Maximum Wave Growth Research Articles

Simulation of Interfacial Waves in Core-Annular Pipe Flow with a Turbulent Annulus

Abstract Numerical simulations are conducted for the wave initiation, growth, and saturation at the oil-water interface in core-annular flow. The focus is on conditions with a turbulent water annulus, but the laminar water annulus is also considered. The simulation results are compared with lab measurements. The growth rate for the linear instability of different wave lengths in the case of a turbulent water annulus is obtained from two-dimensional axisymmetric Reynolds-Averaged Navier-Stokes simulations with the Launder-Sharma low-Reynolds number k-e model. The latter simulation results provide the most unstable wave length for the turbulent water annulus. Our study also shows the following. The maximum wave growth rate for a turbulent water annulus is significantly higher than for a laminar water annulus. The most unstable wave length in the simulations is about 25% smaller than in the experiments. The wave amplitude for the different wave lengths in the simulations is typically 17% lower than in the experiments.

Revisiting the mechanism responsible for the stratified-slug transition in two-phase flows

Linear stability analysis is extensively used for predicting the transition between stratified and slug flow. In the present work, a one-dimensional two-fluid flow is linearly perturbed to evaluate the behavior of the dispersion curves through parameters such as the maximum wave growth rate (ωI,max), the fastest-growing wave (kmax) and the wave that makes the problem permanently stable (kc) as a function of the gas and liquid superficial velocities. The novelty of this article relies upon coupling the analysis of the behavior of transition-related parameters to the physical effects that are responsible for stabilizing and destabilizing the flow interface. The coupling of the transition analysis with the physical parameters showed potential as a reliable way of explaining the obtained transition behavior. By doing this, the stabilizing effects of gravity and surface tension are found to be invariable to the superficial velocities of the phases. On the other hand, the destabilizing effect of inertia increased with phase superficial velocities, while the shear stress increases with the liquid superficial velocity and shows a non-monotonic behavior with the gas superficial velocity. Although the overall trend of ωI,max, kmax and kc was to increase with the superficial velocities of the phases, they were directly affected by the shear stress behavior, also showing a non-monotonic trend.

Effects of liquid and air swirls on the three-dimensional instability of an annular viscous liquid sheet

A three-dimensional temporal linear instability model is proposed for an annular viscous liquid sheet moving between two inviscid gas streams. The model is numerically solved by Chebyshev collocation method. Effects of liquid swirl, inner and outer gas swirls, liquid viscosity, radius ratio, and density ratio on the liquid sheet instability are systematically investigated. The energy budget is calculated to provide insight into the instability driving mechanism. Results show that without gas swirl the liquid swirl has a dual effect on the sheet instability mainly through the axisymmetric mode. For a non-swirling liquid sheet, inner gas swirl induces the maximum wave growth rates of non-axisymmetric modes about 2–3 times higher than those induced by outer gas swirl, suggesting that inner gas swirl is more effective than outer one in promoting the sheet instability. For a swirling liquid sheet, simultaneous inner and outer gas swirls have the greatest destabilizing effect; outer gas swirl shows a slightly greater destabilizing effect than the inner one. Regarding the relative rotating direction, the combination of both co-rotating inner and outer gases is more favored to producing finer droplets. Liquid viscosity is found to have a evident stabilizing effect when Reynolds number is smaller than 500, it dampens the disturbance by dissipating the disturbance energy. Increasing radius ratio decreases the maximum wave growth rates of modes with n=0 and 5⩽n⩽7 but increases those of modes with 1⩽n⩽4. Maximum wave growth rate and optimal wave number are found to first decrease and then increase with increasing density ratio.

Proudman resonance with tides, bathymetry and variable atmospheric forcings

Proudman resonance is a primary amplification mechanism for meteotsunamis, which are shallow-water waves generated by atmospheric forcings. The effect of tides, sloping bathymetry and the speed, amplitude and aspect ratio of the atmospheric forcing on Proudman resonant wave growth are investigated using analytical approximations and numerical models. With tides included, maximum wave growth through Proudman resonance occurred when the atmospheric-forcing speed matched the tidal-wave speed. Growth greater than Proudman resonance occurred with a positive tidal elevation together with a tidal current in the opposite direction to wave propagation, due to linear growth combined with further amplification from wave-flux conservation. Near-Proudman resonant growth occurred when the forced-wave speed or free-wave speed varied by either a small amount, or varied rapidly, around a speed appropriate for Proudman resonance. For a forcing moving at Proudman resonant speed, resultant wave growth was proportional to the total, time-integrated forcing amplitude. Finally, Proudman resonant wave growth was lower for forcings with lower aspect ratios (AP), partly because forced-wave heights are proportional to 1 + AP2, but also because free waves could spread in two dimensions. Whilst the assumptions of strict Proudman resonance are never met, near-Proudman resonant growth may occur over hundreds of kilometres if the effective Froude number is near 1 and the resultant wave propagates predominantly in one dimension.

Synthetic Aperture Radar Remote Sensing of Shear-Driven Atmospheric Internal Gravity Waves in the Vicinity of a Warm Front

Abstract The synthetic aperture radar ocean surface signature of atmospheric internal gravity waves in the vicinity of a synoptic-scale warm front is examined via a classic Kelvin–Helmholtz velocity profile with a rigid lower boundary and a sloping interface. The horizontal distance that the waves extend from the surface warm front is consistent with a bifurcation along the warm frontal inversion from unstable to neutral solutions. Similarity theories are derived for the wave span and the location of maximum growth rate relative to the surface front position. The theoretical maximum wave growth rate is demonstrated to occur near this bifurcation point and, hence, to explain the observed pattern of wave amplitude. Finally, a wave crest-tracing procedure is developed to explain the observed acute orientation of waves with respect to the surface warm front.

Solar wind driving of magnetospheric ULF waves: Pulsations driven by velocity shear at the magnetopause

We present results from global, three‐dimensional magnetohydrodynamic (MHD) simulations of the solar wind/magnetosphere interaction. These MHD simulations are used to study ultra low frequency (ULF) pulsations in the Earth's magnetosphere driven by shear instabilities at the flanks of the magnetopause. We drive the simulations with idealized, constant solar wind input parameters, ensuring that any discrete ULF pulsations generated in the simulation magnetosphere are not due to fluctuations in the solar wind. The simulations presented in this study are driven by purely southward interplanetary magnetic field (IMF) conditions, changing only the solar wind driving velocity while holding all of the other solar wind input parameters constant. We find surface waves near the dawn and dusk flank magnetopause and show that these waves are generated by the Kelvin‐Helmholtz (KH) instability. We also find that two KH modes are generated near the magnetopause boundary. One mode, the magnetopause KH mode, propagates tailward along the magnetopause boundary. The other mode, the inner KH mode, propagates tailward along the inner edge of the boundary layer (IEBL). We find large vortical structures associated with the inner KH mode that are centered on the IEBL. The phase velocities, wavelengths, and frequencies of the two KH modes are computed. The KH waves are found to be fairly monochromatic with well‐defined wavelengths. In addition, the inner and magnetopause KH modes are coupled and lead to a coupled oscillation of the low‐latitude boundary layer. The boundary layer thickness, d, is computed and we find maximum wave growth for kd = 0.5–1.0, where k is the wave number, consistent with the linear theory of the KH instability. We comment briefly on the effectiveness of these KH waves in the energization and transport of radiation belt electrons.

Electromagnetic ion cyclotron waves instability threshold condition of suprathermal protons by kappa distribution

The well‐known generalized Lorentzian (kappa) distribution generally provides a good representation for the high‐energy tail population of natural cosmic suprathermal plasmas. In this study we examine the electromagnetic ion cyclotron waves (EMIC) instability driven by the temperature anisotropy condition (T⊥/T∥ > 1) of suprathermal protons modeled with a typical kappa distribution in a cold multispecies plasma (electron, H+, He+, and O+). Since the EMIC wave instability is found to be significant typically above the O+ band, we apply a linear theory to study the instability threshold condition for the He+ and H+ bands particularly around the geostationary orbit. The instability threshold condition, as in the case for a regular bi‐Maxwellian, is found to follow a typical form T⊥/T∥ − 1 = S/β∥α, with higher values in the He+ band than those in the H+ band in the case of the strong wave instability owing to a lower maximum wave growth in the He+ band. As the spectral index κ increases, the instability threshold condition generally decreases and tends to the lowest limiting values of the bi‐Maxwellian, since the evaluation by the bi‐Maxwellian generally overestimates the maximum wave growth. The densities of the cold components (particularly protons) have impacts on the threshold condition primarily in the H+ band, with a higher density of cold protons leading to a lower value of the threshold condition. The results above may further reveal the nature of this instability threshold condition for the EMIC waves in any other space plasmas where an anisotropic suprathermal ion component and cold multicomponents are present together.

Modeling Density and Anisotropy of Energetic Electrons Along Magnetic Field Lines

The electromagnetic wave growth or damping depends basically on the number density and anisotropy of energetic particles as the resonant interaction takes place between the particles and waves in the magnetosphere. The variance of both the number density and anisotropy along the magnetic field line is evaluated systematically by modeling four typically prescribed distribution functions. It is shown that in the case of ``the positive anisotropy'' (namely, the perpendicular temperature T⊥ exceeds the parallel temperature T||), the number density of energetic electrons always decreases with the magnetic latitude for a regular increasing magnetic field and the maximum wave growth is therefore generally confined to the equator where the resonant energy is minimum, and the number density is the largest. However, the ``loss-cone'' anisotropy of the electrons with a ``pancake'' distribution or kappadistribution keeps invariant or nearly invariant, whereas the ``temperature'' anisotropy with a pure bi-Maxwellian distribution or Ashour-Abdalla and Kennel's distributions decreases with the magnetic latitude. The results may provide a useful approach to evaluating the number density and anisotropy of the energetic electrons at latitudes where the observation information is not available.

On the generation of Alfvén waves by solar energetic particles

A simple analytical theory of Alfven waves amplified by streaming solar energetic particles (SEPs) is studied. It is pointed out that a finite time-integrated net flux of energetic protons has to pass each point in space before we can expect Alfven waves to be significantly modified by the streaming instability. The time-integrated net proton flux needed for the time-integrated wave growth rate (or wave growth, for short) to exceed unity is evaluated. Assuming that protons stream much faster than the waves, we evaluate the wave growth as a function of position and wavenumber for a specified proton injection energy spectrum, dN/dE. The wave growth is found to be proportional to vp dN/dE, where v and p are the particle speed and momentum, and to the local Alfven speed V A . Thus, maximum wave growth is achieved at the location of maximum V A (at a few solar radii), and the minimum value of dN/dE required for the wave growth to exceed unity there is a few times 10 3 2 /vp protons per unit solid angle (in coordinate space) at the solar surface. If dN/dE is below this value, test-particle theory is a valid description of particle transport and acceleration. The value is not exceeded (above 1 MeV energies) in small gradual SEP events having peak 1-MeV proton intensities below ∼10 protons (cm 2 sr s MeV) - 1 at 1 AU. The spatial and momentum dependence of the wave growth can also be used to estimate the maximum emission strength of a moving proton source in the interplanetary medium. For a strong source moving through the solar wind at constant super-Alfvenic speed. the number of escaping particles per unit time and flux-tube cross section is approximately constant in time, predicting a plateau-type time-intensity profile observed ahead of the source. The model reproduces observations of streaming-limited intensities at energies around 10 MeV and explains the double peaked injection profiles observed in large SEP events.

Modeling Shock‐accelerated Solar Energetic Particles Coupled to Interplanetary Alfven Waves

We present an idealized model simulating the coupled evolution of the distributions of multispecies shock-accelerated energetic ions and interplanetary Alfven waves in gradual solar energetic particle (SEP) events. Particle pitch-angle diffusion coefficients are expressed in terms of wave intensities, and wave growth rates in terms of momentum gradients of SEP distributions, by the same quasilinear theory augmented with resonance broadening. The model takes into consideration various physical processes: for SEPs, particle motion, magnetic focusing, scattering by Alfven waves, solar wind convection, and adiabatic deceleration; for the waves, WKB transport and amplification by streaming SEPs. Shock acceleration is heuristically represented by continuous injection of prescribed spectra of SEPs at a moving shock front. We show the model predictions for two contrasting sets of SEP source spectra, fast weakening and softening in one case and long lasting and hard in the other. The results presented include concurrent time histories of multispecies SEP intensities and elemental abundance ratios, as well as sequential snapshots of the following: SEP intensity energy spectra, Alfven wave spectra, particle mean free paths as functions of rigidity, and spatial profiles of SEP intensities and mean free paths. Wave growth plays a key role in both cases, although the magnitude of the wave growth differs greatly, and quite different SEP abundance variations are obtained. In these simulations, the maximum wave growth rate is large, but small relative to the wave frequency, and everywhere the total wave magnetic energy density remains small relative to that of the background magnetic field. The simulations show that, as the energetic protons stream outward, they rapidly amplify the ambient Alfven waves, by several orders of magnitude in the inner heliosphere. Energetic minor ions find themselves traveling through resonant Alfven waves previously amplified by higher velocity protons. The nonuniformly growing wave spectra alter the rigidity dependence of particle scattering, resulting in complex time variations of SEP abundances at large distances from the Sun. The greatly amplified waves travel outward in an expanding and weakening shell, creating an expanding and falling reservoir of SEPs with flat spatial intensity profiles behind, while in and beyond the shell the intensities drop steeply. The wave-particle resonance relation dynamically links the evolving characteristics of the SEP and Alfven wave distributions in this new mode of SEP transport. We conclude that wave amplification, the counterpart to the scattering of streaming particles required by energy conservation, plays an essential role in the transport of SEPs in gradual SEP events. The steep proton-amplified wave spectra just upstream of the shock suggest that they may also be important in determining the elemental abundances of shock-accelerated SEP sources.

Growth and damping of oblique electromagnetic ion cyclotron waves in the Earth's magnetosphere

The dispersive properties of oblique electromagnetic ion cyclotron (EMIC) waves are examined for conditions in the Earth's outer magnetosphere (L = 7) when the energetic particle distribution has a high‐energy tail modeled by a generalized Lorentzian distribution. For small wave normal angles ψ to the ambient magnetic field, the wave growth and damping rates for a generalized Lorentzian distribution are smaller than those for a Maxwellian distribution (typically by a factor of 2), while for larger wave normal angles (ψ ≳ 60°) the corresponding differences in the growth and damping rates are relatively small. For both the generalized Lorentzian and Maxwellian distributions, maximum wave growth due to hot proton temperature anisotropy occurs for parallel propagation, but significant wave growth can occur for wave normal angles |ψ| ≲ 30°. Unstable waves produced near the magnetic equator are expected to be damped at higher latitudes as a result of cyclotron damping by thermal helium (He+) ions near the bi‐ion frequency or near the second harmonic of the helium gyrofrequency. A new physical process identified in this study is the excitation of high‐frequency oblique (ψ ∼ 50°–60°) EMIC waves due to second harmonic resonance with hot anisotropic protons. This leads to significant wave growth at frequencies above the maximum unstable frequency for parallel propagating waves.

Asymmetric instability of a viscid capillary jet in an inviscid media

The instability of asymmetric disturbances in a full circular viscid jet is investigated. It is found that asymmetric disturbances can be dominant for moderate modes and intermediate axial wave numbers for a low Weber number (≊10−5). But no asymmetric disturbance with maximum wave growth higher than the axisymmetric one was found.

Plasma response to the emission of very weak ion beams for spacecraft potential control

Future space missions aiming at the accurate measurement of cold plasmas and DC to very low frequency electric fields will require that the surface potential of their conductive surfaces be actively controlled to be near the ambient plasma potential. To do this, an active ion emission instrument will be flown aboard the upcoming Cluster, Geotail, and Interball satellites that involves the emission of an indium ion beam. To understand the beam‐plasma interaction that can occur for an indium ion beam injected from the satellite into magnetospheric and solar wind plasma, a linear theory parameter search has been carried out to investigate whether particular combinations of ambient plasma parameters and emission modes exist such that plasma instabilities may be excited. A restricted plasma model in the vicinity of spacecraft was used with an indium beam drifting with respect to an ion‐electron background plasma. Since the satellites will fly through different space plasma regions, the linear theory analysis was carried out using background parameters for solar wind, lobe, plasma sheet boundary layer, and central plasma sheet conditions. It was found that a beam‐driven lower hybrid type instability can be excited with maximum wave growth for beam emission and wave propagation transverse to the ambient magnetic field. The stability threshold as a function of the beam emission current, which can be varied during instrument operation, has been found for the various background plasma regions. The instability has the lowest current threshold in the central plasma sheet, and the highest in the lobe. The emission current determines the beam temperature, thus by varying the emission current above or below the threshold determined here, the instability can, in principle, be turned on and off, and the effects on other instruments on the satellite can be evaluated.

Linear stability of the H+ ‐ O+ two‐stream interaction in a magnetized plasma

The H+ ‐ O+ two‐stream instability in a magnetized plasma is examined for application to ion beam heating in the auroral electron acceleration region. In the fluid limit (zero ion temperature) it is found that the most unstable mode coupling is between the obliquely propagating fast O+ and slow H+ cyclotron modes, even for relative ion beam velocities where parallel propagating modes are unstable. The most unstable mode has properties analogous to those of a beam‐plasma instability. Once a relative velocity is reached where obliquely propagating modes dominate, the growth rate becomes independent of the perpendicular wave vector and so there is no particular angle of maximum wave growth. Upon adding a finite ion temperature it is found that the gross features of the H+ ‐ O+ fluid interaction are retained but finite gyroradius effects cause the appearance of an angle of maximum growth. The tangent of this angle is proportional to the velocity difference between the ion beams, for large enough relative velocity, and is also a function of the H+ to O+ density ratio. The qualitative behavior of the maximum growth rate and its frequency and wave vector versus relative velocity and density ratio are examined. The implications of this behavior for H+ and O+ beam heating in the auroral electron acceleration region are discussed.