Dominant Wavenumber Research Articles

Spatial instability of plane liquid sheets

Spatial instability of a thin moving plane liquid sheet is analyzed. It is shown that there exist two independent modes of instability, sinuous and varicose. For the sinuous mode, there is a critical Weber number of unity below which pseudo-absolute instability exists and above which convective instability appears. For the varicose mode, the instability is always convective. The convective mode is induced by the aerodynamic effects, and is related to the corresponding temporal instability through Gaster's relation at the sufficiently high liquid velocity. Perturbation analysis indicates that the spatial amplification rate depends strongly on the gas-to-liquid density ratio ϱ. For the convective instability of the sinuous mode, there exists a critical Weber number of slightly larger than unity, below which liquid viscosity exhibits both stabilizing and destabilizing effects and above which the viscous effects always reduce the growth rate and the dominant wavenumber. For situations where the Weber number, We ⪢ 1, but ϱ We ⩽ 0(1), the growth rate of the sinuous mode is larger than that of the varicose mode under the same flow conditions. If both We ⪢ 1 and ϱ We ⪢ 1, the spatial growth rate of both sinuous and varicose modes is almost the same, except at the small wavenumbers where the sinuous mode has slightly larger growth rate than the corresponding varicose mode. The viscous effects on the varicose mode is always stabilizing. The present results compare favorably with the existing experimental observations.

Mars atmospheric dynamics as simulated by the NASA Ames General Circulation Model: 2. Transient baroclinic eddies

A large set of experiments performed with the NASA Ames Mars general circulation model (GCM) have been analyzed to determine the properties, structure, and dynamics of the simulated transient baroclinic eddies. The Mars GCM simulations span a wide range of seasonal dates and dust loadings and include a number of special sensitivity experiments (e.g., with flat topography). There is strong transient baroclinic eddy activity in the extratropics of the northern hemisphere during the northern autumn, winter, and spring seasons. The eddy activity remains strong for very large dust loadings, though it shifts northward. The eastward propagating eddies are characterized by zonal wavenumbers of 1–4 and periods of ∼2–10 days. In several simulations, the eddy variance is dominated by a single zonal wavenumber and a narrow range of periods. The longer (wavenumbers 1 and 2) transient eddies have a very deep vertical structure, exhibiting a maximum kinetic energy density at the model top (∼45–50 km) in many of the simulations. In a simulation for early northern spring the eddies are quite shallow, however. The transient eddies generate the bulk of their energy baroclincally via large meridional and vertical heat fluxes, at both lower and upper levels. This is despite the fact that their vertical structure is typically close to equivalent barotropic above the lowest 10 km. The eddies also appear to generate a substantial amount of energy barotropically, via large meridional momentum fluxes at both lower and upper levels. In the tropics and northern subtropics at upper levels (∼25–50 km) there are strong transient eddy motions with structures resembling those characteristic of inertially unstable modes. This eddy activity appears to be a response to the forcing of a region of marginal inertial stability by the extratropical transient baroclinic eddies, as the wavenumbers and periods are the same as those in the extratropics. A major surprise is the presence of very weak transient eddy activity in a number of the southern winter simulations. It appears that this is partly a consequence of the stabilizing effects of the zonally symmetric topography in the GCM, but it also must be associated with certain aspects of the zonal‐mean circulation in southern winter. This is indicated by the presence of relatively large amplitude eddies in simulations for early southern autumn and spring and in a southern winter solstice simulation incorporating a different topography (derived from the Mars Digital Terrain Model). This topography differs from that used in most of the GCM simulations in not being characterized by steep symmetric slopes (which are stabilizing to baroclinic instability) in southern high latitudes. It is hypothesized that the very large extent of the southern seasonal polar cap and high elevations in the south both might contribute to weakening the transient eddy activity. Large zonally symmetric topography in the northern hemisphere of the Mars GCM also appears to have a strong impact on the transient eddies, acting to increase the dominant zonal wavenumbers and phase speeds. The properties of the GCM baroclinic eddies in the northern extratropics are compared in detail with analogous properties inferred from Viking Lander meteorology observations. The GCM eddies are found to be very similar to those observed in most respects. A notable exception is that the eddy amplitudes in the highly dusty GCM simulations are much larger than those observed by Viking during the 1977 winter solstice dust storm. This is almost certainly at least partly due to the relatively small latitudinal expansion of the Hadley circulation in the highly dusty GCM experiments.

Hydromagnetic Dynamo in Astrophysical Jets

The origin of a regular magnetic field in astrophysical jets is discussed. It is shown that jet plasma flow can generate a magnetic field provided the streamlines are helical. The dynamo of this type, known as the screw dynamo, generates magnetic fields with the dominant azimuthal wave number m = 1 whose field lines also have a helical shape. The field concentrates into a relatively thin cylindrical shell and its configuration is favorable for the collimation and confinement of the jet plasma.

A multiple-beam, collective scattering technique for spatially resolved measurement of plasma turbulencea)

Collective Thomson scattering has long been utilized to study electrostatic turbulence in magnetic confinement fusion plasmas with a view to demonstrating a causal link to the observed anomolous transport of heat and particles. However, this goal has been severly hampered by the relatively poor spatial resolution available at the dominant turbulence wave numbers using standard experimental techniques. Good spatial resolution is necessary if adequate comparison of theory and experiment is to occur. The present paper describes the basic principles of a new technique utilizing simultaneous, multiple input beams, a detector array, and coherent tomographic inversion algorithm, which can substantially improve the available spatial resolution for these important measurements. The proposed multiple beam scattering diagnostic offers a spatial resolution of ∼10 cm at a fluctuation wave number of 5 cm−1 when the angular envelope of the beams is 0.1 rad. Optical designs suitable for implementing the multiple beam diagnostic on a large tokamak have been developed.

A multiple-beam, collective scattering technique for spatially resolved measurement of plasma turbulence (abstract)a)

Collective Thomson scattering has long been utilized to study electrostatic turbulence in magnetic confinement fusion plasmas with a view to demonstrating a causal link to the observed anomolous transport of heat and particles. However, this goal has been severly hampered by the relatively poor spatial resolution available at the dominant turbulence wave numbers using standard experimental techniques. Good spatial resolution is necessary if adequate comparison of theory and experiment is to occur. The present paper describes the basic principles of a new technique utilizing simultaneous, multiple input beams, a detector array, and coherent tomographic inversion algorithm, which can substantially improve the available spatial resolution for these important measurements. The proposed multiple beam scattering diagnostic offers a spatial resolution of ∼10 cm at a fluctuation wave number of 5 cm−1 when the angular envelope of the beams is 0.1 rad. Optical designs suitable for implementing the multiple beam diagnostic on a large tokamak have been developed.

On the temporal instability of a two-dimensional viscous liquid sheet

This paper reports a temporal instability analysis of a moving thin viscous liquid sheet in an inviscid gas medium. The results show that surface tension always opposes, while surrounding gas and relative velocity between the sheet and gas favour, the onset and development of instability. It is found that there exist two modes of instability for viscous liquid sheets – aerodynamic and viscosity-enhanced instability – in contrast to inviscid liquid sheets for which the only mode of instability is aerodynamic. It is also found that axisymmetrical disturbances control the instability process for small Weber numbers, while antisymmetrical disturbances dominate for large Weber numbers. For antisymmetrical disturbances, liquid viscosity, through the Ohnesorge number, enhances instability at small Weber numbers, while liquid viscosity reduces the growth rate and the dominant wavenumber at large Weber numbers. At the intermediate Weber-number range, Liquid viscosity has complicated effects due to the interaction of viscosity-enhanced and aerodynamic instabilities. In this range, the growth rate curve exhibits two local maxima, one corresponding to aerodynamic instability, for which liquid viscosity has a negligible effect, and the other due to viscosity-enhanced instability, which is influenced by the presence and variation of liquid viscosity. For axisymmetrical disturbances, liquid viscosity always reduces the growth rate and the dominant wavenumber, aerodynamic instability always prevails, and although the regime of viscosity-enhanced instability is always present, its growth rate curve does not possess a local maximum.

The consequences of sampling variability on the estimation of wave number and propagation direction from spaceborne SAR image spectra

Even if it were true that SAR (synthetic aperture radar) ocean surface imagery provides a perfect noiseless representation of ocean surface wave height, each spectrum computed from such imagery would be but a single realization of the ensemble-mean ocean spectrum. There would be sampling variability associated with the parameters of dominant wave number and propagation direction extracted from such a spectrum. The present study addresses two questions: (i) what statistical distribution is applicable to the spectra of SAR ocean images? and (ii) what are the consequences of such statistics on the precision with which wave number and propagation direction can be extracted? An examination is made of spectra computed from ocean imagery acquired during the SIR-B (Shuttle Imaging Radar-B) mission. >

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 & Gollub at Prandtl number 2.5, and by Steinberg, Ahlers & 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.

Convection at finite Rayleigh numbers in large-aspect-ratio containers.

The phase diffusion and mean drift equations which describe the behavior of a convection pattern are derived, a step which is essential for obtaining quantitative comparisons between theory and experiment. The theory recovers the boundaries of the Busse Balloon, agrees closely with the dominant wave numbers observed by Heutmaker and Gollub [Phys. Rev. A 35, 242 (1987)] and Steinberg, Ahlers, and Cannell [Phys. Sci. 30, 534 (1985)] in natural and target patterns, predicts a new instability which is important in facilitating wave-number adjustment in circular target patterns and in initiating time dependence, and predicts the Rayleigh numbers at which loss of spatial correlation due to global defect nucleation will occur.

Experimental investigation of the wave propagation on a point-driven, submerged capped cylinder using K-space analysis

Classically, sparse accelerometer measurements are made on vibrating structures to help in deducing the physics of the vibration. However, the experimenter now has at his command more sophisticated measurement tools, such as nearfield acoustical holography or laser Doppler velocimetry, which provide much more data. Often complete mapping of the acceleration or displacements on the surface of these structures can be acquired. As a result, more sophisticated analysis tools must be developed. Such a tool useful for vibrating structures which are basically cylindrical or planar, is presented herein. The measured spatial velocity is decomposed at fixed temporal frequencies into its helical-wave spectrum, also known as the K-space spectrum. This decomposition contains a great deal of information about the physics of the vibrator. For example, it provides the dominant wavenumbers (the dispersion diagram) of free waves that exist on the shell. These free waves reach maximum amplitudes when close to a resonance frequency. The decomposition also provides an indication of the modal density of the shell. This technique is applied to experimental measurements on a shaker-driven, fluid-loaded, capped cylinder. The resulting helical wavenumber diagrams, plotted at a single frequency of excitation, show strong ‘‘figure 8’’ patterns. By comparison with helical-wave spectra computed from infinite-shell theory, it is shown that these patterns represent the wavenumber loci of the free waves that exist on the experimental shell. Excellent agreement between the measured helical-wave spectrum and predictions from infinite-shell theory are reported. The helical-wave spectra of the acoustic pressure, mapped on a cylindrical contour in the extreme nearfield of the capped cylinder, is also presented. From this spectra, the free waves on the structure can be identified and, in particular, how they participate in acoustic radiation. The frequency region was ka=0.5–2.0.

Topographic Effects on the Mean Tropospheric Flow Patterns around Antarctica

Abstract Topographically induced flows around Antarctica in a rotating tank experiment with both homogeneous and stratified fluid are analyzed and compared with the mean tropospheric circulation. A circular tank of fluid was brought to a state of near-rigid clockwise rotation, and a topographic model of the Antarctic continent was then rotated counterclockwise to simulate a mean westerly zonal wind. Stratification was chosen to give the same ratios of topographic and dynamical length scales as in the atmosphere, as was the Rossby number based on the ratio of rotation rates. After the onset of the relative rotation of the Antarctic model, cyclonic eddies evolved in the coastal areas with, in the homogeneous case, anticyclonic eddies over the Antarctic dome. After about ten tank rotation periods, a dominant wavenumber 3 structure with cyclonic eddies in the Ross and Weddell seas and Prydz Bay is observed as an approximately steady state. Flow over the topography is relatively stagnant, with weak anticycloni...

Measurements of electromagnetic waves in Phaedrus-B: Bench-mark test of ANTENA wave field calculations

It is shown that the predictions of a numerical code (ANTENA) and the data of wave field measurements in the Phaedrus-B tandem mirror are consistent (±25%) for right-handed (B⃗−) wave fields and less so (±40%) for left-handed (B⃗+) wave fields in the plasma core, and that they disagree for B⃗+ fields near the column edge. Shorting out or reduction of the wave azimuthal electric fields by limiters is the probable cause of this discrepancy. The ICRF fluctuating wave B⃗ fields are shown as |B⃗| contour maps in the r-z plane, where the B⃗+ data peak at a smaller radius than predicted. The waves are characterized by different dominant axial wave numbers for the left- and right-handed circularly polarized fields.

Theoretical Study on the Coexistence of Ripples and Dunes

Formation of ripples and dunes on alluvial river bed is studied by river bed stability analysis, basing on two-dimensional shear flow equation and a Meyer-Peter and Wüller type bed load transport equation modified by the inclusion of the lag distance of bed load relative to local bed shear. Dominant dimensionless wave-numbers of sand wave are analyzed. The effect of sediment particle diameter and lag distance on the dominant wavenumbers is examined, and the difference between ripples and dunes, with respect to their cause of occurrence, is discussed.

Continuum properties from interdigital electrode dielectrometry

Using a modal approach, a model is derived that makes the interdigital electrode microdielectrometer developed by S. D. Senturia et al. (J. Adhesion, vol.15, p.69-90, 1982) applicable to measuring continuum parameters in a wide range of heterogeneous media. In this so-called imposed omega -k technique, the medium is excited at the temporal (angular) frequency omega by means of an interdigital electrode structure having a spatial periodicity length lambda =2 pi /k and hence a dominant wavenumber k. Given the surface capacitance density C( omega , k) of any linear system having property gradients perpendicular to the plane of the electrodes, the model predicts the complex gain, taking into account the properties, geometry, and terminal configuration of the interdigital electrode structure. This capability can then be used with an appropriate parameter estimation strategy to determine the continuum properties and/or geometry of the medium. Examples illustrating the application of the technique are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Wavenumber transition and wavenumber vacillation in Eady‐type baroclinic flows

AbstractWavenumber transition and wavenumber vacillation are studied numerically in a ‘maximum simplification’ model of the Eady type with uneven Ekman dissipation. The spectral wave solution contains three waves having adjacent zonal wavenumbers and the gravest meridional mode. The model only allows for wave‐mean‐flow interaction.As the stratification parameter decreases away from the upper symmetric flow, we observe that baroclinic forcing increases faster than dissipation, implying the prevalence of a higher wave energy state. This state is realized by an increase in wave amplitude, number of temporal and/or spatial degrees of freedom, or dominant wavenumber. The first two occur first, but for small enough stratification, the flow transits to a higher wavenumber in order to realize the increased energy.Wavenumber vacillation is a flow regime observed at some parameter settings when the flow has at least two spatial and three temporal degrees of freedom. This regime is usually found in the region of parameter space where wavenumber transitions occur as parameters change.The balance between baroclinic forcing and damping adjusted by wave‐mean‐flow interaction seems to be a basic mechanism for wavenumber transition and wavenumber vacillation. Wave‐wave interaction is not necessary for such phenomena to exist.

Simulation of the electron acoustic instability for a finite‐size electron beam system

Satellite observations at midaltitudes (≈ 20,000 km) near the earth's dayside polar cusp boundary layer indicate that the upward electron beams have a narrow latitudinal width up to 0.1°. In the cusp boundary layer where the electron population consists of a finite‐size electron beam in a background of uniform cold and hot electrons, the electron acoustic mode is unstable inside the electron beam but damped outside the electron beam. Simulations of the electron acoustic instability for a finite‐size beam system are carried out with a particle‐in‐cell code to investigate the heating phenomena associated with the instability and the width of the heating region. The simulations show that the finite‐size electron beam radiates electrostatic electron acoustic waves. The decay length of the electron acoustic waves outside the beam in the simulation agrees with the spatial decay length derived from the linear dispersion equation. The ambient cold electrons in a diffusion region surrounding the beam are heated to a higher temperature by absorbing the radiated electron acoustic waves, with the heating occurring mainly in the parallel direction. In the heat diffusion region, the temperature of the cold electrons decreases with the distance from the beam with a temperature gradient length smaller than the decay length of the wave energy. This feature of electron heating is modeled by using a heat diffusion equation derived from the second‐order theory. In solving the heat diffusion equation, the wave energy inside the beam is assumed to be saturated by trapping the beam and the cold electrons. The dominant wave number is determined by assuming that the wave frequency outside the beam is equal to the frequency corresponding to the maximum growth rate inside the beam. These results are discussed with respect to the DE 1 plasma and wave observations in the polar cusp region. Using the typical parameters in the cusp boundary layer, the spatial decay length of the electron acoustic waves is estimated to be in the range of 1–10 km and the heat diffusion region is about 5 km. The results suggest that the electron acoustic waves generated by cusp upward electron beams are probably confined in a small region surrounding the beams.

The effects of boundary surface inhomogeneities on acoustic scattering. I. Theory

Analytical methods of Marsh [J. Acoust. Soc. Am. 33, 330–333 (1961)] and Kuo [J. Acoust. Soc. Am. 36, 2135–2142 (1964)] are applied to high-frequency acoustic wave scattering from an ocean bottom of roughness amplitude typically quite small in comparison with the acoustic wavelength of interest. The bottom surface variation in acoustic velocity is considered negligible but the bottom surface variation in density is considered appreciable and included in the model. The analytical result indicates the possibility of a new scattering effect in grazing incident directions due to correlated roughness and density variations. When there exists an identifiable dominant roughness wavenumber, the theory predicts discrete backscattering. This phenomenon was observed in an experiment reported by Roderick and Dullea (TD 7181, Naval Underwater Systems Center, 1984). When the dominant roughness wavenumber is zero, the result is that of Rayleigh’s flat boundary. When the dominant roughness wavenumber is low compared to acoustic wavenumber, the result is that of the Kirchhoff approximation.

Bow-tie antennas on a dielectric half-space: Theory and experiment

A new formulation is discussed for the rigorous calculation of the radiation pattern of a bow-tie antenna of finite length and infinitesimal thickness, placed on a lossless dielectric substrate. The analysis is based on a representation of the current density on the metal surface of the antenna as a sum of an imposed (quasistatic) term and a set of current modes with unknown amplitudes. Free-space fields that are expressed in terms of continuous spectra of symmetrized plane waves are matched to the current modes using the method of moments. The resulting set of equations are solved for the unknown current amplitudes. The calculations show that for increasing bow length the antenna impedance spirals rapidly to a value predicted by transmission line theory. The theory also shows that the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> -plane pattern of a two wavelength, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">60\deg</tex> bow-tie antenna is dominated by low-loss current modes propagating at the dielectric wavenumber. As the bow tie narrows, the loss of the modes increases, and the dominant wavenumber tends to the quasistatic value. Pattern measurements made at 94 GHz are shown to agree well with theoretical predictions. Measurements for a long-wire antenna, a linear array of bow-tie elements, and a log-periodic antenna are also presented.

Numerical study of ribbon-induced transition in Blasius flow

The early three-dimensional stages of transition in the Blasius boundary layer are studied by numerical solution of the Navier-Stokes equations. A finite-amplitude two-dimensional wave and low-amplitude three-dimensional random disturbances are introduced. Rapid amplification of the three-dimensional components is observed and leads to transition. For intermediate amplitudes of the two-dimensional wave the breakdown is of subharmonic type, and the dominant spanwise wavenumber increases with the amplitude. For high amplitudes the energy of the fundamental mode is comparable to the energy of the subharmonic mode, but never dominates it; the breakdown is of mixed type. Visualizations, energy histories, and spectra are presented. The sensitivity of the results to various physical and numerical parameters is studied. The agreement with experimental and theoretical results is discussed.

A Comparison of SIR-B Directional Ocean Wave Spectra with Aircraft Scanning Radar Spectra

Directional ocean wave spectra derived from Shuttle Imaging Radar-B (SIR-B) L-band imagery collected off the coast of southern Chile on 11 and 12 October 1984 were compared with independent spectral estimates from two airborne scanning radars. In sea states with significant wave heights ranging from 3 to 5 meters, the SIR-B-derived spectra at 18 degrees and 25 degrees off nadir yielded reasonable estimates of wavelengths, directions, and spectral shapes for all wave systems encountered, including a purely azimuth-traveling system. A SIR-B image intensity variance spectrum containing predominantly range-traveling waves closely resembles an independent aircraft estimate of the slope variance spectrum. The prediction of a U.S. Navy global spectral ocean wave model on 11 October 1984 exhibited no significant bias in dominant wave number but contained a directional bias of about 30 degrees espect to the mean of the aircraft and spacecraft estimates.