Experimental Manifestation of Vortices and Rossby Wave Blocking

ISSN 0021�3640, JETP Letters, 2013, Vol. 97, No. 6, pp. 316–321. © Pleiades Publishing, Inc., 2013.Original Russian Text © A.E. Gledzer, E.B. Gledzer, A.A. Khapaev, O.G. Chkhetiani, 2013, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 97,No. 6, pp. 359–365.

316

Experiments with the flows of a homogeneous fluidin rotating ring channels and cylindrical vessels, whichimitate processes in the atmospheres of rotating plan�ets, are widely used for laboratory investigation ofRossby waves and related perturbations. Rossby waveswhose interaction and dynamics possibly generateatmospheric formations blocking the western trans�port of air masses at middle latitudes appear in theatmosphere because the projection of the angularvelocity of the global rotation on a local verticaldepends on the latitude (the so�called beta effect). Thebeta effect in laboratory experiments is usually simu�lated by a radial change in the thickness of the fluidlayer, e.g., in the presence of an inclined bottom, aswell as the shape of the bottom close to paraboloidal[1]. This follows, for example, from the Charney–Obukhov equation for the potential vorticity. The sim�plest form of this equation is

(1)

where ψ is the function of the current of the diver�genceless two�dimensional velocity field; Δ and[…, …] are the Laplacian and Jacobian in the polar

coordinates, respectively; L0 = /2Ω0 is theObukhov–Rossby scale; Ω0 is the angular velocity of

∂

∂t�� Δψ L0

2–ψ–( ) ψ Δψ,[ ] β

∂ψ

∂r��+ + 0,=

β2Ω0

h0

��dhb

dr��,–=

gh0

the general rotation; h0 is the average depth of thelayer; and hb(r) is the radial dependence of the heightof the (axisymmetric) bottom.

There are several types of such experiments. Themethods for the generation of hydrodynamic flowsvary from the use of thermal convection with the exci�tation of baroclinic Rossby waves (the early best�known works in this field were reported in [2, 3]) to theuse of differential rotation of disks exciting barotropicshear flows [1, 4–7] and to the source–sink method,whose results were reported in many works [8–17].These experiments are performed both with a solid flatcap and with a free surface. The parabolic shape of thefree surface (an increase in the thickness of the fluid tothe outer edge of the vessel owing to the centrifugalforce), as well as a conical bottom (the beta effect inthe case of the solid cap is possible only in the presenceof a bottom with varying geometry), also contributes tothe beta effect. The number of appearing vortex struc�tures and the velocity of their motion specify theregions on the diagrams of flow regimes at variousexternal parameters, which are Reynolds, Taylor, andRossby numbers related to the angular velocity of thegeneral rotation, diameter of the vessel, height of thelayer of the fluid, viscosity, and volume velocity of fluidcirculation when the source–sink method is used. Therecent development of particle image velocimetry(PIV) methods for obtaining the horizontal velocityand vertical vorticity makes it possible to calculate the

Experimental Manifestation of Vortices and Rossby Wave Blocking at the MHD Excitation

of Quasi�Two�Dimensional Flows in a Rotating Cylindrical VesselA. E. Gledzera, E. B. Gledzera, A. A. Khapaeva, and O. G. Chkhetiania, b, *

a Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, 119017 Russia* e�mail: [email protected]

b Space Research Institute, Russian Academy of Sciences, ul. Profsoyuznaya 84/32, Moscow, 117997 RussiaReceived December 24, 2012; in final form, February 14, 2013

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 conduct�ing fluid layer in the field of ring magnets under the action of a radial electric field. The velocity fields havebeen reconstructed by the particle image velocimetry (PIV) method. In the fast rotation regimes with a thinfluid layer, where the Rossby–Obukhov scale does not exceed the characteristic sizes of the vessel, the systemof perturbations appears with almost immobile blocked anticyclones in the outer part of the flow and rapidlymoving cyclones in the main stream. The diagram of regimes is plotted in the variables of the relative angularvelocities of the averaged zonal flow and transfer of vortices about the system rotation axis. Attention isfocused on the results for the regions of the diagram with slow motion of vortices with respect to the rotatingcoordinate system near the parameters for stationary Rossby waves (blocking of circulation). The results arecompared to the results previously obtained in similar experiments using the source–sink method.

DOI: 10.1134/S0021364013060052

JETP LETTERS Vol. 97 No. 6 2013

EXPERIMENTAL MANIFESTATION OF VORTICES 317

corresponding moments of the velocity and vorticityfields [12].

The experiments in a rotating cylindrical vessel atparameters differing from those considered in theseworks were recently reported in [18]. In those experi�ments, the average zonal velocity depending on theshape of the profile of the zonal flow, which is deter�mined by the distribution of sources and sinks and bygeneral rotation, as well as the velocity of the transferof vortex structures, was calculated. Counterpropagat�ing zonal flows in the experiments reported in [18]were excited by inflows (outflows) of the mass at theouter boundary of the vessel and at its central part andsink (source) over the circle in its middle under theaction of the Coriolis force and bottom flow. This cancorrespond to the atmospheric circulation with theeastern polar and equatorial transfers and the westernaverage flow at temperate latitudes. However, forexperiments with a high vessel rotation velocity (withperiods shorter than 3 s), such a scheme for generationof flows is not implemented owing to the prevailingcentrifugal force, which enhances the flow from thecenter to the surface of the fluid layer in the vessel.This is because real circulation in the vessel is signifi�cantly three�dimensional and sinks and sources of thefluid on the bottom cannot completely control flowson the surface of the layer at rapid rotation. In thiscase, there are regimes of the source over one circle onthe bottom (e.g., near the outer boundary) and sinkover another circle inside the vessel, which create onlyflows of one direction.

In this work, to generate counterpropagating flowsin the presence of the beta effect, we used the magne�tohydrodynamic (MHD) method, which appearedefficient at much faster rotation of the system than in[18]. Moreover, the thickness of the layer of the (con�ducting) fluid in the vessel is almost an order of mag�nitude smaller than that for flows with source–sinkexperiments; for this reason, the three�dimensionalcirculation effects in the layer are much weaker. In thefast rotation regime (angular velocity Ω0 = 2π/T,where T ~ 2–3 s) with a thin layer of the fluid (averagedepth of the fluid layer h0 ~ 1–1.5 cm), the Rossby–Obukhov scale, which characterizes the typical size ofvortices in rapidly rotating layers of the fluid, can bestrongly reduced (down to 8–10 cm) as compared tothe source–sink experiments [6]. The experimentswith counterpropagating flows in a system withoutrotation with MHD generation by a radially flowingcurrent were performed at the end of the 1970s (see[19, 20]). This method was used quite recently in [21]for the simulation of individual two�dimensional vor�tex structures (polar vortex).

The results reported below were obtained for rotat�ing cylindrical vessels with a free surface (see Fig. 1).The general rotation at the angular velocity Ω0 = 2π/Tup to Ω0 ≈ 3 s–1 corresponded to the Earth’s rotationin the Southern Hemisphere. The vessel with the inner

radius r0 = 1.5 cm and the outer radius L = 14.5 cm hasa conical plastic bottom 3 with the height hb and slopes = hb/(L – r0) = 0.038. The fluid layer at rest has thedepth h from 0.7 to 1.0 cm near the outer boundary,which ensures the filling with the fluid at all angularvelocities of the general rotation. The conical plasticbottom covers iron conical base 1 with circular, square,or rectangular magnets with the transverse sizes from0.5 to 1.5 cm. The magnets form two coaxial rings withdifferent poles in the upper part (N and S in Fig. 1).The plastic bottom is based on these rings. The motionof the conducting fluid (10% solution of cupric sulfateCuSO4) occurs over the plastic bottom. Because of thedifferent polarities of the magnetic rings, two counter�propagating flows appear; their velocities depend onthe magnitude of the electric current (100–900 mA)supplied to cylindrical electrodes 4 with radii r0 and L.(The flow appearing between plastic bottom 3, base 1,and magnets because the space between 1 and 3 is notleakproof does not affect the main motion over thebottom.) Similar composite magnetic systems weresuccessfully used to simulate quasi�two�dimensionalturbulence [22, 23].

A video camera immobile with respect to the vesselis mounted over it on the rotation axis. This camerarecords the picture of the motion of probe particles onthe surface of the fluid. The PIV method developed atthe Obukhov Institute of Atmospheric Physics, Rus�sian Academy of Sciences, was used on a 75 × 75 meshcovering the image of the vessel on a frame. Themotion of particles followed with steps of 2/25 and4/25 s–1. Since the velocities of particles in the experi�ment were low (less than 1 cm s–1), there was no prob�lem of the “loss” of particles in the calculation of thevelocity. It is also noteworthy that the velocity compo�nents averaged in time and azimuth angle are the mainquantities obtained from the experiments. This

Fig. 1. (a) Scheme of the rotating vessel with the magneticsystem: (1) conical base with permanent magnets S and N,(2) conducting fluid, (3) conical bottom, and (4) cylindri�cal electrodes. (b) Conical base with permanent magnets.

318


GLEDZER et al.

reduces the possible local errors in this method. Fur�thermore, the velocity of the transfer of vortex pertur�bations that was obtained by the PIV method was con�trolled directly in video frames because the positionsof large vortices in them were well determined. In anycase, the accuracy of the determination of quantitiesof interest is 10–15%. It is worth noting that the PIVmethod was tested in some experiments (see, e.g.,[12]) by directly measuring the velocity at one or twopoints in the flow (hot�film probes).

Changing the direction of the current between theelectrodes, one can obtain the counterpropagatingflows in the vessel with different amplitudes of thevelocity field depending on the current magnitude andthe diameters of circles along which magnets werelocated (Fig. 1b). This leads to the formation of a sys�tem of vortices with cyclonic and anticyclonic internalrotations. Figure 2 shows the motion of probe particlesat successive times during the period T in one of theexperiments, where zonal flows near the inner(“polar”) and outer (“temperate latitudes”) bound�aries of the vessel are anticyclonic and cyclonic,respectively. Vortices with anticyclonic rotation direc�tion can be seen in the lower left part of the figure. Therotation directions in vortices are determined whenplotting the velocity vectors.

Figure 3a shows the velocity vectors obtained usingthe PIV method from video images for two times in thesame experiment separated by an interval of 2T/3. Afaster zonal flow in the central region corresponds tothe “crowding” of velocity vector lines. Digit 1 marksanticyclones (with counterclockwise rotation, but inthe “Southern” Hemisphere under the experimentalconditions). These anticyclones in a time of 2T/3 shiftclockwise by a fairly small azimuth angle. Digit 2 indi�

cates a cyclone near the average radius of the vesselshifting in the opposite direction at a higher velocity.Figure 4 shows an example of the velocity field in theregime with five anticyclones whose centers movecounterclockwise at the average radius of the vessel.

The average azimuthal velocity uϕ(r) is determinedfrom the data for the velocity field at the nodes of themesh by averaging in the time of the experiment andthe angular variable ϕ. This velocity was determined in

Fig. 2. Example of the motion of probe particles underaveraging over the period T of the vessel rotation in theexperiment.

Fig. 3. (a) Velocity field u and (b) velocity perturbationfield u' for the experiment shown in Fig. 2.

Fig. 4. Velocity field u for the system of five anticyclonicvortices.



[9] from the calculation of the position of probe parti�cles on photographs; this procedure is used in the PIVmethod. Figure 5 shows the average relative angularvelocity ω(r) = uϕ(r)/( r) as a function of the radiusof the vessel for several experiments with (1) cyclonicand (2) anticyclonic average circulations in the centralpart of the channel. Averaging in radius for each exper�iment determines the radius�average relative angularvelocity ω = . It is one of two circulation�determining parameters, which are used below.

In the experiments where the zonal velocity nearthe inner boundary is much higher than the outerzonal velocity (thick lines in Fig. 5), the internal vortexstructure of flows is poorly identified. It can berevealed by plotting the velocity perturbation field.This field is determined by subtracting the average azi�muthal velocity uϕ(r) from the velocity field u obtainedfrom PIV video images (see, e.g., Fig. 3a) as u' = u –uϕiϕ. Figure 3b shows the velocity field of the perturbedmotion for the same experiment as in Fig. 3a at thesame times. Hidden cyclones near the central regionof the vessel (digits 3 in Fig. 3b) are manifested in theperturbation field after the subtraction of the inner jetflow.

To determine the angular velocity ωv of the transfer

of vortex perturbations at each time, we determine thecoordinates of their centers as points for which thevertical and horizontal components of the velocitychange sign at the surrounding nodes (in this case,saddle points are also determined). Figure 6 shows theangular (azimuthal) coordinates (in the principalvalue sense, i.e., –π/2 < ϕ ≤ π/2) of these centers for

Ω0

ω r( )⟨ ⟩ r

(upper lines C) cyclonic and (lower lines A) anticy�clonic rotations in vortices. The division value on thehorizontal axis corresponds to the vessel rotationperiod T. Figure 6a shows the time picture for theexperiment whose markers and velocity vectors areshown in Figs. 2 and 3, respectively. Figure 6b corre�sponds to the experiment whose vortex motion pictureis shown in Fig. 4. The lines of anticyclones in Fig. 6bshow the time dynamics of five anticyclones that areseen in Fig. 4. There are also two cyclones near theouter boundary of the vessel that move in the samedirection (counterclockwise). The lines of anticy�clones in Fig. 6a show their slow clockwise displace�ment (ϕ values decrease). These anticyclones aremarked by 1 in Fig. 3a. Cyclones 2 in Fig. 3a movecounterclockwise and cyclones 3 in Fig. 3b move rap�idly in the perturbation field u'. These are steep lines inFig. 6a for which ϕ increases. Slow displacement ofsaddle points between anticyclones with the cycloniccirculation in their neighborhood can be seen inFig. 6a. Since the location of anticyclones near theouter boundary of the vessel (Fig. 2) is not strictly uni�form in the angle ϕ (they are in the lower part ofFig. 2), the motion of cyclones in the region of theaverage radius of the vessel in this part is deceleratedowing to the interaction with anticyclones. Thisbehavior is manifested in the bends on steep lines Cnear ϕ = –π/2 in Fig. 6a. The desired quantity ω

v is

determined as the average slope of the lines shown inFig. 6.

The closed squares and diamonds in Fig. 7 are theω and ω

v values calculated for the experiments with

currents from 400 to 900 mA (the horizontal axisshows quantity –ω because “western” transfer is moreoften used, although negative ω values correspond toit). Squares correspond to the experiments in which

Fig. 5. Average relative angular velocity ω(r) for the exper�iments with (1) cyclonic and (2) anticyclonic average cir�culations in the central part of the channel.

Fig. 6. Time dependence of the angular coordinates ϕ ofthe (C) cyclones and (A) anticyclones with the velocityfield shown in Figs. 3a and 4b, respectively.

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GLEDZER et al.

the average angular velocity ω(r) is positive for r nearthe central region of the vessel (as lines 1 in Fig. 5).Diamonds correspond to the experiments with ω(r)similar to lines 2 in Fig. 5. Open squares and diamondsin Fig. 7 correspond to the corresponding source–sinkexperiments reported in [18]. The results of thesource–sink experiments reported in [18] are groupedalong the solid straight lines in Fig. 7.

The linear dependence of the transfer velocity ωv

on the average zonal angular velocity ω can be theoret�ically obtained from the simple formulas for theRossby waves with the zonal wavenumber k for thebarotropic flow on the β plane in the presence of azonal flow at the constant velocity U0, which followfrom Eq. (1) (see [6]). The phase velocity c of the waveis given by the expression

(2)

We introduce the relative quantities

ωv = , ω = – ,

where Rv is the radius of the circle along which vortices

rotate (e.g., cyclones in Fig. 2). According to Eq. (1),the parameter β in the presence of the conical bottomis expressed in terms of the parameters of the experi�

mental setup as β = s [9, 6], where s ≈ hb/L is the

slope of the conical bottom. Defining k = 2π/Lv,

where Lv is the wavelength of perturbations along the

azimuth, disregarding terms with the Rossby–

c U0β U0L0

2–+

k2 L02–+

��.–=

cΩ0 R

v

��U0

Ω0 Rv

��

2 Ω0

h0

��

Obukhov scale L0 (for the parameters indicated, k2 ≈

0.18 cm–2, ≈ 0.03 cm–2, β ≈ 0.15 cm s–1, and

U0 ≈ 0.03 cm s–1 at U0 ≈ 1 cm s–1), we arrive at thefollowing expression independent of the velocity of theexternal rotation:

(3)

Taking Rv = 7 cm, L

v = 14 cm, h0 = 1.5 cm, and hb =

0.5 cm (three cyclones and three anticyclones alongthe circle with the radius R

v) for estimates, we obtain

the line intersecting the horizontal axis at the point⎯ω = 0.033 (dashed line in Fig. 7).

It is noteworthy that some ω and ωv for the MHD

experiments are close to the results of the source–sinkexperiments (see Fig. 7) despite significant differencesin the sizes and methods of the generation of zonalflows. However, the MHD case exhibits an anomalous(for the source–sink method) regime marked by theclosed squares near ω

v = 0 in the negative region of the

horizontal axis in Fig. 7. This regime appears for flowswith zonal flows for which the average angular velocityω(r) is positive for r near the central region of the vessel(lines 1 in Fig. 5), so that the shear flow of counter�propagating flows specifies the cyclonic average circu�lation in this region. In this case, anticyclones (1 inFig. 3a) at the periphery of the flow near the outerboundary of the vessel have ω

v ≈ 0. Their azimuthal

displacement is very small (lines A in Fig. 6a). In someexperiments, these vortices become stationary(blocked) (closed squares at ω

v = 0 in Fig. 7). The

listed differences from source–sink experiments areattributed to the fact that the sizes and azimuthalmotion velocity of vortices generated by the MHDmethod depend on the distance from the system rota�tion axis, whereas the azimuthal motion of vortices insource–sink experiments occurs at an almost constantvelocity. The time diagrams in Fig. 6a show thatcyclones at the middle radius of the vessel and near itsinner boundary move at a large angular velocity,whereas anticyclones near the outer boundary arealmost at rest (blocked).

Thus, the MHD method, along with the source–sink method and the use of rotating disks on the bot�tom or wall of the vessel, can be efficient for the gener�ation of counterpropagating flows and the related vor�tex structures in rotating systems with thin layers offluid. In this method, different arrangements of mag�netic rings and change in the magnetic field and cur�rent make it possible to change the geometry of coun�terpropagating flows, leading to the appearance of vor�tices with different velocities. In the fast rotationregimes with the cyclonic circulation of the averageflow, the system of perturbations appears with almostimmobile anticyclones in the outer part of the flow andrapidly moving cyclones in the middle radius of the

L02–

L02–

ωv

ω– 1

2π2

��hb

L��

Lv

2

h0Rv

�� .–=

Fig. 7. Parameters ω and ωv

for the experiments with the(closed symbols) MHD generation and (open symbols)sources and sinks [17]. The squares and lower solid line arefor experiments with zonal velocities of type 1 in Fig. 5.The diamonds and upper solid line are for experimentswith zonal velocities of type 2 in Fig. 5. The dashed line isplotted according to Eq. (3) for the parameters with zonalvelocities of type 1 in Fig. 5.

ωv



vessel. Anticyclonic vortices blocked at the peripherydo not prevent the motion of cyclones in the mainpart, but reduce their velocity. Bypasses of a blockinganticyclone by cyclones occur in the real atmosphericcirculation at temperate latitudes. As can be seen inFig. 7, blocking of anticyclones in experiments occursfor a wide range of the average relative angular velocityof the zonal flow (–0.11 < –ω < –0.03). This indicatesthe possibility of implementing such regimes at variousexternal parameters.

We are grateful to V.S. Moskalev, leading designerat the Institute for Problems in Mechanics, RussianAcademy of Sciences, for assistance in the fabricationof experimental setups; to G.S. Golitsyn andYu.L. Chernous’ko for permanent interest in this workand consultations; and to an anonymous referee forconstructive remarks. This work was supported by theRussian Foundation for Basic Research (projectnos. 10�05�00457 and 11�05�01206) and by the Pre�sidium of the Russian Academy of Sciences (programno. 19 “Fundamental Problems of Nonlinear Dynam�ics in Mathematics and Physics”).

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Translated by R. Tyapaev

Experimental Manifestation of Vortices and Rossby Wave Blocking

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