Dynamics of Convectively Driven Banded Jets in theLaboratory

Peter L. Read∗, Yasuhiro H. Yamazaki, Stephen R. Lewis†,Paul D. Williams‡, Robin Wordsworth, Kuniko Miki-Yamazaki,

Department of Physics, University of Oxford, Clarendon Laboratory,Parks Road, Oxford, OX1 3PU, UK

Joel Sommeria, Henri DidelleLEGI Coriolis, 21 rue des Martyrs, 38000 Grenoble, France

February 22, 2007

Abstract

The banded organization of clouds and zonal windsin the atmospheres of the outer planets has long fas-cinated observers. Several recent studies in the the-ory and idealized modelling of geostrophic turbulencehave suggested possible explanations for the emergenceof such organized patterns, typically involving highlyanisotropic exchanges of kinetic energy and vorticitywithin the dissipationless inertial ranges of turbulentflows dominated (at least at large scales) by ensem-bles of propagating Rossby waves. Here we present re-sults from an attempt to reproduce such conditions inthe laboratory. Achievement of a distinct inertial rangeturns out to require an experiment on the largest fea-sible scale. Deep, rotating convection on small hori-zontal scales was induced by gently and continuouslyspraying dense, salty water onto the free surface of the13m diameter cylindrical tank on the Coriolis Platformin Grenoble, France. A ‘planetary vorticity gradient’or ‘β-effect’ was obtained by use of a conically-slopingbottom and the whole tank rotated at angular speedsup to 0.15 rad s−1. Over a period of several hours, ahighly barotropic, zonally-banded large-scale flow pat-tern was seen to emerge with up to 5-6 narrow, alternat-ing, zonally-aligned jets across the tank, indicating thedevelopment of an anisotropic field of geostrophic tur-bulence. Using particle image velocimetry (PIV) tech-niques, zonal jets are shown to have arisen from non-linear interactions between barotropic eddies on a scalecomparable to either a Rhines or ‘frictional’ wavelengthwhich scales roughly as(β/Urms)

−1/2. This resultedin an anisotropic kinetic energy spectrum with a signif-

∗p.read1@physics.ox.ac.uk†Now at Dept. of Physics and Astronomy, The Open University,

Walton Hall, Milton Keynes, MK7 6AA, UK‡Now at Dept. of Meteorology, University of Reading, Earley

Gate, Reading, RG6 6BB, UK

icantly steeper slope with wavenumberk for the zonalflow than for the non-zonal eddies, which largely followthe classical Kolmogorovk−5/3 inertial range. Potentialvorticity fields show evidence of Rossby wave breakingand the presence of a ‘hyper-staircase’ with radius, indi-cating instantaneous flows which are super-critical withrespect to the Rayleigh-Kuo instability criterion and ina state of ‘barotropic adjustment’. The implications ofthese results are discussed in light of zonal jets observedin planetary atmospheres and, most recently, in the ter-restrial oceans.

1. Introduction

The banded organization of clouds and zonal winds inthe atmospheres of the outer planets has long fascinatedatmosphere and ocean dynamicists and planetologists,especially with regard to the stability and persistence ofthese patterns. This banded organization, mainly appar-ent in clouds thought to be of of ammonia and NH4SHice, is one of the most striking features of the atmo-sphere of Jupiter. The cloud bands are associated withmultiple zonal jets of alternating sign with latitude (Li-maye 1986; Simon 1999; Garcıa-Melendo and Sanchez-Lavega 2001; Porco et al. 2003), with zonal velocitiesin excess of 100 m s−1 and widths of order 104 km.Similar patterns of multiple zonal jets are also found inthe atmospheres of Saturn (Sanchez-Lavega et al. 2000;Porco et al. 2005; Sanchez-Lavega et al. 2006) and theother gas giants, though on different absolute scales.

A similar pattern of zonation may also occur in theEarth’s oceans, though on a very different scale to thoseof the gas giant planets. Patterns of zonally bandedflow are fairly ubiquitous in numerical model simu-lations of ocean circulation carried out with sufficienthorizontal resolution. In practice this seems to requirea resolution of at least 1/6 in latitude and longitude,

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within which banded flows appear on a scale of 2-4

(Galperin et al. 2004; Maximenko et al. 2005; Richardset al. 2006) which are coherent in the east-west direc-tions for 1000 km or more. Such features often ap-pear to be surface-intensified, but extend coherently togreat depths in an equivalent-barotropic form. The ob-servational evidence for this phenomenon is still quitesketchy, however, though similar features are now be-coming apparent in suitably filtered satellite altimeterdata (Maximenko et al. 2005), at least in medium-longterm time averages.

The dynamical origin of these banded structures re-mains poorly understood, although the fact that the jetsdecay with height in the upper tropospheres and lowerstratospheres of Jupiter and Saturn would suggest thatthey are driven by momentum sources in the weakly-stratified region in the atmosphere below the visibleclouds. The depth to which this pattern of winds pen-etrates below the clouds remains controversial, but mostapproaches towards understanding the processes organ-ising the flow into zonal bands have suggested that thepattern may originate from the anisotropy in a shallowturbulent layer of fluid due to theβ-effect, i.e., owingto the latitudinal variation of the effective planetary vor-ticity (e.g. see Ingersoll et al. (2004) and Vasavada andShowman (2005)).

Until recently, quantitative understanding of this pro-cess has been based principally on high resolution nu-merical simulations of two-dimensional or geostrophicturbulence in stirred rotating fluids (Williams 1978,1979; Vallis and Maltrud 1993; Chekhlov et al. 1996;Galperin et al. 2001; Huang et al. 2001; Sukorianskyet al. 2002; Williams 2003; Galperin et al. 2006). Thishas led to the notional concept of a rotating fluid whichis stirred on small scales and within which kinetic en-ergy cascades to larger scales via nonlinear interactions.The formation of jet-like structures occurs through theeffects of anisotropies introduced by the presence of a‘planetary vorticity gradient’, because of which eddiesof a large enough scale are constrained to propagatezonally in a dispersive manner reminiscent of Rossbywaves. This results eventually in an accumulation of ki-netic energy in zonally-elongated structures, ultimatelyin the form of zonal jets. The mechanism for scaleselection of such jet-like structures, however, has re-mained somewhat obscure, although the scale separat-ing jets which emerge in modelled flows often seems tobe comparable with the so-called Rhines scaleLR ≃(Urms/β)1/2 (Rhines 1975), whereUrms is a measureof the rms total horizontal velocity fluctuations andβ isthe ‘planetary vorticity gradient’ (= (2Ω cosφ)/a for aspherical planet, whereΩ is the planetary rotation rate,a the planetary radius andφ is latitude). Others (e.g.Galperin et al. (2001), Sukoriansky et al. (2002)) haveattributed at least part of the scale selection mechanismto the balance between upscale energy transfer and pro-

cesses removing energy at the largest scales. Undersome circumstances (including systems dominated byEkman bottom friction) this leads to a direct connec-tion between the so-called ‘frictional’ wavenumber (atwhich frictional and dynamical timescales are roughlycomparable) and the Rhines scale, though for other con-ditions the scale selection mechanism is less clear (e.g.see Sukoriansky et al. (2007)).

While such models can provide useful and interest-ing insights into possible mechanisms for banded flows,they are highly idealised and take little account of thevertical structure of realistic geophysical fluid systems.It is sometimes difficult, therefore, to relate results fromsuch studies directly to geophysical problems. Lab-oratory experiments can provide an alternative meansfor studying these processes in a real fluid, for whichapproximations and assumptions underlying simplifiedmodels do not apply. In the present context, aβ-effectcan be emulated by use of conically-sloping topography,at least for quasi-barotropic flow (e.g. Read (2001)).However, previous investigations to date (Mason 1975;Condie and Rhines 1994; Bastin and Read 1998), havebeen unable to access regimes at a sufficiently highReynolds number (or low enough Ekman number) to beable convincingly to demonstrate fully developed, non-linear zonation effects.

Here we report the results of new experiments, con-ducted on the world’s largest rotating platform, whichseek to confirm that multiple zonal jets may indeedbe generated and maintained by internal nonlinearzonation processes. Some preliminary results fromthis study were published by Read et al. (2004), inwhich some of the basic flow patterns were discussedand aspects of the anisotropic spectra presented fortwo of the main experimental cases investigated. Inthis paper, we present a more detailed analysis ofthe results of these experiments, considering not onlythe form of flow pattern which may develop in thepresence of background rotation and sloping or flatbottom topography, but also the detailed interactionsbetween different scales of motion in terms of energyand potential vorticity. Section 2 describes the experi-mental configuration and introduces the main analysismethods. Section 3 presents an overview of the mainscales of motion and the range of phenomenologyobserved during the sequence of experiments. Sections4 and 5 examine diagnoses first of the direct eddy-zonalflow interactions, and then the detailed 2D kineticenergy spectra and anisotropic transfers in Fourierspace. Section 4 considers the observed structure ofthe flow in terms of potential vorticity dynamics andinstabilities. The final section 7 discusses all of theseresults in the context of the geophysical and planetarysystems introduced above.

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2. Apparatus and data analysis

The experiments were performed on the 13 m diameter“Coriolis” rotating platform in Grenoble, France. Theschematic layout of the configuration is illustrated inFig. 1 and the values of the main parameters are listedin Table 1. The experiment consisted of a large circularcylindrical container of diameter 13.0 m and maximumdepthh = 1.5 m, set coaxially on the turntable. Anequivalent topographicβ-effect was obtained by impos-ing a conical slope at the bottom of the circular con-tainer, such that the imposed ‘planetary vorticity gradi-ent’, β was given by

β =2Ω

h

dh(r)

dr, (1)

wherer is the radial coordinate andΩ the rotation rateof the tank. The bottom slope was implemented by us-ing a plastic-coated, fabric sheet, stretched between acircular tubular frame, set just inside the outer radius ofthe tank, and a circular jig close to the centre of the tankwhose height could be adjusted via a clamp attached to apost on the axis. This sheet could also be removed rela-tively easily to enable additional experiments with a flatbottom, in order to compare the flow structure both withand without a topographicβ-effect. The upper surfacewas free and open to the air. This meant that there wasunavoidably a weak centrifugal deformation of the up-per surface, even when the bottom surface was flat. Atthe highest rotation rate used, however, this led to a dif-ference in height across the tank from the centre to therim of only∼ 5 cm. The rotation period of the platform,τr, was set toτr = 40 seconds for most experiments,though some additional experiments were carried outwith a rotation period ofτr = 80 s. The working fluidwas a homogeneous weak salt solution of mean density1026 kg m−3, within which small, neutrally-buoyantparticles (of diameter 0.5 mm) could be suspended forflow visualization and measurement (see below).

2a. Convective forcing

A key objective of this study was to investigate the prop-erties of geostrophic turbulence maintained by a mech-anism which was as ‘natural’ and unconstrained as pos-sible. In particular, it was considered important to applyforcing on a relatively small horizontal scale, but whichwas not fixed in position relative to the rotating refer-ence frame.

To achieve this, convective forcing was applied bygently spraying relatively dense, salty water onto themain fluid layer from a reservoir located above the cen-tre of the tank. A system of spray nozzles was set upalong a radial arm, which could be rotated around thetank in either direction at a speed controlled indepen-dently of the rotation of the main tank. The spacing ofthe nozzles was carefully designed so that, as the arm

Figure 1: Schematic cross-section of the circular con-tainer, showing the rotating spray arm used to providecontinuous convective forcing and the field of view ofthe narrow angle camera. The entire apparatus is set ona circular turntable.

was rotated, the flux of salty water deposited onto thefree upper surface of the tank was as uniform as possi-ble. In practice, tests indicated that the volume flux fromthis system was uniform across the entire radius of thetank to better than±10 %. The flow speed and direc-tion of each jet was set so as to disturb the surface of thewater as little as possible. For most runs carried out, thespray arm was rotated with a period of 30-50 secondsin either the prograde or retrograde direction while de-positing approximately 5-7 litres min−1 of salty water.

By varying the density of the water spray a range ofbuoyancy fluxesFB = g(∆ρ/ρ)FV (whereg is the ac-celeration due to gravity,∆ρ is the difference in densitybetween the working fluid and spray, andFV is the vol-ume flux of the spray) could be obtained ranging fromapproximately 10−9 - 10−7 m2 s−3. For most runs abuoyancy flux of aroundFB = 4.5 × 10−8 m2 s−3 wasobtained. Together with a rotation rate of(1 − 2)π/40rad s−1 and a depthh of around 30-40 cm, this would beexpected to lead to a field of convection with a ‘naturalRossby number’,Ro∗ Fernando et al. (1991); Maxwor-thy and Narimousa (1994); Marshall and Schott (1999),defined by

Ro∗ =

(

FB

f3h2

)1/2

, (2)

(wheref = 2Ω) of around1 − 3 × 10−3. Such a value

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would indicate a convective regime somewhat similar tothat of deep convection in the terrestrial oceans Marshalland Schott (1999), and imply that rotation should exerta strong influence on the form of convection.

2b. Flow measurement

The horizontal velocity fields were obtained from planview CCD images by a Correlation Imaging Velocime-try (CIV) method (Fincham and Spedding 1997; Fin-cham and Delerce 2000). Neutrally-buoyant tracer par-ticles of diameter 0.5 mm were suspended in the flowand illuminated by a beam from a 6 W argon-ion laserlocated at the side of the tank and introduced througha glass panel in the outer wall. The beam was rapidlyscanned in the horizontal to create a diverging lightsheet across a chord through the annular channel, at adepth which could be selected by moving the laser on acomputer-controlled carriage.

The illuminated tracer particles were observed fromabove at a height of approximately 4 m above the sur-face of the tank by two digital cameras, a low-resolution,wide-angle camera with resolution 768× 484 and cov-ering an area 4.5 m× 3.3 m, and a higher resolution,narrow-angle camera with a resolution of 1024× 1024and covering an area 2.5 m× 2.5 m. This allowed theobservation of the largest feasible area of the tank, giventhe limited height of the building in which the rotatingfacility was accommodated, though represented a maxi-mum of just 12% of the total surface area of the annularchannel which was visible simultaneously. By aligningthe long axis of the wide-angle camera with the radiusof the tank however, the optimum coverage of the zonalflow profile across the tank could be obtained. Thisdid mean that a true instantaneous zonal average couldnot be measured directly, though use of time-averagingwould help to recover a more global sample of flowstatistics.

Flow fields were obtained every 20s from groups ofimages obtained by each camera over time interval ofup to 3s. By applying CIV analysis to these bursts ofimages, velocities could be measured to a precision ofbetter than±0.1 mm s−1 (relative to typical velocitiesof a few mm s−1) on a 72× 48 point (wide-angle) or98 × 98 point (narrow-angle) rectangular grid. In or-der to diagnose the azimuthal anisotropy of the evolvingflow, the velocity fields were subsequently decomposedinto radial and zonal (i.e., azimuthal) components andmapped into (r, rθ) coordinates, wherer is the radiusfrom the centre of the tank andθ the azimuthal angle(increasing anticlockwise from above; see Fig. 2(a)).For the purposes of presentation and analysis, we alsomake use of a simple local cartesian system withx inthe (clockwise)θ direction (sox = −rθ, usually mea-sured from the radial centre line of the camera image)andr in the outwards radial direction.

In addition to long series of horizontal flow fields,

r-rθ = x

z

Ω

Figure 2: The coordinate systems used in this paper inrelation to the cylindrical tank. For some purposes, thecylindrical coordinate system(r, θ, z) is used alternatelywith a local cartesian system(x, r, z), for which x =−rθ is in theclockwise(−θ) direction.

a few measurements were made of the flow in theazimuth-height ((rθ, z) or (−x, z)) plane near mid-radius by re-orienting the laser light-sheet into a verticalplane and viewing with the wide-angle camera througha second glass panel in the outer wall. This enabled theflow over a 0.5 m× 0.5 m area to be measured in or-der to examine the flow pattern forced directly by theoverhead spray.

The vertical profile of density in the flow could bemeasured at intervals to an accuracy of approximately±1 kg m−3 using a conductivity probe, mounted on theoperator gantry at mid-radius and which could be drivendownwards into flow from the surface to within 10 cm ofthe bottom of the tank on command via a motor-drivencarriage.

3. Scales and Flow Morphology andVariability

In this section we describe the basic flow structureswhich were obtained over a range of conditions. Be-cause of limited time available for this investigation,only a few combinations of parameters could be ex-plored. Accordingly we concentrate on discussing aset of just three runs whose parameters are listed belowin Table 1. A few additional short experiments werealso carried out at other parameter settings (especiallywith varying buoyancy forcing), though with only lim-ited flow measurements.

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Table 1: Experimental parameters for the three main experiments covering the full range ofβ obtainable.τr is therotation period of the turntable whileτE is the mean Ekman spindown timescale= h/

√νΩ.

Run τr (s) Bottom slope () β (m−1 s−1) FB (×10−8 m2 s−3) Urms (cm s−1) τE (s)I 40 0 0.004 4.6 0.20 1387II 80 4.5 0.035 21.0 0.67 1962III 40 4.5 0.08 4.5 0.34 883

3a. Vertical measurements

As discussed above, the forcing of the flow was effectedby spraying dense, salty water onto the free surface ofthe tank from a rotating arm in the frame of the rotatingtable. This initially produced a shallow layer of densefluid on the top surface of the flow which rapidly be-came statically unstable (on a timescale of a few sec-onds), leading to the formation of discrete plumes ofsalty water which would then descend towards the bot-tom of the tank. This is illustrated in Fig. 3, whichshows two dye visualizations of descending plumes inthe azimuth-height ((rθ, z) or (−x, z)) plane of the tankclose to mid-radius for a case with a flat bottom, a rota-tion period of 80s and buoyancy flux of approximately1.7 × 10−7 m2 s−3.

Fluoroscein dye was introduced at the surface of thetank as the spray arm moved overhead above the fieldof view. This was then entrained into the plumes, whichcould be seen descending and spreading horizontally asthey moved towards the bottom of the tank. It is note-worthy from such images (e.g. in Fig. 3(b)) that, al-though the convective mixing is initially most vigorousin the uppermost 25% of the depth of the tank (Fig.3(a)), individual plumes actually reach the bottom ofthe tank, albeit with some azimuthal tilting, indicativeof deep, penetrative convection.

The velocity field associated with these plumes wasobtained via CIV analysis of sequences of these im-ages, using the dispersion of neutrally-buoyant particlesto trace the flow. A typical example of such a velocityfield is shown in the(x, z) plane at mid-radius in Fig. 4.

The velocity field is evidently dominated by irregu-lar, descending jets of fluid associated with each plume,on a horizontal scale of around 10 cm, again clearlytraversing the entire depth of the tank along a slopingtrajectory. The flow pattern appears to evolve rapidly,on timescales less than the rotation period of 80 s,thus representing a significantly ageostrophic, three-dimensional convective flow on these scales.

Although this convective flow would suggest the pres-ence of a statically unstable stratification, in practice theconductivity probe always measured either a neutral orweakly stabledensity gradient. The vertical density gra-dient was typically found to evolve during the courseof an experiment lasting a few hours, beginning withessentially neutral stratification and then gradually de-

Figure 3: Two visualizations of convective plumes in theazimuth-height plane of the Coriolis tank, rotating witha period of 80 s and with a flat bottom and viewed fromthe side using a wide-angle digital camera. The fieldof view is approximately 0.74 m× 0.55 m in(rθ, z),with the bottom of the tank approximately atz = 0,and plumes are visualized using fluoroscein dye intro-duced at the top surface as the spray arm moves over-head across the field of view. The two images are sepa-rated by approximately 110 s (∼ 1.5 rotation periods).

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Figure 4: Flow field in the presence of convectiveplumes in the azimuth-height plane of the Coriolis tank,rotating with a period of 80 s and with a flat bottom andviewed from the side using a wide-angle digital camera.The maximum vector length corresponds to a velocityof approximately 10 mm s−1.

veloping a weakly stable gradient in the lower part ofthe tank with a maximum value of∂ρ/∂z of around 1-2kg m−4. The upper part of the tank remained approxi-mately neutrally stratified with no clear evidence of un-stable stratification, which was probably confined to athin layer near the top surface, though this could not bemeasured reliably.

These measurements would suggest a crude estimateof an upper limit onN2 of ∼ 5 × 10−3 s−2, where

N2 = − g

ρ0

∂ρ

∂z, (3)

ρ is the fluid density (withρ0 a reference value) andgis the acceleration due to gravity. This would also implya corresponding lower limit on the internal deformationwavenumber,

kd = f/(Nh), (4)

of kd ≥ 13 rad m−1 for cases I and III andkd ≥ 6.5 radm−1 for case II.

3b. Horizontal flow patterns

For each main experiment, the fluid in the tank wasbrought into solid-body rotation with the tank over a pe-riod of at least half a day prior to the start of the exper-iment, without any form of forcing. For the majority ofexperiments carried out, horizontal velocity fields werederived at 20 s intervals, beginning shortly after the ini-tiation of convective forcing, and continued for as longas possible. In at least two cases, the flow continued tobe monitored some time after the convective forcing wasremoved, thereby providing information on the decay of

Figure 5: Horizontal flow field at mid-height in the Cori-olis tank, rotating with a period of 40 s and with a flatbottom. Shaded contours are of azimuthal velocity incm s−1. The maximum vector length corresponds to avelocity of approximately 0.6 cm s−1.

the turbulence field, due mainly to Ekman friction. Thehorizontal flow was typically measured at a level closeto mid-depth (approximately 25 cm below the top sur-face in each case), although in at least two runs (witha flat bottom) a short sequence of measurements weremade in which the horizontal flow field was sampled ata series of 5 different levels spanning the full depth ofthe tank.

i. Flat bottom flows The first case we consider is for aflat bottom, in which the fabric sheet was loosened suffi-ciently to lie at the bottom of the tank. In this case thereis only a weak topographic gradient across the tank, duesolely to the centrifugal distortion of the free upper sur-face. In practice at a rotation periodτr = 40 s this ledto a mean slope in the top surface of around 0.65 and acorresponding value forβ (cf Eq (1)) of∼ 0.0036 m−1

s−1. Although this case corresponds most closely to anf−plane, therefore, there is a weak residualβ−effectwhich should not be ignored. However, the effect iscomparatively weak, and leads to characteristic lengthscales based onβ (such as the ‘Rhines scale’

LR = π (Urms/β)1/2 , (5)

whereUrms is a typical horizontal velocity scale basedon the root mean square velocity fluctuations) which arecomparable with the size of the tank. For Run I, forexample,LR ≃ 2 m, indicating a ‘Rhines wavelength’of around 4 m.

Fig. 5 shows a typical example of an equilibrated hor-izontal flow, taken from run I using the wide-angle cam-era. In this case, the flow is apparently dominated bya single, complex anticyclonic gyre, around 2m across

6

in radius though perhaps more extended in azimuth be-yond the field of view. In addition, there is a generaltendency for broadly retrograde drift in the azimuthaldirection towards the outer radius and prograde drift inthe inner part of the tank. This trend is clearly seen inthe azimuthal mean azimuthal velocity profile in the leftframe of Fig. 5. This overall pattern was seen to persistfor long periods of the experiment, with the anticyclonicgyre remaining almost stationary, though evolving errat-ically in shape and intensity.

It was subsequently noted that the location of the in-clined prograde jet on the inner flank of the gyre (ataroundr =4.5 m andx = 0) coincided with some foldsand irregularities in the fabric sheet which led to someresidual topographic ridges 1-2 cm in height. It waspossible, therefore, that some weak topographic effectswere present to render the main flow pattern nearly sta-tionary relative to the bottom of the tank. The weak dif-ferential azimuthal drift with radius is also in the samesense as might be expected from the residual effects ofwind stress or drag, produced by differential motion be-tween the moving surface of the tank and the stationaryair in the laboratory, though it was difficult to estimatethis effect theoretically. Nevertheless, the influence ofthese experimental imperfections needs to be borne inmind in interpreting these results.

ii. Sloping bottom flows In cases designated as hav-ing a sloping bottom, the fabric sheet was placed undertension between the tubular steel frame along the outerradius and a circular harness attached to the central postof the tank. This enabled a conically-sloping surfaceto be obtained, once the fluid in the tank had spun-up tosolid-body rotation, with a slope angle of approximately5, decreasing the depth of the tank to around 25 cm atthe inner radius of the annular channel (atr = 2.5 m).Several cases were run in this configuration, varying thebuoyancy flux (by changing the density difference be-tween the tank fluid and the spray) over around an or-der of magnitude. Good statistics, however, were onlymeasured for a few cases owing to difficulties with flowvisualisation and control of the spray forcing during theearly phases of the experimental investigation.

For illustration, we present here results from a longduration experiment (case III, lasting some 6 hours) cor-responding to a mean buoyancy flux of 4.5×10−8 m2

s−3 (i.e. comparable to the case I, discussed above), andwith a rotation period of 40 s.

The appearance of the flow in this case is quite dif-ferent to case I discussed above. Overall the flow isdominated by a mixture of small gyre-like eddies oftypical diameter 0.5-1 m interspersed with meandering,azimuthally-oriented currents. The latter also appearclearly in the azimuthal mean azimuthal flow profile inthe leftmost frame of Fig. 6, in which ‘jets’ of alternat-ing sign are found with peak ampltudes of up to 0.5 cm

Figure 6: Horizontal flow field at mid-height in the Cori-olis tank, rotating with a period of 40 s and with a slop-ing bottom. Shaded contours are of azimuthal velocityin cm s−1. The maximum vector length corresponds toa velocity of approximately 0.6 cm s−1.

s−1. This banded organization of the zonal flow is alsoapparent in the shading of Fig. 6, in which clear stripesor bands oriented azimuthally are clearly seen. Flownear the outer boundary, however, is predominantly ret-rograde, much as for case I in Fig. 5.

Eddy gyres of either sign are apparent, though theytypically are found at radii in which the vorticity of theazimuthal currents are of the same sign as the gyres. Asdiscussed below, unlike in case I, these gyres continu-ally evolve and, like other meanders in the azimuthaljets, translate azimuthally across the measurement do-main, moving predominantly in the retrograde direction.Individual eddy gyres typically persisted for longer thanthe transit time across the field of view and formed rea-sonably coherent structures. They had a typical hori-zontal aspect ratio not far from unity, though their shapeand orientation were found to evolve chaotically as theymoved across the field of view.

iii. Time variations of jets As discussed above, thebottom slope was found to have a strong influence onthe time evolution of the nonaxisymmetric eddies andgyres, with a tendency for stationary, non-propagatingstructures with a flat bottom and retrograde-propagatingeddies with a sloping bottom. Such behaviour is sugges-tive of the eddies acting in a manner similar to Rossbywaves, which propagate in a retrograde sense at a speedproportional toβ. In the present case III, large-scale ed-dies with an azimuthal wavelength of around 1 m wereseen to propagate azimuthally at a speed of around 2mm s−1 across the domain at most radii.

During this time interval, however, the zonal flowpattern was also observed to evolve, although the ba-

7

sic qualitative character of the flows shown above werepreserved.

In particular, cases with a sloping bottom were foundto produce fields of relatively small vortices in asso-ciation with an azimuthal mean flow with increasingamounts of structure and variability as the topographicβwas increased. Such a trend is clearly shown in Fig. 7,which presents radius-time contour maps of azimuthalmean azimuthal velocity across the field of view of thenarrow-angle camera. From these maps, it is clear thatthe flow breaks up into patterns of meandering zonal jetsasβ is increased. The position and amplitude of eachjet is not constant, however, as apparent in Fig. 8, whichshows a sequence from the strongestβ case III over alonger time interval. But the flow evidently evolves er-ratically with time, with individual jets appearing to splitand merge. The total number of jets, however, remainsmore or less the same throughout the period of observa-tion, though it may be that some slow, systematic evo-lution is apparent. This might suggest that the jets areemerging during a long transient phase, even though theaverage kinetic energy of the flow reaches an equilib-rium level within 1-2 hours of turning on the forcing.Such a possibility is discussed below in the context ofother model studies.

3c. Vertical flow structure

Although the basic buoyancy-driven forcing of the flowswas essentially baroclinic in character, various theoret-ical and modelling studies suggest (Salmon 1980) thatthe evolution of any upscale nonlinear energy cascadewill lead to the excitation of large-scale flow compo-nents of an essentially barotropic character. Althoughtime did not permit an extensive study of this aspect ofthe flow evolution, a few near-simultaneous measure-ments were made of the horizontal flow field at severallevels in the vertical for one case with a flat bottom.

The flow was allowed to develop for several hourswhilst buoyancy forcing was maintained at a constantvalue of aroundF0 ≃ 4 × 10−8 m2 s−3. Fig. 9 showsresults from the near-simultaneous measurement of hor-izontal velocity at five vertical levels at a particular in-stant, spanning almost the full depth of the tank. Fig.9(a) shows shaded contours of the vertically averagedflow, while (b) shows a map of the standard deviationof KE about that vertical mean. Thus, Fig. 9(a) may beregarded as representing the KE of the barotropic com-ponent of the flow, while Fig. 9(b) represents a measureof the baroclinic velocity field.

From these maps, it is clear that the vertically-averaged (barotropic) flow (i) contains much more ki-netic energy than the standard deviation (baroclinic)components, by a factor of 10 in peak amplitude (notethe difference in contour scale between Figs. 9(a) and(b)). In addition, (ii) the vertically-averaged flow ap-pears to be dominated by relatively large-scale struc-

Figure 9: Instantaneous horizontal kinetic fields in theCoriolis tank, rotating with a period of 40 s and with aflat bottom. Shaded contours are of kinetic energy perunit mass (in units of J kg−1 m−2) and show (a) thekinetic energy of the vertically-averagedflow and (b) thestandard deviation of kinetic energy about the verticalmean.

8

(a)

(b)

(c)

Figure 7: Azimuth-time contour plots of azimuthal mean azimuthal flow at mid-height in the Coriolis tank, for thethree main cases discussed in the text: (a) case I (rotation period 40 s; flat bottom), (b) case II (rotation period 80 s,sloping bottom) and (c) case III (rotation period 40 s, sloping bottom). Shaded contours are of azimuthal velocityin cm s−1.

Figure 8: Azimuth-time contour plot of azimuthal mean azimuthal flow at mid-height in the Coriolis tank, for caseIII (rotation period 40 s, sloping bottom) over a period of approximately 2 hours. Shaded contours are of azimuthalvelocity in cm s−1.

9

tures, whilst the baroclinic standard deviation field isdominated by small-scale patches of KE, with a typi-cal diameter of around 10-20 cm. Such a partitioning ofscale and magnitude between the barotropic and baro-clinic components of the flow is quite striking, and ap-pears to demonstrate strongly the trends suggested fromsimple model studies in forming a strongly barotropiclarge-scale flow from a baroclinic (convective) energyinput.

The typical scale of the baroclinic features presum-ably represents the horizontal size of individual convec-tive plumes in the flow, with reference to Figs 3 and 4above. It is of interest that this scale is comparable tothe scalelrot, defined by

lrot ∼ (3 − 5)(Ro∗)1/2h, (6)

and suggested by Fernando et al. (1991) and Mar-shall and Schott (1999) as a likely horizontal scale forrotationally-dominated convective plumes. Given thevalue of natural Rossby numberRo∗ ∼ 3 × 10−3 andh ≃ 30-40 cm, this would be expected to lead tolrot ∼10-20 cm, much as observed here in Fig. 9(b). More-over, the observed velocity fluctuations inuθ andw onthe scale of the plumes is O(1 cm s−1), again consistentwith the scalings suggested by Fernando et al. (1991)and Marshall and Schott (1999) of|u, w| ∼ Ωlrot ∼ 1 -2 cm s−1.

4. Eddy-zonal flow interactions

In considering the origin of the large-scale barotropicflow discussed above, it was suggested that a form ofup-scale nonlinear cascade of energy could be the mech-anism responsible. In effect, this implies that the patternof slowly-varying, barotropic zonal jets arise through di-rect interactions between non-axisymmetric eddies andthe azimuthally-symmetric component of the flow. Sucha possibility was effectively proposed for Jupiter’s zonaljets by Ingersoll et al. (1981), who presented evidencefrom analyses of cloud-tracked winds derived from im-ages from the Voyager 1 and 2 spacecraft. The evidencefor this mechanism was later questioned by Sromovskyet al. (1982) because of possible biases caused by thenon-uniform distribution of cloud tracers obtained bymanual selection, though this result has subsequentlybeen confirmed from Cassini data by Salyk et al. (2006).Read (1986) also raised additional doubts over the inter-pretation of this result, (a) because it was based on anassumption that barotropic motions dominated the flowin the Jovian troposphere, and (b) because the evidencewas based on the correlation of the horizontal Reynoldsstress componentu′v′ (whereu is the zonal velocitycomponent andv the northward velocity, and primes in-dicate departures from the zonal mean) with the lateralgradient of zonal flow∂u/∂y, which provides a funda-mentally non-local measure of zonal flow interaction.

Moreover, various other modelling studies appeared tosuggest that large-scale non-zonal eddies might derivetheir energy from barotropic instability of the pattern ofzonal jets, thereby linking this with the observed viola-tions of the Rayleigh-Kuo stability criterion in the mea-sured zonal jets around Jupiter’s cloud tops.

4a. Reynolds stress accelerations

Similar issues must surely arise in the present experi-ments, though in this case we have already establishedthat the large-scale flow (including, presumably, themain eddies resolvable in our measurements) is predom-inantly barotropic. Given detailed, accurate and rep-resentative measurements of horizontal velocity fields,well-sampled in time and covering a sufficient area ofthe flow to obtain meaningful statistics, it is feasibledirectly to derive not only the pattern of horizontalReynolds stress at each timestep for which velocitiesare measured, but also the divergence of this Reynoldsstress and hence the implied eddy-induced accelerationof the azimuthal mean zonal flow. In cylindrical geom-etry (e.g. see Pfeffer et al. (1974), Read (1985)), thefull azimuthally-averagedzonal momentum equation re-duces to

∂u/∂t

[1]= (2Ω + u/r) v

[2]

− v∂u/∂r

[3](7)

− w∂u/∂z

[4]

− 1r ∂/∂r(ru′v′)

[5]

+

∂/∂z(w′u′)

[6]

+ F (x)

[7],

(cf Andrews et al. (1987); Eq (3.3.3)) whereu, vand w are now respectively the azimuthal, radial andvertical velocity components.F (x) represents resid-ual forces e.g. due to molecular viscosity and/or windstresses. Term [2] represents the combined Coriolis andcentrifugal acceleration due to zonal mean meridionalflow, while [3] and [4] represent advection of zonal mo-mentum byv. Terms [5] and [6] represent accelera-tions due to the divergence of the Reynolds stresses. Ifthe present experiment develops zonal jets via the samebarotropic nonlinear zonation mechanism as is believedto be responsible for the development of zonal jets in thenumerical simulation studies discussed above, the zonalacceleration [1] should be dominated by the effects ofterm [5] representing the divergence of thehorizontalReynolds stress. Moreover, this balance should be real-ized independently of heightz. Thus, if the zonal flow isindeed a direct result of barotropic eddy-zonal flow in-teractions, it should be possible to compare independentmeasurements of both the horizontal Reynolds stress di-vergence and actual changes in the azimuthal mean flowat any height in the flow.

In Fig. 10 the results of such a comparison are shown,

10

Figure 10: Measurements of the divergence of the horizontalReynolds stress (a) in an equilibrated convectively-driven flow with a sloping bottom (case III), compared with (b) the measured acceleration∂u/∂t of the meanazimuthal flow, as determined by finite time-differencing. Measurements cover a period of approximately 1 hour,sampled twice per rotation period.

for the case III with the most highly developed patternof zonal jets featured above. In Fig. 10(a) shaded con-tours of the instantaneous divergence of the measuredReynolds stress are shown, sampled twice per rotationperiod for a total interval of around 1 hour. By this timethe forcing had been applied continuously for severalhours and the flow seemed to have reached a reason-ably equilibrated level of kinetic energy. In Fig. 10(b),measurements are shown of the acceleration of the az-imuthal mean zonal flow∂u/∂t, obtained by differenc-ing the profile of azimuthal mean flow in successivetimesteps using centred differences. Both fields appearto be quite noisy, with many patches of both positiveand negative accelerations apparent in each field, appar-ently fluctuating back and forth and mostly concentratedin the outer half of the annular channel. Even so, thereare some similarities apparent in the spatial variations ofboth fields, suggesting that they could be correlated.

Fig. 11 shows the result of forming the correlationcoefficientC(r, τ), computed as a function of radius andaveraged over all timesteps in the narrow angle datasetfor case III, between the two quantitites plotted in Fig.10. The definition ofC is given by Eq (8) below

C(r, τ) =< ut(r, t + τ) · (div.u′v′(r, t)) >

[< (ut(r, t))2 >< (div.u′v′(r, t))2 >]1/2,

(8)(where angle brackets denote the time average, thesubscript t denotes time derivative anddiv.[] =1/r∂/∂r(r[])) and allows for the possibility of a timelag in the response between the Reynolds stress diver-gence and the azimuthal mean zonal flow itself. Theresults show a clearly peaked response in correlation

Figure 11: Shaded contour map of the correlation co-efficient between the horizontal Reynolds stress diver-gence and the zonal flow acceleration in an equilibratedconvectively-driven flow with a sloping bottom (caseIII). Covariance is plotted as a function of radius andtime lag between the measurement of Reynolds stressand acceleration. Measurements are averaged over a pe-riod of approximately 1 hour, sampled twice per rotationperiod.

11

close to lag zero (τ = 0) and of magnitude up to 0.5,though with a tendency for the acceleration to respondslightly after the measured Reynolds stress event. Thereis a clear cut-off on the negative lag (time lead) side ofthe origin at all radii, with a more gradual decay in cor-relation with time lag in the positive lag direction. Thisis a clear signature that the horizontal Reynolds stressis strongly correlated with the measured zonal flow ac-celeration, and is entirely consistent with the hypothe-sis that horizontal Reynolds stress plays a major role incontrolling and influencing subsequent changes in thebehaviour of the pattern of azimuthal mean flow. How-ever, it is somewhat surprising that the zonal accelera-tion takes place following a short time delay. The expla-nation for this is not clear in the present work, thoughmay indicate a finite time interval needed for distur-bances detected at the mid-plane to propagate their in-fluence throughout the water column.

The magnitude of the implied conversion of kineticenergy from eddies into the zonal flow, correspond-ing to −u/r∂(ru′v′)∂r integrated over the domain, isaround3.0 × 10−8 W kg−1, which may be comparedwith the computed value for the rate of change of zonalkinetic energy∂(u2/2)/∂t of 3.2 ×10−8 W kg−1 forcase III. Thus, the Reynolds stresses can account formore than 90% of the energy fluctuations in the zonalflow for this case. The missing remainder may representthe effects of measurement errors, but could also reflectother sources of zonal acceleration e.g. due to verti-cal Reynolds stresses associated with baroclinic (con-vective) exchanges on small scales, and also the directeffects of wind stresses. The latter are almost certainlydominant for case I, for which Reynolds stresses are tooweak to account for observed accelerations.

5. Kinetic energy spectra and en-ergy transfers

Numerical studies of small-scale forced, two-dimensional turbulent flows on the sphere orβ-planewhich develop systems of parallel zonal jet streamsare often found to develop strongly anisotropic kineticenergy spectra with various characteristic properties.Given the importance of inertial up-scale energycascades in such flows, the form of the energy spectramay be diagnostic of various important aspects of theturbulent dynamics shaping the equilibrated flow. Inparticular, the logarithmic slope of the energy spectrummay indicate the nature of energy transfers betweenscales in different directions. But for equilibratedspectra with a constant energy injection rateǫ (in unitsof m2 s−3), introduced at small scales, the upscaleenergy-cascading kinetic energy spectrum is expectedto tend to the quasi-universal ‘Kolmogorov 5/3 law’(Kolmogorov 1941),

ER(k) = CKǫ2/3k−5/3 (9)

(later generalized by Kraichnan (1967) and Batche-lor (1969) to two-dimensional turbulence, and hereafterreferred to as ‘KBK’), whereCK is a universal dimen-sionless constant≃ 4 − 6 andk is the total (isotropic)wavenumber (in m−1). In the direction perpendicularto the zonal flow (i.e. increasing latitude on a spheri-cal planet, or cylindrical radius as here), however, vari-ous studies have suggested that the spectrum may equili-brate towards a different, and significantly steeper, form

EZ(k) = CZβ2k−5, (10)

(whereCZ is another universal dimensionless constant≃ 0.3 − 0.5) provided certain conditions are fulfilled(see Galperin et al. (2006) for further discussion). Suchconditions mainly relate to the respective sizes of thescales of energy injection (which need to be smallenough) and those scales affected by large-scale en-ergy removal (i.e at wavenumbers less than the so-called‘frictional wavenumber’kfr; see Galperin et al. (2006)and references therein).

5a. Spectral slopes and anisotropies

In the present series of experiments, the previous sec-tions have indicated that the convectively driven turbu-lent flows result in the formation of jet-like, zonally-elongated structures which appear to be the result of up-scale nonlinear interactions, including strong and coher-ent interactions with the azimuthally-symmetric compo-nent of the zonal flow. It is of interest, therefore, toexamine whether the kinetic energy spectra reveal anysimilarities with those obtained in idealized numericalmodel studies.

Fig. 12 shows kinetic energy spectra for each of thethree main cases discussed above, illustrating the influ-ence of increasingβ on the equilibrated energy spec-tra. In the flat bottom experiment (Fig. 12(a)), the totaland non-zonal spectra appear to follow closely a -5/3slope over the full range of wavenumbers from around5-50 rad m−1, beyond which the spectrum steepens fork > 50. The zonal spectrum in fact follows a slope closeto -8/3, broadly consistent with an isotropic energy spec-trum, with no evidence of any steepening at the lowestwavenumbers accessible in these measurements. Theamplitude of the total energy spectrum is approximately40% below the expected KBK spectrum forCK = 5using the value ofǫ estimated fromǫ ≃ Ek/τE, thoughthe reason for this is not clear.

Fig. 12(b) shows the intermediateβ case with thefull sloping bottom but half the rotation rateΩ. In thiscase the non-zonal and total energy spectra have a some-what steeper slope than -5/3 over the range5 ≤ k ≤ 50rad m−1, though with an amplitude approaching that ofthe expected KBK spectrum aroundk ≤ 10 rad m−1.

12

(a) (b)

1 10 100Wavenumber (rad m-1)

10-10

10-9

10-8

10-7

10-6

10-5

KE

SD

(m

3 s-2)

-5

-5/3

kR kbeta

1 10 100Wavenumber (rad m-1)

10-10

10-9

10-8

10-7

10-6

10-5

KE

SD

(m

3 s-2)

-5

-5/3

kR kbeta

(c)

1 10 100Wavenumber (rad m-1)

10-10

10-9

10-8

10-7

10-6

10-5

KE

SD

(m

3 s-2)

-5

-5/3

kR kbeta

Figure 12: Kinetic energy spectra for the three main cases (a) I, (b) II and (c) III. Graphs show the non-zonalisotropic kinetic energy (solid lines), total kinetic energy (dashed lines) and the kinetic energy of the azimuthalmean flow (dashed-dotted lines), all in units of energy per unit mass per unit wavenumber as a function ofwavenumber (rad m−1). Dotted straight line segments indicate the spectral slopes of -5/3 and -5. The lines la-belled -5 indicate segments of spectra given by Eq (10) and those labelled -5/3 by Eq (9) withCK = 5 in bothslopeand magnitude. Data were obtained by CIV analyses of images from the narrowangle camera.

13

In contrast to case I, however, the zonal spectrum ex-hibits a -8/3 slope only fork > 9 rad m−1, becomingsignificantly steeper for lower values ofk. This indi-cates that the kinetic energy distribution is starting tobecome somewhat anisotropic at low wavenumbers, atleast qualitatively consistently with numerical studieson theβ-plane. It is of interest to note that the steep-ening of the zonal spectrum occurs around a wavenum-ber comparable to or somewhat greater thankβ (∼ 5.5rad m−1 for this flow). At the lowest wavenumbers, thenon-zonal spectrum clearly flattens fork ≤ 5 rad m−1.Spectra obtained from wide-angle images (not shown)indicate a flattening of the non-zonal spectrum aroundk ∼ 3 rad m−1, indicating a value forkfr of around 3rad m−1, which is comparable to the radial wavenumberof the zonal jets.

The strongestβ case in Fig. 12(c) shows some evenclearer signs of anisotropic structure, though evidencefor the classical KBK -5/3 slope is not very clear. How-ever, the total and non-zonal spectra become tangent tothe KBK form aroundk = 10 rad m−1, and in corre-sponding spectra obtained using the wide-angle camera(not shown) there is clearer evidence for a segment ofthe spectrum with a slope of -5/3 for∼ 6 ≤ k ≤ 12rad m−1. The non-zonal and total energy spectra ap-pear to rise to a peak at aroundk = 6 rad m−1, flat-tening or even decreasing ask decreases below∼ 6 radm−1, again suggesting a value forkfr of around 6 radm−1 which, as in case II, corresponds closely to the ra-dial wavenumber of the zonal jets. At smaller scales(k > 50 rad m−1) the spectrum steepens, perhaps ap-proaching the -3 slope characteristic of an enstrophy-cascading range in 2D turbulence (sometimes knownas ‘Kraichnan’s -3 Law’; Kraichnan (1967)). At thesescales, however, corresponding to a horizontal wave-length of< 10 cm, the flow is almost certainly domi-nated by three-dimensional baroclinic effects, and per-haps even viscous effects at the smallest scales. Theazimuthal mean spectrum is significantly steeper than -8/3 for 6 < k < 15 rad m−1, indicating fairly stronglyanisotropic structure biased towards accumulation ofenergy in zonal modes at low wavenumber. We plotin Fig. 12(c) a straight line segment with the slope -5and magnitude consistent with Eq (10)for comparisonwith the measured zonal spectrum. In this case both theslopeand the magnitudeof the zonal spectrum are actu-ally comparable with the theoretical spectrum suggestede.g. from the model studies of Huang et al. (2001) andSukoriansky et al. (2002). Such a trend is somewhat ev-ident for case II, though not at all for case I.

The role of wind stress in adding a directly drivencomponent of the zonal flow in each case here maypartly underly why the effects of anisotropic inertialenergy cascades are difficult to identify in those caseswhereβ-effects are relatively weak. It was mentionedabove that wind stresses may be partly responsible

for the presence of predominantly retrograde azimuthalflow near the outer boundary of the experiment. It isonly in the case with the strongestβ-effect that a possi-ble influence of the anistropic energy cascade becomesapparent in the zonal energy spectrum, even thoughsome weak indication of zonation in the azimuthal flowis apparent in the intermediateβ case II.

5b. Spectral transfer rates

The evolution of the anisotropic spectra discussed aboveis determined by the nature of the nonlinear interactionsbetween various wavenumber components in the flow.Thus far we have concentrated on examining some ofthese interactions in the physical domain, by examin-ing directly the energetic exchanges between the az-imuthal mean (zonally-symmetric) flow and the non-axisymmetric eddies. However, most theoretical treat-ments of energy cascades consider exchanges within thespectral domain, by examining the transfer of energyfrom one region of the spectrum to another in wavenum-ber space. Anisotropy is then revealed in the channellingof energy in particular directions in wavenumber space.

Chekhlov et al. (1996), for example, computed atwo-dimensional enstrophy transfer function in Fourier(wavenumber) space from a sequence of direct numeri-cal simulations ofβ-plane turbulence. Starting with theenstrophy evolution equation,

[

∂

∂t+ 2νk2

]

Ξ(k, t) = TΞ(k, t), (11)

whereΞ(k, t) is the vorticity correlation function,ν isthe kinematic viscosity of the fluid,k is the horizontalwave vector andk its magnitude. The spectral enstrophytransfer,TΞ(k, t), is then given by

TΞ(k, t) = 2πkR[∫

p+q=k

dpdq

4π2×

p × q

|p|2 〈ζ(p, t)ζ(q, t)ζ(−k, t)〉]

,(12)

whereζ(k, t) is the relative vorticity as a function ofwavenumberk at timet, p andq are wave-vector vari-ables of integration,R denotes the real part and an-gle brackets denote the ensemble average. To com-pute a spectral energy transfer map ink-space, a cut-off wavenumberkc was introduced such that the en-ergy injection into modek is taken into account onlyvia triads involving other modes with|k| > kc. Thek-dependent net time-averaged enstrophy transfer intomodek, TΞ(k|kc), was then computed for each modek such that|k| < kc (the so-called ‘explicit’ modes),by integrating Eq (12) over all other modes with wave-vectorsp and q for which |p| and |q| > kc (the‘implicit’ modes), replacing the ensemble average withthe time-mean spectral covariance. The corresponding

14

time-averaged energy transfer functionTE(k|kc) canthen be computed from

TE(k|kc) =TΞ(k|kc)

|k|2 . (13)

Chekhlov et al. (1996) and Galperin et al. (2006) com-puted the integral in Eq (12) for their DNS3 run, and thisclearly showed a strong tendency for energy to accumu-late into particular wavenumbers along theky axis, con-sistent with the formation of strong zonal jets on a scale< kc.

In the present experiments, we have computed thesame spectral energy transfer function in two dimen-sions, using the measured vorticity fields at mid-depthdiscussed above. Fig. 13 shows two examples of suchfunctions, computed from sequences of vorticity fieldsderived from images from the narrow angle camera. Fig.13(a) shows the resulting kinetic energy transfer func-tion for the flat bottom (Case I), which indicates a gen-eral trend for upscale energy transfer towards the lowwavenumber region around|k| = 5-10 rad m−1. Thereis also apparently a weak tendency for the accumulationof kinetic energy into two peaks atk around(∓3,±3)rad m−1. This is consistent with the form of the ve-locity fields within this field of view shown above, withpersistent diagonal flows across the domain in associ-ation with a near-stationary domain-filling eddy with adiameter≥ 2 m.

For the sloping boundary, on the other hand, a clearerpattern is found to emerge. Fig. 13(b) indicates a cleartendency for kinetic energy to accumulate from|k| > kc

into a roughly circular ring in wavenumber space witha radius of around 8-10 rad m−1, but with clear max-ima close to theky axis atky = ±8 rad m−1. Themaxima are roughly twice the strength of the isotropicring in TE(k|kc), indicating a modest accumulation rateof kinetic energy into zonally elongated structures witha wavelength of a little less than 1 m. Such a wave-length is roughly consistent with the development ofthe zonal jets described above, though indicates a muchweaker tendency for strong jet formation than in themuch clearer example presented in the numerical modelsimulations of Chekhlov et al. (1996) and Galperin et al.(2006).

6. Potential vorticity dynamics

Potential vorticity almost certainly plays at least as im-portant a role in the dynamics of the observed jets andvortices in the experiment as their energetics. In partic-ular, the formal stability of the observed flows is gener-ally expressed using criteria formulated in terms of po-tential vorticity and its gradient. In this section, there-fore, we compute diagnostics relating to a potential vor-ticity q based on a barotropic (shallow water) formula-

(a)

-10 0 10kx (rad m-1)

-20

-10

0

10

20

k y (

rad

m-1)

(b)

-10 0 10kx (rad m-1)

-20

-10

0

10

20

k y (

rad

m-1)

Figure 13: Normalized spectral energy transfer func-tions for (a) Case I (flat bottom) and (b) Case III (slop-ing bottom, full β). Functions were computed fromsequences of vorticity fields obtained from narrow an-gle camera images, and integrated over triangles inwavenumber forkc ≃ 20 rad m−1, averaging in timeover approximately 40-50 rotation periods. The field ofTE(k|kc) is normalized in each case by the maximumvalue in the field. Contours are shown with interval 0.1and contours below 0.5 are dashed.

15

(a)

-0.10-0.05 0.00 0.05 0.10 0.15 0.20Zonal velocity (cm s-1)

3.5

4.0

4.5

5.0

r (m

)

(b)

0.25 0.30 0.35 0.40 0.45Scaled SWPV (s-1)

3.5

4.0

4.5

5.0

r (m

)

Figure 14: (a) Radial profile of the azimuthal and timemean azimuthal velocity, and (b) the corresponding pro-file of (shallow water) potential vorticity (see text), av-eraged over a period of approximately 1 hour from ve-locity fields acquired from the narrow angle camera.

tion

q =2Ω + ζ

h(r), (14)

whereζ is the relative vorticity measured at mid-depth.This can be determined relatively easily using the vortic-ity fields measured via CIV and knowledge of the vari-ation of the depth of the tank due to the sloping lowerand upper boundaries.

6a. Rayleigh-Kuo stability

Fig. 14 shows radial profiles of the time and azimuthalmean azimuthal velocity and corresponding potentialvorticity, averaged over approximately one hour of thelate phase of the experiment. From these profiles itis clear that there are some zonal jet-like structureswhich survive the time-averaging, but the potential vor-ticity profile associated with this profile ofu is evi-dently monotonic in radius. Such a result would suggestthat the spatio-temporal mean azimuthal flow is consis-tent with a state of neutral stability with respect to theRayleigh-Kuo criterion for barotropic stability.

On shorter timescales, however, the stability status ofthe azimuthal flow profile is less clear. Figs 15 and 16show the equivalent profiles ofu(r) andq(r) for an ex-ample of an instantaneous flow (Fig. 15) and a shorterterm time average (this time over just 300 s; Fig. 16).In the first case, the instantaneous profile ofu shows astronger and more complex array of zonal jets in asso-ciation with aq profile which is clearly non-monotonic.Indeed∂q/∂r clearly changes sign in Fig. 15(b) severaltimes, indicating positive violations of the Rayleigh-Kuo stability criterion consistent with active barotropicinstability. Sharp positive gradients inq occur in associ-ation with prograde peaks inu, while weak or reversed

(a)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3Zonal velocity (cm s-1)

3.5

4.0

4.5

5.0

r (m

)

(b)

0.25 0.30 0.35 0.40 0.45Scaled SWPV (s-1)

3.5

4.0

4.5

5.0

r (m

)

Figure 15: Radial profiles of the azimuthal mean az-imuthal velocity (a) and the corresponding profile of(shallow water) potential vorticity (b) (see text) for atypical instantaneous flow from Case III in the rotatingtank with sloping bottom. Velocity fields were acquiredfrom the narrow angle camera.

gradients inq are found associated with retrograde az-imuthal jets. This is much as found in idealised modelsof active Rossby wave critical layers (e.g. see Haynes(1989)). Similar traits are also found in the PV structureof Jupiter’s atmosphere (Read et al. 2006) in associationwith the cloud-level zonal jets.

If the profiles are averaged over an intermediate timescale, however, the violations of the Rayleigh-Kuo sta-bility criterion are less marked but still evident, withclear indications of a PV ‘staircase’ with radius. Fig.16(a) and (b) shows profiles averaged over an intervalof just 300 s around the time of the instantaneous flowin Fig. 15(a) and (b). This would seem to indicate a ten-dency for persistent barotropic instability on timescalesof a few hundred seconds, beyond which the instabil-ity is able to eliminate the regions of reversed∂q/∂rin the PV ‘hyperstaircase’ found on short timescales.Such a process would constitute a form of ‘barotropicadjustment’, in which the time-averaged flow on longtimescales approaches a state of near-neutral stabilitywith regard to PV gradients. The mechanism for thisis almost certainly a manifestation of a form of Rossbywave breaking, in which PV gradients become tem-porarily reversed as waves reach a large enough ampli-tude to overturn.

7. Discussion

Our objectives in carrying out this experiment were toseek to create conditions in the laboratory which couldcapture, at least qualitatively, some aspects of the dy-namical regime which might characterize the bandedcirculation observed in the gas giant planets, and per-

16

(a)

-0.2 -0.1 0.0 0.1 0.2Zonal velocity (cm s-1)

3.5

4.0

4.5

5.0

r (m

)

(b)

0.25 0.30 0.35 0.40 0.45Scaled SWPV (s-1)

3.5

4.0

4.5

5.0

r (m

)

Figure 16: Radial profiles of the azimuthal and timemean azimuthal velocity (a) and the corresponding pro-file of (shallow water) potential vorticity (b) (see text)for a flow averaged over a period of 300 s around thetime of the snapshot in Fig. 15 above. Velocity fieldswere acquired from the narrow angle camera.

haps the recently discovered zonally-banded currents inthe Earth’s oceans. Such a goal turns out to be far fromtrivial, since (as we discuss below) the relevant regime isdifficult to achieve on a laboratory scale because of var-ious conflicting scaling requirements. The present ex-periments were made possible as a result of an all-too-brief opportunity (for 6 weeks in 2002) to carry out acollaborative investigation with the team at the Coriolisfacility in Grenoble, which provided the best availableresources to access the conditions necessary for nonlin-ear zonation. However, the difficulties of such an exper-iment should not be underestimated, and our experimen-tal programme in the end was able to sample adequatelyonly a few points in the parameter space of the system.Nevertheless, from the results obtained we were able toobtain clear evidence of the processes anticipated, in as-sociation with a form of small-scale convective forcingwhich is as close as possible to a ‘natural’ means of flowexcitation as could be envisioned within a terrestrial lab-oratory. An analysis of the properties of these flows ob-tained in the laboratory, therefore, ought thereby to pro-vide important insights into the dynamical mechanismsat work within such a regime, and to allow us to putseveral previously proposed mechanisms for zonal jetformation and maintenance to some severe experimen-tal tests.

7a. Zonostrophic regime

Galperin and co-workers (Sukoriansky et al. 2002;Galperin et al. 2006) have attempted to define such aregime, dominated by the nonlinear generation of zonaljets (termed the ‘zonostrophic’ regime), based on exten-sive studies using idealized barotropic numerical model

simulations, as one in which

(a) the flow is forced on relatively small scales (com-pared with those of the physical domain),

(b) an inertial range exists over a large enough rangeof scales such that theβ-effect can exert sufficientinfluence on the flow to allow anisotropic interac-tions to develop fully,

(c) large-scale ‘friction’ (i.e. removal of energy atlarge-scales) is sufficiently small to permit (b) yetenough to prevent the accumulation of energy inthe largest (domain-filling) modes of the system.

The existence of such a ‘universal’ regime has beendisputed e.g. by Danilov and Gurarie (2002), thoughsubsequently Galperin et al. (2006) and Sukorianskyet al. (2007) have claimed that the cases considered byDanilov and Gurarie (2002) did not lie in the appropri-ate region of parameter space. Galperin et al. (2006)proposed a series of chain inequalities as a means offormulating a set of criteria for identifying the range ofparameters over which such a regime might occur. Inthe present context, these can be expressed in the form

kξ ≥ 2kβ ≥ 8kfr ≥30π

L, (15)

wherekξ is a wavenumber (in units of rad m−1) char-acteristic of the scales of energy input/forcing,kβ is de-fined as

kβ =

(

β3

ǫ

)1/5

(16)

(whereǫ is the rate of input of kinetic energy into theupscale cascade by the forcing, in units of m2 s−3) andrepresents the scale at which the Rossby wave period iscomparable with the eddy overturning timescale,kfr isthe ‘frictional’ wavenumber determined by the balancebetween upscale energy transfer and energy removal atlarge scales, andL represents the physical horizontalscale of the domain (in the present case the radius ofthe apparatusL = 6.5 m). Given sufficient time toevolve, flows which satisfy Eq (15) are expected to de-velop self-similar energy spectra which adopt the ‘uni-versal’ forms given in Eqs (9) (over most anglesφ) and(10) (close toφ = ±π/2) over discernible regions oftheir spectra. Under such circumstances, the zonal com-ponent of the energy spectrum (given by Eq (10)) can beintegrated over wavenumber and shown (e.g. Galperinet al. (2001), Galperin et al. (2006)) to lead tokfr ≃ kR

to within a factor O(1), where

kR = (β/Urms)1/2 =

π

LR(17)

is the Rhines wavenumber (cf Eq (5)). Thus, in thisregimekR andkfr are more or less interchangeable asestimates of the meridional scale favoured by the flow

17

for the emergence of banded zonal jets. Most recently,Sukoriansky et al. (2007) have further elucidated thisrelationship betweenkfr and kR, noting that forced-dissipative flows for whichkβ ≥ 2kR will exhibit mostof the salient properties of the zonostrophic regime,whilst those for whichkβ ≤ 1.5kR will lie in the so-called frictionally-dominated regime and will not ex-hibit significant zonostrophic inertial ranges.

Table 2 presents estimates of the main parameters in-voked in Eq (15), derived from the experimental param-eters noted in Table 1 above and the apparatus dimen-sions. kξ is estimated as2π/lrot, wherelrot is com-puted using Eq (6) above, andkβ is obtained from Eq(16). Note that the estimates ofǫ used here were ob-tained asǫ ≃ U2

rms/(2τE), whereτE is the Ekman spin-down timescale(= h/

√νΩ); see Table 1. This typically

results in a value forǫ around 3-15% of the input buoy-ancy fluxFB , which is not unreasonable since it is likelythat only a small fraction of the energy input associatedwith FB will actually be utilised in the nonlinear inverseenergy cascade (cf the estimates used by Galperin et al.(2006) for the oceans and gas giant planets).kfr rep-resents the dominant wavenumber at which large-scalefriction exerts a strong influence on the energy spec-trum, and is estimated from the low wavenumber peakor break in the total or zonal wavenumber spectrum foreach case.

From these estimates, it is apparent that, althougheach experiment satisfies the first inequality in Eq (15),Cases I and II do not rigorously satisfy either of the re-maining two criteria for the zonostrophic regime. CaseIII, however, with the strongestβ-effect, comes the clos-est to fulfilling all of the criteria, withkβ ≃ 2kR. Forcase II, our estimates ofkβ and kR are probably tooclose together to allow much of an inertial range to de-velop between them, placing it on the margins of thezonostropic and frictionally-dominated regimes. As dis-cussed further by Galperin et al. (2006), this section ofthe spectrum is critical for the emergence of the fully-formedk−5 spectrum in the zonal flow, and the separa-tion of kβ andkfr or kR is much greater for the outergas giant planets than could be obtained in our experi-ments here. This may well help to account for why ourbanded zonal jets are much less dominant in the lab-oratory than are typical in the atmospheres of Jupiterand Saturn. Even so, the flows obtained in Case IIIdo show a number of features suggestive of the zonos-trophic regime, indicating that we were able to enter atleast the margins of this class of dynamical behaviour.Galperin et al. (2006) note that the oceans also do notquite satisfy the criteria onkβ/kfr set out in Eq (15),though fulfill them somewhat more clearly than in ourexperiment III.

Also included in Table 2 are estimates of wavenum-ber scales based on both the internal and external de-formation radii. kd is defined in Eq (4), though only a

lower limit can be estimated becauseN is poorly de-fined. The external deformation scale is estimated fromkdext ∼ f/

√

gh, though this wavenumber turns out tobe much less than the smallest wavenumber that can beaccommodated in the annular domain. These estimatessuggest that the external deformation radius does notplay a role in the dynamics of this experiment, but in-ternal baroclinic disturbances could be of significance.However, there is no evidence of scales of motion whichmanifest themselves as different from those representedby kβ , kR or kξ.

7b. Scales and spectral transfers

Insofar as our Experiment III does lie parametrically inthe zonostrophic regime, therefore, the diagnostics com-puted above help to reveal the dynamical mechanismsat work in sustaining a zonally-banded circulation. Wehave shown that, providedβ is strong enough, a flowexcited by stirring at small scales will evolve a large-scale barotropic zonation, accompanied by an ensembleof barotropic vortical structures, many of which prop-agate zonally, at least qualitatively, like Rossby waves.The formation of the zonation in azimuthal flow appearsto be largely as a result of direct eddy-zonal flow in-teractions, with a clear tendency for kinetic energy toaccumulate upscale into the form of zonally-elongatedstructures or jets. Such a tendency is at least con-sistent with the hypothesis of Balk (2005) that suchzonally-elongated structures will be favoured in flowsconstrained simultaneously to conserve the three dy-namical invariants, energy, enstrophy and the geomet-ric invariant noted by Balk (1991) for Rossby wave sys-tems. It would be of interest in further work to investi-gate further the latter invariant in actual diagnostics ofactual flows and numerical simulations, to examine ingreater depth the validity of this hypothesis.

The scale favoured by the zonation process is a sig-nificant factor in determining the effect of zonostrophicanisotropy on flow structure. The Rhines wavenum-ber,kR (as define in Eq (17)) has often been suggestedas representing the likely scale of zonal jet formation.As originally noted by Hide (1966), such a scale rep-resents a rough balance between relative and planetaryvorticity advection, though this may not be the most use-ful balance to consider. Galperin et al. (2001) put for-ward an alternative interpretation, relating the favouredjet scale, represented bykfr, to a balance between up-scale energy transfer and large-scale energy removal.Both of these are effectively internal parameters, de-pendent on flow-dependent quantities, so are not par-ticularly useful as predictive quantities. But they appearto be broadly consistent with the observed scale of jetsfound in the present experiments. However, our exper-iments II and III indicate in this case thatkfr ≃ 1.5kR,which is not quite as suggested by Galperin et al. (2001)and Galperin et al. (2006) for the zonostrophic regime,

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Table 2: Derived parameters for the three main experiments I- III. All wavenumber scales shown are in units ofrad m−1 - see text for explanation of symbols.

Run Ro∗ ǫ (m2 s−3) kξ kβ kfr kR kd kdext

I 2.2×10−3 1.5×10−9 49 2.1 < 3 1.4 > 13 0.13II 1.3×10−2 1.1×10−8 20 5.2 ∼ 3 2.3 > 6.5 0.08III 3.4×10−3 6.6×10−9 61 9.5 ∼ 6 4.9 > 13 0.16

though is close to the form suggested by Sukorianskyet al. (2007). In both cases II and III the scalekfr cor-responds quite closely to the radial wavenumber of theobserved zonal jets, indicating that the scale separat-ing adjacent zonal jets is reasonably well estimated by∼ 1.5kR. The Rhines scalekR also seems to be quiteclosely consistent with observed flows in the outer plan-ets (e.g. Vasavada and Showman (2005)) and the oceans(Richards et al. 2006), which satisfy Eq (15) even moreclosely.

An important aspect of the near-equivalence ofkfr

with kR, asserted by Galperin et al. (2001), Galperinet al. (2006) and Sukoriansky et al. (2007), is the as-sumption that energy removal takes place in a relativelyscale-independent manner. This is consistent, for ex-ample, with Ekman damping, such as would apply toour present experiments, and in the Earth’s oceans. Theprincipal mechanism for energy removal on large scaleson the gas giant planets, however, is not clear at thepresent time. None of the gas giants is believed topossess a solid underlying surface, so the relevance ofEkman damping seems unlikely. But presumably theremust be some means of removing barotropic kinetic en-ergy from the flow on large scales which, given the ap-parent consistency of the observed jets with the Rhineswavenumber, must act comparably on all scales. This isan aspect of the energy flow within the atmospheric cir-culation of these planets which deserves closer attentionin future work.

7c. Convective forcing

The method of forcing used herein seems to work wellas a ‘natural’ means of injecting energy at small scales,though the use of salt-driven convection does pose sig-nificant practical difficulties in sustaining a uniform andsteady input of energy over long periods of time. In fu-ture experiments, it would be useful to explore the pos-sibility of driving convection via imposed thermal con-trasts, although providing sufficient heating and cool-ing (on the order of 10-100 W m−2 on the scale of theCoriolis facility) would be a major undertaking. Theimposition of unstable stratification has clear parallelswith what may be happening in the atmospheres of thegas giant planets, though in their case the role of moistconvective processes needs to be taken into account(Gierasch et al. 2000; Ingersoll et al. 2000). The lat-

ter probably concentrates intense convection into smalland highly intermittent regions, though would certainlyoperate on a small scale compared with that of the planetitself. But an overall upwelling energy input of around5-6 W m−2 from the interior heat source on Jupiter (e.g.Vasavada and Showman (2005)), coupled with a depthfor convection of at least 50 km, would suggest an av-erage natural Rossby number Ro∗ of 0.1 or less, plac-ing it in a qualitatively similar convective regime to thepresent experiments.

Similar types of deep, rotationally-controlled convec-tive forcing may also occur in the oceans (Marshall andSchott 1999), though only in very restricted regions suchas the polar oceans and the Mediterranean. More com-monly, it is baroclinic instability on horizontal scalesof 50-100 km which provides the dominant source of‘stirring’ in the wider ocean basins, and may be respon-sible for the energy input for the putative small-scalezonal jets now being found in ocean models and obser-vations (Galperin et al. 2004; Maximenko et al. 2005;Richards et al. 2006). This was the original concept forexciting geostrophic turbulence by Salmon (1980), andthere is certainly some evidence for zonation in smallscale laboratory experiments on baroclinic instabilityusing sloping endwalls (Mason 1975; Bastin and Read1998), which would support the notion that the zonos-trophic regime does not depend strongly on the mech-anism of small-scale energy injection. But laboratoryexperiments to date on baroclinic instability with slop-ing endwalls have been carried out only on a relativelysmall scale, and almost certainly have not satisfied thecriteria expressed in Eq (15) above. It would be of sig-nificant interest to investigate this further on the scale ofthe Coriolis facility.

7d. Potential vorticity

The potential vorticity structure of the equilibratedbanded zonal flow shows some clear signatures of in-tense interactions with eddies. Although the net effectof such interactions is to accelerate the zonal flow and tomodify its structure, especially in the vicinity of Rossbycritical layers which develop spontaneously as the flowevolves, the resulting instantaneous zonal flow ends upin a state which violates many of the ‘classical’ crite-ria for stability. Instantaneous velocity fields show evi-dence for smoothing out and even reversals of the radial

19

PV gradient, indicative of local Rossby wave breaking.Longer time-averages, however, are much closer to astate of neutral stability. This would seem to suggest thatbarotropic instability acts as an adjustment mechanismon long enough timescales in these experiments, suchthat instantaneous flows are indeed unstable but adjustthe PV distribution over timescales of a few hundredτr

towards marginal stability. This would appear to be incontrast to results from unforced numerical model sim-ulations, which seem to produce jets which are muchcloser to neutral stability, even for instantaneous fields.The continuous forcing must therefore play a major rolein perturbing the zonal flow away from this marginalstate, by an amount which presumably depends on thestrength of forcing.

It has long been noted that the westward zonal jets onthe outer planets tend to violate the classical Rayleigh-Kuo/Charney-Stern instability criteria (Vasavada andShowman 2005; Read et al. 2006) (though may be closerto neutrality with respect to Arnol’d’s second stabilitytheorem (Dowling 1993, 1995)). This may again be asignature of the effect of the processes forcing the jets onJupiter and Saturn, whose origins are still poorly under-stood in detail, though may lead to Rossby wave break-ing as energy cascades upscale from the smaller-scaleconvective forcing. Further studies of the relationshipbetween eddy forcing and stability criteria in experi-ments and models may be a useful source of further in-formation on these mechanisms for driving zonal flowson the outer planets. Finally, we remark that the extentto which the small-scale jets found in ocean observa-tions and models violate the classical instability criteriahas not yet been investigated in detail, but could forman important means of discriminating between zonos-trophic mechanisms for jet formation and alternativemechanisms (e.g. see Berloff (2005)).

Acknowledgement We are grateful to the HY-DRALAB program, funded by the EC contractAccessto Major Research Infrastructures, for their supportduring the experimental design and acquisition phaseof this project. Additional support was provided bythe UK Particle Physics and Astronomy ResearchCouncil. Thanks are also due to the technical team atLEGI/Coriolis for their excellent assistance with thedesign and execution of the experiments, and to ProfsB. Galperin and S. B. Sukoriansky for comments anddiscussions during the course of this study.

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