RIGOROUS DERIVATION OF THE HYDRODYNAMICAL EQUATIONS FOR ROTATING SUPERFLUIDS

Using a modified WKB approach, we present a rigorous semiclassical analysis for solutions of nonlinear Schrödinger equations with rotational forcing. This yields a rigorous justification for the hydrodynamical system of rotating superfluids. In particular, it is shown that global-in-time semiclassical convergence holds whenever the limiting hydrodynamical system has global smooth solutions and we also discuss the semiclassical dynamics of several physical quantities describing rotating superfluids.

On the Hamiltonian approach: Applications to geophysical flows

This paper presents developments of the Harniltonian Approach to problems of fluid dynamics, and also considers some specific applications of the general method to hydrodynamical models. Nonlinear gauge transformations are found to result in a reduction to a minimum number of degrees of freedom, i.e. the number of pairs of canonically conjugated variables used in a given hydrodynamical system. It is shown that any conservative hydrodynamic model with additional fields which are in involution may be always reduced to the canonical Hamiltonian system with three degrees of freedom only. These gauge transformations are associated with the law of helicity conservation. Constraints imposed on the corresponding Clebsch representation are determined for some particular cases, such as, for example. when fluid motions develop in the absence of helicity. For a long time the process of the introduction of canonical variables into hydrodynamics has remained more of an intuitive foresight than a logical finding. The special attention is allocated to the problem of the elaboration of the corresponding regular procedure. The Harniltonian Approach is applied to geophysical models including incompressible (3D and 2D) fluid motion models in curvilinear and lagrangian coordinates. The problems of the canonical description of the Rossby waves on a rotating sphere and of the evolution of a system consisting of N singular vortices are investigated.

A New Type of Chaotic Attractor

An experimentally discovered inverted spiral-type chaotic attractor is reproduced by a model equation. Does there exist a simple equation for the attractor re-cently found both in an electronic and a hydrodynamical system [1]? After a first, unpublished attempt by Sven Sahle to mirror a classical spiral-type attractor using a tube put into the middle, which yielded "messy" equa-tions, a fairly simple ordinary differential equation (ODE) was found and will be presented in the following. An experimental result is reproduced in Figure 1. It Deviation Fig. 1. Experimental attractor obtained with a hydrodynami-cal system, cf. [1]

Vortex structures generated by a coastal current in harbour-like basins at large Reynolds numbers

It has been observed for a long time that under certain conditions a vortex or even a group of vortices forms in bays which have a narrow opening to the sea. What leads to the formation of such vortices confined in a quiet, almost closed bay? Why does their number vary? Can such vortices form in any specific bay with known hydrological conditions, coastal configuration and bottom topography? The answers to these questions are essential in practice because, if several vortices form in a bay, a sort of a ‘vortex cork’ is created which prevents the outflow of pollution from the bay. This pollution will be locked in the bay practically permanently. The formation of vortices can also very strongly modify the topology of the background flow and lead to the formation of structures which intensify such processes as beach drifting, silting, and coastal erosion.This article considers the topology of the vortex regimes generated in harbour-like basins by the external potential longshore current at large Reynolds numbers. The theory discusses the issues of what solution compatible with the Prandtl–Batchelor theorem for inviscid fluids, and under what conditions, may be realized as an asymptotic state of the open hydrodynamical system. The analysis is developed based on the variational principle, the most appropriate fundamental method of modern physics in this case, modified for the open degenerated hydrodynamical system. It is shown that the steady state corresponds to the circulational regime in which the system has minimal energy and enstrophy. This state is fixed by the Reynolds number. The relation between the Reynolds number, the geometry factor and the topological number, characterizing the number of vortex cells, is found.