A. A. Khapaev

Vortex patterns in quasi-two-dimensional flows of a viscous rotating fluid

A modified von Kármán problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flows is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the model problem is given.

Barotropic blocking of the motion of vortices in laboratory experiments with a rotating circular channel

647 1. Experiments with flows of a homogeneous fluid in rotating circular channels forced by the source sink method are widespread in the laboratory modeling of Rossby waves and related perturbations simulating processes in the atmospheres of rotating planets. The beta effect known in geophysical hydrodynamics appears in the experiments owing to the radial varia tion in the thickness of the fluid layer, for example, in the case of the conical sloping bottom. In the real atmospheric dynamics, this effect manifests itself when the Coriolis parameter changes with the lati tude. The major part of such experiments has a demon stration effect to show the capability of the forcing in large scale atmospheric dynamics, which manifests itself in sufficiently small size laboratory table setups [1–4]. The maximum possible information that can be obtained from the quantitative characteristics is the number of appearing vortex structures and possibly their motion velocity. These data are used to construct the diagrams of current regimes in the circular chan nels as functions of the external parameters. In recent years, when the PIV visualization methods for the fluid particle motion were developed, the possibility appeared to measure the entire velocity field and its vorticity during the motion in a channel. For example, the measurements described in [4] allowed us to per form a sufficiently complete statistical description including construction of the probability distribution functions and estimates of the indicators that charac terize the highest momentums of the velocity field. Two types of experiments with rotating circular channels are known using the source sink method: one with a transparent upper rigid lid and the other with a free surface (there are also a large number of experimental works in which the fluid motion is gen erated by thermal forcing, in particular, the authors of well known work [5] present the results of earlier research in this field). In the first case, which is more difficult for visualization, surface gravity waves are not generated in the fluid, but slip conditions at the lid change the velocity field in the upper part of the tank. In the second case, the application of the PIV method is simpler, but the generated gravity waves (which are undesirable if one is interested in the beta effect) can also contribute to the measured velocity field. The par abolic form of the free surface (an increase in the fluid thickness to the outer edge of the channel due to the centrifugal forces) gives its contribution to the beta effect, as well as the conical bottom usually used in both types of experiments (if a rigid lid is used, the beta effect is possible only if the geometry of the bot tom is variable).

Experimental manifestation of vortices and Rossby wave blocking at the MHD excitation of quasi-two-dimensional flows in a rotating cylindrical vessel

Experiments on the excitation of counterpropagating zonal flows by the magnetohydrodynamic (MHD) method in a rotating cylindrical vessel with a conic bottom have been performed. Flows appear in a conducting fluid layer in the field of ring magnets under the action of a radial electric field. The velocity fields have been reconstructed by the particle image velocimetry (PIV) method. In the fast rotation regimes with a thin fluid layer, where the Rossby-Obukhov scale does not exceed the characteristic sizes of the vessel, the system of perturbations appears with almost immobile blocked anticyclones in the outer part of the flow and rapidly moving cyclones in the main stream. The diagram of regimes is plotted in the variables of the relative angular velocities of the averaged zonal flow and transfer of vortices about the system rotation axis. Attention is focused on the results for the regions of the diagram with slow motion of vortices with respect to the rotating coordinate system near the parameters for stationary Rossby waves (blocking of circulation). The results are compared to the results previously obtained in similar experiments using the source-sink method.

Zonal flows, Rossby waves, and vortex transport in laboratory experiments with rotating annular channel

Results of experiments are considered for flows generated by different sources-sinks of mass in the rotating annular channel with beta-effect simulation using the inclined bottom. Diagrams of regimes are presented in parameters of the dimensionless angular velocity of the zonal flow averaged over the channel width and the dimensionless angular velocity of transport of vortex perturbations of cyclonic and anticyclonic types. In experiments and the simplest linear theories, most attention is paid to diagram regions with a slow motion of vortices relative to the rotating coordinate system near the parameters for stationary Rossby waves.

On the decay law of quasi-two-dimensional turbulence

Laboratory measurements of decaying quasi-two-dimensional turbulence in thin fluid layers with various depths have been performed. It has been shown that decay at large Reynolds numbers corresponds to a non-linear bottom friction with the coefficient satisfying the law λ ∼ (ν/h2)1/2|K|1/4 following from theoretical estimates, where K is the Okubo-Weiss function depending on the enstrophy and degree of ellipticity of vortices. It has been shown that the structure of the flow changes in the decay process.

Effect of three-dimensional structures on the dynamics of turbulence in thin layers of fluid in a laboratory experiment

The results of experiments on turbulent flows excited by the Ampere force in a thin layer of a conducting fluid over a solid surface upon the passage of a current and the action of a spatially periodic magnetic field are considered. Third-order longitudinal structure functions of the velocity field are shown to be approximately linear in the spatial shift and negative even at horizontal scales that exceed the layer thickness by an order of magnitude. This is how the three-dimensional dynamics is manifested as a result of the dominant contribution of energy dissipation when the no-slip boundary condition is satisfied on the lower surface. Dissipation and the main summands of energy production have been estimated for the energy-balance equation.

On the Structures Observed in Thin Rotating Layers of a Conductive Fluid and the Anomalies of the Geomagnetic Field

The results of the laboratory and numerical experiments in circular rotating trays with thin layers of a conductive fluid under the MHD generation of small-scale velocity fields are presented. The configurations of constant magnets for MHD generation were determined based on the numerical calculations with shallow water equations. Both the laboratory and numerical experiments with rotating trays demonstrate the emergence of nonaxisymmetric structures and large-scale near-circular vortices caused by the energy transfer from the system of the externally generated small-scale vortices to the large-scale velocity fields under the action of the Coriolis force. The near-circular vortex has areas with differential rotation when the angular velocity of rotation decreases with the radius. The single large-scale vortices and wide jet flows arise in the regimes of subrotation and superrotation relative to the external rotation depending on its angular velocity. The emergence of the flow structures with the azimuthal wave number m = 2 is demonstrated, and their probable relation to the anomalies of the geomagnetic field observed on the Earth’s surface is considered.

Vertical Helicity Flux as an Index of General Atmospheric Circulation

As an index of the general atmospheric circulation over the hemisphere, it is proposed to calculate the hemisphere-area-averaged (poleward of the latitude 20°) product of the Coriolis parameter by the wind velocity squared at the upper boundary of the planetary boundary layer. In practical calculations, data on the wind velocity at an isobaric level of 850 hPa were used. Control calculations for the 900 hPa level gave similar results. It is shown that the index introduced adequately characterizes the seasonal and interannual variability of the general atmospheric circulation over both hemispheres.

Emergence of sub(super)-rotation and jet streams from small-scale quasi-two-dimensional vortices in laboratory experiments

This paper presents the results of experiments in a circular stationary and rotating channels with thin layers of conductive fluid for configurations consisting of a large number of permanent magnets and providing the MHD generation of small-scale velocity fields. The alternating radial configurations of magnets were chosen in such a way as to ensure the conservation of a discrete symmetry of their mutual arrangement relative to rotations of the circular channel around a central axis and were formed on the basis of numerical calculations with the shallow-water equations. Both in numerical and laboratory experiments, large-scale nearly circular vortices were obtained as a result of the energy transfer from the system of externally generated small-scale vortices to large-scale velocity fields (inverse cascade) under the influence of the Coriolis force in the rotating case. Single large-scale vortices and wide jet streams appear in subrotation and superrotation modes relative to external rotation, depending on its angular velocity. Rotation in a nearly circular vortex has a differential character with a decrease in the angular velocity of rotation with the radius in most area of the channel.

The 3D circulation and cyclone-anticyclone asymmetry in the shallow layers of viscous rotating fluid

A modified von Karman problem that describes steady vortex flow in a rotating thin viscous fluid layer is solved. An analysis of the effect of bottom friction on the behavior of cyclonic and anticyclonic vortices at arbitrary values of the Rossby number is presented. Several anticyclonic flow patterns are examined. An approximate analytical solution obtained for steady flow is compared with numerical computations of a time-dependent problem. Experimental results on cyclonic and anticyclonic vortices in multiple-vortex quasi-turbulent flow are presented, and their interpretation based on the solution of the numerical model is given.