phase trace were demonstrated by their appearance in this experiment. Good agreement is obtained between theory and the experimental results concerning the side lobes and the phase gaps.

Solitary waves in a magnetic flux tube
View Description Hide DescriptionMagnetic flux tubes (slabs or cylinders) are of considerable physical interest because of their occurrence in the sun’s atmosphere. The propagation of weakly nonlinear, long‐wavelength (weakly dispersive), sound waves in both a magnetic slab and a magnetic cylinder is discussed. The slab geometry leads to the Benjamin–Ono equation, viz., ∂v/∂t+c _{ T }(∂v/∂z) +βv(∂v/∂z) +(α/π)(∂^{2}/∂z ^{2}) ×∫/^{∞} _{−∞}[v(s,t)d s/(s−z)]=0, and the cylindrical geometry yields its allied form, viz., ∂v/∂t+c _{ T }(∂v/∂z)+βv(∂v/∂z) +α’(∂^{3}/∂z ^{3})∫^{∞} _{−∞}{v(s,t)d s/ [λ^{2}+(s−z)^{2}]^{1} ^{/} ^{2}}=0, an equation known also to arise in waves on concentrated vortices in fluids. The magnetic flux tube thus provides a simple illustration of these two equations, hitherto only considered in separate physical systems.

Nonlinear saturation of single‐mode trapped electron instability
View Description Hide DescriptionThe trapped electron instability, like other types of drift waves, may appear as a single mode in many linear plasma devices. Recognizing the significant variation of density gradient over the mode localization region, a nonlocal linear analysis in slab model yields eigenfunctions for Weber‐like equations. Using three‐wave, nonresonant, two‐dimensional mode coupling based on these Weber‐like eigenfunctions, two mode‐coupling equations for the fundamental and higher‐order radial harmonics are derived. The solution of these coupled equations yields the nonlinear saturation level of the unstable mode. Mode coupling to the damped higher‐order radial modes is the principal physical saturation mechanism of the unstable fundamental mode.

Kelvin–Helmholtz instabilities of high‐velocity magnetized, anisotropic shear layers
View Description Hide DescriptionA study is made of the linear stability of finite‐thickness, anisotropic compressible shear layers v=ẑv _{ z }(x), with a parallel uniform magnetic field, B=ẑB _{0}. The stability of such sheared flows, described by the double adiabatic equations of Chew, Goldberger, and Low [Proc. R. Soc. London Ser. A 2 3 6, 112 (1956)], and involving the nonlocal coupling of the firehose and mirror modes caused by the velocity shear, is relevant in a number of astrophysical, geophysical, and space plasma configurations. The scalar perturbation quantities have the form f (x) exp[i(k _{ z } z −ωt)]. The dimensionless variables characterizing a shear layer with a given velocity profile, assumed to be a linear profile in the present work, are the sonic Mach number, M ≡ (2v _{ z m }/S _{⊥}), the ratio of the magnetic field energy density to the perpendicular thermal energy density, q ^{2} ≡ (v _{A}/S _{⊥})^{2}, and the anisotropy parameter, r ^{2} ≡ (S _{∥}/S _{⊥})^{2}. Here, v _{ z }(x=±∞)=±v _{ z m }, S _{∥}, and S _{⊥} are the sound speeds parallel and perpendicular to the magnetic field, and v _{A} is the Alfvén speed. The dimensionless variable characterizing the perturbation is the wavenumber B ≡ k _{ z } L, where L is the shear layer thickness. It is shown that the resonance of the sound and firehose modes drives unstable standing and traveling waves for a shear layer having a vortex sheet profile [where v _{ z }(x) is a step function]. For the vortex sheet, the unstable standing wave modes first appear at M=2(3)^{1/2} r for r>(1+q ^{2})^{1/2}/2, and the unstable traveling wave modes first appear at M=0 for r<((1+q ^{2})^{1/2}/2). Numerical methods are used to generate values of ω_{ r } and ω_{ i }>0 (corresponding to unstable wave motion) for the ‘‘linear’’ shear layer in the (B, M) plane for various values of q and r.
The coverage of the (B, M) plane is for B≤5, M≤10, and for discrete values of q≤0.5 and r≤2. Two regimes of instability are found to occur in the (B, M) plane with the structure of the unstable modes of the ‘‘linear’’ layer being very different from that for the anisotropic vortex sheet. The unstable modes are standing waves with ω_{ r }=0, and traveling waves with ω_{ r }≠0.

Direct conversion of a fast wave into ion Bernstein modes caused by density fluctuations
View Description Hide DescriptionAn analytical study is presented of the interaction between a fast Alfvén wave having a frequency larger than the ion‐cyclotron frequency, and low frequency density fluctuations. When appropriate k, ω matching conditions are satisfied, the interaction results in a beat current that acts as an i n‐s i t u antenna, exciting ion Bernstein modes resonantly. Since such modes have relatively small damping for large parallel phase velocities, their excitation can significantly enhance the damping of fast waves having small k _{∥}, particularly in those regions of the plasma where the temperature is relatively low. The interaction with coherent, as well as with broad band density fluctuations is investigated, and the role of plasma nonuniformities is examined. Specific features of relevance to ion‐cyclotron range of frequency heating of tokamak plasmas are considered.

Statistical theory for magnetohydrodynamic turbulent shear flows
View Description Hide DescriptionA statistical theory is developed for magnetohydrodynamic turbulent shear flows in the presence of both mean velocity and magnetic gradients. In the case of magnetic Prandtl number unity, the correlations between fluctuating velocity and magnetic fields, which play a crucially important role in the equations for the mean velocity and magnetic fields, are calculated with the aid of the two‐scale, direct‐interaction approximation. The electromotive force expressing the effect of turbulent fields in the equation for mean magnetic field is found from the results. Particularly, the electromotive force related linearly to the mean magnetic field is shown to originate from the statistical inhomogeneity of fluctuating fields, which is sustained by the mean field gradients.

Stability of a hot‐electron‐stabilized symmetric tandem mirror with thermal barrier
View Description Hide DescriptionIt is shown that in general the stability of a hot‐electron‐stabilized symmetric tandem mirror is the same as the stability of its barrier‐plug cell alone, plus an additional limitation on the center‐cell length such that the center‐cell Alfvén frequency is higher than the average barrier‐plug E×B drift and the hot‐electron precession frequency. The stability of the hot‐electron barrier‐plug cell is studied extensively.