Nonlinear Critical Layer Research Articles

The evolution of free-disturbances in a two-dimensional nonlinear critical layer

This paper investigates the evolution of (relatively) long Rayleigh waves on an inflectional two-dimensional boundary layer such as may occur when a flow encounters a small surface mounted obstacle. Under the assumption that the flow remains essentially two-dimensional a coupled set of evolution equations are derived that describe the nonlinear growth of an essentially arbitrary (2D) disturbance to the base flow. Numerical solutions are presented for a representative initial condition.

The evolution of free disturbances on an initially two-dimensional inflectional boundary-layer flow

In a recent paper Savin, Smith and Allen (Proc. R. Soc. A 455 (1999)) performed a nonlinear critical layer analysis based upon the long wavelength instability of an inflectional sublayer flow. This limit made it possible for the authors to consider the evolution of general disturbances to the base flow for a number of possible transition paths. The present analysis follows one of those paths and derives, in closed form, the amplitude evolution equation for a viscous non‐equilibrium critical layer. This equation is then solved numerically in the inviscid limit for a representative initial condition.

Wave–Mean‐Flow Interactions in a Forced Rossby Wave Packet Critical Layer

This paper describes the nonlinear critical layer evolution of a zonally localized Rossby wave packet forced in mid‐latitudes and propagating horizontally on a beta plane in a zonal shear flow. The wave packet has an amplitude that varies slowly in the zonal direction. Numerical solutions of the governing nonlinear equations show that the wave–mean‐flow interactions differ from those that would result with a monochromatic forcing. With the localized forcing, the net absorption of the disturbance at the critical layer continues for large time, because there is an outward flux of momentum in the zonal direction. Further insight into the mechanism for this and other aspects of the evolution of the critical layer is obtained through an approximate asymptotic analysis which is valid for large time.

Interactions between pairs of oblique waves on a mixing layer

Abstract We consider the weakly nonlinear spatial evolution of two pairs of oblique waves superimposed on an inviscid mixing layer, with each wave being slightly amplified on a linear basis. One pair of waves is assumed to be inclined at an angle θ 1 to the plane of the mixing layer, the other at an angle θ 2 . A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the pairs as well as within each pair. These equations are discussed and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.

A non-resonant triad on a mixing layer

We consider the weakly nonlinear spatial evolution of a disturbance consisting of a plane wave and a pair of oblique waves inclined at an angle θ other than 60° superimposed on an inviscid mixing layer, with each wave being slightly amplified on a linear basis. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the oblique and plane waves as well as the usual interaction within the pair of oblique waves. These equations are discussed and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.

Stuart vortices in a stratified mixing layer: the Holmboe model

Asymptotic techniques are used to model the quasi-steady state vortices that have been observed in two-dimensional simulations of vortex roll-up in stratified shear layers. A time-independent nonlinear critical layer analysis is used to find a family of steady-state finite amplitude vortices in the Holmboe model of an inviscid stratified shear layer, with the vorticity inside closed streamlines based on the Stuart vortex. The vortices are compared to results of simulations and also an alternative model where the vorticity was constant inside closed streamlines.

Modal interactions in a Bickley jet: comparison of theory with direct numerical simulation

This study is concerned with three-dimensional transition in plane wakes and jets. Nonlinear stability theory has predicted that interactions between different instability modes can play an important role in that transition. In this paper, we consider interactions between varicose and sinuous oblique disturbances in the Bickley jet, using both nonlinear stability theory (in its nonlinear critical layer form) and direct numerical simulation using a spectral method. We will first present an overview of the nonlinear stability theory which indicates that a (nonlinear) interaction between the modes should occur, and then present simulations that appear to support the theory.

Stuart Vortices In A Stratified Mixing Layer: The Holmboe Model

Asymptotic techniques are used to model the quasi-steady state vortices that have been observed in two-dimensional simulations of vortex roll-up in stratified shear layers. A time-independent nonlinear critical layer analysis is used to find a family of steady-state finite amplitude vortices in the Holmboe model of an inviscid stratified shear layer, with the vorticity inside closed streamlines based on the Stuart vortex. The vortices are compared to results of simulations and also an alternative model where the vorticity was constant inside closed streamlines.

Subcritical Rossby Waves in Zonal Shear Flows with Nonlinear Critical Layers

It was shown by Benney and Bergeron [1] that singular neutral modes with nonlinear critical layers are mathematically possible in a variety of shear flows. These are usually subcritical modes; i.e., they occur at values of the flow parameters where their linear, viscous counterparts would be damped. One question raised then is how such modes might be generated.This article treats the problem of Rossby waves propagating in a mixing layer with velocity profile ū = tanh y. The beta parameter, which is a measure of the stabilizing Coriolis force, is taken to be large enough so that linear instability cannot occur. First, computed dispersion curves are presented for singular modes with nonlinear critical layers. Then, full numerical simulations are employed to illustrate how these modes can be generated by resonant interaction with conventional nonsingular Rossby waves, even when the singular mode is absent initially.

Nonlinear evolution of singular disturbances to a tanh3y mixing layer

AbstractWe consider the nonlinear evolution of a disturbance to a mixing layer, with the base profile given by u0(y) = tanh3y rather than the more usual tanh y, so that the first two derivatives of u0 vanish at y = 0. This flow admits three neutral modes, each of which is singular at the critical layer. Using a non-equilibrium nonlinear critical layer analysis, equations governing the evolution of the disturbance are derived and discussed. We find that the disturbance cannot exist on a linear basis, but that nonlinear effects inside the critical layer do permit the disturbance to exist. We also present results of a direct numerical simulation of this flow and briefly discuss the connection between the theory and the simulation.

On Disturbances to a Stratified Mixing Layer

Using a nonlinear critical layer analysis, we examine the behavior of disturbances to the Holmboe model of a stratified shear layer for Richardson numbers 0<J≤1/4, regarding J as an ϑ (1) quantity rather than a small quantity. By examining the structure of the solution inside the critical layer, we demonstrate that, for this particular flow, disturbances with a phase change of the form predicted by linear stability theory cannot exist on a purely linear basis, and because of this, an instability wave will have no phase change across the critical layer, and that instead we should regard the Richardson number J as a small quantity within the framework of a nonlinear analysis. We argue further that when nonlinear effects are included, disturbances with a phase change can exist, but this leads to such a slow time scale that the flow in question is unlikely to arise, except possibly as a secondary instability.

A numerical simulation of the nonlinear critical layer evolution of a forced Rossby wave packet in a zonal shear flow

We investigate the nonlinear development of a forced Rossby wave packet in the presence of a critical layer in a zonal shear flow. Most previous analyses of this phenomenon have dealt with spatially periodic, monochromatic waves. These studies observed that in the initial linear stages, the disturbance is absorbed at the critical layer and as a consequence a discontinuity in the wave-induced Reynolds stress occurs across the critical layer. Subsequently, the linear theory breaks down and nonlinear phenomena such as wave breaking and reflection result. For a more realistic representation of wave activities in the atmosphere, we employ a forcing in the form of a spatially-localized wave packet rather than a monochromatic wave, and solve the nonlinear equations numerically using Fourier transform methods and a high order compact finite difference scheme. It is found that the spatial localization delays the onset of the nonlinear breakdown in the critical layer and that there is an outward flux of momentum in the zonal direction.

Forced Disturbances in a Zero Absolute Vorticity Gradient Environment

Observations show the presence of localized regions in the atmosphere with diminished potential vorticity gradients, an example being the tropical upper troposphere where convective heating plays an important role. The present work investigates the effect of forcing on the evolution of Rossby waves in a zero potential vorticity gradient environment. As a preliminary investigation, the barotropic case is studied, where the analog of potential vorticity is absolute vorticity. A forcing term is employed to model thermal forcing in the real atmosphere. The analytic solution of the linearized problem shows that the streamfunction grows in time and eventually develops a nonlinear critical layer. The presence of forcing within the critical layer is found to drive the dynamics. The numerical solution of the nonlinear problem within the critical layer shows that the nonlinearity and the forcing act together to halt the growth as coherent vortices develop in a nonlinear oscillatory regime. At long times, the critical layer solution settles down to a quasi steady state consisting of relatively large amplitude stationary vortices with a set of small amplitude steadily propagating vortices superimposed. These results are contrasted with the results of previous unforced problems.

Non-axisymmetric vortices in two-dimensional flows

Slightly non-axisymmetric vortices are analysed by asymptotic methods in the context of incompressible large-Reynolds-number two-dimensional flows. The structure of the non-axisymmetric correction generated by an external rotating multipolar strain field to a vortex with a Gaussian vorticity profile is first studied. It is shown that when the angular frequency w of the external field is in the range of the angular velocity of the vortex, the non-axisymmetric correction exhibits a critical-point singularity which requires the introduction of viscosity or nonlinearity to be smoothed. The nature of the critical layer, which depends on the parameter h = 1/(Re ε3/2), where ε is the amplitude of the non-axisymmetric correction and Re the Reynolds number based on the circulation of the vortex, is found to govern the entire structure of the correction. Numerous properties are analysed as w and h vary for a multipolar strain field of order n = 2, 3, 4 and 5. In the second part of the paper, the problem of the existence of a non-axisymmetric correction which can survive without external field due to the presence of a nonlinear critical layer is addressed. For a family of vorticity profiles ranging from Gaussian to top-hat, such a correction is shown to exist for particular values of the angular frequency. The resulting non-axisymmetric vortices are analysed in detail and compared to recent computations by Rossi, Lingevitch & Bernoff (1997) and Dritschel (1998) of non-axisymmetric vortices. The results are also discussed in the context of electron columns where similar non-axisymmetric structures were observed (Driscoll & Fine 1990).

Nonlinear stability of a stratified shear flow in the regime with an unsteady critical layer. Part 2. Arbitrary stratification of asymmetric flow

A weakly nonlinear analysis of the downstream evolution of weakly unstable disturbances in a stably stratified mixing layer with a large Reynolds number is carried out. No other requirements are imposed upon velocity and density profiles, thus making it possible to overcome the restrictions placed in earlier studies (Brown & Stewartson 1978; Brown et al. 1981; Churilov & Shukhman 1987, 1988) by a particular choice of weakly supercritical flow models assuming symmetry. For each of the two critical layer regimes possible here, viscous and unsteady, evolution equations are obtained, their solutions and competition between nonlinearities in the course of instability development are analysed, and evolution scenarios for unstable disturbances are constructed for different levels of their supercriticality. It is established that the regime with a nonlinear critical layer does not arise in an evolutionary manner, except for the previously studied case of a weak stratification (Shukhman & Churilov 1997). It is shown that while in the viscous critical layer regime the relaxation of assumptions of the symmetry and weak supercriticality of the flow produces no fundamental changes in the theory, in the unsteady critical layer regime a new (non-dissipative cubic) nonlinearity appears which governs the instability development on equal terms with two already known nonlinearities. Results are illustrated by calculations for two families of flow models with a controlled degree of asymmetry.

Nonlinear Evolution of Singular Disturbances to the Bickley Jet

We consider the nonlinear evolution of a disturbance to the Bickley jet, and the critical layer for the disturbance is located very close to the nose of the jet rather than at the inflection points. Using a nonlinear critical layer analysis, equations governing the evolution of the disturbance are derived and discussed. When the critical layer is located exactly at the nose of the jet, we find that the disturbance cannot exist on a linear basis, even with weak viscosity present, but that nonlinear effects inside the critical layer do permit the disturbance to exist if both modes are present. However, when the phase velocity of the disturbance is perturbed sufficiently away from unity, so that we have a pair of critical layers slightly above and below the nose rather than a single critical layer, we find that the waves can exist on a linear basis, and again we derive equations governing the nonlinear evolution of the disturbance.

Interactions between pairs of oblique waves in a Bickley jet

Abstract We consider the weakly nonlinear spatial evolution of a pair of varicose oblique waves and a pair of sinuous oblique waves superimposed on an inviscid Bickley jet, with each wave being slightly amplified on a linear basis. The two pairs are assumed to both be inclined at the same angle to the plane of the jet. A nonlinear critical layer analysis is employed to derive equations governing the evolution of the instability wave amplitudes, which contain a coupling between the modes. These equations are discussed and solved numerically, and it is shown that, as in related work for other flows, these equations may develop a singularity at a finite distance downstream.

Nonlinear evolution of Rayleigh waves in an initial value context: non‐symmetric input and cross‐flow

In recent papers the present authors considered the effects of small cross-flow on the evolution of two unequal oblique waves. In these studies the relative size of the crossflow meant that a diffusion (or buffer) layer was required around the critical layer to smooth out the algebraic growth in the mean-flow distortion generated by the nonlinear critical-layer interactions. The present analysis increases the cross-flow to an order of magnitude such that the buffer and critical layers coalesce. In this instance the nonlinear critical layer contains viscous as well as nonequilibrium effects. The resulting amplitude equations are solved for perturbations initiated at a fixed station in the flow.

Interactions between a pair of helical waves on an axisymmetric jet

Using a nonlinear critical layer analysis, we derive a nonlinear evolution equation governing the spatial growth of a pair of helical instability waves that are superimposed on an inviscid cylindrical jet. We consider the stage of evolution in which the growth first becomes nonlinear, with the nonlinearity appearing inside the critical layer. This evolution equation is of the same form as that for a pair of oblique waves superimposed on a shear layer (Goldstein and Choi, 1989. J. Fluid Mech. 207, 97–120), namely an integro-differential equation with a cubic nonlinearity. As shown in related work for other flows, these equations may develop a singularity at a finite distance downstream.

A nonlinear critical layer generated by the interaction of free Rossby waves

Two free waves propagating in a parallel shear flow generate a critical layer when their nonlinear interaction induces a perturbation whose phase velocity matches the basic-state velocity somewhere in the flow domain. The condition necessary for this to occur may be interpreted as a resonance condition for a triad formed by the two waves and a (singular) mode of the continuous spectrum associated with the shear. The formation of the critical layer is investigated in the case of freely propagating Rossby waves in a two-dimensional inviscid flow in a β-channel.A weakly nonlinear analysis based on a normal-mode expansion in terms of Rossby waves and modes of the continuous spectrum is developed; it leads to a system of amplitude equations describing the evolution of the two Rossby waves and of the modes of the continuous spectrum excited during the interaction. The assumption of weak nonlinearity is not however self-consistent: it breaks down because nonlinearity always becomes strong within the critical layer, however small the initial amplitudes of the Rossby waves. This demonstrates the relevance of nonlinear critical layers to monotonic, stable, unforced shear flows which sustain wave propagation.A nonlinear critical-layer theory is developed that is analogous to the well-known theory for forced critical layers. Differences arise because of the presence of the Rossby waves: the vorticity in the critical layer is advected in the cross-stream direction by the oscillatory velocity field due to the Rossby waves. An equation is derived which governs the modification of the Rossby waves that results from their interaction; it indicates that the two Rossby waves are undisturbed at leading order. An analogue of the Stewartson–Warn–Warn analytical solution is also considered.