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Diabatic heating by convection in the eyewall often produces an annular region of high potential vorticity (PV) around the relatively low PV eye in a strong tropical cyclone (TC). Such a PV ring is barotropically unstable and can encourage exponentially growing PV waves. In this study, such instability and the subsequent nonlinear evolutions of three TC-like vortices with different degrees of hollowness PV ring on an f-plane are first examined using an unforced, inviscid shallow-water-equation model. Results show that the simulated eyewalls evolve similarly to those in the nondivergent barotropic model. It is also found that the polygonal eyewall structure can be decomposed into vortex Rossby waves (VRWs) of different wavenumbers with different amplitudes, allowing for wave-wave interaction to produce complicated behaviors of mesovortices in the TC eyewall. The same set of PV rings has been examined on a beta-plane. Although the beta effect has been rendered unimportant to the eyewall evolution due to the relatively small-scale of inner-core circulation, it is shown in this study that the beta effect may erode the coherent structure of mesovortices in the eyewall of an initially hollow PV-ring vortex. Mesovortices modeled on the beta-plane with greater beta parameter tends to experience an earlier breakdown along with the enhanced radial gradients of the basic-state (azimuthal mean) angular velocity, followed by wave-wave and wave-flow interactions, leading to earlier merger and axisymmetrization processes. This implies that the beta effect could be one of the forcings that shorten the lifetime of quasi-steady mesovortices in the eyewall of real TCs.

Numerical simulations of steady and unsteady viscous flows are presented by adopting two different numerical methodologies: the Smoothed Particle Hydrodynamics formulation implemented in the open-source code DualSPHysics and an in-house lattice Boltzmann code based on a concise central-moments scheme. Both methods employ a weakly compressible assumption to simulate incompressible flow, which means the fluid is assumed barotropic and the density and pressure are related through an equation of state. The accuracy of the two approaches is evaluated against well-defined and consolidated benchmark tests. Advantages and disadvantages of the two methodologies are discussed and substantiated by quantitative comparisons that focus on accuracy and efficacy of the two methodologies against other well-established computational methods. Overall, both formulations proposed herein are able to capture the relevant flow physics with a good level of accuracy when compared to other more affirmed techniques. Remarkably, this is observed in spite of the proposed two methods lacking key strategies commonly used in grid-based methods, such as adaptive mesh refinement.

Atmospheric transient eddy dynamical forcing (TEDF)-driven midlatitude unstable air–sea interaction has recently been recognized as a crucial positive feedback for the maintenance of the extratropical decadal variabilities. Our recent theoretical work (Chen et al., Clim Dyn 10.1007/s00382-020-05405-0, 2020) has characterized such an interaction through building an analytical midlatitude barotropic atmospheric model coupled to a simplified upper oceanic model. This study extends the analytical model to including a two-layer quasi-geostrophic baroclinic atmospheric model and then identifies the roles of vertical distributions of atmospheric TEDF and diabatic heating in midlatitude unstable air–sea interaction. It is found that midlatitude air–sea coupling with more realistic vertical profiles of atmospheric TEDF and diabatic heating destabilizes oceanic Rossby wave modes over the entire range of zonal wavelengths, in which the most unstable coupled mode features an equivalent barotropic atmospheric low (high) pressure over a cold (warm) oceanic surface. Spatial structure and period of the most unstable mode are more consistent with the observation than those from in previous model. Although either TEDF or diabatic heating alone can lead to a destabilized coupled mode, the former makes a dominant contribution to the instability. The increase of low-layer TEDF stimulates the instability more effectively if the TEDF in upper layer is larger than in lower layer, while the TEDF in either high or low layers can individually cause the instability. The surface heating always destabilizes the air–sea interaction, while the mid-level heating always decays the coupled mode. The results of this study further confirm the TEDF-driven positive feedback mechanism in midlatitude air–sea interaction proposed by recent observational and numerical experiment studies.

We consider the homogenization limit of the compressible barotropic Navier–Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is decreasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy’s law. The range of sizes of particles we consider is exactly the same which leads to Darcy’s law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovskiĭ operator. This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents [Formula: see text].

The possible existence of distinct regimes of barotropic circulation in closed annular channels at the same external parameters governing the flow dynamics is investigated both experimentally and numerically. Transitions between the regimes are realized by means of varying the value of the main parameter determining the velocity field energy (for example, the current controlling the Ampere force in the case of MHD generation of a velocity field) with subsequent reconstruction of the former parameter value. Depending on the channel rotation period or the configurations of magnet locations in the case of MHD generation or sources and sinks in numerical experiments the following results are possible. (1) The initial and final regimes differ quantitavely in the number of cyclonic or anticyclonic vortices generatted. (2) The number of vortex formations does not change but their localization in space, for example, the angular coordinates of their centers, varies. (3) After the change and reconstruction of the original value of the governing parameter the flow returns to the regime almost undistinguishable from the original regime. The flow patterns and the corresponding diagrams for laboratory experiments and numerical simulations based on shallow water equations are presented.

The aim of this paper is to investigate the regime of high Mach number flows for compressible barotropic fluids of Korteweg type with density dependent viscosity. In particular we consider the models for isothermal capillary and quantum compressible fluids. For the capillary case we prove the existence of weak solutions and related properties for the system without pressure, and the convergence of the solution in the high Mach number limit. This latter is proved also in the quantum case for which a weak-strong uniqueness analysis is also discussed in the framework of the so-called “augmented” version of the system. Moreover, as byproduct of our results, in the case of a capillary fluid with a special choice of the initial velocity datum, we obtain an interesting property concerning the propagation of vacuum zones.

The effect of equatorially trapped waves on the movement of tropical cyclones (TC) is studied numerically based on a two-dimensional barotropic model in a beta-plane approximation. According to recent studies, equatorially trapped waves contribute to the genesis of TCs. It is thus natural to assume that these waves affect also the movement of the TC. The effect of three types of equatorially trapped waves, namely Kelvin, Rossby, and n = 0 eastward inertio-Gravity (EIG) waves, on the TC trajectory is investigated with a focus on the sensitivity on some key physical parameters such as the wavenumber and wavespeed. Using a simple barotropic model forced by a prescribed baroclinic flow, the barotropic response to equatorially trapped waves is simulated for a period of 50 days, under various wave parameter configurations. This response is then used as a background flow where TC’s can evolve and propagate. TC-like flows are injected into this wavefield background at arbitrary times during the simulation, and the TC trajectories are tracked and recorded for 48h after the injection time. The resulting TC trajectory patterns with respect to the injection times and wave parameters appear to be stochastic and the mean paths and the associated standard deviations are calculated and reported here. The statistics are different for different wave types. Kelvin waves make shorter length of TC trajectories and small divergence of direction. On the contrary, Rossby waves cause rather dramatic changes in the TC path and yield longer trajectories. Meanwhile, TCs in EIG waves maintain fairly the same direction and typically have longer trajectories though less dramatic. A robustness test using a random forcing instead has also been conducted.

In this paper, we study the stability of a barotropic polar vortex cap on a rotating sphere. We present the linear stability analysis of the polar vortex cap, approximating the piecewise-continuous vorticity distribution by zonal bands of uniform vorticity. We investigate the dependence of the flow stability on the location of the vortex cap, modes of perturbation and rotation speed. The linear stability analysis shows that the polar vortex cap is always stable when the angular speed of the rotating body and the vorticity constant of the north are of the same sign, but is unstable when they are of different signs, regardless of the location of the cap boundary. We also compute the nonlinear evolution of the vortex cap on the rotating sphere for the unstable cases and compare it with the linear stability analysis. The flow away from the boundary of the vortex cap is the most unstable, which is in accordance with the eigenvector corresponding to the unstable eigenvalue. It is found that the solid body rotation has stabilizing effects on the evolution of the polar vortex cap.

The variations in the wave energy and the amplitude along the energy dispersion paths of the barotropic Rossby waves in zonally symmetric basic flow are studied by solving the wave energy equation, which expresses that the wave energy variability is determined by the divergence of the group velocity and the energy budget from the basic flow. The results suggest that both the wave energy and the amplitude of a leading wave increase significantly in the propagating region that is located south of the jet axis and enclosed by a southern critical line and a northern turning latitude. The leading wave gains the barotropic energy from the basic flow by eddy activities. The amplitude continuously climbs up a peak at the turning latitude due to increasing wave energy and enlarging horizontal scale (shrinking total wavenumber). Both the wave energy and the amplitude eventually decrease when the trailing wave continuously approaches southward to the critical line. The trailing wave decays and its energy is continuously absorbed by the basic flow. Furthermore, both the wave energy and the amplitude oscillate with a limited range in the propagating region that is located near the jet axis and enclosed by two turning latitudes. Both the leading and trailing waves neither develop nor decay significantly. The jet works as a waveguide to allow the waves to propagate a long distance.

We describe a simple classroom demonstration of a fluid-dynamic instability. The demonstration requires only a bucket of water, a piece of string and some used tealeaves or coffee grounds. We argue that the mechanism for the instability, at least in its later stages, is two-dimensional barotropic (shear-flow) instability and we present evidence in support of this. We show results of an equivalent basic two-dimensional numerical non-linear model, which simulates behavior comparable to that observed in the bucket demonstration. Modified simulations show that the instability does not depend on the curvature of the domain, but rather on the velocity profile., 22 pages, 8 figures. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in American Journal of Physics 88, Issue 12, 1041 (2020) and may be found at https://doi.org/10.1119/10.0002438

The interaction of a barotropic flow with topography generates baroclinic motion that exerts a stress on the barotropic flow. Here, explicit solutions are calculated for the spatial-mean flow (i.e., the barotropic tide) resulting from a spatially uniform but time-varying body force (i.e., astronomical forcing) acting over rough topography. This approach of prescribing the force contrasts with that of previous authors who have prescribed the barotropic flow. It is found that the topographic stress, and thus the impact on the spatial-mean flow, depend on the nature of the baroclinic motion that is generated. Two types of stress are identified: (i) a “wave drag” force associated with propagating wave motion, which extracts energy from the spatial-mean flow, and (ii) a topographic “spring” force associated with standing motion at the seafloor, including bottom-trapped internal tides and propagating low-mode internal tides, which significantly damps the time-mean kinetic energy of the spatial-mean flow but extracts no energy in the time-mean. The topographic spring force is shown to be analogous to the force exerted by a mechanical spring in a forced-dissipative harmonic oscillator. Expressions for the topographic stresses appropriate for implementation as baroclinic drag parameterizations in global models are presented.

The results of the experiments on the influence of a positive-buoyancy-particle layer on the process of breaking and regularization of the standing gravity Faraday wave on free water surface in a rectangular vessel are discussed. The effect of increase in the particle layer thickness on the limit steepness of the regular wave and its dissipative properties is considered. It is shown that the use of highly concentrated polystyrene particle suspension as the upper layer modifies significantly the barotropic wave mode dynamics and ensures regularization of the waves with total suppression of their breaking mechanisms.

In this paper, we consider a region occupied by viscous or inviscid compressible magnetohydrodynamic fluids, and surrounded by vacuum. It is shown that the fluid region will expand at least linearly in time as soon as there are no singularities. The expanding rate is proportional to initial total energy and is inversely proportional to initial mass. The result indicates an interesting fact that the expansion of the viscous monatomic fluids seems similar to the inviscid fluids.

While recent observational studies have shown the critical role of atmospheric transient eddy (TE) activities in midlatitude unstable air-sea interaction, there is still a lack of a theoretical framework characterizing such an interaction. In this study, an analytical coupled air-sea model with inclusion of the TE dynamical forcing is developed to investigate the role of such a forcing in midlatitude unstable air-sea interaction. In this model, the atmosphere is governed by a barotropic quasi-geostrophic potential vorticity equation forced by surface diabatic heating and TE vorticity forcing. The ocean is governed by a baroclinic Rossby wave equation driven by wind stress. Sea surface temperature (SST) is determined by mixing layer physics. Based on detailed observational analyses, a parameterized linear relationship between TE vorticity forcing and meridional second-order derivative of SST is proposed to close the equations. Analytical solutions of the coupled model show that the midlatitude air-sea interaction with atmospheric TE dynamical forcing can destabilize the oceanic Rossby wave within a wide range of wavelengths. For the most unstable growing mode, characteristic atmospheric streamfunction anomalies are nearly in phase with their oceanic counterparts and both have a northeastward phase shift relative to SST anomalies, as the observed. Although both surface diabatic heating and TE vorticity forcing can lead to unstable air-sea interaction, the latter has a dominant contribution to the unstable growth. Sensitivity analyses further show that the growth rate of the unstable coupled mode is also influenced by the background zonal wind and the air–sea coupling strength. Such an unstable air-sea interaction provides a key positive feedback mechanism for midlatitude coupled climate variabilities.

Seamounts have been theorized to act as “stirring rods,” converting barotropic flow into an unsteady wake, turbulence, and diapycnal mixing. The energetics of these processes are not well understood, but they may have implications for basin-scale mixing calculations. This study presents the results of a series of simulations for idealized seamounts in steady barotropic flow, with varying degrees of stratification and rotation. The kinetic energy within each simulation domain is decomposed into the mean kinetic energy, unsteady eddy energy, and turbulent kinetic energy; evolution equations are derived for each. Within the evolution equations, energy exchange terms arise, which relate the various forms of kinetic energy and potential energy. Key exchange terms, such as the rate at which the mean flow is converted into eddy energy, are compared across the Froude–Rossby parameter space. It is shown that the conversion terms associated with mesoscale motions are a function of the Burger number, which is consistent with a quasigeostrophic flow regime. Conversely, conversion terms associated with turbulent processes scale with the product of the Froude and Rossby number. The amount of energy extracted from the mean flows suggests that wake effects may be significant for the parameter range and model assumptions studied. These results suggest that some seamounts may indeed act as oceanic stirring rods.

In the present work the possibility of turbulence closure applying to improve barotropic jet instability simulation at coarse grid resolutions is considered. This problem is analogous to situations occurring in eddy-permitting ocean models when Rossby radius of deformation is partly resolved on a computational grid. We show that the instability is slowed down at coarse resolutions. As follows from the spectral analysis of linearized equations, the slowdown is caused by the small-scale normal modes damping arising due to numerical approximation errors and nonzero eddy viscosity. In order to accelerate instability growth, stochastic and deterministic kinetic energy backscatter (KEBs) parameterizations and scale-similarity model were applied. Their utilization led to increase of the growth rates of normal modes and thus improve characteristic time and spatial structure of the instability.

Two common approximations to the full Shallow Water Equations (SWEs) are non-divergence and quasi-geostrophy, and the degree to which these approximations lead to biases in numerical solutions are explored using the testbed of barotropic instability. Specifically, we examine the linear stability of strong polar and equatorial jets and compare the growth rates obtained from the SWEs along with those obtained from the Non-Divergent barotropic vorticity (ND) equation and the Quasi-Geostrophic (QG) equation. The main result of this paper is that the depth over which a layer is barotropically unstable is a crucial parameter in controlling the growth rate of small-amplitude perturbations and this dependence is completely lost in the ND equation and is overly weak in the QG system. Only for depths of 30 km or more are the growth rates predicted by the ND and QG systems a good approximation to those of the SWEs, and such a convergence for deep layers can be explained using theoretical considerations. However, for smaller depths, the growth rates predicted by the SWEs become smaller than those of the ND and QG systems and for depths of between 5 and 10 km they can be smaller by more than 50%. For polar jets, and for depths below 2 km the mean height in geostrophic balance with the strong zonal jet becomes negative and hence the barotropic instability problem is ill-defined. While in the SWEs an equatorial jet becomes stable for layer depths smaller than ~3-4 km, in the QG and ND approximation it is unstable for layer depths down to 1 km. These results may have implications for the importance of barotropic instability in Earth's upper stratosphere and perhaps also other planets such as Venus.

In this study, the intensity change of tropical cyclone (TC) Noru (1705) is investigated by using numerical simulations. TC Noru experienced a weakening stage during the binary interaction with TC Kulap (1706). In the presence of Kulap, the model captures well the observed intensity change. On the contrary, once Kulap is removed, it fails to reproduce the weakening stage of Noru. Possible mechanisms are proposed as follows: During the mutual interaction, the competition of moisture supplies will lead to a dry air layer wrapped around Noru. Meanwhile, due to the circulation of nearby Kulap, the vertical shear is relatively larger in the adjacent of Noru. As proposed by previous studies, the negative impacts by dry air layer will be enhanced through two possible pathways. The first is that the dry air directly impacts the TC inner core convection under vertical wind shears. The second is through the downward entropy flux into boundary layer, by which reducing the boundary layer entropy and thus convection. Dynamically, along with the approach of Kulap, it will induce barotropic instability at the outer region, by which greatly enhances the inner-core asymmetry. The asymmetric component will grow at the expense of the mean flow, and thus TC intensity weakens.

Transient cavities generated from unsteady leading-edge cavitation may undergo aggressive collapses which are responsible for cavitation erosion. In this paper, we studied the hydrodynamic mechanisms of these events in the leading edge cavitation formed over a modified NACA0009 hydrofoil using experimental and numerical methods. In the experimental investigation, high-speed visualization (HSV) and paint test are employed to study the behavior of the cavitating flow at sigma = 1.25, beta = 5 degrees, U-infinity = 20 m/s. In the numerical part, the same cavitating flow is simulated using an inviscid density-based compressible solver with a barotropic cavitation model. The numerical results are first compared with the experimental HSV to show that the simulation is able to reproduce the main features of the cavitating flow. Then, as the compressible solver is capable of capturing the shock wave upon the collapse of cavities, the location of collapse events with high erosion potential are determined. The location of these collapse events are compared with the paint test results with a qualitatively good agreement. It is clearly observed, in both the experiments and the numerical simulation, that there exists four distinct regions along the hydrofoil with higher risks of erosion: (1) A very narrow strip at the leading edge, (2) an area of accumulated collapses at around 60 percent of the sheet cavity maximum length, (3) an area around the closure line of the sheet cavity with the highest erosion damage, and (4) a wide area close to the trailing edge with dispersed collapse events. A combined analysis of the experimental and numerical results reveals that the small-scale structures generated by secondary shedding are more aggressive than the large-scale cloud cavities (primary shedding). It is also observed that the high risk of cavitation erosion in regions 2 and 3 is mainly due to the collapses of the small cavity structures that are formed around the sheet cavity closure line or the rolling cloud cavity.

We study in the linear approximation barotropic Rossby waves in the ocean covered by compressed ice. We derive the dispersion relation and analyse its dependence on the degree of ice compression an...

The effects of topography on the linear stability of both barotropic vortices and two-layer, baroclinic vortices are examined by considering cylindrical topography and vortices with stepwise relative vorticity profiles in the quasigeostrophic approximation. Four vortex configurations are considered, classified by the number of relative vorticity steps in the horizontal and the number of layers in the vertical: barotropic one-step vortex (Rankine vortex), barotropic two-step vortex, and their two-layer, baroclinic counterparts with the vorticity steps in the upper layer. In the barotropic calculation, the vortex is destabilized by topography having an oppositely signed potential vorticity jump while stabilized by topography of same-signed jump, that is, anticyclones are destabilized by seamounts while stabilized by depressions. Further, topography of appropriate sign and magnitude can excite a mode-1 instability for a two-step vortex, especially relevant for topographic encounters of an otherwise stable vortex. The baroclinic calculation is in general consistent with the barotropic calculation except that the growth rate weakens and, for a two-step vortex, becomes less sensitive to topography (sign and magnitude) as baroclinicity increases. The smaller growth rate for a baroclinic vortex is consistent with previous findings that vortices with sufficient baroclinic structure could cross the topography relatively easily. Nonlinear contour dynamics simulations are conducted to confirm the linear stability analysis and to describe the subsequent evolution.

We consider the energy conditions for a dissipative matter distribution. The conditions can be expressed as a system of equations for the matter variables. The energy conditions are then generalised for a composite matter distribution, a combination of viscous barotropic fluid, null dust and a null string fluid is also found in a spherically symmetric spacetime. This new system of equations comprises the energy conditions that are satisfied by a Type I fluid. The energy conditions for a Type II fluid are also presented, which are reducible to the Type I fluid only for a particular function. This treatment will assist in studying the complexity of composite relativistic fluids in particular self-gravitating systems.

We numerically investigate the effect of non-condensable gas inside a vapor bubble on bubble dynamics, collapse pressure and pressure impact of spherical and aspherical bubble collapses. Free gas inside a vapor bubble has a damping effect that can weaken the pressure wave and enhance the bubble rebound. To estimate this effect numerically, we derive and validate a multi-component model for vapor bubbles containing gas. For the cavitating liquid and the non-condensable gas, we employ a homogeneous mixture model with a coupled equation of state for all components. The cavitation model for the cavitating liquid is a barotropic thermodynamic equilibrium model. Compressibility of all phases is considered in order to capture the shock wave of the bubble collapse. After validating the model with an analytical energy partitioning model, simulations of collapsing wall-attached bubbles with different stand-off distances are performed. The effect of the non-condensable gas on rebound and damping of the emitted shock wave is well captured., To be published in Physics of Fluids

This study examines the long-wavelength instabilities of an inviscid parallel time-dependent current on the beta-plane. The basic flow is represented by the Kolmogorov pattern, the amplitude of which is modulated in time. Particular attention is given to the regime in which the corresponding steady flows are stable according to the Rayleigh–Kuo criterion. It is shown that the presence of a fluctuating component, regardless of how weak it may be, always renders the basic current linearly unstable. The destabilization is attributed to the resonant forcing of large-scale Rossby waves. The analysis is based on the asymptotic multiscale model, which is validated by numerical simulations. Since most geophysical flows are inherently time-dependent, the associated shear instabilities could represent a significant and ubiquitous source of barotropic turbulence.

We examine the equilibrium of a magnetized neutron-star-crust. We calculate axially symmetric models in which an elastic force balances solenoidal motion driven by a Lorentz force. A large variety of equilibrium models are allowed by incorporating the elastic shear deformation; in addition, toroidal-magnetic-field dominated models are available. These results remarkably differ from those in barotropic fluid stars. We demonstrate some models wherein the magnetic energy exceeds the elastic energy. The excess comes from the fact that a large amount of magnetic energy is associated with the irrotational part of the magnetic force, which is balanced with gravity and pressure. It is sufficient for equilibrium models that the minor solenoidal part is balanced by a weak elastic force. We find that the elasticity in the crust plays an important role on the magnetic-field confinement. Further, we present the spatial distribution of the shear-stress at the elastic limit, by which the crust-fracture location can be identified. The result has useful implications for realistic crust-quake models., 13 pages, 9 figures

We consider the interactions of internal tides (ITs) with a dynamic, rotational and heterogeneous ocean, and spatially varying topography. The IT fields are expanded using vertical modal basis functions, whose amplitudes vary horizontally and temporally. We obtain the evolution equations of modal amplitudes and energy including simultaneous three-way interactions with the mean flow, buoyancy and topography. We apply these equations to a set of idealized and two realistic data-assimilative primitive equation simulations. These simulations reveal that significant interactions of ITs with the background fields occur at topographic features and strong currents, in particular when the scales of the background and ITs are similar. In local hot spots, the new three-way interaction terms, when compared to the total modal conversion, are found to reach up to 10 %–30 % at steep topography and approximately 50 % in the Gulf Stream. We provide a dimensional analysis to guide the diagnosis of such strong interactions. When IT interactions are with a large-scale barotropic current (without topographic effects), our modal energy equation reduces to the conservation of modal wave action under a Wentzel–Kramers–Brillouin consideration. We further derive analytical solutions of the modulation of wavenumber and energy of an IT propagating into a collinear current. For ITs propagating along the flow direction, the wavelength is stretched and the amplitude is reduced, with the degree of modulation determined by exists, below and above which the waves show remarkably different behaviours. The critical opposing current speed which triggers the wave focusing/blocking phenomenon is obtained and its implication for the propagation and dissipation of ITs is discussed.

Taking into account the action of inhomogeneous zonal wind (shear flow), nonlinear dynamic equations describing the propagation of planetary ULF magnetized Rossby waves in the ionospheric D-, E-, and F-layers are obtained and investigated. The influence of existence of charged particles through Hall and Pedersen conductivities on such dynamic equations is studied in detail. It is shown that the existence of shear flow and Pedersen conductivity can be considered as the presence of an external energy source. The possibility of a barotropic instability of the magnetized Rossby waves is shown. Based on the Rayleigh's theorem, the appropriate stability conditions are defined in case of the ionospheric D- and E-layers. It is indicated that magnetized Rossby waves under the action of shear zonal flow correspond to states with negative energy. Some exponentially localized vortical solutions are found for the ionospheric D- and E-layers. Joint Call of Shota Rustaveli National Science Foundation of Georgia [04/01]; Scientific and Technological Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) The carried out investigation is supported through the Grant No. 04/01 of 2017 Joint Call of Shota Rustaveli National Science Foundation of Georgia and the Scientific and Technological Research Council of Turkey (TUBITAK).

Two zero-order bulk models (ZOMs) are developed for the velocity, buoyancy, and moisture of a cloud-free barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. The models differ in the entrainment closure assumption: in the first one, termed the ‘‘energetics-based model,’’ the negative and positive areas of the buoyancy flux are assumed to match between the model and the actual CBL; in the second one, termed the ‘‘geometric-based model,’’ the modeled CBL depth is assumed to match different definitions of the actual CBL depth. Parameterizations for these properties derived from direct numerical simulation (DNS) are employed as entrainment closure equations. These parameterizations, and hence the resulting models, are free from the potential singularity at finite wind strength that has been a major limitation in previous bulk models. The proposed ZOMs are verified using the DNS data. Model results show that the CBL depths obtained from the energetics-based model and previous ZOMs correspond to the height that marks the transition from the lower to the upper entrainment-zone sublayer; this reference height is few hundred meters above the height of the minimum buoyancy flux. It is also argued that ZOMs, despite their simplicity compared to higher-order models, can accurately represent CBL bulk properties when the relevant features of the actual entrainment zone are considered in the entrainment closures. The vertical structure of the actual entrainment zone, if required, can be constructed a posteriori using the available relationships between the predicted zero-order CBL depth and various definitions of the actual CBL depth.

In this work, we study the small oscillations of a system formed by an elastic container with negligible density and a heavy barotropic gas (or a compressible fluid) filling the container. By means of an auxiliary problem, that requires a careful mathematical study, we deduce the problem to a problem for a gas only. From its variational formulation, we prove that is a classical vibration problem.

The results of the experiments on studying the effect of the upper layer of a viscous fluid on the process of breaking and regularization of the standing gravity Faraday wave on the free surface of a two-layer fluid in a rectangular vessel are discussed. The situation considered is compared with the case of standing gravity wave on the free surface of homogeneous fluids whose viscosity differs significantly, namely, water and seed oil. The effect of increase in the upper layer thickness on the limit steepness of the regular wave and its dissipative properties is considered. The crucial role of emulsion formed under intense oscillations of immiscible fluids for suppression of the standing wave breaking mechanism is demonstrated.

The anomalous transport of passive admixture is studied when unsteady chains of wave structures with closed streamlines are excited in a barotropic jet flow that models the zonal flows in the Earth’s atmosphere and ocean, as well as in the laboratory experiments. The analysis is performed within a dynamical model describing saturation of the barotropic instability of the jet current in a plane channel with rigid parallel walls with account for the beta effect and external friction. The equations of a barotropic current are solved numerically using the pseudospectral approach. It is found that the generation of high modes in a jet with a “two-hump” velocity profile leads to the accelerated transition to the complex dynamics. A multiharmonic regime with a discrete spectrum appears in the initial stage of this motion. The exponents of the power dependence on the time of the average (over the ensemble) displacement of tracer particles and its dispersion are found for the basic generation regimes of the wave structure, which confirm the occurrence of the anomalous diffusion of the admixture. A self-similar probability density function of tracer displacements has been found and the dependence of the transition to the complex dynamics on the number of vortices in the chain and on the intensity of the beta effect has been revealed. Numerical estimates are presented which confirm the possibility of generating unsteady vortex chains and the related anomalous transport of the passive admixture.

Barotropic geostrophic flows are investigated using rectangular and polar coordinate systems. It is shown that, for the analysis of geostrophic flows with radial symmetry, the use of a polar coordinate system is preferable. Relations between the hydrodynamic characteristics, including the expressions of the fluid particle velocity components and vorticity through pressure, are obtained. It is established that, in the case of stationary barotropic flows, isobaths coincide with current lines. Stationary radially symmetric geostrophic flows are considered.

We consider two-dimensional quasi-geostrophic annular flows around a circular island with a radial offshore bottom slope. Since the conical bottom topography causes a certain beta effect, by analogy with the conventional beta plane we term our model a beta cone. Our focus is on the flows with zero total circulation, which are composed of two concentric rings of uniform potential vorticity (PV) attached to the island. The linear stability of such flows on a beta cone was investigated in a previous publication of ours. In the present paper, we study numerically the nonlinear evolution of weakly viscous flows, whose parameters are fitted so as to guarantee the highest instability of the azimuthal mode $m=1,\ldots ,6$. We study the production of vortices and Rossby waves due to the instability, consider the effect of waves on the emerging vortices and the interaction between the vortices. As in the flat-bottom case, at $m\geqslant 2$, the instability at weak bottom slopes normally leads to the emission of $m$ dipoles. However, a fundamental difference between the flat-bottom and beta-cone cases is observed in the trajectories of the dipoles as the latter recede from the island. When the flow is initially counterclockwise, the conical beta effect may force the dipoles to make a complete turn, come back to the island and rearrange in new couples that again leave the island and return. This quasi-periodic process gradually fades due to filamentation, wave radiation and viscous dissipation. Another possible outcome is symmetrical settling of $m$ dipoles in a circular orbit around the island, in which they move counterclockwise. This behaviour is reminiscent of the adaptation of strongly tilted beta-plane modons (dipoles) to the eastward movement. If the initial flow is clockwise, the emerged dipoles usually disintegrate, but sometimes, the orbital arrangement is possible. At a moderate slope, the evolution of an unstable flow, which is initially clockwise, may end up in the formation of a counterclockwise flow. At steeper slopes, a clockwise flow may transform into a quasi-stationary vortex multipole. When the slope is sufficiently steep, the topographic Rossby waves developing outside of the PV rings can smooth away the instability crests and troughs at the outer edge of the main flow, thus preventing the vortex production but allowing the formation of a new quasi-stationary pattern, a doubly connected coherent PV structure possessing $m$-fold symmetry. Such an $m$-fold pattern can be steady only if it rotates counterclockwise, otherwise it radiates Rossby waves and transforms eventually into a circularly symmetric flow.

In this paper, an extended nonlinear multiscale interaction model of blocking events in the equivalent barotropic atmosphere is used to investigate the effect of a slowly varying zonal wind in the meridional direction on dipole blocking that is regarded as a nonlinear Rossby wave packet. It is shown that the meridional gradient of potential vorticity (PVy=∂PV/∂y) prior to the blocking onset, which is related to the background zonal wind and its nonuniform meridional shear, can significantly affect the lifetime, intensity, and north–south asymmetry of dipole blocking, while the blocking dipole itself is driven by preexisting incident synoptic-scale eddies. The magnitude of the background PVy determines the energy dispersion and nonlinearity of blocking. It is revealed that a small background PVy is a prerequisite for strong and long-lived eddy-driven blocking that behaves as a persistent meandering westerly jet stream, while the blocking establishment further reduces the PVy within the blocking region, resulting in a positive feedback between blocking and PVy. When the core of the background westerly jet shifts from higher to lower latitudes, the blocking shows a northwest–southeast-oriented dipole with a strong anticyclonic anomaly to the northwest and a weak cyclonic anomaly to the southeast as its northern pole moves westward more rapidly and has weaker energy dispersion and stronger nonlinearity than its southern pole because of the smaller PVy in higher latitudes. The opposite is true when the background jet shifts toward higher latitudes. The asymmetry of dipole blocking vanishes when the background jet shows a symmetric double-peak structure. Thus, a small prior PVy is a favorable precursor for the occurrence of long-lived and large-amplitude blocking.

Author(s): Diaz, M; Boos, WR | Abstract: This study makes the case that monsoon depressions over South Asia can form from a variant of moist barotropic instability. Using an idealized numerical framework in which the atmosphere is partitioned into a basic state and a perturbation, we simulate vortices resembling monsoon depressions that draw energy from the meridional shear of the monsoon trough and amplify when they interact with precipitating ascent. The influence of the basic state vertical shear on the vortex induces upward velocity, which couples precipitation with a Rossby-wave-like mode arising from dry barotropic growth, allowing the vortex to intensify. Sensitivity experiments reveal that both the sheared basic state and latent heating are necessary to achieve positive growth rates and that this process requires a sufficiently large initial perturbation. Trajectory analyses suggest that the combined flow of the vortex and the large-scale monsoon transport diabatically generated potential vorticity from southwest of the vortex into the vortex center, thus enabling growth. In contrast with tropical cyclones, this mechanism does not require a feedback between surface wind speed and surface heat and moisture fluxes, though this feedback does ultimately result in a slightly stronger vortex.

The problem of the stability of a jet flow with a piecewise linear velocity profile in a stratified rotating atmosphere is considered. The linearized system of equations for perturbations is reduced to a single equation with respect to the amplitude of the longitudinal velocity component containing the turning points. In terms of Airy functions, an asymptotic solution of the equation is constructed that is valid for small values of the Rossby number. It is shown that the flow becomes unstable due to radiation of inertial-gravitational waves. An analytical expression is obtained for the growth rate of perturbations.

The Charney model is reexamined using a new mathematical tool, the multiscale window transform (MWT), and the MWT-based localized multiscale energetics analysis developed by Liang and Robinson to deal with realistic geophysical fluid flow processes. Traditionally, though this model has been taken as a prototype of baroclinic instability, it actually undergoes a mixed one. While baroclinic instability explains the bottom-trapped feature of the perturbation, the second extreme center in the perturbation field can only be explained by a new barotropic instability when the Charney–Green number γ ≪ 1, which takes place throughout the fluid column, and is maximized at a height where its baroclinic counterpart stops functioning. The giving way of the baroclinic instability to a barotropic one at this height corresponds well to the rectification of the tilting found on the maps of perturbation velocity and pressure. Also established in this study is the relative importance of barotropic instability to baroclinic instability in terms of γ. When γ ≫ 1, barotropic instability is negligible and hence the system can be viewed as purely baroclinic; when γ ≪ 1, however, barotropic and baroclinic instabilities are of the same order; in fact, barotropic instability can be even stronger. The implication of these results has been discussed in linking them to real atmospheric processes.

A simplified test case of a centrifugal pump blade, a hydrofoil with circular leading edge, is investigated. Cloud cavitation as a particularly aggressive type of cavitation is studied using a compressible 3D flow solver with barotropic cavitation model. The results are compared to measurement data for different Reynolds and cavitation numbers. The shedding frequency is well predicted. A statistical evaluation of a multitude of single collapsing voids is performed. Collapse frequency and magnitude of the pressure peaks are evaluated to obtain wall load collectives. Material damage on the hydrofoil suction surface obtained in erosion measurements is compared to the local flow aggressiveness distribution in the simulation. Erosion sensitive wall zones are identified in good agreement with measurement data. A relationship between cloud cavitation structures and erosion sensitive wall zones is demonstrated that enables a simple and approximate erosion assessment by 3D flow simulation.

The analytic wave solutions obtained by Matsuno (1966) in his seminal work on equatorial waves provide a simple and informative way of assessing the performance of atmospheric models by measuring the accuracy with which they simulate these waves. These solutions approximate the solutions of the shallow-water equations on the sphere for low gravity-wave speeds such as those of the baroclinic modes in the atmosphere. This is in contrast to the solutions of the non-divergent barotropic vorticity equation, used in the Rossby–Haurwitz test case, which are only accurate for high gravity-wave speeds such as those of the barotropic mode. The proposed test case assigns specific values to the wave parameters (gravity-wave speed, zonal wave number, meridional wave mode and wave amplitude) for both planetary and inertia-gravity waves, and suggests simple assessment criteria suitable for zonally propagating wave solutions. The test is successfully applied to a spherical shallow-water model in an equatorial channel and to a global-scale model. By adding a small perturbation to the initial fields it is demonstrated that the chosen initial waves remain stable for at least 100 wave periods. The proposed test case can also be used as a resolution convergence test.

Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371–385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic,$\unicode[STIX]{x1D6FD}$, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.

This paper investigates the stability of a jet flow with a piecewise linear velocity profile in a rotating stratified atmosphere. The linearized set of perturbation equations is reduced to one longitudinal-velocity amplitude equation with turning points. An asymptotic solution of the equation, valid at small Rossby numbers, is derived in terms of the Airy functions. A flow that is stable in a quasi-geostrophic approximation is shown to become unstable due to the emission of inertia–gravity waves. An analytic expression for the growth increment of the perturbations is derived.

The Gent and McWilliams (1990, GM) tracer transport scheme has long been a cornerstone of global-scale ocean modeling, and is well-known for its ability to mimic the effects of baroclinic turbulence by reducing the available potential energy of resolved flows. However, theory predicts that baroclinic turbulence also acts to transfer this potential energy to kinetic energy, which can then be fluxed toward larger scales by an inverse energy cascade. These dynamical processes represent a separate branch of the large-scale ocean energy cycle which GM presently does not account for. In this paper a framework is developed which allows the potential energy removed by the GM scheme to be transferred to resolved motions. In this framework, which is labeled GM＋E (or “GM plus energetics”), the energy transfer occurs by employing GM alongside a parameterization of the barotropic Reynolds stresses in the momentum equations, which takes the form of a negative harmonic viscosity acting on the barotropic velocities. The Reynolds stress parameterization is considered as being complementary to GM, in that its energetics are explicitly tied to those of GM while being independent of the choice of transfer coefficient or form of the GM closure. A set of idealized baroclinic spindown simulations is used to compare GM＋E against standard GM, and is shown to substantially improve the kinetic energy spectra, probability distributions of flow structures, and restratification rate within the front relative to an eddy-resolving truth simulation.

In this paper we are concerned with the problem of anomalous transport in a quasi-two-dimensional (barotropic) shear layer localized in the horizontal channel with rigid walls and subjected to the beta-effect. The flow may be regarded as a model of a zonal shear layer in the Earth’s atmosphere and ocean as well as in laboratory experiments. External (bottom) friction is taken into account by adding Rayleigh damping to two-dimensional equations. The attention is focused on a new mechanism of anomalous wave transport, when the disturbances producing stochastic layers in the vicinity of a vortex chain are caused by the transition to dynamical chaos. The problem is solved numerically using a pseudospectral method. The supercriticality is defined as the ratio of the characteristic flow velocity to the critical one. The critical wavenumber of the linear eigenvalue problem is such that 7 vortices are excited on a flow period, while the transition to chaos begins when 5 vortices are excited. The anomalous transport is examined at the first stage of the chaos onset, when the frequency spectra of spatial harmonics are quite narrow, thereby the trajectories of tracer particles are formed like in the kinematic model. The self-similarity property of variance in the ensemble of tracers is confirmed and the exponents of time power law are obtained. The Poincare map is drawn and the average displacements of the tracers are computed.

Diabatic heating from deep moist convection in the hurricane eyewall produces a towering annular structure of elevated potential vorticity (PV), known as a hollow PV tower. For sufficiently thin annular structures, eddies can extract energy from the mean flow, leading to hollow tower breakdown with significant changes in vortex structure and intensity. A forced primitive equation model in normalized isentropic coordinates is used to understand the role of diabatic heating in the generation, maintenance, and breakdown of the hurricane PV tower. It is shown that the generation of hollow PV towers is due to the combined effects of diabatic heating and the radial and vertical PV advection associated with the induced secondary circulation of the vortex. If the diabatic forcing makes the eyewall thin enough, then the PV tower can become dynamically unstable and cause air parcels with high PV to be mixed preferentially into the eye at lower levels, where unstable PV wave growth rates are largest. The breakdown of the hollow PV tower leads to a transient break in vortex intensification, a decrease in minimum central pressure, and an inward shift and tilt of absolute angular momentum surface. It is shown that the maintenance of the PV tower structure depends on the strength of the heating-induced secondary circulation and the interaction between diabatic heating and barotropic instability.

In this paper, we present the results of numerical simulations matched to laboratory experiments of tidal, starting-jet vortices forming at idealized, barotropic inlets. The laboratory experiments are for sinusoidally varying inflow/outflow along a wide, flat basin. Inlet configurations include simple inlets with negligible channel length as well as a longer inlet channels through a long barrier island and jettied inlet. Laboratory observations are made using particle image velocimetry, and we analyze the laboratory and numerical model velocity fields to compare starting-jet trajectory, diameter, and total circulation. The numerical simulations use the Fine Resolution Environmental Hydrodynamic model, solving the three-dimensional, hydrostatic, barotropic equations of motion. Our goal is to identify the minimum model attributes necessary to match the vortex properties. This is accomplished using a third-order upwind scheme for advection, a constant bottom friction drag coefficient, and a one-equation turbulence model for transport of turbulent kinetic energy. The optimal value of the bottom drag coefficient matches the time-average value at the basin inflow/outflow boundaries, and the optimal dissipation coefficient, $$C_{me}$$, was an order of magnitude smaller than literature values for a steady open-channel flow. We show that the drag coefficient mainly affects the advection speeds of the outer coastal waters, which control the penetration of the tidal vortices away from the inlet, and the lower turbulence dissipation rate results from the low turbulent dissipation in the large, shallow starting-jet vortices.

The mixing induced by breaking internal gravity waves is an important contributor to the ocean’s energy budget, shaping, inter alia, nutrient supply, water mass transformation and the large-scale overturning circulation. Much of the energy input into the internal wave field is supplied by the conversion of barotropic tides at rough bottom topography, which hence needs to be described realistically in internal gravity wave models and mixing parametrisations based thereon. A new semi-analytical method to describe this internal wave forcing, calculating not only the total conversion but also the direction of this energy flux, is presented. It is based on linear theory for variable stratification and finite depth, that is, it computes the energy flux into the different vertical modes for two-dimensional, subcritical, small-amplitude topography and small tidal excursion. A practical advantage over earlier semi-analytical approaches is that the new one gives a positive definite conversion field. Sensitivity studies using both idealised and realistic topography allow the identification of suitable numerical parameter settings and corroborate the accuracy of the method. This motivates the application to the global ocean in order to better account for the geographical distribution of diapycnal mixing induced by low-mode internal gravity waves, which can propagate over large distances before breaking. The first results highlight the significant differences of energy flux magnitudes with direction, confirming the relevance of this more detailed approach for energetically consistent mixing parametrisations in ocean models. The method used here should be applicable to any physical system that is described by the standard wave equation with a very wide field of sources.

Flow blockage by large wind farms leads to an upward displacement of the boundary layer, which may excite atmospheric gravity waves in the free atmosphere and on the interface between the boundary layer and the free atmosphere. In the current study, we assess the sensitivity of wind-farm gravity-wave excitation to important dimensionless groups and investigate the feedback of gravity-wave-induced pressure fields to wind-farm energy extraction. The sensitivity analysis is performed using a fast boundary-layer model that is developed to this end. It is based on a three-layer representation of the atmosphere in an idealised barotropic environment, and is coupled with an analytical wake model to account for turbine wake interactions. We first validate the model in two-dimensional mode with data from previous large-eddy simulations of ‘infinitely’ wide wind farms, and then use the model to investigate the sensitivity of wind-farm-induced gravity waves to atmospheric state and wind-farm configuration. We find that the inversion layer induces flow physics similar to shallow-water flow and that the corresponding Froude number plays a crucial role. Gravity-wave excitation is maximal at a critical Froude number equal to one, but the feedback on energy extraction is highest when the Froude number is slightly below one due to a trade-off between amplitude and upstream impact of gravity waves. The effect of surface friction and internal gravity waves is to reduce the flow perturbation and the related power loss by dissipating or dispersing perturbation energy. With respect to the wind-farm configuration, we find that gravity-wave-induced power loss increases with wind-farm size and turbine height. Moreover, we find that gravity-wave effects are small for very wide or very long wind farms and attain a maximum at a width-to-depth ratio of approximately $3/2$.

We find all solutions to the submodel of vortex, steady, planar, barotropic, entropic flows of an ideal gas and show that possible motions are exhausted by rectilinear motions under a constant pressure and motions along concentric circles. We present a group classification of the model of planar, vortex, entropic, nonbarotropic flows, examine invariant submodels, and propose a physical interpretation of certain solutions.

The problem of parametrization of the eddy fluxes of a potential vorticity is discussed. Traditional diffusion parameterization is complemented by the inclusion of a rotational component. For the analysis of the new scheme, a quasi-geostrophic model of the dynamics of the barotropic flow in a zonal channel with a non-uniform bottom is used. An analytical solution of the problem is found and the influence of topography on the flow disturbances is discussed. It is shown that the equation for the eddy potential enstrophy allows to relate diffusion and «rotational» coefficients.