Eugene S. Benilov

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• BSc, MSc (Moscow Institute of Physics and Technology), PhD (P. P. Shirshov Institute of Oceanology)
• Professor at University of Limerick

We investigate the stability of thin liquid curtains with respect to two-dimensional perturbations. The dynamics of perturbations with wavelengths exceeding (or comparable to) the curtain's thickness are examined using the lubrication approximation (or a kind of geometric optics). It is shown that, contrary to the previous theoretical results, but...

We examine the stability of a vertical liquid bridge between two vertically vibrating, coaxial disks. Assuming that the vibration amplitude and period are much smaller than the mean distance between the disks and the global timescale, respectively, we employ the method of multiple scales to derive a set of asymptotic equations. The set is then used...

We examine two- and three-dimensional drops steadily sliding down an inclined plate. The contact line of the drop is governed by a model based on the Navier-slip boundary condition and a prescribed value for the contact angle. The drop is thin, so the lubrication approximation can be used. In the three-dimensional case, we also assume that the drop...

We examine two-dimensional flows of a viscous liquid on an inclined plate. If the upstream depth h(-) of the liquid is larger than its downstream depth h(+), a smooth hydraulic jump (bore) forms and starts propagating down the slope. If the inclination angle of the plate is small, the bore can be described by the so-called lubrication theory. In th...

We examine the evolution of a liquid drop on an inclined substrate oscillating vertically. The oscillations are weak and slow, which makes the liquid's inertia and viscosity negligible (so that the drop's shape is determined by a balance of surface tension, gravity, and vibration-induced inertial force). No assumptions are made about the drop's thi...

It is shown that most of the existing versions of the Bhatnagar-Gross-Krook model—those whose coefficient are independent of the molecular velocity—do not satisfy the Onsager relations. This circumstance poses a problem when calibrating these models, making their transport properties match those of a specific fluid.

It is argued that the van der Waals force exerted by the liquid and vapor/air on the molecules escaping from one phase into the other strongly affects the characteristics of evaporation. This is shown using two distinct descriptions of the van der Waals force: the Vlasov and diffuse-interface models, each of which is applied to two distinct setting...

The classical model of evaporation of liquids hinges on Maxwell's assumption that the air near the liquid's surface is saturated. It allows one to find the evaporative flux without considering the interface separating liquid and air. Maxwell's hypothesis is based on an implicit assumption that the vapour-emission capacity of the interface exceeds t...

Evaporation of a liquid layer on a substrate is examined without the often-used isothermality assumption, i.e., temperature variations are accounted for. Qualitative estimates show that nonisothermality makes the evaporation rate depend on the conditions at which the substrate is maintained. If it is thermally insulated, evaporative cooling dramati...

Oblique (non-vertical) liquid curtains are examined under the assumption that the Froude number is large. As shown previously (Benilov 2019, J. Fluid Mech. 861, 328), their structure depends on the Weber number: if We < 1 (strong surface tension), the Navier–Stokes equations admit asymptotic solutions describing curtains bending upwards, i.e., agai...

Fundamental properties of the multicomponent diffuse-interface model (DIM), such as the maximum entropy principle and conservation laws, are used to explore the basic interfacial dynamics and phase transitions in fluids. Flat interfaces with monotonically changing densities of the components are proved to be stable. A liquid layer in contact with o...

The dynamics of saturated vapor between two intersecting walls is examined. It is shown that, if the angle ϕ, between the walls is sufficiently small, the vapor becomes unstable, and spontaneous condensation occurs in the corner, similar to the so-called capillary condensation of vapor into a porous medium. As a result, an ever-growing liquid menis...

Evaporation of a liquid drop surrounded by either vapor of the same fluid, or vapor and air, is usually attributed to vapor diffusion, which, however, does not apply to the former setting, as pure fluids do not diffuse. The present paper puts forward an additional mechanism, one that applies to both settings. It is shown that disparities between th...

This paper examines the shape of a steady jet with a swirling component, ejected from a circular orifice at an angle to the horizontal. Assuming the Froude number to be large, we derive a set of asymptotic equations for a slender jet. In the inviscid limit, the solutions of the set predict that, if the swirling velocity of the flow exceeds a certai...

It is well known that liquid and saturated vapor, separated by a flat interface in an unbounded space, are in equilibrium. One would similarly expect a liquid drop, sitting on a flat substrate, to be in equilibrium with the vapor surrounding it. Yet, it is not: as shown in this work, the drop evaporates. Mathematically, this conclusion is deduced u...

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow is, to leading order, shearless and viscosity, negligible. The Froude number is large, so that the radius of the curtain's curvature exceeds its thickness. The Weber number is close to unity, so that the fo...

The dynamics of a thin layer of liquid between a flat solid substrate and an infinitely thick layer of saturated vapor is examined. The liquid and vapor are two phases of the same fluid governed by the diffuse-interface model. The substrate is maintained at a fixed temperature, but in the bulk of the fluid, the temperature is allowed to vary. The s...

The diffuse-interface model (DIM) is a widely used tool for modeling fluid phenomena involving interfaces, such as sessile drops (liquid drops on a solid substrate, surrounded by saturated vapor) and liquid ridges (two-dimensional sessile drops). In this work, it is proved that, surprisingly, the DIM does not admit solutions describing static liqui...

The diffuse-interface model (DIM) is a tool for studying interfacial dynamics. In particular, it is used for modeling contact lines, i.e., curves where a liquid, gas, and solid are in simultaneous contact. As well as all other models of contact lines, the DIM implies an additional assumption: that the flow near the liquid-gas interface is isotherma...

Four results associated with the diffuse-interface model (DIM) for contact lines are reported in this paper. First, a boundary condition is derived, which states that the fluid near a solid wall must have a certain density ρ0 depending on the solid's properties. Unlike previous derivations, the one presented here is based on the same physics as the...

The Enskog–Vlasov (EV) equation is a semi-empiric kinetic model describing gas–liquid phase transitions. In the framework of the EV equation, these correspond to an instability with respect to infinitely long perturbations, developing in a gas state when the temperature drops below (or density rises above) a certain threshold. In this paper, we sho...

To elucidate the role played by surface tension on the formation and on the structure of a circular hydraulic jump, the results from three different approaches are compared: the shallow-water (SW) equations without considering surface tension effects, the depth-averaged model (DAM) of the SW equations for a flow with a parabolic velocity profile, a...

This paper examines two-dimensional liquid curtains ejected at an angle to the horizontal and affected by gravity and surface tension. The flow in the curtain is, generally, sheared. The Froude number based on the injection velocity and the outlet’s width is assumed large; as a result, the streamwise scale of the curtain exceeds its thickness. A se...

An observation is presented that the ratio of the critical and triple-point temperatures Tcr/Ttp of neon, argon, krypton, and xenon fit within a narrow interval, Tcr/Ttp=1.803±0.5%, and the same applies to the density ratio, ncr/ntp=0.3782±1.7% (of the two remaining noble gases, helium does not have a triple point and, for radon, ntp is unknown). W...

The Enskog-Vlasov (EV) equation is a widely used semiphenomenological model of gas-liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H theorem as well, it can be used to...

Observations show that radii of oceanic eddies often exceed the Rossby radius of deformation, whereas theoretical studies suggest that such vortices should be unstable. The present paper resolves this paradox by presenting a wide class of large geostrophic vortices with a sign-definite gradient of potential vorticity (which makes them stable), in a...

Previous theoretical studies have indicated that liquid bridges close to the Plateau-Rayleigh instability limit can be stabilized when the upper supporting disk vibrates at a very high frequency and with a very small amplitude. The major effect of the vibration-induced pressure field is to straighten the liquid bridge free surface to compensate for...

We derive a kinetic equation for rarefied diatomic gases whose molecules have a permanent dipole moment. Estimating typical parameters of such gases, we show that quantum effects cannot be neglected when describing the rotation of molecules, which we thus approximate by quantum rotators. The intermolecular potential is assumed to involve an unspeci...

This paper presents a model which yields examples of stable vortices in a continuously stratified rotating fluid, thus providing a possible explanation of the observed longevity of oceanic eddies. The model is based on two assumptions. Firstly, the ocean comprises a thin upper (active) layer and a thick lower (passive) one, with large and small ver...

This paper uses asymptotic techniques to find the shape of a two dimensional liquid bridge suspended between two vertical walls. We model the equilibrium bridge shape using the Laplace–Young equation. We use the Bond number as a small parameter to deduce an asymptotic solution which is then compared with numerical solutions. The perturbation approa...

We examine a rarefied gas with inter-molecular attraction. It is argued that the attraction force amplifies random density fluctuations by pulling molecules from lower-density regions into high-density regions and thus may give rise to an instability. To describe this effect, we use a kinetic equation where the attraction force is taken into accoun...

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number Re remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as Re → 0 and Re → ∞, resp...

We examine the dynamics of a layer of viscous liquid on an inclined plate. If the layer's upstream depth h_{-} exceeds the downstream depth h_{+}, a smooth hydraulic jump (bore) forms and starts propagating down the slope. If the ratio η=h_{+}/h_{-} is sufficiently small and/or the plate's inclination angle is sufficiently large, the bore overturns...

We consider thin liquid films on the inside (rimming flows) or outside (coating flows) of a cylinder with horizontal axis, rotating about this axis. If the liquid’s net volume is small, such films are known to evolve towards a steady state with a smooth surface, whereas, for larger amounts, the flow develops a ‘shock’ similar to a tidal bore. In th...

We study the stability of a static liquid column rising from an infinite pool, with its top attached to a horizontal plate suspended at a certain height above the pool’s surface. Two different models are employed for the column’s contact line. Model 1 assumes that the contact angle always equals Young’s equilibrium value. Model 2 assumes a function...

We examine the existence and stability of frozen waves in diblock copolymers with local conservation of the order parameter, which are described by the modified Cahn--Hilliard model. It is shown that a range of stable waves exists and each can emerge from a `general' initial condition (not only the one with the lowest density of free energy). We di...

This work builds on the foundation laid by Benney & Timson (Stud. Appl. Maths, vol. 63, 1980, pp. 93-98), who examined the flow near a contact line and showed that, if the contact angle is 180°, the usual contact-line singularity does not arise. Their local analysis, however, does not allow one to determine the velocity of the contact line and thei...

Stout beers show the counter-intuitive phenomena of sinking bubbles while the beer is settling. Previous research suggests that this phenomena is due the small size of the bubbles in these beers and the presence of a circulatory current, directed downwards near the side of the wall and upwards in the interior of the glass. The mechanism by which su...

It is well known that internal or Rossby waves propagating across a jet can be amplified, a phenomenon usually referred to as over-reflection. In some cases, over-reflection can be infinitely strong – physically, this means that the reflected and transmitted waves can exist without an incident one, i.e. they are spontaneously emitted by the mean fl...

We examine the evolution of a liquid drop on an inclined substrate oscillating vertically. The drop is assumed thin, and the oscillations are assumed weak and slow (the latter makes the liquid's inertia and viscosity negligible, so the drop's shape is determined by a balance of surface tension, gravity, and vibration-induced inertial force). On the...

We examine rimming flows, i.e. flows of a liquid film on the inside of a horizontal rotating cylinder. So far this problem has mostly been explored using the so-called lubrication approximation (LA). It was shown that, if the volume of the liquid in the cylinder exceeds a certain threshold, then a shock similar to a tidal bore appears in the lower...

We consider a viscous flow on an inclined plate, such that the liquid's depth far upstream is larger than that far downstream, resulting in a "smoothed-shock wave" steadily propagating downstream. Our numerical simulations show that in a large section of the problem's parameter space all initial conditions overturn (i.e., the liquid's surface becom...

Recent experiments by Brunet, Eggers & Deegan (Phys. Rev. Lett., vol. 99, 2007, p. 144501 and Eur. Phys. J., vol. 166, 2009, p. 11) have demonstrated that drops of liquid placed on an inclined plane oscillating vertically are able to climb uphill. In the present paper, we show that a two-dimensional shallow-water model incorporating surface tension...

This paper examines two related problems from liquid-film theory. Firstly, a steady-state flow of a liquid film down a pre-wetted plate is considered, in which there is a precursor film in front of the main film. Assuming the former to be thin, a full asymptotic description of the problem is developed and simple analytical estimates for the extent...

We study the stability of a static liquid column rising from an infinite pool, with its top attached to a horizontal plate suspended at a certain height above the pool's surface. Two different models are employed for the column's contact line. Model 1 assumes that the contact angle always equals Young's equilibrium value. Model 2 assumes a function...

We examine the dynamics of a two-dimensional drop on an inclined substrate vibrating vertically. The drop is assumed to be driven by its contact lines, while its shape is determined by a quasistatic balance of surface tension, gravity, and vibration-induced inertial force. It is shown that, if the dependence of the inertial force on time involves n...

This paper is concerned with liquid films on horizontally vibrating substrates. Using an equation derived by Shklyaev [Phys. Rev. E 79, 051603 (2009)], we show that all periodic and solitary-wave solutions of this equation are unstable regardless of their parameters. Some of the solitary waves, however, are metastable--i.e., still unstable, but wit...

We examine the linear stability of a capillary rivulet under the assumption that it is shallow enough to be described by the lubrication approximation. It is shown that rivulets on a sloping plate are stable regardless of their parameters, whereas rivulets on the underside of a plate can be either stable or unstable, depending on their widths and t...

This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ɛ, then taking the limit ɛ → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equations (e.g. waves of...

We present a numerical study of stability of two-layer isolated vortices on the f-plane with respect to normal modes, i.e. disturbances with harmonic dependence on the azimuthal angle and time. Two types of vortices are considered: compensated vortices (for which the lower-layer is at rest), and vortices with uniform potential vorticity in the lowe...

We consider an infinite plate being withdrawn (at an angle α to the horizontal, with a constant velocity U ) from an infinite pool of viscous liquid. Assuming that the effects of inertia and surface tension are weak, Derjaguin ( C. R. Dokl. Acad. Sci. URSS , vol. 39, 1943, p. 13.) conjectured that the ‘load’ l , i.e. the thickness of the liquid fil...

We examine steady flows of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. If the amount of fluid in the cylinder is sufficiently small, all of it is entrained by rotation and the film is distributed more or less evenly. For medium amounts, the fluid accumulates on the ‘rising’ side of the cy...

Alfvén waves, made to propagate along an ambient magnetic field and polarized transverse to a gravitational field g, with wave amplitude stratified along g, are shown to reduce the growth rate of interchange instability by increasing the effective inertia by a factor of 1+(y′/zkz)2, where z is the ambient magnetic field, kz is the wavenumber, and y...

A set of asymptotic equations is derived, describing the dynamics of the flute mode in a magnetized plasma with cold ions, under a ``local'' approximation (i.e., near a particular point). The asymptotic set is then used to calculate the growth rate of interchange instability in the slab model. It is shown that, unlike the magnetohydrodynamic orderi...

Wave packets in a smoothly inhomogeneous medium are governed by a nonlinear Schrödinger (NLS) equation with variable coefficients. There are two spatial scales in the problem: the spatial scale of the inhomogeneities and the distance over which nonlinearity and dispersion affect the packet. Accordingly, there are two limits where the problem can be...

We examine the stability of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic models, one of which was examined by Benilov [J. Fluid Mech. 501:105–124 (2004)]. It turned out that the li...

We examine the linear stability of a thin film of viscous fluid on the inside of a cylinder with horizontal axis, rotating about this axis. Unlike previous models, both axial and azimuthal components of the hydrostatic pressure gradient are taken into account, which yields solutions which collapse in both dimensions. Two types of such solutions are...

This paper attempts to resolve the long-standing contradiction between the observed longevity of oceanic vortices and their theoretical instability. Using the model of quasigeostrophic, two-layer ocean, we show that a vortex in the upper layer can be stabilised by a circulation in the lower layer, such that the potential vorticity (PV) there is uni...

Weakly nonlinear packets of surface gravity waves over topography are governed by a nonlinear Schrödinger equation with variable coefficients. Using this equation and assuming that the horizontal scale of topography is much larger than the width of the packet, we show that, counter-intuitively, the amplitude of a shoaling packet decays, while its w...

We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. The stability of the film has been previously explored using the leading-order lubrication approximation, under which it was found to be neutrally stable. In the present paper, we examine how the stability of t...

We examine the stability of a thin film of viscous fluid inside a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic equations, one of which was examined by Benilov, O'Brien, and Sazonov (J. Fluid Mech. 2003 497, 201–224). It turned out that...

This paper examines the stability of vortices in a two-layer ocean on the f-plane. The mean depth h of the upper layer is assumed to be much smaller than the depth h of the lower layer. Using the primitive equations, we derive an asymptotic criterion for baroclinic instability of compensated (i.e. confined to the upper layer) vortices. Surprisingly...

The stability of magnetically confined plasmas is sometimes examined using the so-called “slab” model, where the toroidal geometry of the problem is approximated locally by the Cartesian one. In the present paper, a (more accurate) cylindrical approximation is considered and shown to yield results which are qualitatively different from those of the...

The stability of a quasigeostrophic vortex over a radially symmetric topographic feature (elevation or depression) in a two-layer ocean on the f plane is examined. This article's concern is with compensated vortices, that is, those in which the lower layer is at rest (the disturbances, however, are present in both layers). Through numerical solutio...

The stability of a barotropic zonal jet aligned with zonal topography on the beta-plane is investigated. The topography is assumed to be spatially periodic, with a period much smaller than the width of the jet. The problem is examined both by linear normal-mode analysis and by direct numerical simulations. The following results are obtained. If the...

It is well-known that oceanic vortices exist for years, whereas almost all theoretical studies indicate that they must be unstable. A rare exception is the work by Dewar & Killworth (1995), who demonstrated that a Gaussian vortex in the upper layer of a two-layer ocean becomes stable if accompanied by a weak co-rotating circulation in the lower lay...

We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Using the so-called lubrication approximation, we derive an asymptotic equation for three-dimensional motion of the film and use this equation to examine its linear stability. It is demonstrated that: (i) there...

We examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating around this axis. An asymptotic equation is derived, which takes into account two orders of the so-called lubrication approximation. The equation admits a solution describing a steady-state distribution of film around the cylin...

We examine the stability of jets over topography on the so-called barotropic beta plane (which models oceanic currents in mid-latitudes). Attention is focused on disturbances with a large wave number, for which an asymptotic solution of the normal-mode eigenvalue problem is presented. It is demonstrated that short-wave modes, if they exist, are loc...

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast...

We examine the stability of sheared flows in an inversely stratified fluid (where the density increases upward). As demonstrated by Kuo (1963), if the shear and Väsälä frequency are both constant (i.e., if the velocity and density profiles are both linear), the shear suppresses the Rayleigh-Taylor instability that would affect the fluid in the abse...

Using the QG approximation, the stability of two-layer zonal flows on the beta plane over bottom topography is examined. The topography is assumed to be one-dimensional, with the isobaths being directed at a fixed angle to the streamlines of the flow. The horizontal spatial scale of bottom irregularities is assumed to be much shorter than the defor...

We examine linear waves on the beta-plane over topography which consists of isolated radially-symmetric irregularities. We assume that the radii of those are much smaller than the characteristic distance between neighbouring topographic features, and that the latter parameter is much smaller than the wavelength.

This paper deals with linear waves on the beta-plane over topography. The main assumption is that the topography consists of isolated radially symmetric irregularities (random or periodic), such that their characteristic radii are much smaller than the distances between them. This approximation allows one to obtain the dispersion relation for t...

The evolution of an intense barotropic vortex on the -gyres) accelerating the vortex. During the second stage, Tw [less-than-or-eq, slant] t [less-than-or-eq, slant] Td, the quadrupole and secondary axisymmetric components are intensified; the vortex decelerates. During the last, third, stage the vortex decays and is destroyed. Our main attention i...

The dynamics of a near-surface vortex are examined in a two-layer setting on the beta-plane. Initially, the vortex is radially symmetric and localized in the upper layer. Two non-dimensional parameters govern its evolution and translation: the ratio δ of the thickness of the vortex to the total depth of the fluid, and the non-dimensional beta-effec...

The author considers the stability of a barotropic jet on the beta plane, using the model of a 'rough-bottomed ocean' (i.e., assuming that the horizontal scale of bottom irregularities is much smaller than the width of the jet). An equation is derived, which governs disturbances in a sheared flow over one-dimensional bottom topography, such that th...

The linear approximation of density and velocity profiles is compared to more realistic models with vertically inhomogeneous density gradients and nonzero anomalous vorticity (i.e., the nonplanetary part of potential vorticity). Calculations based on the parameters of ''real life'' currents in the Northern Pacific demonstrate that these effects, ac...

The evolution of tracer "injected" into an equivalent barotropic eddy on the beta-plane is examined numerically. The eddy is governed by the standard quasigeostrophic equation, and the concentration of tracer is governed by the advection equation with diffusion. At the initial moment of time, the streamfunction and distribution of tracer are both r...

The stability of continuously stratified vortices with large displacement of isopycnal surfaces on the f -plane is examined both analytically and numerically. Using an appropriate asymptotic set of equations, we demonstrated that sufficiently large vortices (i.e. those with small values of the Rossby number) are unstable. Remarkably, the growth rat...

We examine the interaction of near-surface and near- bottom flows over bottom topography. A set of asymptotic equations for geostrophic currents in a three-layer fluid is derived. The depths of the active (top/bottom) layers are assumed small, the slope of the bottom is weak, the interfacial displacement is comparable to the depths of the thinner l...

Strong eddies on the β plane are considered within the framework of the one-layer primitive equations. The attention is focused on calculating the β-induced changes to the spatial structure of the eddy, as well as the speed of its translation. In contrast to the earlier studies, the results of this paper are valid for eddies with Ro ∼ 1 and are app...

We examine two-layer geostrophic flows over a flat bottom on the -plane. If the displacement of the interface is of the order of the depth of the upper layer, the dynamics of the flow depends on the following non-dimensional parameters: (i) the Rossby number ε, (ii) the ratio δ of the depth of the upper layer to the total depth of the fluid, (iii)...

In this paper the dynamics of geostrophic flows localized in a thin layer of continuously stratified fluid, which overrides a thick homogeneous layer are studied. The displacement of isopycnal surfaces is assumed large; the β-effect is strong, i.e. \[(R_0/R_e)\cot\theta\gtrsim\epsilon, \] where ε is the Rossby number, θ is the latitude; Re is the E...

This paper examines the baroclinic instability of a quasi-geostrophic flow with vertical shear in a continuously stratified fluid. The flow and density stratification are both localized in a thin upper layer. (i) Disturbances whose wavelength is much smaller than the deformation radius (based on the depth of the upper layer) are demonstrated to sat...

This paper examines the stability of two-layer geostrophic flows with large displacement of the interface and strong β-effect. Attention is focused on flows with non-monotonic interface profiles which are not covered by the Rayleigh-style stability theorems proved by Benilov (1992a, b) and Benilov & Cushman-Roisin (1994). For such flows the coeffic...

The two-layer model of baroclinic instability is modified to include small continuous variations of the velocity/density profile in the upper layer. It is demonstrated that, if the difference between the average upper-layer velocity and the velocity in the lower layer is negative (westward), the flow is unstable. The instability takes place in the...

We examine the stability of two-layer geostrophic flows with large displacement of the interface. The depth of the upper layer is assumed much smaller than the total depth of the fluid. It is proven that all westward and weak eastward flows are stable with respect to disturbances whose wavelengths is of the order of, or longer than the width of the...

This paper examines the dynamics of geostrophic flows with large displacement of isopycnal surfaces. The β-effect is assumed strong i.e. the parameter (Rd cot θ)/Re (where θ is the latitude, Rd is the deformation radius, Re is the Earth's radius) is of the order of, or greater than, the Rossby number. A system of asymptotic equations is derived, wi...

We construct oscillatory solitary wave solutions of a fifth-order Korteweg-de Vries equation, where the oscillations decay at infinity. These waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide. Our approach is a wave-packet analysis about this wavenumber which le...

We consider a fifth-order KdV equation, where the fifth-order derivative term is multiplied by a small parameter. It has been conjectured that this equation admits a non-local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core. We prove that this solution cannot be exactly steady, and inste...

The propagation of nonlinear waves in random media is an important aspect of nonlinear wave theory and has a long and informative history. This paper describes the basic ideas of the approaches that have been applied. The average-field method, which has been used most extensively in linear problems, is considered. This approach is then shown to be...

This paper examines the large-scale dynamics of a layer of stratified fluid on the β-plane. A three-dimensional asymptotic system is derived which governs geostrophic flows with large displacement of isopycnal surfaces. This is then reduced to a two-dimensional set of equations which describe the interaction of a baroclinic ‘quasi-mode’ with arbitr...

It is shown that all one-layer geostrophic isolated fronts are stable regardless of their profiles.

This paper is concerned with large-amplitude flows of a two-layer fluid on the -plane. The Rossby number is small, while the displacement of the interface and the depth of the upper layer are both of the order of the total depth of the fluid. Two systems of equations are derived, corresponding to two asymptotic ranges of the parameter / (where is t...

This paper examines surface gravity waves in a shallow channel with periodic or random bottom irregularities. Three types of bottom topography are distinguished, allowing a simplification of the basic shallow-water-wave equations. For two of them, asymptotic equations of the Korteweg-de Vries type are derived (the third type has been considered ear...

Interaction of a zonal jet and small-amplitude Rossby-wave turbulence is studied within the framework of the barotropic β-plane model. It is demonstrated that turbulent-laminar interaction in this case transfers energy from the wave turbulence to the laminar flow (the effect of negative friction). We derive a conclusion that, as the geophysical tur...

An asymptotic theory, describing turbulent diffusion due to wave-induced random motion in incompressible or compressible fluids, is constructed. It is shown that even weakly nonlinear waves cause irreversible stretching of material lines. The results obtained are applied to the Rossby-wave-induced motion in the atmosphere or ocean. An expression fo...