On the algebraic perturbations in atmospheric boundary layer

A simple model for the development of submesoscale perturbations in the atmospheric boundary layer (ABL) is proposed. The growth of perturbations is associated with the shear algebraic instability of the wind velocity profile in the atmospheric boundary layer (ABL). For the scales of optimal perturbations (streaks) in the lower part of the ABL, estimates of their sizes were obtained about 100-200 m vertically and 300-600 m horizontally. Similar scales are noted for experimental data on the structure of the wind field in the lower part of the ABL, obtained in 2017, 2018 in the summer at the Tsimlyansk Scientific Station at the acoustic sounding of the atmosphere by the Doppler three-component minisodar of high resolution.

Preferential concentration driven instability of sheared gas–solid suspensions

We examine the linear stability of a homogeneous gas–solid suspension of small Stokes number particles, with a moderate mass loading, subject to a simple shear flow. The modulation of the gravitational force exerted on the suspension, due to preferential concentration of particles in regions of low vorticity, in response to an imposed velocity perturbation, can lead to an algebraic instability. Since the fastest growing modes have wavelengths small compared with the characteristic length scale ($U_{g}/{\it\Gamma}$) and oscillate with frequencies large compared with ${\it\Gamma}$, $U_{g}$ being the settling velocity and ${\it\Gamma}$ the shear rate, we apply the WKB method, a multiple scale technique. This analysis reveals the existence of a number density mode which travels due to the settling of the particles and a momentum mode which travels due to the cross-streamline momentum transport caused by settling. These modes are coupled at a turning point which occurs when the wavevector is nearly horizontal and the most amplified perturbations are those in which a momentum wave upstream of the turning point creates a downstream number density wave. The particle number density perturbations reach a finite, but large amplitude that persists after the wave becomes aligned with the velocity gradient. The growth of the amplitude of particle concentration and fluid velocity disturbances is characterised as a function of the wavenumber and Reynolds number ($\mathit{Re}=U_{g}^{2}/{\it\Gamma}{\it\nu}$) using both asymptotic theory and a numerical solution of the linearised equations.

An inviscid modal interpretation of the ‘lift-up’ effect

In this paper, we give a modal interpretation of the lift-up effect, one of two well-known mechanisms that lead to an algebraic instability in parallel shearing flows, the other being the Orr mechanism. To this end, we first obtain the two families of continuous spectrum modes that make up the complete spectrum for a non-inflectional velocity profile. One of these families consists of modified versions of the vortex-sheet eigenmodes originally found by Case (Phys. Fluids, vol. 3, 1960, pp. 143–148) for plane Couette flow, while the second family consists of singular jet modes first found by Sazonov (Izv. Acad. Nauk SSSR Atmos. Ocean. Phys., vol. 32, 1996, pp. 21–28), again for Couette flow. The two families are used to construct the modal superposition for an arbitrary three-dimensional distribution of vorticity at the initial instant. The so-called non-modal growth that underlies the lift-up effect is associated with an initial condition consisting of rolls, aligned with the streamwise direction, and with a spanwise modulation (that is, a modulation along the vorticity direction of the base-state shearing flow). This growth is shown to arise from an appropriate superposition of the aforementioned continuous spectrum mode families. The modal superposition is then generalized to an inflectional velocity profile by including additional discrete modes associated with the inflection points. Finally, the non-trivial connection between an inviscid eigenmode and the viscous eigenmodes for large but finite Reynolds number, and the relation between the corresponding modal superpositions, is highlighted.

Degenerate continuous spectra producing localized secular instability – an example in a non-neutral plasma

Fluctuations in ambient shear flow exhibit interesting transient phenomena. Shear flow produces not only Kelvin–Helmholtz modes (global exponential instabilities represented by point spectra) but also local algebraic instabilities associated with multiple continuous spectra. Since the generating operator is non-Hermitian, the orthogonality of eigenmodes is broken, and unresolvable mode couplings (resonances) bring about secular behavior (algebraic instability). We analyze electrostatic fluctuations in a magnetized non-neutral (single species) plasma where the electrostatic potential parallels the stream function. This secular behavior is reproduced by solving the initial value problem with a renormalization method.

Time-Dependent Optimal Perturbations for the Algebraic Instability in the Nonlinear Regime

The aim of the present study is to extend the linear unsteady optimal-perturbation analysis of (Luchini 2000) to the nonlinear regime. In order to account for the nonlinear interactions, a Fourier expansion is applied in the streamwise direction and in time and the solution is decomposed in Fourier modes along both z and t. The optimal unsteady spanwise-sinusoidal leading-edge excitation that provides the maximum energy growth for a given initial energy and frequency can thus be determined. Of interest will be that the optimal growth decreases with both.

Algebraic Growth and Streak Formation in Shear Flows

By examination of the long-term behavior of an initial three-dimensional and localized disturbance in an inflection-free shear flow a detailed study of the algebraic instability mechanism of an inviscid shear flow (Landahl, 1980) is carried out. It is shown that the vertical velocity component will tend to zero at least as fast as 1/t whereas, as a result of a nonzero liftup of the fluid elements, the streamwise disturbence velocity component will tend to a limiting finite value in a convected frame of reference. For an initial disturbence having a nonzero net vertical momentum along a streamline, the streamwise dimension of the disturbed region is found to grow indefinitely at a rate set by the difference between the maximum and minimum velocities in the parallel flow. The total kinetic energy of the disturbence therefore grows linearly in time through the formation of continuously elongating high-speed or low-speed regions. In these, internal shear layers are formed that intensify through the mechanism of spanwise stretching of the mean vorticity. The effect of a small viscosity is felt primarily in the shear layers so as to make them diffuse and eventually cause the disturbence to decay on a viscous time scale. For the streaky structures near a wall the horizontal pressure gradients are found to be small, making possible a simple approximate treatment of nonlinearty. Such an analysis suggests the possibility of the appearance of a rapid outflow event (“bursting”) from the wall that may occur at a finite time inversely proportional to the amplitude of the initial disturbance. On basis of the analysis presented it is proposed that algebraic growth is the primary mechanism for the formation of streaks in laminar and turbulent shear flows.