Устойчивость правильных вихревых многоугольников в конденсате Бозе-Эйнштейна

Рассматривается задача об устойчивости вращающихся правильных вихревых $N$-угольников (томсоновских конфигурации) в конденсате Бозе-Эйнштейна в гармонической ловушке. Получена зависимость скорости вращения $\omega$ томсоновской конфигурации вокруг центра ловушки в зависимости от количества вихрей $N$ и радиуса конфигурации $R$. Выполнен анализ устойчивости движения таких конфигураций в линейном приближении. Для $N \leqslant 6$ построены области орбитальной устойчивости конфигураций в пространстве параметров. Показано, что вихревые $N$-угольники для $N > 6$ при любых параметрах системы неустойчивы

Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices

This paper deal with the motion of a polygonal ring of identical vortex points that are equally spaced at a line of latitude on a sphere with vortex points fixed at the both poles, which we call "N-ring". We give not only all the eigenvalues but also all the eigenvectors corresponding to them for the linearized steationary N-ring. Then, we also reduce the equations to those for a pair of two vortex points, when N is even. As a consequence of the mathematical and numerical studies of the reduced system, we obtain a transition of global periodic motions of the perturbed N-ring and the stability of these periodic motions

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Stability of Relative Equilibria in the Planar N-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.Comment: 23 pages, 3 figure

Predictions for the final equilibrium state of flows on the sphere

Fall 1999.Also issued as author's dissertation (Ph.D.) -- Colorado State University, 2000.Includes bibliographical references.Taking as motivation the experimental evidence, both observational and numerical, that un¬ forced high-Reynolds number flows have a tendency to an equilibrium state with dominant coherent structures, two theories that predict the end-state of the flow are extended to the case of a spherical domain in order to apply them to large-scale meteorological problems. The maximum entropy theory is a statistical mechanics approach that abandons the idea of following the precise changes of the fluid, predicting instead a "macroscopic" state, based on the assumption that the most probable macroscopic state is the one that corresponds to the final equilibrium state of the flow; the specification of the actual values of total kinetic energy, angular momentum and circulation of the flow define the structure of the equilibrium state. The minimum enstrophy theory is based on results of low diffusion numerical experiments, where the values of the domain integrated kinetic energy and circulation are approximately constant over time, while the value of the enstrophy (i.e., one-half the domain integrated squared vorticity) has a considerable decay with time. Using the tools of the calculus of variations, the equilibrium state of the flow is predicted by minimizing the enstrophy while keeping constant the initial value of either total kinetic energy or total angular momentum. A nonlinear barotropic non-divergent numerical model on the sphere is used to perform long­ time integrations of barotropically unstable initial conditions. The type of flows studied are northern hemisphere stratospheric polar vortices, tropical shear layers and alternating zonal jets, the last being integrated both on a rotating and non-rotating sphere. Predictions of the zonally independent equilibrium state are compared with the zonal average of the direct numerical integration after 100 days of evolution for the stratospheric polar vortex experiments. Maximum entropy theory has good predictive skill for the zonal wind and absolute vorticity profiles, as well as for the statistical distribution of traced air parcels. Minimum enstrophy theory has good skill for the zonal wind and absolute vorticity profiles, but just for the cases where mixing is confined to a polar cap, failing in cases where mixing is global or inside a latitude belt; for the latter case a second version of the minimum enstrophy theory, the two-edges problem, shows considerable improvement compared with the one edge solution. For the tropical shear layer experiments, the minimum enstrophy theory with two-edges and constant energy captures a northward displacement of the easterly wind maximum, as well as the flattening of the absolute vorticity profile in tropical regions, behavior which is consistent with the direct numerical integration. Maximum entropy theory qualitatively captures changes of the flow in the southern hemisphere but shows strong sensitivity to small variations of the scale and strength of the initial condition. Predictions from minimum enstrophy theory with two edges and constant angular momentum, with one edge and constant energy, and with one edge and constant angular momentum, show little skill predicting the end-state; the weak decay of absolute enstrophy observed in the direct numerical integration for these cases is a major factor in the predictive skill of the theory. Two-dimensional predictions of equilibrium states were found using maximum entropy theory for an initial condition with alternating zonal jets. For the rotating sphere, a maximum entropy solution was found which contains a number of zonally elongated coherent structures that resemble the direct numerical integration after 150 days of evolution. The non-rotating sphere case reveals the possibility of having more than one equilibrium state, and that the end-state chosen by the nonlinear evolution might be a linear combination of quasi-orthogonal maximum entropy states.Sponsored by the National Autonomous University of Mexico (UNAM) scholarship through the Dirección General de Asuntos del Personal Académico and by NSF under grant ATM-9729970 and by NOAA under grant NA67RJ0152

Analytical and numerical studies of the thermocapillary flow in a uniformly floating zone

The microgravity environment of an orbiting vehicle permits crystal growth experiments in the presence of greatly reduced buoyant convection in the liquid melt. Crystals grown in ground-based laboratories do not achieve their potential properties because of dopant variations caused by flow in the melt. The floating zone crystal growing system is widely used to produce crystals of silicon and other materials. However, in this system the temperature gradient on the free sidewall surface of the melt is the source of a thermocapillary flow which does not disappear in the low-gravity environment. The idea of using a uniform rotation of the floating zone system to confine the thermocapillary flow to the melt sidewall leaving the interior of the melt passive is examined. A cylinder of fluid with an axial temperature gradient imposed on the cylindrical sidewall is considered. A half zone and the linearized, axisymmetric flow in the absence of crystal growth is examined. Rotation is found to confine the linear thermocapillary flow. A simplified model is extended to a full zone and both linear and nonlinear thermocapillary flows are studied theoretically. Analytical and numerical methods are used for the linear flows and numerical methods for the nonlinear flows. It was found that the linear flows in the full zone have more complicated and thicker boundary layer structures than in the half zone, and that these flows are also confined by the rotation. However, for the simplified model considered and for realistic values for silicon, the thermocapillary flow is not linear. The fully nonlinear flow is strong and unsteady (a weak oscillation is present) and it penetrates the interior. Some non-rotating flow results are also presented. Since silicon as a large value of thermal conductivity, one would expect the temperature fields to be determined by conduction alone. This is true for the linear and weakly nonlinear flows, but for the stronger nonlinear flow the results show that temperature advection is also important. Uniform rotation may still be a means of confining the flow and the results obtained define the procedure to be used to examine this hypothesis

Determination of stable branches of relative equilibria of the NN-vortex problem on the sphere

We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with angular velocity $\omega$. Each such ring contains $m$ vortices placed at the vertices of a concentric regular polygon and we allow the presence of additional vortices at the poles. We develop a framework to prove existence and orbital stability of branches of RE of this type parametrised by $\omega$. Such framework is implemented to rigorously determine and prove stability of segments of branches using computer-assisted proofs. This approach circumvents the analytical complexities that arise when the number of rings $n\geq 2$ and allows us to give several new rigorous results. We exemplify our method providing new contributions consisting in the determination of enclosures and proofs of stability of several equilibria and RE for $5\leq N\leq 12$.Comment: 60 page

Turbulent Shear Flow in a Rapidly Rotating Spherical Annulus

This dissertation presents experimental measurements of torque, wall shear stress, pressure, and velocity in the boundary-driven turbulent flow of water between concentric, independently rotating spheres, commonly known as spherical Couette flow. The spheres' radius ratio is 0.35, geometrically similar to that of Earth's core. The measurements are performed at unprecedented Reynolds number for this geometry, as high as fifty-six million. The role of rapid overall rotation on the turbulence is investigated. A number of different turbulent flow states are possible, selected by the Rossby number, a dimensionless measure of the differential rotation. In certain ranges of the Rossby number near state borders, bistable co-existence of states is possible. In these ranges the flow undergoes intermittent transitions between neighboring states. At fixed Rossby number, the flow properties vary with Reynolds number in a way similar to that of other turbulent flows. At most parameters investigated, the large scales of the turbulent flow are characterized by system-wide spatial and temporal correlations that co-exist with intense broadband velocity fluctuations. Some of these wave-like motions are identifiable as inertial modes. All waves are consistent with slowly drifting large scale patterns of vorticity, which include Rossby waves and inertial modes as a subset. The observed waves are generally very energetic, and imply significant inhomogeneity in the turbulent flow. Increasing rapidity of rotation as the Ekman number is lowered intensifies those waves identified as inertial modes with respect to other velocity fluctuations. The turbulent scaling of the torque on inner sphere is a focus of this dissertation. The Rossby-number dependence of the torque is complicated. We normalize the torque at a given Reynolds number in the rotating states by that when the outer sphere is stationary. We find that this normalized quantity can be considered a Rossby-dependent friction factor that expresses the effect of the self-organized flow geometry on the turbulent drag. We predict that this Rossby-dependence will change considerably in different physical geometries, but should be an important quantity in expressing the parameter dependence of other rapidly rotating shear flows