E. S. Benilov

Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg-de Vries equation

Abstract 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 leads in the first instance to a higher-order nonlinear Schrodinger equation, from which we then obtain the steady solitary wave solution. We then describe a complementary normal-form analysis which leads to the same result. In addition we derive the nonlinear Schrodinger equation for all wavenumbers, and list all the various anomalous cases.

The generation of radiating waves in a singularly-perturbed Korteweg-de Vries equation

Abstract 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 instead the amplitude of the central core decays due to the energy flux generated in the oscillatory tail. The decay rate is calculated in the limit as the parameter tends to zero. In order to verify the analytical results, we have developed a high-precision spectral method for numerical integration of this equation. The analytical and numerical result show good agreement.

Inertial instability of a liquid film inside a rotating horizontal cylinder

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 the film is affected by higher-order corrections, such as inertia (described by the material derivatives in the Navier–Stokes equations), surface tension, and the hydrostatic pressure gradient. Assuming that these effects are weak, we derive an asymptotic equation which takes them into account as perturbations. The equation is used to examine the stability of the steady-state distribution of film around the cylinder (rimming flow) with respect to linear disturbances with harmonic dependence on time (normal modes). It is shown that hydrostatic pressure gradient does not affect those at all, and the effect of surface tension is weak—whereas inertia always causes in...

Steady rimming flows with surface tension

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 cylinder and, for large ones, pools at the cylinder’s bottom. The paper examines rimming flows with a pool affected by weak surface tension. Using the lubrication approximation and the method of matched asymptotics, we find a solution describing the pool, the ‘outer’ region, and two transitional regions, one of which includes a variable (depending on the small parameter) number of asymptotic zones.

Stability of vortices in a two-layer ocean with uniform potential vorticity in the lower layer

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 layer. Note that this paper assumed the lower-layer circulation to have the same profile as the main (upper-layer) vortex. The present paper considers the case where the profile of the circulation in the lower layer is determined by the condition that potential vorticity (PV) there is constant – which models a vortex surrounded by water of a different origin. Given that most oceanic vortices are shed by frontal currents, such model appears to be more realistic than any ad hoc choice. The stability of vortices with uniform lower-layer PV is examined for both quasigeostrophic and ageostrophic cases, numerically and asymptotically, assuming that the upper layer of the ocean is thin. It is shown that such vortices are stable for a wide range of parameters. The effect of vortex stabilization is interpreted through representation of the unstable disturbance by two phase-locked Rossby waves, rotating around the vortex in the upper and lower layers. Then, if the lower-layer PV gradient is zero, it cannot support the corresponding wave, which inhibits the instability.

Instability of quasi-geostrophic vortices in a two-layer ocean with a thin upper layer

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 (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

On the stability of large-amplitude vortices in a continuously stratified fluid on the f-plane

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 rate of the unstable disturbance is a function of the spatial coordinates. At the same time, the corresponding boundary-value problem for normal modes has no smooth square-integrable solutions, which would normally be regarded as stability. We conclude that (potentially) stable vortices can be found only among ageostrophic vortices. Since this assumption cannot be verified analytically due to complexity of the primitive equations, we verify it numerically for the particular case of two-layer stratification.

Explosive instability in a linear system with neutrally stable eigenmodes. Part 2. Multi-dimensional disturbances

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 are infinitely many normal modes (harmonic in the axial variable and time), which are all neutrally stable and their eigenfunctions form a complete set; (ii) but the film is nonetheless unstable with respect to non-harmonic disturbances, which develop singularities in a finite time

Drops climbing uphill on an oscillating substrate

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 and inertia can reproduce qualitatively the main features of these experiments. We find that the motion of the drop is controlled by the interaction of a ‘swaying’ (odd) mode driven by the in-plane acceleration and a ‘spreading’ (even) mode driven by the cross-plane acceleration. Both modes need to be present to make the drop climb uphill, and the effect is strongest when they are in phase with each other.

On the stability of shallow rivulets

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 the plate’s slope. For the case of a horizontal plate, sufficiently narrow rivulets are shown to be stable and sufficiently wide ones unstable, with the threshold width being π/2 ( σ/ gρ) 1/2 (ρ and σ are the liquid’s density and surface tension, g is the acceleration due to gravity). It is also shown that, even though the plate’s slope induces in a rivulet a sheared flow (which would normally be viewed as a source of instability) – in the present problem, it is a stabilizing factor. The corresponding stability criterion involving the rivulet’s width and the plate’s slope is computed, and it is demonstrated that, if the latter is sufficiently strong, all rivulets are stable regardless of their widths.