Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

Occurrence and transition probabilities of omega and high-over-low blocking in the Euro-Atlantic region

Stationary, long-lasting blocked weather patterns can lead to extreme conditions such as anomalously high temperatures or heavy rainfall. The exact locations of such extremes depend on the location of the vortices that form the block. There are two main types of blocking: (i) a high-over-low block with a high located poleward of an isolated low and (ii) an omega block with two lows that lie southeast and southwest of the blocking high in the Northern Hemisphere. In this work, we refine a novel method based on the kinematic vorticity number and the point vortex theory that allows us to distinguish between these two blocking types. Based on the National Centers for Environmental Prediction–Department of Energy (NCEP–DOE) Reanalysis 2 data, we study the trends of the occurrence probability and the onset (formation), decay (offset) and transition probabilities of high-over-low and omega blocking in the 30-year period from 1990 to 2019 in the Northern Hemisphere (90∘ W–90∘ E) and in the Euro-Atlantic sector (40∘ W–30∘ E). First, we use logistic regression to investigate long-term changes in blocking probabilities for full years, seasons and months. While trends are small for annual values, changes in occurrence probability are more visible and also more diverse when broken down to seasonal and monthly resolution, showing a prominent increase in February and March and a decrease in December. A three-state multinomial regression describing the occurrence of omega and high-over-low blocking reveals different trends for both types. Particularly the February and December changes are dominated by the omega blocking type. Additionally, we use Markov models to describe transition probabilities for a two-state (unblocked, blocked) and a three-state (unblocked, omega block, high-over-low block) Markov model. We find the largest changes in transition probabilities in the summer season, where the transition probabilities towards omega blocks significantly increase, while the unblocked state becomes less probable. Prominent in winter are decreasing probabilities for transitions from omega to high-over-low and persistence of the latter. Moreover, we show that omega blocking is more likely to occur and to be more persistent than the high-over-low blocking pattern.

A SEMI-ANALYTICAL METHOD FOR A FLOW OF NON-VISCOUS FLUID AROUND AN AIRFOIL WITH A SHARP TRAILING EDGE

In the present work we propose a method to study a two-dimensional flow of non-viscous fluid around an airfoil with a sharp trailing edge, by the double-layer potential theory. The circulation of velocity vector is modeled by the potential of a point vortex whose center is located inside the boundary contour. The magnitude of the circulation is defined on the basis of the Joukowski-Chaplygin postulate. There are presented some results for a Joukowski rudde, as well as for the airfoil in the form of a pair of interacting circles. It is performed a comparison of the circulation with its theoretical value.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such “rotors” are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either ideal vortices in an inviscid fluid, or driven rotors in a viscous membrane, spontaneously self assembles. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in distinct structural states. We find that the rotationally invariant interactions isotropically suppress long wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.

The control of the displacement of a passive particle in a point vortex flow

This work reports numerical explorations in advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z0 to the final destination zf, and (iii) the maximum absolute value umax that is imposed on the control variables. The latter consist in staircase controls, i.e., the control is written as a finite linear combination of characteristic functions on the real interval. The resulting optimization problems are solved numerically. The numerical results shows the existence nearly/quasi optimal control for the cases of N=1, N=2, N=3, and N=4 vortices.