Nonlinear Kelvin Research Articles

Nonlinear Kelvin—Helmholtz instability in magnetic fluids of finite thickness: effects of a tangential magnetic field

The nonlinear stability of a horizontal interface separating two streaming magnetic fluids of finite thickenss is investigated in two dimensions. The fluids are considered to be inviscid and incompressible. The magnetic field is applied along the direction of streaming. The method of multiple scales, in both space and time, is used to examine the stability properties of the system arising from second-harmonic resonance. A pair of partial differential equations for the amplitude of the wave and its second harmonic are derived. These describe the evolution of the wave train up to cubic order, and may be regarded as the counterparts of the single nonlinear Schrödinger equation that occurs in the non-resonant case. The stability condition of this equation is discussed both analytically and numerically, and stability diagrams are obtained. Regions of stability and instability are identified. The nonlinear cut-off wavenumber separating the regions of stability from those of instability is obtained. The equation governing the evolution of the amplitude at the critical point is also obtained, which leads to a nonlinear Klein—Gordon equation.

An experimental and numerical study of nonlinear internal waves

Nonlinear internal waves were measured on the large rotating platform at the Institut de Mécanique de Grenoble (I.M.G.). The experimental data complement the results presented in Renouard et al. [J. Fluid Mech. 177, 381 (1987)] and support the assumption that the solitary Kelvin wave is accompanied by Poincaré waves. Based on the assumption of weak nonlinear, dispersive, and rotational effects, governing equations of the Boussinesq type are derived to model the evolution of an initial disturbance in a two-layer rotating fluid. The numerical study is based on these equations which are analogous to the Boussinesq equations of shallow-water theory and are not constrained to almost unidirectional propagation. Comparison of numerical solutions of the equations and experimental results are very good for moderately nonlinear conditions. These results provide supporting evidence for the resonant interaction of nonlinear Kelvin waves and linear Poincaré waves, as described by Melville et al. [J. Fluid Mech. 206, 1 (1989)].

The Dynamics of Coastally Trapped Mesoscale Ridges in the Lower Atmosphere

Abstract The dynamics of coastally trapped ridges that propagate in the marine layers of western North America and southeastern Australia is examined. A nonlinear semigeostrophic theory shows that the coastal ridges develop initially as an alongshore intrusion of denser marine air that is driven by the synoptic-scale pressure gradient. Nonlinear Kelvin waves evolve with the intruding flow on a slower time scale governed by the dynamic parameters. If dispersive effects balance the nonlinearities, then these waves evolve into solitary form. Otherwise, the nonlinear waves steepen so that the leading edge of the ridge eventually propagates as a shock.The theory is applied to two ridging events in California and one in southeastern Australia. In each case, good agreement is found between theory and observations of the evolution times and propagation speeds of the coastal ridges. The theory also explains the observed behavior of the events at prominent convex bends and gaps in the coastal topography.

Geostrophic adjustment in a channel: nonlinear and dispersive effects

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq-type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincaré waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincaré waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincaré waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

Coastal Kelvin waves in the presence of a slowly varying topography

The evolution equation is derived for a weakly nonlinear coastal Kelvin wave propagating in slowly varying topography in anf-plane ocean. For weak transverse variations in the topography, the wave evolution is governed by a perturbed Korteweg–de Vries equation. In the absence of transverse variation, wave dispersion vanishes and the evolution equation reduces to a nonlinear advection equation with variable coefficients. As a general property of these equations, the total mass flux associated with the Kelvin wave is not conserved; residual mass must be generated. It is shown by an asymptotic analysis that this residual mass field is in balance with a mean geostrophic current long after the passage of the Kelvin wave. This result is verified using a numerical model. The physical mechanism evolved in the generation of the residual mass can be understood in terms of potential vorticity conservation.

On the generation of nonlinear internal Kelvin waves in a rotating channel

Recent observations have shown that islands and constrictions may be the sites of significant internal wave activity in sea straits. In this paper we examine one generation mechanism: resonant forcing by transcritical flow past topography. Experiments were conducted on the large rotating platform at the Coriolis Laboratory, Institut de Mécanique de Grenoble, Grenoble, France. A slender body was towed through a two‐layer stratified rotating channel, simulating the flow past an island or constriction in a strait. For a range of Froude numbers, blockage coefficients, and rotation rates, nonlinear internal Kelvin waves were generated upstream. The dependence of the wave parameters on the Froude number, Rossby number, and blockage coefficient was measured. The transition between subcritical and supercritical flow is found to occur at Froude numbers greater than unity, to depend on the blockage, but to be essentially independent of the internal Rossby radius of deformation. The results are compared with recent similar measurements of single‐layer flows in nonrotating channels, and good agreement is obtained. The nonlinear Kelvin waves which are generated upstream are found to be similar in all respects to those described by Renouard et al. (1987).

Nonlinear electrohydrodynamic Kelvin–Helmholtz instability conditions of an interface between two fluids under the effect of a normal periodic electric field

The electrohydrodynamic Kelvin–Helmholtz instability conditions of an interface separating two dielectric streaming fluids, stressed by a normal electric field in absence of surface charges on the interface, are studied. We use the method of multiple scales to solve nonlinear equations. In the first-order problem we obtained Mathieu's differential equation. For the second-order problem, we obtain the nonhomogeneous Mathieu equation and we use the method of multiple scales to obtain a sequence of equations. In the third-order problem, we obtain the second-order differential equation of periodic coefficients; we also obtain a formula for surface elevation, and we determine the instability conditions.

On the stability of Kelvin waves

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincaré modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincaré waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d'Hières & Zhang (1987).

Evolution of an isolated eddy near a coast and its relevance to the “Kyucho”

The evolution of an isolated meso-scale eddy near a coast is studied numerically. In particular, it is found that the translation speed of the adjusted eddy is estimated well by the mutual induction mechanism adapted to a rotating stratified fluid. The nonlinear Kelvin wave excited during the adjustment process is also discussed in connection with the “Kyucho”, the sudden warming of coastal waters associated with swift currents.

Weakly nonlinear Kelvin–Helmholtz waves

The Lagrangian L for gravity waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates qv(t)≡ {qnv(t)} of the N + 1 interfaces, where the qnv are the Fourier coefficients in the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The explicit expansion of L is constructed through fourth-order in the qnν and . Progressive interfacial waves and Kelvin–Helmholtz instability in a two-layer fluid are examined, and the earlier results of Drazin (1970), Nayfeh & Saric (1972) and Weissman (1979) are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin–Helmholtz instability, which is supercritical for infinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin–Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The effect of surface tension is examined in an Appendix. Finally, the wave motion forced by flow over a sinusoidal bottom (as in Thorpe's tilting tank) is examined and the corresponding resonance curves and Hopf bifurcations determined. Numerical integrations reveal that stable limit cycles exist in some parametric neighbourhoods of these bifurcations. Period doubling was observed but did not lead to chaotic motion.

An experimental study on in vivo dynamic mechanical properties of the knees of cats in valgus loading

In these experiments an attempt was made to investigate in vivo dynamic mechanical properties of the knee in valgus loading. Fifteen adult cats were used. Sinusoidal bending oscillation of 0.3, 1.0, 1.6, 3.3, 5.0, 8.6 and 10.0 Hz was transmitted to the knee-leg complex under each pre-bending strain in the valgus direction, and the bending moment was measured. Dynamic mechanical properties of the knee-leg complex were given as a complex bending elastic coefficient. Bending elastic and bending damping coefficient of the knee could be calculated by the least square method. As a first approximation, the knee was represented as a non-linear Kelvin model. Both bending elastic and bending damping coefficient of the knee increased as pre-bending moment was increased, and decreased as amplitude of bending oscillation was increased. Mechanical loss tangent of the knee-leg complex decreased within the range of pre-bending moment larger than 1.0 × 10 6 dyn cm as bending moment was increased, and decreased as amplitude of bending oscillation was increased.

Electrohydrodynamic solitons in Kelvin-Helmholtz flow: the case of a normal field in the absence of surface charges

Nonlinear electrohydrodynamic Kelvin—Helmholtz instability conditions are investigated. A charge-free surface separating two semi-infinite dielectric streaming fluids influenced by a normal electric field is subjected to nonlinear deformations. The method of multiple-scale perturbations is used in order to obtain two nonlinear Schrödinger (NLS) equations describing the behavior of the disturbed system. The stability and instability conditions of the perturbed system are discussed analytically. One of the two NLS equations is used to obtain the electrohydrodynamic (EHD) nonlinear cutoff wave number separating stable and unstable disturbances while the other equation is used to describe analytically the necessary condition for stability and instability for the system. For unstable cases, the solution starting with a solitary wave degenerates into a finite number of EHD solitons.

Equatorial solitary waves Part 4. Kelvin solitons in a shear flow

It is shown that a mean flow with shear makes the Kelvin wave dispersive. This in turn modifies its nonlinear behavior and makes it necessary to replace the one-dimensional advection equation derived in an earlier work of the author's by the Korteweg-deVries equation instead. The frontogenesis predicted in the earlier paper will still occur, but the wave breaking will not. Instead, once a steep front has formed, it will disintegrate into a train of solitary waves. These then propagate towards the east at a faster-than-linear rate. It is also shown that Kelvin solitary waves will have much smaller zonal widths than Rossby solitons of the same height; “round” Kelvin solitary waves (equal zonal and latitudinal width) are to be expected, and are fully consistent with the small amplitude, weak dispersion theory. An interesting implication of the Korteweg-deVries model is that the peak signal from a nonlinear Kelvin wave packet may be roughly double that of a linear Kelvin wavetrain.

On inertial flow over topography. Part 1. Semigeostrophic adjustment to an obstacle

The nonlinear time-dependent adjustment of a homogeneous rotating-channel flow to the sudden obtrusion of an obstacle is studied. Solutions are obtained using a Lax–Wendroff numerical scheme which allows rotating breaking bores and jumps to form and be maintained. The flow upstream of the obstacle is found to be completely blocked, partially blocked (and hydraulically controlled), or unobstructed depending upon the height of the obstacle. Partial blockage is accomplished through the excitation of a combination of nonlinear Kelvin waves, some of which steepen into interfacial shocks. Riemann invariants for the Kelvin waves are found, and jump conditions on mass, momentum and potential vorticity for the shocks are discussed. The shocks are surrounded by dispersive regions of Rossby deformation scale, and the potential vorticity of passing fluid is altered at a rate proportional to the differential rate of energy dissipation along the line of breakage. For the special case of initially uniform potential vorticity the asymptotic state is found as a function of the initial conditions.

Numerical studies of surface-tension effects in nonlinear Kelvin–Helmholtz and Rayleigh–Taylor instability

We consider the behaviour of an interface between two immiscible inviscid incompressible fluids of different density moving under the action of gravity, inertial and interfacial tension forces. A vortex-sheet model of the exact nonlinear two-dimensional motion of this interface is formulated which includes expressions for an appropriate set of integral invariants. A numerical method for solving the vortex-sheet initial-value equations is developed, and is used to study the nonlinear growth of finite-amplitude normal modes for both Kelvin-Helmholtz and Rayleigh-Taylor instability. In the absence of an interfacial or surface-tension term in the integral-differential equation that describes the evolution of the circulation distribution on the vortex sheet, it is found that chaotic motion of, or the appearance of curvature singularities in, the discretized interface profiles prevent the simulations from proceeding to the late-time highly nonlinear phase of the motion. This unphysical behaviour is interpreted as a numerical manifestation of possible ill-posedness in the initial-value equations equivalent to the infinite growth rate of infinitesimal-wavelength disturbances in the linearized stability theory. The inclusion of an interfacial tension term in the circulation equation (which stabilizes linearized short-wavelength perturbations) was found to smooth profile irregularities but only for finite times. While coherent interfacial motion could then be followed well into the nonlinear regime for both the Kelvin-Helmholtz and Rayleigh-Taylor modes, locally irregular behaviour eventually reappeared and resisted subsequent attempts at numerical smoothing or suppression. Although several numerical and/or physical mechanisms are discussed that might produce irregular behaviour of the discretized interface in the presence of an interfacial-tension term, the basic cause of this instability remains unknown. The final description of the nonlinear interface motion thus awaits further research.

An existence theorem for a nonlinear evolution equation in viscoelasticity

We establish the existence of weak global solutions of initial-boundary value problems for $$u_{tt} - \sum\limits_{i = 1}^N {\frac{d}{{dx_i }}a} ,(x,t,u_{x_i } ) - \Delta _N u_t = f$$ which occours as the equation of motion in nonlinear Kelvin solids with stress components σi=αi(x,t,uxi)+uxti=1,2...,N.. Each functiona i is required to be sufficiently smooth and must satisfy the following conditions:a)∣αi(x,t,η)∣⩽K0{∣η∣p-1+1}a)αi(x,t,η)≶K1∣η∣p-2η η≶0c)(∂/∂t)αi(x,t,η)≶K2(t){∣η∣p-2+1}η≶0d)[αi(x,t,η)—αi(x,t,ξ)].(η—ξ⩾0 for somep≥2, some positive constantsK 0,K 1, some non negative functionK 2∈L 1(0,T) and for allx∈Ω, t∈[0, T], ξ and ν∈R.

The Nonlinear Equatorial Kelvin Wave

Using the method of strained coordinates, a uniformly valid approximation to the nonlinear equatorial Kelvin wave is derived. It is shown that nonlinear effects are negligible for the Kelvin waves associated with the Gulf of Guinea upwelling. The Kelvin waves involved in El Nino, however, are significantly distorted both in shape and speed. The leading edge is smoothed and expanded rather than steepened, but the trailing edge will form sharp fronts and eventually break.

Nonlinear Kelvin–Helmholtz instability in hydromagnetics

Nonlinear Kelvin–Helmholtz instability for an incompressible plasma under the influence of gravity is investigated theoretically, using the method of multiple time-scales. The effect of nonlinearity is to reduce the growth rate for the linear instability and to narrow the unstable wavenumber domain. There are two cases, in one of which the effect is strong and in the other slight. The difference between these cases is studied.

Wind-forced linear and nonlinear Kelvin waves along an irregular coastline

Using a normal and tangential co-ordinate approach, a perturbation theory is developed for wind-forced linear and nonlinear Kelvin waves propagating along an irregular coastline. The theory is valid for coastline curvatures which are non-dimensionally small, the curvature being non-dimensionalized with respect to the reciprocal of the boundary-layer trapping scale, i.e. the reciprocal of the radius of deformation. According to linear theory, the main effect of a coastline of small curvature is to cause a phase-speed change in the wave (from – c to – c(1 – ½k(s)), where k(s) is the non-dimensional curvature a distance s along the coast from the origin) and to make the offshore Ekman transport change more rapidly along the coast, the latter effect implying a more ‘wavelike’ ocean or lake response. Two discernible nonlinear effects were found to be an increase (decrease) in the linear-solution longshore gradients in regions of positive (negative) isopycnal displacement and a tendency for increased (decreased) isopyncal displacement at capes (bays).

Coastally trapped waves in the atmosphere

AbstractIt is suggested that the ‘coastal lows’ observed off southern Africa are similar in structure to ‘coastally trapped waves’ in the ocean. Inversion conditions typical of the area prevent escape of energy upwards and Coriolis effects keep energy trapped against the high escarpment which borders the continent. Coastal disturbances are produced because low‐level flow in the synoptic‐scale systems cannot cross the barrier provided by the escarpment. Nonlinear effects cause lows to be more prominent than highs. A simple nonlinear forced Kelvin wave model is found to reproduce many of the features observed.