Absolute instability in shock-containing jets

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

Counting equilibria of large complex systems by instability index

We consider a nonlinear autonomous system of N≫1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.

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

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.

Convective and absolute instability of falling viscoelastic liquid jets surrounded by a gas

We examine the convective and absolute instability of a 2D axisymmetric viscoelastic liquid jet falling vertically in a medium of an inviscid gas under the influence of gravity. We use the upper-convected Maxwell model to describe the viscoelastic liquid jet and together with an asymptotic approach, based on the slenderness of the jet, we obtain steady-state solutions. By considering travelling wave modes, and using linear instability analysis, the dispersion relation, relating the frequency to wavenumber of disturbances, is derived. We solve this dispersion relation numerically using the Newton–Raphson method and explore regions of instability in parameter space. In particular, we investigate the influence of gravity, the effect of changing the gas-to-liquid density ratio, the Weber number and the Deborah number on convective and absolute instability. In this paper, we utilize a mapping technique developed by Afzaal (2014, Breakup and instability analysis of compound liquid jets. Doctoral Dissertation, University of Birmingham) to find the cusp point in the complex frequency plane and its corresponding first-order saddle point (the pinch point) in the complex wavenumber plane for absolute instability. The convective/absolute instability boundary is identified for various parameter regimes along the axial length of the jet.

African Easterly Waves in an Idealized General Circulation Model: Instability and Wavepacket Diagnostics

We examine the group dynamic of African easterly waves (AEW) 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 wavepacket that disperses upstream and downstream. The inclusion of a crude representation of boundary layer damping stabilizes the waves in most cases. 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 the wavepacket in almost all cases remains within the AEJ 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 wavepacket 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. 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.