Fluid Particle Trajectories Research Articles

Diffusive effects in local instabilities of a baroclinic axisymmetric vortex

We present a local stability analysis of an idealized model of the stratified vortices that appear in geophysical settings. The base flow comprises an axisymmetric vortex with background rotation and an out-of-plane stable stratification, and a radial stratification in the thermal wind balance with the out-of-plane momentum gradient. Solving the local stability equations along fluid particle trajectories in the base flow, the dependence of short-wavelength instabilities on the Schmidt number $Sc$ (ratio between momentum and mass diffusivities) is studied, in the presence of curvature effects. In the diffusion-free limit, the well-known symmetric instability is recovered. In the viscous, double-diffusive regime, instability characteristics are shown to depend on three non-dimensional parameters (including $Sc$ ), and two different instabilities are identified: (i) a monotonic instability (same as symmetric instability at $Sc = 1$ ), and (ii) an oscillatory instability (absent at $Sc = 1$ ). Separating the base flow and perturbation characteristics, two each of base flow and perturbation parameters (apart from $Sc$ ) are identified, and the entire parameter space is explored for the aforementioned instabilities. In comparison with $Sc = 1$ , monotonic and oscillatory instabilities are shown to significantly expand the instability region in the space of base flow parameters as $Sc$ moves away from unity. Neutral stability boundaries on the plane of $Sc$ and a modified gradient Richardson number are then identified for both these instabilities. In the absence of curvature effects, our results are shown to be consistent with previous studies based on normal mode analysis, thus establishing that the local stability approach is well suited to capturing symmetric and double-diffusive instabilities. The paper concludes with a discussion of curvature effects, and the likelihood of monotonic and oscillatory instabilities in typical oceanic settings.

An experimental and numerical study of slurry erosion behavior in a horizontal elbow and elbows in series

Erosion in elbows is one of the most important issues in pipeline flow assurance for the oil and gas and mining industries. In this study, slurry erosion in a horizontal elbow and elbows in series are numerically and experimentally investigated. The maximum erosion location of horizontal elbow is revealed by conducting slurry erosion painting experiment. The erosion process is then modeled by using computational fluid dynamics (CFD) techniques. The accuracy and stability of the CFD-based erosion model are carefully examined using experimental results. Using the validated numerical methods, the effects of particle concentration, geometry parameter, inlet conditions are investigated. The erosion processes under these different conditions are further analyzed by examining fluid flow development, particle trajectories and impact information. The results of this CFD study present some cautious guidelines for conducting erosion studies of liquid-solid flows and pipeline system designs.

Numerical study of acoustophoretic manipulation of particles in microfluidic channels.

The microfluidic technology based on surface acoustic waves (SAW) has been developing rapidly, as it can precisely manipulate fluid flow and particle motion at microscales. We hereby present a numerical study of the transient motion of suspended particles in a microchannel. In conventional studies, only the microchannel's bottom surface generates SAW and only the final positions of the particles are analyzed. In our study, the microchannel is sandwiched by two identical SAW transducers at both the bottom and top surfaces while the channel's sidewalls are made of poly-dimethylsiloxane (PDMS). Based on the perturbation theory, the suspended particles are subject to two types of forces, namely the Acoustic Radiation Force (ARF) and the Stokes Drag Force (SDF), which correspond to the first-order acoustic field and the second-order streaming field, respectively. We use the Finite Element Method (FEM) to compute the fluid responses and particle trajectories. Our numerical model is shown to be accurate by verifying against previous experimental and numerical results. We have determined the threshold particle size that divides the SDF-dominated regime and the ARF-dominated regime. By examining the time scale of the particle movement, we provide guidelines on the device design and operation.

Numerical simulation of critical particle size in asymmetrical deterministic lateral displacement

Microfluidics devices are widely used for particle separation. Deterministic Lateral Displacement (DLD) is a passive method for particle separation. DLD devices mainly separate particles based on their sizes. There are two main modes of movement in DLD arrays; the small particles move in a zigzag path, and the larger particles separate in the displacement mode. It is therefore important to estimate the critical particle size for the transition of modes before the fabrication of DLD devices. Asymmetry in the design of the arrays can affect the fluid behavior and the critical particle size. In this study, we investigate the effects of the asymmetry caused by changing the downstream gap size to the lateral gap size ratio on the fluid behavior and particle trajectories in DLD devices. We used two dimensional (2D) Finite Element Method (FEM) to study the variations in the flow lane's widths and combined the fluid analysis with structural mechanics to model the contact between the particles and the posts in DLD arrays. We simulated the spherical particles' trajectories with diameters ranging from 1.4 to 19.2 μm in circular post DLD arrays with a lateral gap size of 20μm. In contrast to the previous works, in these simulations, the effect of particle movement on the fluid flow profiles was considered. We evaluated the particle movement mode in seven different values of the downstream gap size to the lateral gap size ratio (ranging from 0.5 to 2) and eight different row shift fraction (ranging from 0.025 to 0.3). Our simulations showed that increasing the value of the downstream gap while the lateral gap is fixed increases the veering flow rate and width. By finding the particle with the largest diameter in the zigzag mode and the particle with the smallest diameter in the displacement mode, we estimated the critical particle diameter for each value of shift fraction in different values of the downstream gap to the lateral gap size ratio. Using these data, a curve was fitted for predicting the critical particle diameter in each ratio. Finally, a more general form of the formula for the critical particle diameter was proposed, which considers an extra parameter compared to the previous ones. The results of this study can lead to a better understanding of DLD devices' functions and, thus, save time and costs for better designs and experiments.

Investigation of the weak solubility of the fractional Voigt alpha-model

This paper is devoted to investigating the weak solubility of the alpha-model for a fractional viscoelastic Voigt medium. The model involves the Voigt rheological relation with a left Riemann–Liouville fractional derivative, which accounts for the medium’s memory. The memory is considered along the trajectories of fluid particles determined by the velocity field. Since the velocity field is not smooth enough to uniquely determine the trajectories for every initial value, we introduce weak solutions of this problem using regular Lagrangian flows. On the basis of the approximation-topological approach to the study of hydrodynamical problems, we prove the existence of weak solutions of the alpha-model and establish the convergence of solutions of the alpha-model to solutions of the original model as the parameter tends to zero.

Uncertainty Quantification of Trajectory Clustering Applied to Ocean Ensemble Forecasts

Partitioning ocean flows into regions dynamically distinct from their surroundings based on material transport can assist search-and-rescue planning by reducing the search domain. The spectral clustering method partitions the domain by identifying fluid particle trajectories that are similar. The partitioning validity depends on the accuracy of the ocean forecasting, which is subject to several sources of uncertainty: model initialization, limited knowledge of the physical processes, boundary conditions, and forcing terms. Instead of a single model output, multiple realizations are produced spanning a range of potential outcomes, and trajectory clustering is used to identify robust features and quantify the uncertainty of the ensemble-averaged results. First, ensemble statistics are used to investigate the cluster sensitivity to the spectral clustering method free-parameters and the forecast parameters for the analytic Bickley jet, a geostrophic flow model. Then, we analyze an operational coastal ocean ensemble forecast and compare the clustering results to drifter trajectories south of Martha’s Vineyard. This approach identifies regions of low uncertainty where drifters released within a cluster predominantly remain there throughout the window of analysis. Drifters released in regions of high uncertainty tend to either enter neighboring clusters or deviate from all predicted outcomes.

Hydrophilic properties of soot particles exposed to OH radicals: A possible new mechanism involved in the contrail formation

This work investigates the role of soot particles in the early steps of formation of condensation trails (contrails). Contrails are thin linear clouds that form behind cruising aircrafts and can evolve into persistent clouds, and therefore can contribute to the radiative forcing of the atmosphere. Mitigating their contribution is considered an efficient way for decreasing the radiative forcing. Condensation freezing is proposed as a major pathway leading to the heterogeneous nucleation of ice particles in the aircraft exhausts. Soot particles, freshly produced in the flames and not significantly oxidized in the post-flame, are hydrophobic. Therefore, many theoretical and experimental approaches that investigate contrails formation introduce prior activation by surface adsorption of sulfur compounds issued from kerosene combustion. Even though the role of sulfur content of the fuel is unclear, and may only weakly impacts contrails formation, many simulations still rely on this assumption. In this work, to elucidate the role of sulfur compounds on the activation of soot chemically aged with OH radicals, a comparative study is performed with soot sampled from a turbulent jet flame burning a sulfur-containing fuel (kerosene) and a sulfur-free fuel (diesel). Soot aging experiments are performed in the atmospheric simulation chamber CESAM, with controlled generation of OH radicals. Soot activation (ratio between the number of nucleated droplets and seeding particles) is measured in water supersaturation conditions. OH exposure significantly enhances soot activation regardless of the sulfur presence in the fuel. The possibility for soot oxidation by OH radicals formed in aeronautical engines to be sufficient to promote soot activation is explored. Large eddy simulation in the high pressure turbine is used to model the fluid particle and OH trajectories and their chemical evolution. Residence time in the turbine is found sufficient to activate soot particles, opening a new possible route to explain ice particles formation.

Physics of Magnetohydrodynamic Rossby Waves in the Sun

Abstract Evidence of the existence of hydrodynamic and MHD Rossby waves in the Sun is accumulating rapidly. We employ an MHD Rossby wave model for the Sun in simplified Cartesian geometry, with a uniform toroidal field and no differential rotation, to analyze the role of each force that contributes to Rossby wave dynamics, and compute fluid particle trajectories followed in these waves. This analysis goes well beyond the traditional formulation of Rossby waves in terms of conservation of vorticity. Hydrodynamic Rossby waves propagate retrograde relative to the rotation of the reference frame, while MHD Rossby waves can be both prograde and retrograde. Fluid particle trajectories are either clockwise or counterclockwise spirals, depending on where in the wave pattern they are initiated, that track generally in the direction of wave propagation. Retrograde propagating MHD Rossby waves move faster than their hydrodynamic counterparts of the same wavelength, becoming Alfvén waves at very high field strengths. Prograde MHD Rossby waves, which have no hydrodynamic counterpart, move more slowly eastward than retrograde MHD Rossby waves for the same toroidal field, but with a speed that increases with toroidal field, in the high field limit again becoming Alfvén waves. The longitude and latitude structures of all these waves, as seen in their velocity streamlines and perturbation field lines as well as fluid particle trajectories, are remarkably similar for different toroidal fields, rotation, longitudinal wavelength, and direction of propagation.

Theoretical and experimental study of trapping PM2.5 particles via magnetic confinement effect in a multi-electric field ESP

The capture of fine particulate matter (PM2.5) is the central function of dust removal systems in coal-fired power plants. To improve the collection efficiency of multi-electric field electrostatic precipitators (ESPs), a magnetic field was applied to investigate its effect on the dust removal performance of an experimental ESP. Particle image velocimetry (PIV) technology was used to test the effect of magnetic confinement on fluid field distribution and particle trajectory. A numerical model was developed to describe the physical processes occurring in the multi-electric field ESP, including airflow, corona discharge, particle charging and particle dynamics. The effect of magnetic field on the trapping efficiency of PM2.5 under various working voltages was analysed. The PIV experiment and numerical simulation showed that particle trajectories exposed to a magnetic field obeyed the same laws of motion as those not exposed to a magnetic field. The introduction of a magnetic field resulted in more complex spiral movement of particles in the ESP and increased the number of particles deflecting towards the dust collecting plate. Thus, magnetic confinement improved the efficiency of PM2.5 collection in the multi-electric field ESP. Increasing magnetic induction not only improved dust removal to the same extent as that achieved by increasing the voltage but also reduced power consumption. At lower working voltages, weaker magnetic induction intensities or lower collection efficiencies, the magnetic field enhanced the PM2.5 trapping performance more evidently. These results provide a theoretical basis and a technical reference for the design of novel and efficient multi-electric field ESPs to reduce costs and protect the environment.

Two-phase flow measurement of sub-millimeter sized particles falling in water with grid-generated turbulence

Sediment transport is one key process in sedimentation problems. The dynamics of sediments in rivers will lead to an ecological change of the catchment. Among all complicated processes in sediment transport, the settling dynamics of solid particles in the water contribute to the water quality and geomorphology of a river channel directly. This paper aims at an investigation of the interaction between particle movement and water flow, with application to hydro-environment situations such as transport of fine sand sediment in natural streams. The Stokes number in this type of application is much smaller than unity at which enhancement of particle settling velocity by strong ambient turbulence has been observed in many numerical and laboratory studies. The present laboratory study is conducted to study the settling of sub-millimeter sized heavy particles in water with a relatively weak grid-generated turbulence. The two-phase flow is measured with the whole-field imaging techniques of particle image velocimetry (PIV) and particle tracking velocimetry (PTV). A two-camera PIV/PTV technique is used to obtain the instantaneous settling velocities of the solid particles and the turbulent water velocity field. Phase separation is based on an effective optical distinction of the two light scattering signals with fluorescent tagging on the solid particles. With this technique, it is found that in most cases of grid-generated turbulence the settling velocity of the heavy particles is slightly lower than the still water value. The falling heavy particles are also found to cause additional turbulence in the water with low ambient turbulence intensities. At one particular case of grid turbulence, the particles are found to fall with a depth-averaged settling velocity evidently higher than the still water value. The instantaneous turbulent fluid motions and particle trajectories tend to support the fast tracking effect to be related to the higher particle falling velocities.

Von Kármán-Howarth and Corrsin equations closures through Liouville theorem

In this communication, the closure formulas of von Kármán-Howarth and Corrsin equations are obtained through the Liouville theorem and the hypothesis of homogeneous isotropic incompressible turbulence. Such closures, based on the concept that, in fully developed turbulence, contiguous fluid particles trajectories continuously diverge, are of non-diffusive nature, and express a correlations spatial propagation phenomenon between the several scales which occurs with a propagation speed depending on length scale and velocity standard deviation. These closure formulas coincide with those just obtained in previous works through the finite scale Lyapunov analysis of the fluid act of motion. Here, unlike the other articles, the present study does not use the Lyapunov theory, and provides the closures showing first an exact relationship between the pair spatial correlations calculated with the velocity distribution function and those obtained using the material separation line distribution function. As this analysis does not adopt the Lyapunov theory, this does not need the definition and/or the existence of the Lyapunov exponents. Accordingly, the present proof of the closures results to be more general and rigorous than that presented in the other works, corroborating the previous results. Finally, the conditions of existence of invariants in isotropic turbulence are studied by means of the proposed closures. In the presence of such invariants and self-similarity, the sole evolution of velocity and temperature standard deviations and of the correlation scales is shown to be adequate to fairly describe the isotropic turbulence.

In-Vitro Simulation of the Blood Flow in an Axisymmetric Abdominal Aortic Aneurysm

We investigated the blood flow patterns and the hemodynamics associated with an abdominal aortic aneurysm detected in an in vitro measurement campaign performed in a laboratory model of an aneurysm with rigid walls and an axisymmetric shape. Experiments were run in steady flow conditions and by varying the Reynolds number in the range 410 < Re < 2650. High spatial and temporal resolution 2D optical measurements of the velocity field were obtained through a particle tracking technique known as Hybrid Lagrangian Particle Tracking. Conversely to classical Particle Image Velocimetry, both the fluid particle trajectories and the instantaneous and time-averaged velocity fields are provided without constraints on the grid size and very close to the vessel boundary. All the most relevant quantities needed to investigate the flow features were evaluated, and in particular, we focused on the wall shear stress distribution both in the healthy aortic portion and within the aneurysm. Results show that the recirculation zone in correspondence of the cavity moves downstream, and this displacement is found to increase with Re. Very low wall shear stress values are recovered in correspondence of the aneurysmal cavity, while a sharp peak occurs in correspondence of the reattachment point. In agreement with the literature data, the peak value is found to decrease with Re and to be about equal to twice the upstream value.

Calculation of flow in incompressible regenerative turbo-machines with bucket form blades based on the geometry of flow path

Purpose Regenerative flow pumps are dynamic machines with the ability to develop high heads at low flow rates. Simplicity, compactness, stable features and low manufacturing costs make them interesting for many applications in industries. The purpose of this study is to present a new method for calculating the flow through regenerative pumps with bucket form blades to predict the performance curves by a cheap and easy-to-use way. Design/methodology/approach The analysis was carried out based on the geometric shape of a fluid particle trajectory in a regenerative turbomachine. The fluid particle path was assumed to be a helix wrapped into a torus. Loss models were considered and the results of predictions were compared with computational fluid dynamics (CFD) data. Findings The overall trend of performance curves resulted from presented model looked consistent with CFD data. However, there were slight differences in high and low flow coefficients. The results showed that the predicted geometric shape of the flow path with the presented model (a helix wrapped into a torus) was not consistent with CFD results at high flow coefficients. Due to the complexity and turbulence of the fluid flow and errors in the calculation of losses, as well as slip factor, there was a discrepancy between the results of the presented model and numerical simulation, especially in high and low flow coefficients. Originality/value The analysis was carried out based on the geometric shape of a fluid particle trajectory in a regenerative turbomachine with bucket form blades. The fluid particle path was assumed to be a helix wrapped into a torus.

Lagrangian investigations of velocity gradients in compressible turbulence: lifetime of flow-field topologies

We perform Lagrangian investigations of the dynamics of velocity gradients in compressible decaying turbulence. Specifically, we examine the evolution of the invariants of the velocity-gradient tensor. We employ well-resolved direct numerical simulations over a range of Mach number along with a Lagrangian particle tracker to examine trajectories of fluid particles in the space of the invariants of the velocity gradient tensor. This allows us to accurately measure the lifetimes of major topologies of compressible turbulence and provide an explanation of why some selective topologies tend to exist longer than the others. Further, the influence of dilatation on the lifetime of various topologies is examined. Finally, we explain why the so-called conditional mean trajectories (CMT) used previously by several researchers fail to predict the lifetime of topologies accurately.

Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference.

КВАЗИ-ЛАГРАНЖЕВОЕ ИНТЕГРИРОВАНИЕ ДВУМЕРНЫХ УРАВНЕНИЙ ПРАНДТЛЯ И БУССИНЕСКА В НЕВЯЗКОМ ПРЕДЕЛЕ

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Numerical investigation of the isothermal flow field and particle deposition behaviour in a rotating fluidized bed solar receiver

A numerical analysis of the isothermal flow field within a directly irradiated Rotating Fluidized Bed Receiver (RFBR), is presented to provide a systematic assessment of the influence of key receiver control parameters, namely fluidized bed rotational speed and radial fluidizing gas velocity, on the flow field inside the receiver and particle deposition onto the receiver window. To achieve these aims, a Computational Fluid Dynamics (CFD) model of the RFBR was developed and coupled with Discrete Phase Model (DPM) to analyse the fluid flow and particle trajectory in the receiver cavity due to systematic variations in the key control parameters. The fluid flow modelling approach was partially verified by comparing the numerical predictions with previously published experimental flow measurements in a rotating vortex flow device that is geometrically similar to the RFBR. Using the reported modelling approach, the sensitivity of the flow field and particle deposition to the variations in the key control parameters was determined. Flow features and physical mechanisms linked to particle deposition onto the receiver window were identified with the view to better understand the operation of the RFBR and determine suitable operating regimes that achieve a low risk of particle deposition.

Tunable diffusion in wave-driven two-dimensional turbulence

We report an abrupt change in the diffusive transport of inertial objects in wave-driven turbulence as a function of the object size. In these non-equilibrium two-dimensional flows, the turbulent diffusion coefficient$D$of finite-size objects undergoes a sharp change for values of the object size$r_{p}$close to the flow forcing scale$L_{f}$. For objects larger than the forcing scale ($r_{p}>L_{f}$), the diffusion coefficient is proportional to the flow energy$U^{2}$and inversely proportional to the size$r_{p}$. This behaviour,$D\sim U^{2}/r_{p}$, observed in a chaotic macroscopic system is reminiscent of a fluctuation–dissipation relation. In contrast, the diffusion coefficient of smaller objects ($r_{p}<L_{f}$) follows$D\sim U/r_{p}^{0.35}$. This result does not allow simple analogies to be drawn but instead it reflects strong coupling of the small objects with the fabric and memory of the out-of-equilibrium flow. In these turbulent flows, the flow structure is dominated by transient but long-living bundles of fluid particle trajectories executing random walk. The characteristic widths of the bundles are close to$L_{f}$. We propose a simple phenomenology in which large objects interact with many bundles. This interaction with many degrees of freedom is the source of the fluctuation–dissipation-like relation. In contrast, smaller objects are advected within coherent bundles, resulting in diffusion properties closely related to those of fluid tracers.

Three-dimensional small-scale instabilities of plane internal gravity waves

We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.

Wind generated equatorial Gerstner-type waves

A class of non-stationary surface gravity waves propagating in the zonal direction in the equatorial region is described in the f-plane approximation. These waves are described by exact solutions of the equations of hydrodynamics in Lagrangian formulation and are generalizations of Gerstner waves. The wave shape and non-uniform pressure distribution on a free surface depend on two arbitrary functions. The trajectories of fluid particles are circumferences. The solutions admit a variable meridional current. The dynamics of a single breather on the background of a Gerstner wave is studied as an example.