Volume 31, Issue 1, January 1988
Index of content:

The anomalous behavior of the runup of cnoidal waves
View Description Hide DescriptionA new solution to the linearized shallow‐water wave equations is introduced for the case of cnoidal waves climbing up a plane beach. The solution is used to calculate the maximum runup. It is shown that the maximum relative runup of cnoidal waves is significantly larger than the runup of monochromatic waves with the same wave height and wavelength far from the shore. It is also shown that the maximum relative runup of cnoidal waves is not a monotonically varying function of the normalized wavelength.

Chaotic behavior of interacting elliptical instability modes
View Description Hide DescriptionA three‐mode projection of the Navier–Stokes equations for nonlinear perturbations to an elliptical vortex is studied numerically. It is found that, as the Reynolds number increases, the perturbations undergo a sequence of period doublings leading to chaos according to the Feigenbaum scenario [J. Statis. Phys. 1 9, 25 (1978); Phys. Lett. 7 4 A, 375 (1979)].

The effects of electric field fluctuations on bootstrap current and resistivity in toroidal plasmas
View Description Hide DescriptionThe effects of electric field fluctuations on bootstrap current and resistivity in toroidal plasmas are studied in the plateau regime. A tokamak is discussed as an example. The bootstrap current density is more sensitive to the fluctuations than the plasma resistivity because the bootstrap current density depends on the poloidal mode number of the fluctuations, while plasma resistivity depends on the parallel wave vector.

Transport scaling in the collisionless‐detrapping regime in stellarators
View Description Hide DescriptionStellarator transport scalings with electric field, geometry, and collision frequency in the reactor‐relevant collisionless‐detrapping regime are determined from numerical solutions of the drift kinetic equation. A new geometrical scaling, proportional to ε^{3/2} _{ t } rather than ε_{ t }ε^{1/2} _{ h }, is found, where ε_{ t } is the inverse aspect ratio and ε_{ h } is the helical ripple. With the new scaling, no reduction in energy confinement time is associated with large helical ripple, which provides design flexibility. Integral expressions for the particle and heat fluxes that are useful for transport simulations are given.

Determination of the macroscopic (partial) slip boundary condition for a viscous flow over a randomly rough surface with a perfect slip microscopic boundary condition
View Description Hide DescriptionConsider a viscous fluid, at zero Reynolds number, moving over a solid surface flat except for a random array of microscopic defects having a small area fraction c. Assuming a microscopic boundary condition of perfect slip, the macroscopic boundary condition is determined from first principles. The asymptotic structure of the solution for a random surface with finite slope is quite different from those of earlier studies in the limit of an ‘‘almost flat’’ surface. The results of this study show that very small amounts of roughness can well approximate a no‐slip boundary condition macroscopically, for example, one defect of the order of 10^{−} ^{9} m per (10^{−} ^{7} m)^{2} gives a slip length of only 10^{−} ^{5} m.

Bubble motion in a Hele–Shaw cell
View Description Hide DescriptionThe shapes and motion of immiscible bubbles in a Hele–Shaw cell driven by the motion of the surrounding fluid were studied. Six classes of steady shapes, some of which are remarkable, were observed. Multiple steady states exist over some ranges of parameters and the shape as a function of speed may slow hysteresis. The observed translational velocities do not agree with available theory, but some of the shapes are in qualitative agreement with those computed by Tanveer [Phys. Fluids 2 9, 3537 (1986)].

Solutal convection in the melt during solidification of a binary alloy
View Description Hide DescriptionNonlinear solutal convection in the melt under a planar solidifying surface is investigated in the limit of small segregation. Stationary solutions of the nonlinear problem are obtained and the preferred mode of convection is determined by a stability analysis. It is found that down‐hexagons (where motion is downward at the cells’ centers) are stable for sufficiently small amplitude Ε, while both down‐hexagons and squares are stable in a range of larger Ε. The dependence of various flow features on the segregation coefficient and the effective depth of the melt is discussed.

Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side
View Description Hide DescriptionThe stability of laminar natural convection in a shallow cavity has been studied theoretically. The flow is driven by a horizontal temperature gradient between isothermal vertical sidewalls of the cavity, the top and bottom of which are insulated. It was found that for a Prandtl number (Pr) less than 0.033, shear instability causes stationary transverse cells to be formed in the flow. For larger values of Prandtl number the instability sets in as oscillating longitudinal rolls in the range 0.033<Pr<0.2; and as stationary longitudinal rolls for larger values of Pr. The importance of three‐dimensional disturbances was investigated for Pr=0.02 and they were found to be less critical than two‐dimensional ones. Finite amplitude motions of the stationary transverse cells were simulated by solving the nonlinear equations numerically with the use of pseudospectral methods. This simulation supports the calculations of the onset of the instability by linear theory, and explains why the Nusselt number decreases when secondary flows are present.

Free‐streamline analysis of deformation and dislodging by wind force of drops on a surface
View Description Hide DescriptionFree‐streamline theory is used to analyze the deformation and dislodging by wind pressure of drops of liquid adhered by surface tension to a solid surface. The critical Weber number for droplets to be dislodged is determined as a function of advancing and receding contact angle. Graphical results for drop shape are in good agreement with observation.

Model functions of Reynolds stress models
View Description Hide DescriptionA Reynolds stress (Re‐stress) model for the prediction of shear flows is presented. Special attention is paid to the determination of the model functions. First, the general modeling assumptions are discussed and the modeling of the equations is then described in detail. It is shown that by successive analysis of different types of flows a set of model functions with a high level of universality can be deduced. Finally, prediction/measurement comparisons in homogeneous shear flows, round jets, plane jets, and wakes are presented.

Three‐dimensional centrifugal‐type instabilities in inviscid two‐dimensional flows
View Description Hide DescriptionIn this paper the classical Rayleigh centrifugal instability theory is extended to general inviscid two‐dimensional flows. Sufficient conditions for centrifugal instability are that the streamlines be convex closed curves in some region of the flow, with the magnitude of the circulation decreasing outward. If these conditions are satisfied, a class of three‐dimensional short‐wave instabilities can be constructed, which are localized near the streamline on which the exponent of a certain matrix Floquet problem is maximized.

Magnetohydrodynamic flow in a curved pipe
View Description Hide DescriptionThe flow of conductive fluids in highly conductive curved pipes is studied analytically in this paper. The flow is assumed to be steady state, laminar, and fully developed. Coupled continuity, Navier–Stokes, and appropriate Maxwell equations are solved in toroidal coordinates. The dimensionless parameters of the problem are Dean number K and Hartmann number Ha. For low Hartmann numbers [Ha^{2}∼θ(1)], the solution is expanded in a power series of K and Ha^{2}. For intermediate Hartmann numbers [Ha^{2}∼θ(1000)], the solution is expressed as a power series of K. The axial velocity contours are shown to be shifted towards the outer wall. For low Ha, these contours are nearly circular. The effect of a strong transverse magnetic field is to enhance the compression of fluid towards the outer wall. The secondary flow field comprises a symmetric pair of counter‐rotating vortices. A strong magnetic field is found to confine the secondary flow streamlines to a thin layer near the tube wall. The secondary flow rate in the near‐wall boundary layer is increased by the magnetic field. This increase in flow rate raises the possibility of efficient convective cooling of curved first wall tubes in magnetic confinement fusion reactors (MFCR).

Stability of Bernstein–Greene–Kruskal plasma equilibria. Numerical experiments over a long time
View Description Hide DescriptionBernstein–Greene–Kruskal (BGK) equilibria for a Vlasov plasma consisting of a periodic structure exhibiting depressions or ‘‘holes’’ in phase space are under consideration. Marginal stability analysis indicates that such structures are unstable when the system contains at least two holes. An Eulerian numerical code is developed allowing noiseless information on the long time phase space behavior (about 10^{3}ω^{−1} _{ p }) to be obtained. Starting with equilibria with up to six holes, it is shown that the final state is given by a structure with only one large hole, the initial instability inducing coalescences of the different holes. On the other hand, starting with a homogeneous two‐stream plasma it is shown that, in a first step, a BGK periodic structure appears with a number of holes proportional to the length of the system, followed, in a second step, by a coalescence of the holes to always end up with the above mentioned one large hole structure.

Soliton decay of nonlinear Alfvén waves: Numerical studies
View Description Hide DescriptionThe derivative nonlinear Schrödinger equation is numerically solved for arbitrary initial conditions by an extension of the Ablowitz–Ladik scheme [Stud. Appl. Math 5 7, 1 (1977)]. The numerical nonlinear difference code, which takes advantage of the inverse scattering method, simulates the original differential equation reproducing common features, like solitons and an infinite set of constants of motion. The long‐time behavior is analyzed in terms of the sign of one of the constants of motion. The formation of a soliton train is seen whenever the constant has a negative value. This fact is the global expression of the Mj≂lhus local criterion to distinguish between modulationally stable and unstable cases.

Ponderomotive force in a nonisothermal plasma
View Description Hide DescriptionIn this paper a formula is derived for the ponderomotive force of electromagnetic fields in collisionless and inhomogeneous plasma when the temperature varies slowly in space and time. This result, compared to the current formula for the isothermal case, has an additional term that involves the temperature gradient. The new result is valid in anisotropic medium and better suited for application to both laboratory and space plasma situations in which the plasma is anisotropic.

Generalization of the synchrotron absorption–emission coefficient in plasmas to arbitrary direction of propagation
View Description Hide DescriptionA new analytical formulation of the synchrotron absorption and emission coefficient in plasmas, for wave propagation perpendicular to the magnetic field, has been derived previously [Phys. Fluids 2 9, 3275 (1986)]. These coefficients were shown to be both accurate and analytically simple. Their generalization to arbitrary directions of propagation with respect to the magnetic field is presented here. Based on the analytical formulas for perpendicular propagation and on the exact sum–integral formulation of the coefficients for arbitrary propagation, this generalization is achieved by empirical data fit. Hereby the exact formulation serves as target function for the fitting procedure and subsequently as reference for determination of the accuracy of the results. The generalized analytical representations for the coefficients are presented here, one for each polarization mode and for the low and high temperature domain (above and below a few tens of kilo‐electron‐volts).

Synchrotron radiation from dynamically evolving runaway tails
View Description Hide DescriptionA quasilinear code (2‐D in velocity and wave vector space) based on the Ritz–Galerkin method and finite elements is used to study the s y n c h r o t r o n e m i s s i o n from a dynamically evolving runaway tail. The evolution of runaway tails was studied recently using the same code [Phys. Fluids 2 8, 3356 (1985)]. The velocity distribution derived from the above code is used to estimate the synchrotron radiation in different time steps. The applicability of these results to the Princeton Large Torus [Phys. Rev. Lett. 4 9, 1255 (1982)] and solar microwave bursts is also commented on.

Weakly relativistic dispersion of Bernstein waves
View Description Hide DescriptionWeakly relativistic effects on the dispersion of Bernstein waves are investigated for waves propagating nearly perpendicular to a uniform magnetic field in a Maxwellian plasma. Attention is focused on those large‐wave‐vector branches that are either weakly damped or join continuously onto weakly damped branches since these are the modes of most interest in applications. The transition between dispersion at perpendicular and oblique propagation is examined and major weakly relativistic effects on dispersion at perpendicular propagation are found to persist at other angles; these effects can dominate even in low‐temperature plasmas. A number of simple analytic criteria are obtained which delimit the ranges of harmonic number and propagation angle within which various types of weakly damped Bernstein modes can exist.

Wave–plasma interaction near the second electron cyclotron harmonic
View Description Hide DescriptionThe linear wave propagation and spatial damping near the second electron cyclotron harmonic in a weakly relativistic Maxwellian plasma for an arbitrary angle of wave incidence are investigated. It is found that mode interaction with linear conversion of the extraordinary mode into a quasilongitudinal mode occurs over a relatively large range of plasma parameters. The absorption properties of the extraordinary and ordinary modes and their scaling laws are examined. The noninductive current drive by electron cyclotron waves is also discussed. The results obtained indicate quite favorable features of the wave–plasma interaction near the second electron cyclotron harmonic for plasma heating, current drive, and diagnostics applications.