Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

Bifurcation to Mean Flows in Convection

Convection with a strip plate in the middle is studied in this paper. Simultaneous instability of two convection modes of different vertical structures with a same horizontal wave number is possible in this system. It is found that the interaction of these two modes generates mean flows similar to those observed by Krishnamurti and Howard [9] in a turbulent convection experiment. Coupled nonlinear equations are derived for the amplitudes of the two modes. Traveling wave solutions and more complicated time-dependent solutions are also found near the onset of convection.     

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.     

On the Nonlinear Initial Value Problem for Vortex–Wave Interactions in Shear Flows

Small-amplitude wave systems interacting nonlinearly can produce 0(1)amplitude streamwise vortex structures through the vortex–wave interaction mechanism described, for example, by [1–3]. The key feature of the interaction is that the spanwise velocity component of a vortex is small as compared to the streamwise component so that a nonlinear wave system driving the spanwise velocity component through Reynolds stresses can provoke a 0(1) response of the vortex. The wave system can correspond to either a Rayleigh or Tollmien–Schlichting wave disturbance, but previous work on the initiation of the process has been confined to Rayleigh waves (see, for example, [5, 6]). Here, we address the nonlinear initial value problem for Tollmien–Schlichting wave–vortex interactions in channel flows. The evolution of the disturbances is accounted for using the phase equation approach of [7]. We determine the circumstances, if any, under which the finite amplitude vortex–wave equilibrium states of [4] are generated. Our discussion of the nonlinear evolution of a wave system points toward a possible mechanism for the experimentally observed breakup of three-dimensional instabilities into shorter streamwise scales.     

Finite-amplitude Rotational Waves in Viscous Shear Flows

Growing finite-amplitude initially spanwise-independent two-dimensional rotational waves and their nonlinear interaction with unidirectional viscous shear flows of various strengths are considered. Both primary and secondary instabilities are studied, but only secondary instabilities are permitted to vary in the spanwise direction. A generalized Lagrangian-mean formulation is employed to describe wave-mean interactions, and a separate theory is constructed to account for the back effect of the developing mean flow on the wave field. Viscosity is seen to significantly complicate calculation of the back effect. The primary instability is seen to act as a platform for, and catalyst to, secondary instabilities. The analysis leads to an eigenvalue problem for the initial growth of the secondary instability, this being a generalization of the eigenvalue problem constructed by Craik for inviscid neutral waves. Two inviscid secondary instability mechanisms to longitudinal vortex form are observed: the first has as its basis the Craik–Leibovich type 2 mechanism. The second, which is as yet unproven, requires that both the wave and flow field distort in concert at all levels of shear. Both mechanisms excite exponential growth on a convective rather than diffusive scale in the presence of neutral waves, but growing waves alter that growth rate.     

A Family of Wave-Mean Shear Interactions and Their Instability to Longitudinal Vortex Form

The inviscid instability of O(ε) two-dimensional periodic flows to spanwiseperiodic longitudinal vortex modes in parallel O(1) shear flows of the form ū = ± |z|^q is considered. Here the mean velocity ū is relative to the wave and q is a constant. Such shear flows admit neutral Rayleigh waves with amplitudes that either diminish or diverge with |αz|; both are considered. Of particular interest are streamwise α and spanwise l wavenumbers in the range l² ? α², α = O(1), as it is here that the most analytical progress can be made. A generalized Lagrangian-mean formulation is used to describe the effect of fluctuations upon the mean state and, because the developing mean flow acts to distort the waves, a further equation, the Rayleigh-Craik equation, is employed to complete the specification. It is shown that instability to longitudinal vortex form is likely for both classes of waves in many physically interesting situations, from simple mixing layers to atmospheric boundary layers over undulating surfaces.     

Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

Critical Layers m Parallel Flows

Finite amplitude neutrally stable two-dimensional disturbances in parallel flows are determined for large Reynolds numbers when both nonlinearity and viscosity are important in the critical layer. The phase shift across the critical layer depends on the local vertical Reynolds number in the critical layer, and it varies monotonically between the value zero of the nonlinear theory and — π of the viscous theory. An O(?^½) distortion of both the mean and fundamental harmonic is shown to be necessary. The eigenvalue problem is solved for long waves yielding neutral modes which link the two previous theories. In particular certain difficulties of the nonlinear theory are resolved.     

Resonant instability and nonlinear wave interactions in electrically forced jets

We investigate the problem of linear temporal instability of the modes that satisfy the dyad resonance conditions and the associated nonlinear wave interactions in jets driven by either a constant or a variable external electric field. A mathematical model, which is developed and used for the temporally growing modes with resonance and their nonlinear wave interactions in electrically driven jet flows, leads to equations for the unknown amplitudes of such waves. These equations are solved for both water and glycerol jet cases, and the expressions for the dependent variables of the corresponding modes are determined. The results of the generated data for these dependent variables versus time indicate, in particular, that the instability resulted from the nonlinear interactions of such modes is mostly quite strong but can also lead to significant reduction in the jet radius.     

Inhibiting Shear Instability Induced by Large Amplitude Internal Solitary Waves in Two-Layer Flows with a Free Surface

We consider a strongly nonlinear long wave model for large amplitude internal waves in two-layer flows with the top free surface. It is shown that the model suffers from the Kelvin–Helmholtz (KH) instability so that any given shear (even if arbitrarily small) between the layers makes short waves unstable. Because a jump in tangential velocity is induced when the interface is deformed, the applicability of the model to describe the dynamics of internal waves is expected to remain rather limited. To overcome this major difficulty, the model is written in terms of the horizontal velocities at the bottom and the interface, instead of the depth-averaged velocities, which makes the system linearly stable for perturbations of arbitrary wavelengths as long as the shear does not exceed a certain critical value.     

Exact Free Surface Flows for Shallow Water Equations I: The Incompressible Case

A comprehensive exact treatment of free surface flows governed by shallow water equations (in sigma variables) is given. Several new families of exact solutions of the governing PDEs are found and are shown to embed the well-known self-similar or traveling wave solutions which themselves are governed by reduced ODEs. The classes of solutions found here are explicit in contrast to those found earlier in an implicit form. The height of the free surface for each family of solutions is found explicitly. For the traveling or simple wave, the free surface is governed by a nonlinear wave equation, but is arbitrary otherwise. For other types of solutions, the height of the free surface is constant either on lines of constant acceleration or on lines of constant speed; in another case, the free surface is a horizontal plane while the flow underneath is a sine wave. The existence of simple waves on shear flows is analytically proved. The interaction of large amplitude progressive waves with shear flow is also studied.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Zonal Jet Creation from Secondary Instability of Drift Waves for Plasma Edge Turbulence

A new strategy is presented to explain the creation and persistence of zonal flows widely observed in plasma edge turbulence. The core physics in the edge regime of the magnetic-fusion tokamaks can be described qualitatively by the one-state modified Hasegawa-Mima (MHM for short) model, which creates enhanced zonal flows and more physically relevant features in comparison with the familiar Charney-Hasegawa-Mima (CHM for short) model for both plasma and geophysical flows. The generation mechanism of zonal jets is displayed from the secondary instability analysis via nonlinear interactions with a background base state. Strong exponential growth in the zonal modes is induced due to a non-zonal drift wave base state in the MHM model, while stabilizing damping effect is shown with a zonal flow base state. Together with the selective decay effect from the dissipation, the secondary instability offers a complete characterization of the convergence process to the purely zonal structure. Direct numerical simulations with and without dissipation are carried out to confirm the instability theory. It shows clearly the emergence of a dominant zonal flow from pure non-zonal drift waves with small perturbation in the initial configuration. In comparison, the CHM model does not create instability in the zonal modes and usually converges to homogeneous turbulence.     

Nonlinearly Coupled KdV Equations Describing the Interaction of Equatorial and Midlatitude Rossby Waves

Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions （teleconnections） between the tropical and midlatitude troposphere. An overview of that derivation is nonlinear wave theory, but not in atmospheric presented and geared to readers versed in sciences. In the course of the derivation, two other sets of asymptotic equations are presented： the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show （i） the interaction of two solitons, one from either mode, （ii） and （iii） the interaction of a soliton in the presence of different mean wind shears, and （iv） the interaction of two solitons mediated by the presence of a mean wind shear.     

A New Class of Nonlinear Waves in Parallel Flows

Waves in parallel shear flows are found to have different characteristics depending on whether nonlinear or viscous effects dominate near the critical layer. In this paper a nonlinear theory is developed which gives rise to a class of disturbances not found in the classical viscous theory. It is suggested that the modes found from such an analysis may be of importance in the breakdown of laminar flow due to free stream disturbances.     

A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves

Summary We study a model equation describing the temporal evolution of nonlinear finite-amplitude waves on a density front in a rotating fluid. The linear spectrum includes an unstable interval where exponential growth of the amplitude is expected. It is shown that the length scale of the waves in the nonlinear situation is determined by the linear instabilities; the effect of the nonlinearities is to limit the amplitude's growth, leaving the wavelength unchanged. When linearly stable waves are prescribed as initial data, a short interval of rapid decrease in amplitude is encountered first, followed by a transfer of energy to the unstable part of the spectrum, where the fastest growing mode starts to dominate. A localized disturbance is broken up into its Fourier components, the linearly unstable modes grow at the expense of all other modes, and final amplitudes are determined by the nonlinear term. Periodic evolution of linearly unstable waves in the nonlinear situation is also observed. Based on the numerical results, the existence of low-order chaos in the partial differential equation governing weakly nonlinear wave evolution is conjectured.     

The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex

We consider in this paper the propagation of neutral modes along a vortex with velocity profile being the radial coordinate. In the linear stability theory governing such flows, the boundary in parameter space separating stable and unstable regions is usually comprised of modes that are singular at some value of r denoted r_c , the critical point. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications, nonlinearity is essential, but it develops that viscosity, treated here as a small perturbation, still plays a subtle role. After first presenting the scaling for the general case, we formulate a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small. Solutions are found having no phase change across the critical layer thus permitting the existence of modes not possible in a linear theory. It is found that both the axial and azimuthal mean vorticity are different on either side of the critical layer as a result of the wave–mean flow interaction. A long wave analysis with O (1) vorticity leads to similar conclusions.     

The Evolution in Space and Time of Nonlinear Waves in Parallel Shear Flows

It is shown, using a quite general formulation, that the amplitude evolution equation for slowly varying finite amplitude waves is usually first order in both space and time. One advantage of the present formulation is that it becomes possible to easily identify, from their linear eigensolutions, interesting exceptional cases in which the amplitude evolves according to a partial differential equation that is second order in either space or time. The theory is applied to a number of specific problems, including flows with broken line profiles, and inviscid shear flows having nonlinear critical layers.     

Weakly 3D Transonic Flows in Narrow Channels

Steady transonic flows through channels so slender that the classical boundary layer approach fails are considered. The resulting viscous inviscid interaction problem for weakly three-dimensional laminar flows is formulated for perfect gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes two-dimensional in the leading order approximation. The behavior of the flow upstream of a surface mounted three-dimensional obstacle will be demonstrated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Stabilizing the Strongly Nonlinear Internal Wave Model

A strongly nonlinear asymptotic model describing the evolution of large amplitude internal waves in a two-layer system is studied numerically. While the steady model has been demonstrated to capture correctly the characteristics of large amplitude internal solitary waves, a local stability analysis shows that the time-dependent inviscid model suffers from the Kelvin–Helmholtz instability due to a tangential velocity discontinuity across the interface accompanied by the interfacial deformation. An attempt to represent the viscous effect that is missing in the model is made with eddy viscosity, but this simple ad hoc model is shown to fail to suppress unstable short waves. Alternatively, when a smooth low-pass Fourier filter is applied, it is found that a large amplitude internal solitary wave propagates stably without change of form, and mass and energy are conserved well. The head-on collision of two counter-propagating solitary waves is studied using the filtered strongly nonlinear model and its numerical solution is compared with the weakly nonlinear asymptotic solution.