Conditions For Absolute Instability Research Articles

African easterly waves in an idealized general circulation model: instability and wave packet diagnostics

Abstract. We examine the group dynamic of African easterly waves (AEWs) generated in a realistic, spatially non-homogeneous African easterly jet (AEJ) using an idealized general circulation model. Our objective is to investigate whether the limited zonal extent of the AEJ is an impediment to AEW development. We construct a series of basic states using global reanalysis fields and initialize waves via transient heating over West Africa. The dominant response is a localized, near-stationary wave packet that disperses upstream and downstream. The inclusion of a crude representation of boundary layer damping stabilizes the waves in most cases, consistent with other studies in the past. In some basic states, however, exponential growth occurs even in the presence of damping. This shows that AEWs can occasionally emerge spontaneously. The key result is that, whether triggered by an external forcing or generated internally, the wave packet can remain within the AEJ for multiple wave periods instead of being swept away. Drawing from other studies, this also suggests that even the damped waves can grow if coupled with additional sources of energy such as moist convection and dust radiative feedback. The wave packet in the localized AEJ appears to satisfy a condition for absolute instability, a form of spatial hydrodynamic instability. However, this needs to be verified more rigorously. We conclude that the limited zonal extent of the AEJ is not an impediment. Our results also suggest that the intermittent nature of AEWs is mediated, not by transitions between convective and absolute instability, but likely by external sources such as propagating equatorial wave modes.

Acoustic modes in jet and wake stability

Motivated by recent studies that have revealed the existence of trapped acoustic waves in subsonic jets (Towneet al.,J. Fluid Mech., vol. 825, 2017, pp. 1113–1152), we undertake a more general exploration of the physics associated with acoustic modes in jets and wakes, using a double vortex-sheet model. These acoustic modes are associated with eigenvalues of the vortex-sheet dispersion relation; they are discrete modes, guided by the vortex sheet; they may be either propagative or evanescent; and under certain conditions they behave in the manner of acoustic-duct modes. By analysing these modes we show how jets and wakes may both behave as waveguides under certain conditions, emulating ducts with soft or hard walls, with the vortex-sheet impedance providing effective ‘wall’ conditions. We consider, in particular, the role that upstream-travelling acoustic modes play in the dispersion-relation saddle points that underpin the onset of absolute instability. The analysis illustrates how departure from duct-like behaviour is a necessary condition for absolute instability, and this provides a new perspective on the stabilising and destabilising effects of reverse flow, temperature ratio and compressibility; it also clarifies the differing symmetries of jet (symmetric) and wake (antisymmetric) instabilities. An energy balance, based on the vortex-sheet impedance, is used to determine stability conditions for the acoustic modes: these may become unstable in supersonic flow due to an energy influx through the shear layers. Finally, we construct the impulse response of flows with zero and finite shear-layer thickness. This allows us to show how the long-time wavepacket behaviour is indeed determined by interaction between Kelvin–Helmholtz and acoustic modes.

Spatial modes in one-dimensional models for capillary jets.

One-dimensional (1D) models are widely employed to simplify the analysis of axisymmetric capillary jets. These models postulate that, for slender deformations of the free surface, the radial profile of the axial velocity can be approximated as uniform (viscous slice, averaged, and Cosserat models) or parabolic (parabolic model). In classical works on spatial stability analysis with 1D models, considerable misinterpretation was generated about the modes yielded by each model. The already existing physical analysis of three-dimensional (3D) axisymmetric spatial modes enables us to relate these 1D spatial modes to the exact 3D counterparts. To do so, we address the surface stimulation problem, which can be treated as linear, by considering the effect of normal and tangential stresses to perturb the jet. A Green's function for a spatially local stimulation having a harmonic time dependence provides the general formalism to describe any time-periodic stimulation. The Green's function of this signaling problem is known to be a superposition of the spatial modes, but in fact these modes are of fundamental nature, i.e., not restricted to the surface stimulation problem. The smallness of the wave number associated with each mode is the criterion to validate or invalidate the 1D approaches. The proposed axial-velocity profiles (planar or parabolic) also have a remarkable influence on the outcomes of each 1D model. We also compare with the classical 3D results for (i) conditions for absolute instability, and (ii) the amplitude of the unstable mode resulting from both normal and tangential surface stress stimulation. Incidentally, as a previous task, we need to re-deduce 1D models in order to include eventual stresses of various possible origins (electrohydrodynamic, thermocapillary, etc.) applied on the free surface, which were not considered in the previous general formulations.

Two‐fluid spatial instability with velocity defect in the basic flow

AbstractThe initial state of liquid atomization by a fast gas stream is considered by viscous linear spatial stability analysis for parallel two‐fluid flow. The unbounded basic velocity profile is characterized by boundary layers near the interface and different asymptotic velocities. By computing generalized spatial branches we identify conditions for absolute instability when a velocity deficit is introduced to account for the conditions near the nozzle. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ondes variqueuses absolument instables dans un canal élastique

Conditions for absolute instability of the elastic walls of a plane channel with potential flow are investigated. The absolute instability threshold is given in closed form in the thin film approximation for walls made of plates or membranes. The more general case is solved numerically and is compared with more classical results for a single wall in the presence of a semi-infinite fluid domain. It is found that absolute instability is strongly influenced by the membrane or plate nature of the wall stiffness.

Note on the Kelvin–Helmholtz instability of stratified fluids

In this paper the conditions under which the Kelvin–Helmholtz instability in stratified fluids is absolute or convective are investigated. It is first shown that at transition from convective to absolute instability two double roots of the dispersion relation coalesce. Based on this property an analytical condition for absolute instability is derived. It is found that the instability is absolute for almost all values of the flow velocity and stratification. Convective instability is found only for a narrow range of flow velocities over the instability threshold when the density ratio parameter, r=ρ1/(ρ1+ρ2), lies in the range 1/3<r<1/2. The different behavior near the instability threshold can be related to the signs of the group velocities of the two waves which coalesce to create the instability: For r<1/3, the group velocities of the two waves have opposite signs, and the resulting instability is absolute, whereas for 1/3<r<1/2, the two waves have group velocities with the same sign, and the instability is convective. This result is also shown to be reflected in the form of the amplitude equation at the instability threshold, which is the linearly unstable Klein–Gordon equation.

Physical condition for absolute instability in inviscid hydroelastic coupling

In this paper the instability of infinitely long cylindrical structures submerged in a uniform inviscid flow is investigated. When the flow velocity is zero, two families of hydroelastic waves exist, with phase velocities that are equal in magnitude and have opposite signs. Positive values of the fluid velocity increase the magnitude of the phase velocity of the waves that propagate in the same direction as the flow, and decrease the magnitude of the phase velocity of the waves propagating opposite to the flow. It is shown by constructing an energy conservation law for the linearized equations of motion that both types of waves have initially positive energy, in the sense that they both require an energy source to be sustained. After a critical value of fluid velocity is exceeded the slow waves acquire negative energy, in the sense that they require an energy sink to be sustained. Coalescence between the ‘‘fast’’ (positive energy) and the ‘‘slow’’ (negative energy) waves creates waves with zero energy which do not require neither an energy source nor a sink to be sustained, and grow exponentially in time. It is further shown that the front end of any initially localized disturbance will propagate with a speed equal to the minimum group velocity of the fast waves and the back end will propagate with speed equal to the maximum group velocity of the slow waves. Consequently, if those two characteristic speeds have opposite signs, the instability is absolute, whereas if they have the same sign the instability is convective. It is shown that this simple physical criterion applies to a more general class of instabilities that involve the interaction of positive and negative energy waves.

Conditions for absolute instability in the cyclotron resonance maser

In this paper exact analytic conditions for absolute instability in the cyclotron resonance maser are presented for the case of a cold-beam, lossless circular waveguide and infinite interaction length in the axial direction. The conditions are expressed in terms of the parallel beam velocity, the applied magnetic field strength, and the strength of the coupling between the beam and waveguide modes. The results are applicable to both the gyrotron and cyclotron autoresonance maser operating regimes.

Stimulated Brillouin scattering in a field of opposite light waves

A theoretical and experimental investigation is made of the conditions for absolute instability in stimulated Brillouin scattering of two opposite optical beams. It is shown that the presence of an opposite pump wave with an optimal wave mismatch can reduce the stimulated Brillouin scattering threshold severalfold.

Convective and Absolute Instabilities in Semiconductors Exhibiting Negative Differential Mobility

It is shown that, in a semiconductor exhibiting negative differential mobility, the coupling between diffusion effects and the tendency for space-charge accumulation can lead to temporal growth rather than spatial amplification if the dielectric relaxation frequency exceeds a certain threshold value. This conclusion results from an analysis of the dispersion relation for longitudinal waves in the semiconductor. The criterion for temporal growth is a condition for absolute instability, as opposed to convective instability, which would indicate rather spatial amplification.