Speaker-wire vortices in stratified anabatic Prandtl slope flows and their secondary instabilities

Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows due to the dominance of the normal component of the buoyancy force over flow shear. Here, we further identify self pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha=3^{\circ}$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps explain the relative stability of a single vortex pair compared to the vortex dynamics in presence of two or an even number of pairs.These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via direct numerical simulations with which we are able to track the nonlinear temporal evolution of these instabilities

Impact of stratification mechanisms on turbulent characteristics of stable flows over flat surfaces

Flow over a surface can be stratified by imposing a fixed mean vertical temperature (density) gradient profile throughout or via cooling at the surface. These two distinct mechanisms can act simultaneously as well to establish a stable stratification in a flow. Here, we perform a series of direct numerical simulations of open channel flows to study adaptation of a neutrally stratified turbulent flow under the combined or independent action of the aforementioned stratification mechanisms. When both stratification mechanisms are active, the dimensionless stratification perturbation number enters the picture as an external flow control parameter, in addition to the Reynolds, Froude, and Prandtl numbers. Additionally, we force the fully developed flow with constant mass flow rate. This alternative way of forcing the flow enables us to keep the bulk Reynolds number constant throughout our investigation and avoid complications arising from the acceleration of the bulk flow when a constant pressure gradient approach to drive the flow were to be adopted instead. We demonstrate that significant deviations from the original Monin-Obukhov similarity formulation are possible when both stratification mechanisms are active within an otherwise weakly stable flow with contiguous turbulence, even when the flux Richardson number is well below 0.2.Independent of active stratification mechanisms, the degree of deviation from neutral dimensionless shear as a function of the vertical coordinate emerges as a good measure for the strength of stable stratification for the six different cases investigated in this study. An extended version of the Monin-Obukhov similarity also shows promise.Comment: submitted to the Journal of the Atmospheric Science

Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the $n$-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.Comment: 12 pages, 0 figur

Asymmetric nested pitchfork bifurcation in stratified anabatic flows in idealized valleys

We characterize the full structure of steady laminar anabatic flows in a stably stratified V-shaped valley using a dynamical systems approach. Our approach is based on the discovery of a quiescent conduction state from which a unique asymmetric nested pitchfork bifurcation emerges. We characterize the flow via the stratification perturbation parameter, $\Pi_s$, which is a measure of the surface heat flux relative to the strength of the background stable stratification. At very low $\Pi_s$ values, the pure conduction state remains stable. Beyond a threshold $\Pi_s$ value, it bifurcates into asymmetric and symmetric circulation patterns, with the critical value for the asymmetric state being slightly lower than that of the symmetric state. The asymmetric instability manifests as a perfect mirror image of a clockwise and counterclockwise circulation in the valley. The symmetric instability gives rise to an upslope and a downslope convection patterns which are not mirror images of each other. Linear modal analysis and numerical simulations show that these two symmetric states are linearly unstable and will transition to the asymmetric state under the slightest perturbation.Comment: 10 pages, 6 figure

Statistical modeling of texture sketch

Abstract. Recent results on sparse coding and independent component analysis suggest that human vision first represents a visual image by a linear superposition of a relatively small number of localized, elongate, oriented image bases. With this representation, the sketch of an image consists of the locations, orientations, and elongations of the image bases, and the sketch can be visually illustrated by depicting each image base by a linelet of the same length and orientation. Built on the insight of sparse and independent component analysis, we propose a two-level generative model for textures. At the bottom-level, the texture image is represented by a linear superposition of image bases. At the top-level, a Markov model is assumed for the placement of the image bases or the sketch, and the model is characterized by a set of simple geometrical feature statistics