An advanced investigation of interaction of allocated quasi-two-dimensional vortices

Abstract

This work should be regarded as a natural development of the investigations by Dolzhanskii, Krymov and Manin [Sov. Phys. Usp. 33, 495-520 (1990); J. Fluid Mech. 241, 705-722 (1992); Russ. J. Comput. Model. 1, 107-118 (1993)] of quasi-two-dimensional (Q2D) flows in which the linear and weakly nonlinear stability theory based on the 2D hydrodynamic equations with the Rayleigh (Ekman) friction term imitating the influence of the bottom on the motion of upper fluid layers was corroborated with laboratory and observational data. The applicability of the Q2D approach to describe self-oscillating supercritical regimes was even more vague as Batchaev's experiments [Izv. AN SSSR Fiz. Atmos. Okeana 25, 434-439 (1989); Z. Prikl. Mech. Tech. Fiz., No. 4, 85-91 (1990)] on modeling the four vortex self-oscillations in a thin fluid layer by the magnetohydrodynamics method (the so called hydrodynamical clock [Obukhov, Dolzhanskii and Batchaev, Topological Fluid Mechanics, Proceedings of the IUTAM Symposium, Cambridge, 13-18 August 1989 (Cambridge University Press, Cambridge, 1989), pp. 304-314]) did not find an appropriate theoretical explanation. To remove earlier uncontrolled effects the supplementary detailed measurements of the experimental flow characteristics were implemented, including the spatial spectral composition of the external vorticity sources and free surface 2D velocity fields. Satisfactory agreement is found between experimentally measured flow characteristics and the results of numerical simulations. The frequency of self-oscillations was found to be greatly susceptible to the spectral composition of the external vorticity sources and fluid layer thickness, which should be taken into account in designing laboratory experiments to simulate the natural Q2D processes observed in the ocean and atmosphere. Applicability conditions of the Q2D approach and the influence of geometrical parameters of vortices on their nonlinear interplay are also discussed. (c) 1996 American Institute of Physics.

The three-dimensional structure of an electromagnetically generated dipolar vortex in a shallow fluid layer

“…The interaction of the current density and the magnetic field induces a Lorentz force that sets the fluid in motion. Our experimental setup is similar to that used in several other studies by, e.g., Dolzhanskii et al, 5 Danilov et al, 6 and Tabeling and co-workers 16,17 ͑although the latter authors used a stably stratified two-layer system͒. A schematic of the setup is depicted in Fig.…”

“…[3][4][5][6][7] In the latter case, it is commonly assumed that these shallow flows behave in a 2D fashion when the vertical length scale H is much smaller than the horizontal length scale L. The rationale behind this thin-layer configuration is that although vertical threedimensional ͑3D͒ motions are present, their magnitude, assumed to be proportional to H / L, is much smaller than the dominant ͑2D͒ horizontal flow speeds. Moreover, the effect of the bottom friction can be parametrized by adding a linear friction term −␣v H to the 2D Navier-Stokes equation ͑usu-ally referred to as "Rayleigh friction"͒ under the assumption of a Poiseuille-like profile in the vertical direction.…”

“…This thin-layer configuration has been applied in flowing soap-film experiments [9][10][11][12] and shallow fluid experiments. [3][4][5][6][7] In some experiments a single fluid layer is utilized, see Refs.…”

Three-dimensional structure and decay properties of vortices in shallow fluid layers

“…As was mentioned in the Introduction, it is also a commonly used parametrization in studies of Q2D turbulent flows in shallow fluid layers. 12,13 In that case, it could be expected that the behavior and the decay properties of very small scales ͑so that HϳL͒ are not described correctly, which could be a drawback of this formulation. This point will be discussed in Sec.…”

“…[8][9][10] Besides the experiments on soap films, it is also possible to study Q2D flows in a thin layer of fluid inside a container, for example, the experiments on vortex interactions performed by Antonova et al 11 and the experiments on freely decaying Q2D turbulence by Tabeling et al 12 In the latter experiment, but also in several other studies of this type, the flow is forced electromagnetically. Other examples are the experiments on the interaction of allocated vortices performed by Danilov et al 13 and the experimental study of Q2D shear flows by Dolzhan-skii et al 14,15 In the experimental studies on thin-layer flows of this type, usually a single layer of fluid is used. Recently, in the experiments of Paret and Tabeling, 16 a system of saltstratified fluid layers was used.…”