The Okubo-Weiss criterion in hydrodynamic flows:Geometric aspects and further extension

The Okubo [5]-Weiss [6] criterion has been extensively used as a diagnostic tool to divide a two-dimensional (2D) hydrodynamical flow field into hyperbolic and elliptic regions and to serve as a useful qualitative guide to the complex quantitative criteria. The Okubo-Weiss criterion is frequently validated on empirical grounds by the results ensuing its application. So, we will explore topological implications into the Okubo-Weiss criterion and show the Okubo-Weiss parameter is, to within a positive multiplicative factor, the negative of the Gaussian curvature of the underlying vorticity manifold. The Okubo-Weiss criterion is reformulated in polar coordinates, and is validated via several examples including the Lamb- Oseen vortex, and the Burgers vortex. These developments are then extended to 2D quasi- geostrophic (QG) flows. The Okubo-Weiss parameter is shown to remain robust under the -plane approximation to the Coriolis parameter. The Okubo-Weiss criterion is shown to be able to separate the 2D flow-field into coherent elliptic structures and hyperbolic flow configurations very well via numerical simulations of quasi-stationary vortices in QG flows. An Okubo-Weiss type criterion is formulated for 3D axisymmetric flows, and is validated via application to the round Landau-Squire Laminar jet flow.

Vortices in Rotating and Gravitating Gas Disk and in a Protoplanetary Disk

Nonlinear equations describing dynamics of 2D vortices are very important in the physics of the ocean and the atmosphere and in plasma physics and Astrophysics. Here linear and nonlinear 2D vortex perturbations of gravitating and light gaseous disks are examined in the geostrophic and post-geostrophic approximations. In the frame of geostrophic approximation, it is shown that the vortex with positive velocity circulation is characterized by low pressure with negative excess mass density of substance. Vortex with negative circulation has higher pressure and is a relatively tight formation with the positive excess mass density. In the post-geostrophic approximation, structures of the isolated monopole and dipole vortex (modons) solutions of these equations are studied. Two types of mass distributions in dipole vortices are found. The first type of modon is characterized by an asymmetrically positioned single circular densification and one rarefaction. The second type is characterized by two asymmetrically positioned densifications and two rarefactions, where the second densification-rarefaction pair is crescent shaped. The constant density contours of a dipole vortex in a light gas disk coincide with the streamlines of the vortex; in a self-gravitating disk, the constant density contours in the vortex do not coincide with streamlines. Possible manifestations of monopole and dipole vortices in astrophysical objects are discussed. Vortices play decisive role in the process of planet formation. Gas in a protoplanetary disk practically moves on sub-Keplerian speeds. Rigid particles, under the action of a head wind drags, lose the angular momentum and energy. As a result, the ~10 cm to meter-sized particles drift to the central star for hundreds of years. Long-lived vortical structures in gas disk are a possible way to concentrate the ~10 cm to meter sized particles and to grow up them in planetesimal. Here the effect of anticyclonic Burgers vortex on formation of planetesimals in a protoplanetary dusty disc in local approach is also considered. It is shown that the Burgers vortex with homogeneously rotating kernel and a converging radial stream of substance can effectively accumulate in its nuclear area the meter-sized rigid particles of total mass ∼1028 g for characteristic time ∼106 year.

A Simple Approach for Vortex Core Visualization

We propose a method to visualize vortex cores based on manipulation of the pressure field produced by isolated vortices in incompressible flow. Under ideal conditions, the function D=2|∇p|/∇2p yields an approximate distance to vortex centerlines. As opposed to local methods to identify coherent structures, isosurfaces of D produce a field of vortex tubes equidistant to the vortex core center which, ideally, are independent of vortex intensity or size. In contrast to other line-vortex identification methods, which typically rely on algorithms to detect vortex core lines and frequently need complex implementations, the proposed method can be computed from the local Eulerian velocity and pressure fields as easily as vortex identification methods such as the Q and λ2 criteria. D=2|∇p|/∇2p results in the exact distance to the core center for a Rankine vortex and is in general valid for the region of a vortex where there is pure rotation, yielding an approximation to the distance farther from the core in other simple one-dimensional vortex models. The methodology performs well in all tests we attempted, though limitations are presented and discussed. The method is demonstrated for a canonical Burgers vortex, a Bodewadt vortex, homogeneous isotropic turbulent flow, the wake of a propeller, a heaving plate, and a turning containership. The proposed method helps to better visualize vortical flow fields by displaying vortex cores, complementing methods like Q and λ2 which display vortical volumes.

Turbulence-induced cloud voids: observation and interpretation

The phenomenon of “cloud voids”, i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Clustering and segregation effects in a vortex tube are discussed for reasonable cloud conditions.

Turbulence Induced Cloud Voids: Observation and Interpretation

The phenomenon of cloud voids, i.e., elongated volumes inside a cloud that are devoid of droplets, was observed with laser sheet photography in clouds at a mountain-top station. Two experimental cases, similar in turbulence conditions yet with diverse droplet size distributions and cloud void prevalence, are reported. A theoretical explanation is proposed based on the study of heavy inertial sedimenting particles inside a Burgers vortex. A general conclusion regarding void appearance is drawn from theoretical analysis. Numerical simulations of polydisperse droplet motion with realistic vortex parameters and Mie scattering visual effects accounted for can explain the presence of voids with sizes similar to that of the observed ones. Preferential concentration and sorting effects in a vortex tube are discussed for reasonable cloud conditions.

Tracer particle dispersion around elementary flow patterns

The motion of tracer particles is kinematically simulated around three elementary flow patterns; a Burgers vortex, a shear-layer structure with coincident vortices and a node-saddle topology. These patterns are representative for their broader class of coherent structures in turbulence. Therefore, examining the dispersion in these elementary structures can improve the general understanding of turbulent dispersion at short time scales. The shear-layer structure and the node-saddle topology exhibit similar pair dispersion statistics compared to the actual turbulent flow for times up to $3{-}10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, where, $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ is the Kolmogorov time scale. However, oscillations are observed for the pair dispersion in the Burgers vortex. Furthermore, all three structures exhibit Batchelor’s scaling. Richardson’s scaling was observed for initial particle pair separations $r_{0}\leqslant 4\unicode[STIX]{x1D702}$ for the shear-layer topology and the node-saddle topology and was related to the formation of the particle sheets. Moreover, the material line orientation statistics for the shear-layer and node-saddle topology are similar to the actual turbulent flow statistics, up to at least $4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$. However, only the shear-layer structure can explain the non-perpendicular preferential alignment between the material lines and the direction of the most compressive strain, as observed in actual turbulence. This behaviour is due to shear-layer vorticity, which rotates the particle sheet generated by straining motions and causes the particles to spread in the direction of compressive strain also. The material line statistics in the Burgers vortex clearly differ, due to the presence of two compressive principal straining directions as opposed to two stretching directions in the shear-layer structure and the node-saddle topology. The tetrad dispersion statistics for the shear-layer structure qualitatively capture the behaviour of the shape parameters as observed in actual turbulence. In particular, it shows the initial development towards planar shapes followed by a return to more volumetric tetrads at approximately $10\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, which is associated with the particles approaching the vortices inside the shear layer. However, a large deviation is observed in such behaviour in the node-saddle topology and the Burgers vortex. It is concluded that the results for the Burgers vortex deviated the most from actual turbulence and the node-saddle topology dispersion exhibits some similarities, but does not capture the geometrical features associated with material lines and tetrad dispersion. Finally, the dispersion around the shear-layer structure shows many quantitative (until 2–$4\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) and qualitative (until $20\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$) similarities to the actual turbulence.