Predictions Of Linear Stability Theory Research Articles

Onset of fuzz formation in plasma-facing tungsten as a surface morphological instability

Based on simulation results and theoretical analysis according to an atomistically informed continuum-scale model for the surface morphological response of tungsten plasma-facing component (PFC) in nuclear fusion devices, we demonstrate that the onset of the well-known surface nanostructure (known as ``fuzz'') formation in PFC tungsten is the outcome of a stress-induced surface morphological instability. Specifically, we show that formation and growth of nanotendrils emanating from the exposed surface of PFC tungsten, a precursor to fuzz formation, is caused by a long-wavelength surface morphological instability triggered by compressive stress in the PFC near-surface layer due to over-pressurized helium (He) bubbles forming in this near-surface region through implantation of low-energy He ions from the plasma. Using linear stability theory (LST), we predict the onset of surface growth in response to low-amplitude perturbations from a planar PFC surface morphology and calculate the average spacing between nanotendrils growing from the PFC surface, in excellent agreement with self-consistent dynamical simulations of surface morphological response according to the fully nonlinear surface evolution model starting with random fluctuations from the planar surface morphology that result in nanoscale surface roughness. In addition, we examine the morphological response of the PFC surface to low-amplitude perturbations of very long wavelengths from its planar morphology and interpret fully the simulation results on the basis of a weakly nonlinear tip-splitting instability theory, which predicts a post-instability nanotendril pattern formation with nanotendril separation consistent with the LST predictions regardless of the initial surface perturbation. Finally, we compare our simulation predictions for surface nanotendril growth with experimental measurements of fuzz layer growth on the exposed surface of PFC tungsten. Our simulation results are in very good agreement with the surface growth measurements at the early stages of fuzz growth, further establishing the onset of fuzz formation in PFC tungsten as the outcome of a stress-driven PFC surface morphological instability.

Boundary-Layer Instabilities in Supersonic Expansion Corner Flows

Experiments show that a sudden expansion of supersonic flow may strongly stabilize or even relaminarize the boundary layer. Only few theoretical studies addressed this effect considering evolution of small disturbances in cases of Mach number less than 4, where the first mode is most unstable. This paper addresses stability of a hypersonic (Mach 6) flow over the planar expansion corners of 0° (flat plate case), 5°, and 10°, where the Mack second mode is dominant. The linear analysis and direct numerical simulations of three-dimensional wave packets are performed to investigate details and understand mechanisms behind the stabilization effect. The theoretical and numerical results are compared demonstrating applicability of the linear stability theory to hypersonic expansion-corner configurations. No new modes of boundary layer are found over the expansion corners but the second Mack mode that is relevant to the flat plate boundary layer. Downstream the corner line, the second mode instability region shifts to lower frequencies due to an abrupt thickening of the boundary layer that causes the stabilization effect. The results of direct computations agree well with predictions of linear stability theory.

Direct numerical simulation and stability analysis of the transitional boundary layer on a marine propeller blade

It is generally assumed that the cross-flow instability exists on propeller blade boundary layers due to their similarity with the rotating disk flow. However, to the best of our knowledge, there is no experimental or numerical evidence that directly shows the existence of the cross-flow instability on propellers. In this paper, we present a direct numerical simulation of the boundary layer flow around a marine propeller blade. For the investigated case, the cross-flow velocity inside the attached laminar boundary layer is smaller than the rotating disk case, but it is large enough to lead to the cross-flow instability. The wave-numbers and wave-direction of the cross-flow vortices observed agree well with the prediction of linear stability theory. Linearized Navier–Stokes simulations show that the boundary layer is absolutely unstable in the radial direction but convectively unstable in the streamwise direction. A significant flow direction deviation between laminar and turbulent regions is also observed on the blade surface. In the separation bubble, there is an unusually large radial velocity (around 50% of the inviscid streamwise velocity). Both the flow direction deviation and the large radial velocity in the separation bubble can be explained by the relationship between Coriolis forces and circumferential velocities.

Stability of liquid film flow laden with the soluble surfactant sodium dodecyl sulphate: predictions versus experimental data

Gravity-driven liquid film flows laden with a soluble surfactant are considered, and aqueous solutions of sodium dodecyl sulphate (SDS) are taken as a case-study. Literature measurements of the critical Reynolds number for the onset of instability are set in perspective with predictions of linear stability theory. The theory is based on a Frumkin model of adsorption equilibrium, modified by the inclusion of finite compressibility of the adsorbed monolayer. Quantitative comparison between data and theory is first attempted in the limit of infinite wavelength. Though wave characteristics are satisfactorily predicted, the theoretical critical Reynolds number is an order of magnitude below measurements. This discrepancy is understood in terms of the large difference between momentum and mass diffusivities and indicates that the assumption of infinite wavelength is far more restrictive for the mass transfer than for the flow problem. Finite-wavelength effects are taken into account by numerical solution of the Orr–Sommerfeld eigenvalue problem, leading to predictions of maximum stabilization in good agreement with the measurements. Introduction of realistic values of monolayer compressibility improves further the agreement at high surfactant loadings. Finally, a strong stabilizing effect of salinity is confirmed.

Transition in a separation bubble under tonal and broadband acoustic excitation

Transition and flow development in a separation bubble formed on an airfoil are studied experimentally. The effects of tonal and broadband acoustic excitation are considered since such acoustic emissions commonly result from airfoil self-noise and can influence flow development via a feedback loop. This interdependence is decoupled, and the problem is studied in a controlled manner through the use of an external acoustic source. The flow field is assessed using time-resolved, two-component particle image velocimetry, the results of which show that, for equivalent energy input levels, tonal and broadband excitation can produce equivalent changes in the mean separation bubble topology. These changes in topology result from the influence of excitation on transition and the subsequent development of coherent structures in the bubble. Both tonal and broadband excitation lead to earlier shear layer roll-up and thus reduce the bubble size and advance mean reattachment upstream, while the shed vortices tend to persist farther downstream of mean reattachment in the case of tonal excitation. Through a cross-examination of linear stability theory (LST) predictions and measured disturbance characteristics, nonlinear disturbance interactions are shown to play a crucial role in the transition process, leading to significantly different disturbance development for the tonal and broadband excited flows. Specifically, tonal excitation results in transition being dominated by the excited mode, which grows in strong accordance with linear theory and damps the growth of all other disturbances. On the other hand, disturbance amplitudes are more moderate for the natural and broadband excited flows, and so all unstable disturbances initially grow in accordance with LST. For all cases, a rapid redistribution of perturbation energy to a broad range of frequencies follows, with the phenomenon occurring earliest for the broadband excitation case. By taking nonlinear effects into consideration, important ramifications are made clear in regards to comparing LST predictions and experimental or numerical results, thus explaining the trends reported in recent investigations. These findings offer new insights into the influence of tonal and broadband noise emissions, resulting from airfoil self-noise or otherwise, on transition and flow development within a separation bubble.

Prograde, retrograde, and oscillatory modes in rotating Rayleigh–Bénard convection

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number $Pr=6.4$, corresponding to water, in a cylinder with a diameter-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=2$; second, a fluid with $Pr=0.8$, corresponding to $\text{SF}_{6}$ or air, confined in a slender cylinder with $\unicode[STIX]{x1D6E4}=0.5$; and third, the main focus of this paper, a fluid with $Pr=0.025$, corresponding to a liquid metal, in a cylinder with $\unicode[STIX]{x1D6E4}=1.87$. The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate $Pr$ a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number $Ra$, the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For $Pr=0.8$, DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low-$Pr$ fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher $Ra$, the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate $Pr$, the quasi-steady Taylor vortices, is entirely absent in low-$Pr$ flows.

Interface coupling and growth rate measurements in multilayer Rayleigh-Taylor instabilities

Using magnetic levitation, we measure the dispersion relationship for a 3-layer (semi-infinite oil / aqueous / semi-infinite oil) fluid system as a function of the height of the middle layer. The experimental results are compared with predictions of linear stability theory.

Response of a laminar separation bubble to impulsive forcing

The spatial and temporal response characteristics of a laminar separation bubble to impulsive forcing are investigated by means of time-resolved particle image velocimetry and linear stability theory. A two-dimensional impulsive disturbance is introduced with an alternating current dielectric barrier discharge plasma actuator, exciting pertinent instability modes and ensuring flow development under environmental disturbances. Phase-averaged velocity measurements are employed to analyse the effect of imposed disturbances at different amplitudes on the laminar separation bubble. The impulsive disturbance develops into a wave packet that causes rapid shrinkage of the bubble in both upstream and downstream directions. This is followed by bubble bursting, during which the bubble elongates significantly, while vortex shedding in the aft part ceases. Duration of recovery of the bubble to its unforced state is independent of the forcing amplitude. Quasi-steady linear stability analysis is performed at each individual phase, demonstrating reduction of growth rate and frequency of the most unstable modes with increasing forcing amplitude. Throughout the recovery, amplification rates are directly proportional to the shape factor. This indicates that bursting and flapping mechanisms are driven by altered stability characteristics due to variations in incoming disturbances. The emerging wave packet is characterised in terms of frequency, convective speed and growth rate, with remarkable agreement between linear stability theory predictions and measurements. The wave packet assumes a frequency close to the natural shedding frequency, while its convective speed remains invariant for all forcing amplitudes. The stability of the flow changes only when disturbances interact with the shear layer breakdown and reattachment processes, supporting the notion of a closed feedback loop. The results of this study shed light on the response of laminar separation bubbles to impulsive forcing, providing insight into the attendant changes of flow dynamics and the underlying stability mechanisms.

Nonlinear Convection in a Partitioned Porous Layer

Convection in a partitioned porous layer is considered where the thin partition causes a mechanical isolation of the two identical sublayers from one another, but heat may neveretheless conduct freely. An unsteady solver that employs the multigrid method is employed to determine steady-state strongly nonlinear for values of the Darcy–Rayleigh number up to eight times its critical value. The predictions of linear stability theory are confirmed and the accuracy of the computations are carefully monitored and controlled. It is found that the wavenumber for which the maximum rate of heat transfer is attained at any chosen value of the Darcy–Rayleigh number, Ra increases quite strongly from roughly 2.33 at onset to 6.25 when Ra = 200 . It is also found that convection generally cannot take place with wavenumbers which are close to the left-hand branch of the neutral stability curve because nonlinear interactions favour modes selected from higher harmonics.

Double-diffusive instability in core–annular pipe flow

The instability in a pressure-driven core–annular flow of two miscible fluids having the same densities, but different viscosities, in the presence of two scalars diffusing at different rates (double-diffusive effect) is investigated via linear stability analysis and axisymmetric direct numerical simulation. It is found that the double-diffusive flow in a cylindrical pipe exhibits strikingly different stability characteristics compared to the double-diffusive flow in a planar channel and the equivalent single-component flow (wherein viscosity stratification is achieved due to the variation of one scalar) in a cylindrical pipe. The flow which is stable in the context of single-component systems now becomes unstable in the presence of two scalars diffusing at different rates. It is shown that increasing the diffusivity ratio enhances the instability. In contrast to the single fluid flow through a pipe (the Hagen–Poiseuille flow), the faster growing axisymmetric eigenmode is found to be more unstable than the corresponding corkscrew mode for the parameter values considered, for which the equivalent single-component flow is stable to both the axisymmetric and corkscrew modes. Unlike single-component flows of two miscible fluids in a cylindrical pipe, it is shown that the diffusivity and the radial location of the mixed layer have non-monotonic influences on the instability characteristics. An attempt is made to understand the underlying mechanism of this instability by conducting the energy budget and inviscid stability analyses. The investigation of linear instability due to the double-diffusive phenomenon is extended to the nonlinear regime via axisymmetric direct numerical simulations. It is found that in the nonlinear regime the flow becomes unstable in the presence of double-diffusive effect, which is consistent with the predictions of linear stability theory. A new type of instability pattern of an elliptical shape is observed in the nonlinear simulations in the presence of double-diffusive effect.

Separated shear layer transition over an airfoil at a low Reynolds number

Shear layer development over a NACA 0018 airfoil at a chord Reynolds number of 100 000 was investigated using a combination of flow visualization, velocity field mapping, surface pressure fluctuation measurements, and stability analysis. The results provide a detailed description of shear layer transition on an airfoil at low Reynolds numbers. An extensive comparison of measured surface pressure and velocity fluctuations demonstrated that time-resolved surface pressure sensor arrays can be used to identify the presence of flow separation, estimate the extent of the separated flow region, and measure disturbance growth rate spectra in significantly less time than is required by conventional techniques. Surface pressure sensor measurements of disturbance growth rate, wave number, and convection speed are found to compare well with predictions of linear stability theory, supporting the claim that convection speeds measured in separation bubbles over low Reynolds number airfoils are associated with wave packets of growing disturbances propagating through the shear layer. Through a comparison of measured convection speeds in this investigation and prior low Reynolds number airfoil experiments, it is shown that disturbance convection speeds of between 30% and 50% of the edge velocity are typical for this type of flow, consistent with phase speed estimates from previous analytical studies on transitional separation bubbles. Modal RMS velocity profiles were measured and found to be reasonably predicted by stability theory. The results suggest that, even for the relatively thick NACA 0018 airfoil profile, disturbance development over the majority of the laminar separated shear layer is primarily governed by a linear inviscid mechanism.

Parametric study of separation and transition characteristics over an airfoil at low Reynolds numbers

Time-resolved surface pressure measurements are used to experimentally investigate characteristics of separation and transition over a NACA 0018 airfoil for the relatively wide range of chord Reynolds numbers from 50,000 to 250,000 and angles of attack from 0° to 21°. The results provide a comprehensive data set of characteristic parameters for separated shear layer development and reveal important dependencies of these quantities on flow conditions. Mean surface pressure measurements are used to explore the variation in separation bubble position, edge velocity in the separated shear layer, and lift coefficients with angle of attack and Reynolds number. Consistent with previous studies, the separation bubble is found to move upstream and decrease in length as the Reynolds number and angle of attack increase. Above a certain angle of attack, the proximity of the separation bubble to the location of the suction peak results in a reduced lift slope compared to that observed at lower angles. Simultaneous measurements of the time-varying component of surface pressure at various spatial locations on the model are used to estimate the frequency of shear layer instability, maximum root-mean-square (RMS) surface pressure, spatial amplification rates of RMS surface pressure, and convection speeds of the pressure fluctuations in the separation bubble. A power-law correlation between the shear layer instability frequency and Reynolds number is shown to provide an order of magnitude estimate of the central frequency of disturbance amplification for various airfoil geometries at low Reynolds numbers. Maximum RMS surface pressures are found to agree with values measured in separation bubbles over geometries other than airfoils, when normalized by the dynamic pressure based on edge velocity. Spatial amplification rates in the separation bubble increase with both Reynolds number and angle of attack, causing the accompanying decrease in separation bubble length. Values of the convection speed of pressure fluctuations in the separated shear layer are measured to be between 35 and 50% of the edge velocity, consistent with predictions of linear stability theory for separated shear layers.

Formation and ordering of topological defect arrays produced by dilatational strain and shear flow in smectic-Aliquid crystals

A microscale shear cell is used to study the formation of parabolic focal conic defects in the thermotropic smectic-A liquid crystal 8CB (4-octyl-4'-cyanobiphenyl). Defects are produced by four distinct methods: by the application of dilatational strain alone, by shear flow alone, by dilatational strain and subsequent shear flow, and by the simultaneous application of dilatational strain and shear flow. We confirm that defects originate within the bulk, consistent with the previously suggested undulation instability mechanism. In the presence of a shear flow, we observe that defect formation requires micrometer-level dilatations, whose magnitude depends on the sample thickness. The size and ordering of both disordered and ordered defect arrays is quantified using a pair distribution function. Deviations from the predictions of linear stability theory are observed that have not been reported previously. For example, defects form a square array with greater ordering in the principal flow direction. Ordering due to shear flow does not change the average defect size. It has been shown previously that the principal defect sizes of ordered defects scale differently with sample thickness than the wavelength of the small amplitude undulations. We find that disordered defects show a similar deviation from this predicted wavelength.

Differential frost heave model for patterned ground formation: Corroboration with observations along a North American arctic transect

Frost boils in the Arctic are a manifestation of patterned ground in the form of nonsorted circles. Active frost boils involve convection of water through the soil that can bring basic salts from depth to the surface. As such, active frost boils can mitigate acidification and thereby strongly influence the type of vegetation supported by Arctic soils. The presence or absence of active frost boils is thought to play a pivotal role in establishing the sharp demarcation between moist nonacidic tundra (MNT) and moist acidic tundra (MAT) in the Arctic. The focus of this paper is to corroborate the predictions of a mathematical model that relates observable patterned ground features to ecosystem parameters with observations at the field sites along the North American Arctic Transect (NAAT) established by the Biocomplexity of Patterned‐Ground Ecosystems Project. Model predictions indicate that recurrent one‐dimensional frost heave can become unstable and evolve into multidimensional differential frost heave (DFH). A laboratory frost heave simulation produced a 28‐cm pattern in an active layer of 10 cm, which agrees with linear stability theory predictions. A finite element solution predicts three‐dimensional patterns with approximately 3‐m spacing develop in a 1.0‐m active layer with a surface n factor of 0.35, which agrees well with field observations from the NAAT. The lack of significant frost boil activity in the MAT is a result of suppression of DFH owing to denser surface vegetation characterized by low n factors. Prominent active frost boils are observed in the MNT at higher latitudes with more sparse vegetation characterized by higher n factors that promote DFH. However, at the northernmost field sites frost boils cannot be generated even though the n factors are relatively high owing to very rapid freezing conditions that mitigate DFH.

Receptivity of a hypersonic boundary layer over a flat plate with a porous coating

Two-dimensional direct numerical simulation (DNS) of receptivity to acoustic disturbances radiating onto a flat plate with a sharp leading edge in the Mach 6 free stream is carried out. Numerical data obtained for fast and slow acoustic waves of zero angle of incidence are consistent with the asymptotic theory. Numerical experiments with acoustic waves of non-zero angles of incidence reveal new features of the disturbance field near the plate leading edge. The shock wave, which is formed near the leading edge owing to viscous–inviscid interaction, produces a profound effect on the acoustic near field and excitation of boundary-layer modes. DNS of the porous coating effect on stability and receptivity of the hypersonic boundary layer is carried out. A porous coating of regular porosity (equally spaced cylindrical blind micro-holes) effectively diminishes the second-mode growth rate in accordance with the predictions of linear stability theory, while weakly affecting acoustic waves. The coating end effects, associated with junctures between solid and porous surfaces, are investigated.

On two-dimensional linear waves in Blasius boundary layer over viscoelastic layers

This work concerns the direct numerical simulation of small-amplitude two-dimensional ribbon-excited waves in Blasius boundary layer over viscoelastic compliant layers of finite length. A vorticity-streamfunction formulation is used, which assures divergence-free solutions for the evolving flow fields. Waves in the compliant panels are governed by the viscoelastic Navier's equations. The study shows that Tollmien–Schlichting (TS) waves and compliance-induced flow instability (CIFI) waves that are predicted by linear stability theory frequently coexist on viscoelastic layers of finite length. In general, the behaviour of the waves is consistent with the predictions of linear stability theory. The edges of the compliant panels, where abrupt changes in wall property occur, are an important source of waves when they are subjected to periodic excitation by the flow. The numerical results indicate that the non-parallel effect of boundary-layer growth is destabilizing on the TS instability. It is further demonstrated that viscoelastic layers with suitable properties are able to reduce the amplification of the TS waves, and that high levels of material damping are effective in controlling the propagating CIFI.

Experiments on stability and transition at Mach 3

The results of an experimental study of stability, receptivity and transition of the flat-plate laminar boundary layer at Mach 3 are discussed. With a relatively low free-stream disturbance level (∼0.1%), spectra, growth rates and amplitude distributions of naturally occurring boundary layer waves were measured using hot wires. Physical (mass-flux) amplitudes in the boundary layer and free stream are reported and provide stability and receptivity results against which predictions can be directly compared. Comparisons are made between measurements of growth rates of unstable high-frequency waves and theoretical predictions based on a non-parallel, mode-averaging stability theory and receptivity assumptions; good agreement is found. In contrast, it was found that linear stability theory does not account for the measured growth of low-frequency disturbances. A detailed investigation of the disturbance fields in the free stream and on the nozzle walls provides the basis for a discussion of the source and the development of the measured boundary layer waves. Attention is drawn to the close matching in streamwise wavelengths for instability waves and the free-stream acoustic disturbances. It was also found that a calibration of the hot wire in the free stream yields a double-peak boundary layer disturbance amplitude distribution, as has been found by previous investigators, which is not consistent with the predictions of linear stability theory. This double peak was found to be an experimental anomaly which resulted from assumptions that are frequently made in the free-stream calibration procedure. A single-peak amplitude distribution across the boundary layer was established only when the hot-wire voltage was calibrated against the mean boundary layer profile. Finally, the late stages of transition, at a higher Reynolds number with a higher free-stream disturbance level, were explored. Calibrated amplitude levels are provided at locations where nonlinearities are first detected and where the mean boundary layer profile is first observed to depart from the laminar similarity solution. A qualitative discussion of the character of ensuing nonlinearities is also included.

Nonlinear dynamics of an interface in an inclined channel

Here we consider the gravity-driven dynamics of two superposed, incompressible, immiscible, viscous fluids flowing in a parallel-wall channel. The fluids are governed by the Navier–Stokes equations with surface tension along the interface. We use the front tracking numerical method to determine the solutions. Both stable and unstable solutions are obtained. The unstable solutions are characterized by a finger of lower fluid rising into the upper fluid. Our results are consistent with the predictions of linear stability theory. In addition, we attempt to characterize the growth of the unstable solutions as a function of the Reynolds and the Bond number.

A numerical study of dynamics of a temporally evolving swirling jet

Direct numerical simulation (DNS) of a swirling jet near the outlet of a nozzle with axisymmetric and non-axisymmetric disturbances is performed to investigate the dynamic characteristics of the flow. The early (linear) stage of the jet evolution agrees well with the predictions of linear stability theory. In the nonlinear stage, the axisymmetric DNS shows that the interaction between the primary vortex ring and the streamwise columnar vortex creates a secondary vortex structure with opposite azimuthal vorticity near the columnar vortex. Then a vortex pair consisting of the primary and secondary vortices forms and travels radially away from the symmetry axis, causing a rapid increase of the thickness of mixing layer. The non-axisymmetric DNS shows that the streamwise vortex layer developed in the early stage of evolution due to azimuthal instability breakdowns into small eddies under the joint stretch of the axial and azimuthal shear. The results reveal several mechanisms of mixing enhancement by swirl, i.e., the radial motion of vortex ring pairs, the rapid growth of streamwise vorticity, and the creation of three-dimensional small eddies. They are all favorable for fluid entrainment in swirling jets.

The Life Cycle of a Mesoscale Gravity Wave as Observed by a Network of Doppler Wind Profilers

Abstract For the first time, an analysis has been made of the evolving vertical structure of a long-lived mesoscale gravity wave that exerted a strong influence upon the precipitation distribution across a large area. This paper describes this gravity wave system on 14 February 1992, which was observed using a combination of a surface mesonetwork, digital satellite and radar imagery, and several Doppler wind profilers. The resulting vertical structures are compared to the predictions of linear stability theory. Since the signature of the gravity waves in the profiler vertical beam data was often complicated by the presence of precipitation, a kinematic method was developed for estimating the vertical air motions during these periods. The resultant time–height fields show vertical and horizontal winds that are consistent with a gravity wave conceptual model, the microbarograph traces, and the cloud and precipitation patterns. In the early stages of development, a strong vertically erect wave of depression ...