Neutral Disturbances Research Articles

Linear two-dimensional stability of a Burgers vortex layer

A linear stability analysis is presented for a stationary Burgers vortex layer in irrotational straining flow, to normal mode disturbances invariant in the direction of main flow vorticity. The whole neutral curve is calculated by combining numerical and asymptotic analysis. It is similar to that for free mixing layers which are always unstable, except that there is unconditional stability below a critical Reynolds number, in agreement with the long-wave asymptotic result by Neu [J. Fluid Mech. 143, 253 (1984)]. The Reynolds number compares shear flow vorticity versus stretching rate and diffusion, so both latter factors are stabilizing if strong enough. Neutral disturbances represent standing waves.

Stability of thermoviscous Hele-Shaw flow

Viscous fingering can occur as a three-dimensional disturbance to plane flow of a hot thermoviscous liquid in a Hele-Shaw cell with cold isothermal walls. This work assumes the principle of exchange of stabilities, and uses a temporal stability analysis to find the critical viscosity ratio and finger spacing as functions of channel length, Lc. Viscous heating is taken as negligible, so the liquid cools with distance (x) downstream. Because the base flow is spatially developing, the disturbance equations are not fully separable. They admit, however, an exact solution for a liquid whose viscosity and specific heats are arbitrary functions of temperature. This solution describes the neutral disturbances in terms of the base flow and an amplitude, A(x). The stability of a given (computed) base flow is determined by solving an eigenvalue problem for A(x), and the critical finger spacing. The theory is illustrated by using it to map the instability for variable-viscosity flow with constant specific heat. Two fingering modes are predicted, one being a turning-point instability. The preferred mode depends on Lc. Finger spacing is comparable with the thermal entry length in a long channel, and is even larger in short channels. When applied to magmatic systems, the results suggest that fingering will occur on geological scales only if the system is about freeze.

An experimental investigation of interfacial instability in superposed flow of viscoelastic fluids in a converging/diverging channel geometry

The interfacial stability of superposed flow of polypropylene (PP) and high-density polyethylene (HDPE) in a converging-diverging channel geometry is experimentally investigated. Using the apparatus and procedures developed in our previous study [20], we have investigated the role of base flow kinematics on the interfacial stability of the HDPE/PP system. A comparison of experimentally determined neutral stability contours and disturbance growth rates in parallel, converging and diverging channel flows has revealed that channel convergence tends to stabilize interfacial disturbances while channel divergence has a destabilizing influence. Moreover, we have shown that the most dangerous wave number when made dimensionless with respect to the thickness of the more elastic and/or viscous component is of 0(1). In addition, based on a simple analysis, we have shown that the mechanism of interfacial instability in two-layer viscoelastic flows is due to the coupling between the perturbation velocity field and the polymeric stress gradient in the base flow.

Stability of Thermal Convection in a Rapidly Rotating Sphere

Linear stability of thermal convection driven by self-gravity and internal heat source in a rotating sphere is analyzed by a singular perturbation method at large Taylor number T a . The neutral disturbance localized in the radial direction generally tilts from that direction. The radial structure of such oblique disturbance takes a Gaussian form spreading over length-scale of O ( T a -1/12 ). The localness of disturbance becomes weaker as the tilt angle decreases. In the special case of uniform heat source and linear gravity force it is proved that the critical Rayleigh number R a c * should be attained by such disturbances that have vanishing tilt angle (F. H. Busse, J. Fluid Mech. 44 (1970) 441). Since, however, no disturbance without tilting can be localized in the radial direction, this critical Rayleigh number is nominal. At the Rayleigh number slightly greater than the critical ( R a * - R a c * = O ( T a -1/9 )), the disturbance tilts only slightly (by O ( T a -1/18 )) and spreads over length-scale of O ( T a -1/9 ) in the radial direction.

Secondary instability of a gas-fluidized bed

It is known that stationary fluid with density that varies sinusoidally with small amplitude and wavenumber k in the vertical direction is unstable to disturbances that are sinusoidal in a horizontal direction with wavenumber α. Small values of α/k are the most unstable in the sense that a neutral disturbance exists at sufficiently small α/k however small the Rayleigh number may be. The non-uniformity of density in the undisturbed state may be regarded as being a consequence of non-uniformity of concentration of extremely small solid particles in fluid. The paper is concerned with the corresponding instability of such a non-uniform dispersion when the particle size is not so small that the fall speed relative to the fluid is negligible. In the undisturbed state, which is an outcome of the well-known primary instability of a uniform fluidized bed with particle volume fraction Φ 0 , the sinusoidal distribution of concentration propagates vertically, and in the steady state relative to this kinematic wave particles fall with speed V(=|ΦdU/dΦ| Φ0 ), where U(Φ) is the mean speed of fall particles, relative to zero-volume-flux axes, in a uniform dispersion with volume fraction Φ. This particle convection with speed V transports particle volume and momentum and tends to even out variations of a disturbance in the vertical direction and thereby to suppress a disturbance, especially one with small α/k. Analysis of the behaviour of a disturbance is based on the equation of motion of the mixture of particles and fluid and an assumption that the disturbance velocities of the particles and the fluid are equal (as is suggested by the relatively small relaxation time of particles). The method of solution used in the associated pure-fluid problem is also applicable, and values of the Rayleigh number as a function of α/k for a neutral disturbance and a given value of the new non-dimensional parameter involving V are found. Particle convection with only modest values of V stabilizes all disturbances for which α/k 1. It appears that under practical conditions disturbances with α/k above unity are unstable, although ignorance of the values of parameters characterizing a fluidized bed hinders quantitative conclusions

Instability of stratified fluid in a vertical cylinder

In a previous paper we analysed the stability to small disturbances of stationary stratified fluid which is unbounded. Various forms of the undisturbed density distribution were considered, including a sinusoidal profile and a function of the vertical coordinate z which is constant outside a central horizontal layer. Both these types of stratification are so unstable that the critical Rayleigh number is zero. In this sequel we make the study more complete and more useful by taking account of the effect of a vertical circular cylindrical boundary of radius a which is rigid and impermeable. As in the previous paper we assume that the undisturbed density distribution is steady.The case of fluid in a vertical tube with a uniform density gradient is useful for comparison, and so we review and extend the available results, in particular obtaining growth rates for a disturbance which is neither z-independent nor axisymmetric. A numerical finite-difference method is then developed for the case in which dρ/dz = ρ0kA cos kz. When ka [Lt ] 1 the relation between growth rate and Rayleigh number approximates to that for a uniform density gradient of magnitude ρ0kA; and when ka [Gt ] 1 the tilting-sliding mechanism identified in the previous paper is relevant and the results approximate to those for an unbounded fluid, except that the smallest Rayleigh number for a neutral disturbance is not zero but is of order (ka-1. In the case of an undisturbed density which varies only in a central layer of thickness l, the same mechanism is at work when the horizontal lengthscale of the disturbance is large compared with l, resulting in high growth rates and a critical Rayleigh number which vanishes with l/a. Estimates of the growth rate are given for some particular density profiles.

Secondary thermocapillary motions of soliton type

An exact analytic solution is obtained for the problem of the stability of the axisymmetric thermocapillary motion due to a point heat source of constant power located on the horizontal free surface of a viscous fluid. Analytic expressions are found for monotonic neutral disturbances of hydrodynamic and thermal type. The critical values of the dimensionless source power for disturbances with arbitrary “quantum” numbersl andm are determined, together with the secondary motions near the stability threshold. An exact solution of the problem of the axisymmetric thermocapillary motion due to a spherical heat source is presented and its stability is investigated. It is shown that it is always possible to select physical heater properties such that for arbitrarily small source power, the axisymmetric motion is unstable relative to the vortex motion. A comparison is made with experiment.

Magnetospheric substorms and nighttime height changes of the F2 region at middle and low latitudes

Using the published plots of 15‐min interval values of h′F2, the minimum virtual height of the ionospheric F2 region, at five Japanese stations spread over 26°–45°N (15°–35°N geomagnetic) and 128°–142°E, it is shown that there are frequent rapid increases and less frequent decreases of h′F2 by 100–200 km in about 2 hours with near‐simultaneous start and end at all five stations in the postmidnight hours and at three to five stations in the premidnight hours. In the months of September and December 1988 and March 1989, about 80 h′F2 perturbations of the above type are identified. The near simultaneity of the h′F2 disturbances at the five stations suggests that an plasma drift due to an east‐west electric field is the cause of the h′F2 disturbances. This interpretation is validated by showing that there is a simultaneous presence of bay disturbances in auroral latitude magnetograms over a wide longitude range during almost all the h′F2 increases. It is shown that neutral wave disturbances propagating equatorward even with a speed of 750 m/s cannot account for the h′F2 disturbances, The study reveals two categories of h′F2 perturbations: (1) h′F2 increases starting near the beginning of the geomagnetic negative bays at auroral latitudes and (2) rapid h′F2 increases starting near the maximum intensity phase of the auroral bays and preceded by h′F2 decreases starting at the onset time of the bay disturbance or a little earlier. Using the vast amount of h′F2 data collected all over the globe, it is possible to quantify and characterize the substorm‐related electric fields and their variabilities. Such characterization is very useful in understanding the magnetospheric and the magnetosphere‐ionosphere interaction processes involved in the generation of magnetospheric substorms. It is also pointed out that the understanding of F2 region changes during magnetic storms may improve greatly if the large F2 region changes caused by electric fields on the substorm time scale are taken into account. The h′F2 disturbances reported in this study are similar in time scale and appearance to the h′F2 disturbances caused by long‐period atmospheric waves; and this suggests the possibility of a mixup of the two effects and the consequent wrong identification on some occasions.

Nonlinear Critical Layers in Barotropic Stability

Abstract Applying the method of matched asymptotic expansions (MAE) to the shallow water equations on a rotating sphere, the structure of critical layers that occur in the linear and inviscid analysis of neutral disturbances of barotropic zonal flows is investigated, assuming that the critical layers are controlled by nonlinearity rather than viscosity or nonparallel flow effects. It turns out that nonlinearity is insufficient to resolve the critical layer singularity completely. It suffices however to connect linear and nondissipative solutions across critical latitudes.

Instability of stationary unbounded stratified fluid

Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.

Neutral winds and thermosphere/ionosphere coupling and energetics during the geomagnetic disturbances of March 6–10, 1989

Observations of electron density made using the fully steerable 46‐m‐diameter antenna at Millstone Hill have been used to derive the peak electron density (NmF2) and the peak height of the F2 region (hmF2) as a function of latitude during the March 6–10, 1989, period. This period was characterized by varying levels of geomagnetic activity, with a magnetic storm commencing near 1800 UT on March 8. The radar data set presented for this period provides a detailed example of the mid‐latitude ionospheric response to geomagnetic disturbances. The derived hmF2 values are combined with measurements of electric field‐induced ion drifts and the MSIS‐86 model to estimate the meridional neutral winds at thermospheric heights over the geodetic latitude range 30° to 56°N. Strong postmidnight surges in the neutral wind were observed on March 7, 9, and 10 which reached well equatorward of Millstone Hill. The nighttime electron density trough was above Millstone Hill during the disturbances and hmF2 exceeded 500 km in the trough on March 7 and 9. A dusk enhancement in NmF2 followed the magnetic storm commencement on March 8. This is associated with a large increase in westward ion velocity due to the equatorward penetration of magnetospheric electric fields. Large daytime decreases in NmF2, apparently due to a neutral composition disturbance zone, were observed on March 9 and 10, with a sharp gradient on March 9, and a stronger equatorward penetration of the NmF2 decreases on March 10. The Joule heating as a function of latitude is estimated for March 7 and March 9 from calculations of height‐integrated Pedersen conductivity and incoherent scatter electric field measurements. In spite of considerably more Joule heating input at high latitudes on March 9, the postmidnight surge is stronger on March 7. This is explained by a combination of Coriolis and ion drag effects.

Traveling disturbances in rotating-disk flow

The stability curves for traveling disturbances in rotating-disk flow are computed using the sixth-order system of incompressible linear stability equations. It is found that the neutral curve has two minima for disturbances with positive frequencies as found earlier by Malik (1986) for stationary disturbances. The upper branch minimum occurs at ω=−2.9, R=283.6 while the lower branch minimum occurs at ω=7.9, R=64.46 where R is the Reynolds number. There exists a critical angle of approximately −35.34° (which is about 15° from the direction of maximum wall shear) below which all the waves are linearly damped. The results also show that at high frequencies the wave number for lower branch neutral disturbances varies with Reynolds number like R−1 while for stationary waves it behaves like R−1/2. The eigenfunction distribution suggests that the structure of the nonstationary high-frequency lower branch neutral disturbances are different from the structure of the viscous stationary disturbances.

Asymptotic structure of inviscid disturbances in a thin shock layer

The WKB method, used in [4] to analyze the short-wave instability of a supersonic mixing layer, is employed to investigate various types of inviscid three-dimensional short-wave disturbances in a thin shock layer of perfect gas with arbitrary velocity and temperature distributions across the layer. Simple analytic expressions for the dispersion relations are obtained for neutral disturbances. The results of an asymptotic analysis are compared with direct numerical calculations for a simple model of the shock layer.

Large‐ and small‐scale organization of electrons in the high‐latitude mesosphere: Implications of the STATE data

The electron density fluctuation spectra measured during the first and third flights of the STATE experiments (STATE 1 and STATE 3) are compared to present theories for the structuring of mesospheric electrons by neutral atmospheric turbulence. The STATE 1 electron turbulence occurs in a region of vertical electron density gradient just above the mesopause. The spectra show a nearly classic spectral form with evidence for buoyancy, inertial, and viscous subranges. A close inspection, however, reveals that the electron gas is characterized by a much smaller microscale than can reasonably be expected for neutral turbulence. Although at first very surprising, it is important to note that the very existence of the observed intense summer polar mesospheric 50‐MHz radar backscatter virtually requires such a result. We explain the extension of electron structure to smaller scales than the neutral gas by invoking a high effective Schmidt number for the passive scalar electrons. This dimensionless number is the ratio of the kinematic viscosity of the neutrals to the diffusion coefficient of the electrons and can only be significantly larger than unity if D is anomalously small. We suggest that these conditions may occur and may be due to heavily hydrated positive ions near the region of extremely low temperature at the polar mesopause. The maximum in the STATE 1 electron density fluctuation strength was found just above the mesopause and seemed to be organized with a vertical scale of several hundred meters. This variation in fluctuation spectral density correlates with the outer scale of the electron density profile itself. This suggests that the turbulence in this case was dependent on the phase of a wavelike neutral gas disturbance. The STATE 3 turbulence, on the other hand, was centered on a deep electron density “bite‐out” and was correlated in strength with the steep gradients in that region. The STATE 3 spectrum does not display a classic inertial subrange, but it does have a viscous range which is nearly identical to the STATE 3 data. Both data sets imply neutral density fluctuations of the order of 1%.

Nonlinear spiral waves in rotating pipe flow

A numerical investigation of finite-amplitude, non-axisymmetric disturbances, in the form of travelling spiral waves, is made in pipe flow with superimposed solid-body rotation. Rotating pipe flow is found to be supercritically unstable both in the rapid and slow-rotation regimes. Earlier weakly nonlinear calculations, suggesting subcritical instability in the slow-rotation limit, are shown to be in error. Bifurcating neutral waves are calculated for various axial and azimuthal Reynolds numbers and wavenumbers. For fixed axial mean pressure gradient, the axial mean flow induced by these waves gives rise to a significant flux defect, in certain cases as large as 40-50% of the undisturbed mass flux; the possible relevance of this finding to the phenomenon of vortex breakdown is pointed out. In non-rotating pipe flow, no neutral disturbances in the assumed form of spiral waves are found for moderate Reynolds numbers; this indicates that previous conjectures, regarding a possible connection between nonlinear spiral waves in slowly rotating pipe flow and the asymptotic neutral states of Smith & Bodonyi (1982) in non-rotating pipe flow, are not valid.

Stability bounds on turbulent Poiseuille flow

For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes to the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.

Simulations of the seasonal and universal time variations of the high-latitude thermosphere and ionosphere using a coupled, three-dimensional, model

The University College London Global Thermospheric Model and the Sheffield University High-Latitude Ionospheric Convection Model have been integrated and improved to simulate the self-consistent interaction of the thermosphere and ionosphere at high latitudes. For mid- and low-latitudes, equatorward of 65 degrees geomagnetic, the neutral thermospheric code maintains the use of an empirical description of plasma densities. The neutral thermospheric wind velocity, composition, density, and energy budget are computed, including their full interactions with the high-latitude ion drift and the evolution of the plasma densities of O+ , H+ , NO+ , N 2 + , and O 2 + . Two 24 hr Universal Time (UT) simulations have been performed at high solar activity, for a level of moderate geomagnetic activity, at the June and December solstices, to investigate the UT and seasonal response of the coupled system. During winter, the diurnal migration of the polar convection pattern into and out of sunlight, together with ion transport, plays a major role in the plasma density structure at F-region altitudes. Only during those UT periods, when the entire geomagnetic polar region is in total darkness, is the effect of auroral oval precipitation imprinted on the F-region. In summer, the increase in the proportion of molecular to atomic species, created by the global seasonal circulation and augmented by the geomagnetic forcing, is effective in controlling the plasma densities at all Universal Times. The increased destruction of atomic oxygen ions in summer reduces the mean level of F-region ionization to similar mean levels seen in winter, despite the increased level of solar insolation. The UT variation exceeds the seasonal differences, implying a longitudinal dependency in what can be described as a high-latitude winter ionospheric anomaly. Below 200 km summer plasma densities exceed winter values at all times, and are responsible for the larger summer conductivities, Joule heating, and consequently, increased neutral winds and composition disturbance. The summer F-region ion density profile is a broader, flatter feature than in winter, the peak spanning a wider altitude range.

Global convection electric field and current: Comparisons between model's predictions and data from STARE, Saint‐Santin, and magnetometers

Ionospheric electric field and ground magnetic field measurements at high, middle and low latitudes are used to study the global pattern of the convection electric field on March 26, 1979. The IMF Bz component was southward and relatively steady over a prolonged period of time. The data set is thus representative of a steady magnetospheric convection. A semi‐analytical time‐dependent convection model is used to interpret the observations. During the daytime the model steady state results roughly reproduces the mean observed convection pattern. Fluctuations of the observed electric field from the model predictions are interpreted as resulting from (1) the transient responses of the ionosphere to small time variations of the IMF, (2) substorm effects, (3) neutral wind disturbances. Only the first type of fluctuations can be reproduced by the initial state prediction of the convection model.

Saint-Santin observations of electric fields and neutral winds in the dynamo layer during the GTMS period of June 1984

Incoherent scatter measurements of ion drift velocities obtained by the quadristatic incoherent scatter sounder of Saint-Santin have been used to analyse midlatitude electrodynamic during the GTMS period of June 1984. June, 26, 1984 (first day of this period) is used as a magnetically quiet reference day in this study. The following results are obtained : 1) On June 27, the electric field and neutral wind disturbances could be interpreted by the sole action of the ionospheric disturbance dynamo process, 2) On June 28, two physical processes are invoked to interpret the observations : the ionospheric disturbance dynamo and the direct penetration of magnetospheric convection electric fields.

Electrostatic effects in the coupling of upper atmospheric waves to ionospheric plasma

Neutral atmosphere waves cause ion motion by neutral‐ion collisions. If the ion motion leads to space charge perturbation, then the neutral‐plasma coupling must be calculated with a self‐consistent electrostatic field. We illustrate such self‐consistent solutions for a variety of examples involving both acoustic and acoustic gravity waves. We find that the electrostatic field introduces three phenomena. First, the plasma perturbation within the neutral disturbance differs from that which would be calculated without an electric field. Second, image plasma perturbations appear above the neutral disturbance. Third, image plasma perturbations appear in the conjugate ionosphere, at least for low L shells. We emphasize the latter two image effects.