J. Fluid Mech. (2013), vol. 732, pp. 485–509. c© Cambridge University Press 2013 485doi:10.1017/jfm.2013.413

Inertial instability of intense stratifiedanticyclones. Part 2. Laboratory experiments

Ayah Lazar1,2,†, A. Stegner2,3, R. Caldeira4, C. Dong5, H. Didelle6

and S. Viboud6

1Department of Geophysics and Planetary Science, Tel Aviv University,Tel Aviv, Tel Aviv 69978, Israel

2Laboratoire de Meteorologie Dynamique, Ecole Polytechnique, 91128 Palaiseau CEDEX, France3UME, ENSTA Centre de l’Yvette, Chemin de la Huniere, 91761 Palaiseau CEDEX, France

4CIIMAR – Interdisciplinary Centre of Marine and Environmental Research, 289 Rua dos Bragas,4050-123 Porto, Portugal

5Institute of Geophysics and Planetary Physics, University of California, 603 Charles E. YoungDrive, East, Los Angeles, CA 90095-1567, USA

6LEGI/Coriolis, 21 Avenue des Martyrs, 38000 Grenoble, France

(Received 26 October 2012; revised 19 May 2013; accepted 7 August 2013)

Large-scale laboratory experiments were performed on the Coriolis rotating platformto study the stability of intense vortices in a thin stratified layer. A linear saltstratification was set in the upper layer on top of a thick barotropic layer, and acylinder was towed in the upper layer to produce shallow cyclones and anticyclones ofsimilar size and intensity. We focus our investigations on submesoscale eddies, wherethe radius is smaller than the baroclinic deformation radius. Towing speed, cylindersize and stratification were changed in order to cover a large range of the parameterspace, staying in a relatively high horizontal Reynolds number (Re= 2000–7000). TheRayleigh criterion states that inertial instabilities should strongly destabilize intenseanticyclonic eddies if the vorticity in the vortex core is negative enough ζ0/f < −1,where ζ0 is the relative vorticity in the core of the vortex, and f is the Coriolisparameter. However, we found that some anticyclones remain stable even for veryintense negative vorticity values, up to ζ0/f = −3.5, when the Burger number is largeenough. This is in agreement with the linear stability analysis performed in part 1(J. Fluid Mech., vol. 732, 2013, pp. 457–484), which shows that the combined effectof a strong stratification and a moderate vertical dissipation may stabilize even veryintense anticyclones, and the unstable eddies we found were located close to themarginal stability limit. Hence, these experimental results agree well with the simplestability diagram proposed in the Rossby, Burger and Ekman parameter space forinertial destabilization of viscous anticyclones within a shallow and stratified layer.

Key words: stratified flows, vortex instability, wakes/jets

† Email address for correspondence: ayahlazar@gmail.com

486 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

1. IntroductionIn part 1 of this study (Lazar, Stegner & Heifetz 2013), we performed a linear

stability analysis for various submesoscale circular vortices to three-dimensionalinertial perturbations. We provided a first estimate of the stability diagram of intenseanticyclones at the submesoscale, and found a stability criterion, which is moresuitable for geophysical vortices, where stratification and dissipation are important.The main goal of this sequel study is to complete this stability analysis with large-scale laboratory experiments that simulate these conditions.

The increased spatial resolution of oceanic numerical modelling reveals a widevariety of small-scale structures smaller than the deformation radius (Capet et al. 2008;Klein et al. 2008). These submesoscale structures control the transition from the large-scale quasi-geostrophic flow to the microscale dynamics, which is three-dimensionaland highly turbulent. On the one hand, quasi-geostrophic flows are driven by aninverse energy cascade where the vortex merging process transfers energy towardslarge scales. On the other hand, in scales smaller than the deformation radius, whereageostrophic motions become frequent, direct cascade toward small-scale dissipationmay occur (Capet et al. 2008; Molemaker, McWilliams & Capet 2010). Thesenon-standard paths toward dissipation and mixing are crucial to understanding theenergetics of the ocean. The inertial instability, which breaks the geostrophic balanceand generates a local transient three-dimensional flow, is one of these direct routes ofdissipation and mixing. Therefore, as inertial instability may be of importance in theocean, the experimental setup is made to imitate oceanic conditions, with a shallowstratified layer, mimicking the upper oceanic pycnocline.

A large number of studies have been devoted to the inertial or symmetricinstability of intense currents or fronts (Hua, Moore & Le Gentil 1997; Griffiths2003; D’Orgeville & Hua 2005; Griffiths 2008; Taylor & Ferrari 2009; Plougonven &Zeitlin 2009). However, only a few papers (Smyth & McWilliams 1998; Kloosterziel,Carnevale & Orlandi 2007; Lazar et al. 2013) have investigated the inertial–centrifugaldestabilization of intense and stratified vortices. For inviscid and circular vortices, thegeneralized Rayleigh criterion (Kloosterziel & van Heijst 1991; Mutabazi, Normand& Wesfreid 1992) asserts that all anticyclonic vortex columns are unstable to three-dimensional perturbations if somewhere in the flow the Rayleigh discriminant isnegative, namely χ(r) = [ζ(r) + f ][2V(r)/r + f ] < 0, where V(r) is the azimuthalvelocity profile, f = 2Ω0 is the Coriolis parameter, with Ω0 the angular rotation ofthe fluid layer, and ζ(r) = ∂rV + V/r is the relative vorticity. The WKB methodapplied to short vertical wave instabilities demonstrates that this criterion is indeed asufficient condition for inviscid three-dimensional instabilities (Sipp, Jacquin & Cossu2000; Billant & Gallaire 2005). According to this widely used criterion the stabilityof a circular anticyclone depends crucially on its velocity (or vorticity) profile. Inorder for an axisymmetric vortex with a monotonic vorticity profile to be unstablethe relative core vorticity ζ0 = ζ(r = 0) should be smaller than −f . The growth ratesof this instability will increase with the steepness of the velocity profile (Gallaire& Chomaz 2003). Hence, if the dissipation is weak, a region of negative absolutevorticity ζ/f + 1 < 0 (i.e. negative potential vorticity for barotropic velocity) is anecessary condition for inertial instability. One of the main results of Lazar et al.(2013) was to create a stronger condition, which reduces significantly the unstablearea in the parameter space for barotropic anticyclonic eddies confined in a thin(δ = h/rmax 1) and strongly stratified (N/f 1) layer where h is the thickness ofthe eddy and rmax its typical radius. Close to the marginal stability limit, where thegrowth rates are strongly controlled by the vertical dissipation, the stability analysis

Inertial instability of intense stratified anticyclones. Part 2 487

reveals that the instability is not sensitive to the velocity or the vorticity profile if theintensity of the vortex is characterized by the vortex Rossby number Ro= Vmax/(f rmax)

instead of the relative core vorticity ζ0/f . An analytical marginal stability limit wasderived for the Rankine vortex, which depends on the vortex intensity (i.e. the vortexRossby number Ro), the relative vortex size in comparison with the deformation radiusRd, and the vertical dissipation (i.e. the Ekman number Ek). As expected, this curvedelineates a smaller region in the parameter space than the inviscid Rayleigh criterion.However, surprisingly, the marginal stability curves of various other vortices (parabolic,conical, Gaussian, etc.) appear to be very close to the analytical one. Hence, themarginal stability equation derived for the Rankine vortex could be relevant to awide variety of vortices and can be used to build a ‘first guess’ stability diagram forinertial–centrifugal destabilization of viscous and stratified anticyclones. In this paperwe will test this criterion with laboratory experiments.

The pioneering experiments of Kloosterziel & van Heijst (1991) and Afanasyev &Peltier (1998) investigated the three-dimensional destabilization of circular anticyclonicflow, and the three-dimensional instability of elliptical anticyclones were studied byAfanasyev (2002), Stegner, Pichon & Beunier (2005) and Le Bars, Le Dizes & LeGal (2007). However, all these experiments were conducted in a thick barotropic layerwith a finite (δ ∼ 1) or large (δ 1) aspect ratio leading to a rapid destabilizationof anticyclonic vortices. The recent experiments of Teinturier et al. (2010) performedin a thin barotropic layer with typical aspect ratio of δ ∼ 0.1 show that unstableanticyclones may remain coherent, and intense negative core vorticity (ζ0/f ' −2)can hold for several rotation periods. The small-scale disturbances strongly affectthe annular region where the Rayleigh discriminant is negative, but seem to have amuch weaker influence on the vortex core. For such non-stratified experiments theanticyclonic structures appears to be unstable once the absolute vorticity becomesnegative, as if the flow were non-dissipative or almost inviscid. However, to thebest of our knowledge, the impact of a strong stratification (N/f 1) on theinertial–centrifugal destabilization of anticyclonic eddies has not yet been studied inthe laboratory.

We conducted our experiments at the LEGI-Coriolis platform in France, to addressthis question precisely. Our setup (described in detail in § 2) produces intense eddiesof both signs, cyclonic and anticyclonic, in a shallow stratified layer at high Reynoldsnumbers (avoiding excessive dissipation). What will be the combined effect on thestability diagram of the stratification, the shallow layer confinement, and moderatedissipation? Is the generalized Rayleigh criterion still relevant in such a case? Howaccurate is the marginal stability equation proposed by Lazar et al. (2013)? Whatwill be the nonlinear evolution and the destabilization process of unstable eddies?These are the main questions we address in this experimental study. We start, in§ 2, with a description of the experimental setup and measurement techniques, andthe main dynamical parameters. In § 3, we quantify the initial velocity profiles ofthe various vortices generated in the shallow and stratified wake. In § 4, we followthe typical geostrophic evolution of low Rossby number vortices. These cyclonic andanticyclonic structures correspond to stable cases and will be used as a reference casein comparison with the unstable ones. We then describe, in § 5, the typical evolution ofan unstable circular anticyclone at high Rossby number. To do so, we need to identifythe main signatures of the inertial–centrifugal instability on the surface velocity andvorticity fields of the vortex. In § 6 we test the accuracy of the stability diagramproposed by Lazar et al. (2013) with ten different experiments. Finally, in the last

488 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

section, we summarize the main results of the paper and discuss the consequences ofour study to some oceanic vortices.

2. Experimental setup and physical parametersWe performed the experiments on the 14 m diameter rotating platform at the LEGI-

Coriolis in Grenoble. The large scale allows us to reach small aspect ratios of thevortex height to radius, and a linear salt stratification was set in the top ∼5 cm layer,above a thick barotropic layer. We generate cyclonic and anticyclonic eddies withsimilar size and strength by dragging a cylinder in a rotating tank and creating avon Karman wake (see figure 1a). As we are interested in the stability of circularvortices, we focus our analysis on the far-wake area of the von Karman street, wherethe vortices are circular and have a very weak interaction between them (this occursroughly at the bottom of figure 1b). Without rotation the wake should be symmetric,and the cyclones and anticyclones should evolve in a similar way. However, in ourexperiments, we see the effect of rotation when the anticyclones exhibit inertialinstability. We give in tables 1 and 2 the values of the initial conditions and thedimensionless parameters (described in 2.2) of the vortices generated in our differentwake experiments (and whether they are stable, according to the method we develop in§§ 4 and 5).

2.1. Eddy generationThe turntable had an anticlockwise rotation (like planetary rotation) with a fixedrotation period of T0 = 90 s, corresponding to a Coriolis parameter of f = 4π/T0 =2Ω0 = 0.139 s−1. We towed a cylinder of diameter D = 50 cm or D = 25 cm andheight hc = 5–7 cm in the top thin stratified layer. The height of the submergedcylinder hc was slightly smaller than the upper stratified layer hs. We assume that themotion of the cylinder transfers momentum mainly to the upper layer and not to thedeep 60–70 cm lower layer, so that the dynamic is governed by the first baroclinicmode (the barotropic mode is negligible). This is reasonable when hs ∼ hc H. Asimilar technique was used by Perret et al. (2006) and Teinturier et al. (2010) tostudy the island wake dynamics in a rotating shallow water layer, where indeed nosignificant motion was detected in the deep lower layer.

2.2. Eddy parametersIn each experiment, the cylinder was towed at a constant velocity Vc, varying from 1to 6 cm s−1. To quantify the vortex intensity we use the vortex Rossby number, andnot the normalized maximal vorticity, as in Lazar et al. (2013):

Ro= Vmax

frmax, (2.1)

where rmax is the radius corresponding to the maximum azimuthal velocity Vmax of theeddy. The typical vortex size usually scales with the cylinder radius rmax ' D/2. Hence,by changing either the drifting speed Vc or the cylinder diameter D, we reached a widerange of finite Rossby numbers, |Ro| = 0.3–2.

We also introduce a horizontal Reynolds number:

Re= Vmaxrmax

ν, (2.2)

where ν is the molecular viscosity of water at ambient temperature. Thus, the towingspeed and the cylinder size also determine the horizontal Reynolds number, which

Inertial instability of intense stratified anticyclones. Part 2 489

CCD camera

Buoyant particles

Bottom lights

D

H

(a)

(b)

FIGURE 1. (a) Side view of the experimental setup, and (b) top view of the particlevisualization of the surface wake.

remains large (Re = 2000–8000) in all our experiments. Such values correspond toa weak horizontal momentum dissipation. However, according to previous stabilityanalysis (Kloosterziel et al. 2007; Plougonven & Zeitlin 2009; Lazar et al. 2013),strong stratification enhances the impact of the vertical dissipation on inertiallyunstable modes, by constraining the modes to small vertical scales. Thus, the verticaldissipation within the surface stratified layer plays a crucial role in the wavelengthselection and the vortex stability. We therefore introduce the vertical Ekman number:

Ek = ν

f h2c

. (2.3)

490 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

Exp Exp label N (s−1) Rd (cm) rmax (cm) Vmax (cm s−1)

1 C50 001 ∼ 1.4 50 30 2.02 C50 003 ∼ 1.1 47 23 1.03 C50 004 ∼ 1.1 47 30 2.44 C50 005 ∼ 1.1 47 26 2.95 C50 006 ∼ 1.1 47 22 2.96 C25 004 ∼ 2.0 97 11 0.97 C25 005 ∼ 2.0 97 15 2.58 C25 010 ∼ 1.7 84 15 4.29 C25 011 ∼ 1.4 69 12 1.410 C25 012 ∼ 1.4 68 14 3.5

TABLE 1. Physical dimensions, measured at the initial stage, of the anticyclones studied inthe laboratory experiment.

Without precise measurements of the vertical structure of the vortices (see § 2.4),we assume here that the typical height of the eddies scales with the height of thesubmerged cylinder hc. In what follows we will often use 1/Ek instead of the Ekmannumber, because it evolves as the Reynolds number. Small dissipation corresponds tolarge value of 1/Ek according to

1Ek= Re

δ2

Ro, (2.4)

where δ = hc/rmax is the aspect ratio parameter. In our experiments, δ remainssmall and varies between δ ' 0.15 and δ ' 0.6. The typical value of the Ekmannumber remains almost constant, and for most of the experiments corresponds to1/Ek ' 500–650.

2.3. Vertical stratificationIn order to mimic the oceanic density stratification we used a salt stratification. Wefirst filled the tank with a deep (∼50 cm) salty layer, ρbottom = 1040 g l−1. Due tothe slow Ekman recirculation, it took one day for this thick layer to reach a solidbody rotation. Then we used the double bucket technique (Oster 1965) to createa thin (∼5 cm) surface layer with linear stratification. In order to avoid residualmotions it was then necessary to wait at least one or two hours between consecutiveexperiments. We used a conductivity and temperature profiler (125MicroScale, http://www.pme.com) to measure accurately the vertical density profile and quantify theBrunt–Vaisala frequency N(z) = √−g∂zρ/ρ profile (see figure 2). In what followswe will use an averaged value over the height of the stratified layer as theBrunt–Vaisala frequency, i.e. N = ∫ 0

−h N(z) dz/h. After several days the free surfaceevaporation together with the molecular diffusion of salt tends to smooth theinitial stratification. Hence, the surface density and the upper layer thickness mayincrease from (ρ = 1005 g l−1, hs = 6 cm) to (ρ = 1020 g l−1, hs = 9 cm), leading toa significant decay of the mean Brunt–Vaisala frequency. However, it remains muchstronger than the Coriolis frequency (N ' 2 to N ' 1 s−1), and the experiments stillcorrespond to a strong stratification regime where N/f = 7–15.

As the motion is primarily in the upper layer (see § 2.1), the barotropic mode isnegligible, and the radius associated with the first baroclinic mode is the relevanthorizontal length scale. If we consider a purely linear stratification in the upper layer,

Inertial instability of intense stratified anticyclones. Part 2 491

Exp Ro ζ0/f Bu δ Re 1/Ek Stable

1 −0.48 −1.8 2.8 ∼0.15 6000 ∼350 32 −0.31 −1.1 4.3 ∼0.25 2200 ∼500 33 −0.56 −2.5 2.4 ∼0.20 7100 ∼500 34 −0.81 −2.5 3.7 ∼0.25 7000 ∼500 35 −0.95 −3.5 4.5 ∼0.25 6400 ∼500 X6 −0.70 −2.2 76 ∼0.60 990 ∼630 37 −1.20 −3.1 40 ∼0.45 3700 ∼630 38 −2.00 −6.2 31 ∼0.45 6300 ∼630 X9 −0.85 −2.6 31 ∼0.45 1700 ∼630 310 −1.77 −5.9 28 ∼0.50 4800 ∼630 X

TABLE 2. Initial dimensional parameters of the anticyclones and their stability after 2–6rotation periods.

15 20 25 30 0 0.5 1.0 1.5 2.035 40 45

–40

–30

–20

–10

–50

–40

–30

–20

–10

0

–50

0(a) (b)

FIGURE 2. (a) A typical density anomaly induced by the salt stratification, and (b) thecorresponding Brunt–Vaisala frequency.

the deformation radius of the first baroclinic mode is defined by Rd = Nhc/f . Thetypical values of Rd = 45–95 cm are larger than the typical vortex radius and lead toBurger numbers in the following range, from small to large:

Bu=(

Rd

rmax

)2

=(

N

f

)2( hc

rmax

)2

≈ 3.5–60. (2.5)

Therefore, the displacement of the isopycnal interface between the thin stratified layerand the deep barotropic layer is expected to be small or moderate (see appendix A).

2.4. Particle image velocimetry measurementsUnlike previous experiments of island wake flows with a two-layer stratification(Perret et al. 2006; Teinturier et al. 2010), we cannot use a horizontal laser sheetfor standard particle image velocimetry (PIV) because the upper layer is stratified,and the optical index gradient induced by the salt stratification will bend the laserbeam. In our large-scale experiments (the typical visualization area is 2 m × 2 m),

492 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

the laser sheet will curve out from the shallow stratified layer. Therefore, to performquantitative velocity measurements we used several powerful waterproof lamps (usedfor swimming pools) to illuminate small plastic particles of a buoyancy correspondingto 5–10 mm below the surface. Hence, we only measure the horizontal velocity field atthat surface and we have no information on the exact vertical structure of the velocityfield. Nevertheless, we expect to find the highest values of the velocity and vorticityfields at the free surface. The bottom of the tank was painted black to enhance thecontrast (figure 1b).

The particle motions were recorded with two 1024 × 1024 pixels CCD camerasand the frame rate was adjusted to the towing velocity of the cylinder to optimizethe PIV processing (1–5 frames per second). The velocities were analysed usingLaVision PIV software with successive overlap cross-correlation boxes yielding finally106 × 106 horizontal velocity vectors, corresponding to a velocity field resolution ofabout 1x ' 2 cm. In order to reduce the noise, the vorticity fields were derived froman average of ten consecutive velocity field frames. This does not affect the valuesdrastically, as the eddies drift slowly in the rotating frame, and this averaging periodcorresponds to roughly 0.05 of the time it takes an eddy to drift the length of oneradius τ = rmax/Vdrift.

3. Vortex characteristicsThe rotating von Karman street created by the cylinder has three different areas

(Boyer & Kmetz 1983; Tarbouriech & Renouard 1996; Stegner et al. 2005; Teinturieret al. 2010), which are subject to different types of instability: first, immediatelydownstream of the obstacle, the attached boundary layer, which may induce an intenseshear flow; second, a near wake, where the detached boundary layer shear rolls up intoelliptical vortices and the strain rate, D = 1/2(∂xv + ∂yu), can be significant; and thirdthe far wake, where the vortices are circular. In the near wake, the elliptical vorticesmight be unstable to elliptical instability if 2D − (2f − ζ ) > 0 (Cambon et al. 2006).However, this transient elliptical structure evolves into quasi-circular vortices after afew rotation periods. We start the analysis at this stage, in the far wake (at eightdiameter lengths downstream), where the vortices have no significant ellipticity. Weexclude cases where strong interactions between the vortices are detected, and restrictour analysis to vortices that are circular.

Both the azimuthal velocity V(r) profile and the corresponding vorticityζ(r) = ∂rV + V/r are needed to compute the Rayleigh discriminant χ(r) =[ζ(r)+ f ] [2V(r)/r + f ] and locate the unstable region where the inertial perturbationswill first grow and destabilize the circular vortices (Sipp et al. 2000; Billant & Gallaire2005; Kloosterziel et al. 2007). Hence, the main goal of the PIV measurements wasto quantify accurately the velocity profiles and the vorticity profiles for the variousvortices studied in this experiments. We plot in figure 3 the dimensionless azimuthalaveraging of the velocity and vorticity fields for a circular geostrophic anticyclone(Ro = −0.4) and a cyclogeostrophic anticyclone (Ro = −1.7). These measurementswere made at the initial stage of the vortex formation around t = 0.4− 1.5T0, just afterthe eddy detached from the cylinder.

As expected, the von Karman vortex street is made up of non-isolated vorticeswith monotonic vorticity profiles. The core vorticity of the anticyclones (cyclones) isnegative (positive) and tends to zero at the vortex edge. Such structures correspondto non-isolated vortices with typical velocity profiles decaying like 1/r outsidethe vortex core. Unlike the standard Rankine vortex, the core vorticity of theseexperimental wake vortices are not constant and they seem closer to a parabolic

Inertial instability of intense stratified anticyclones. Part 2 493

–1.2

–1.0

–0.8

–0.6

–0.4

–0.2

0

–2.0

–1.5

–1.0

–0.5

0

–1.2

–1.0

–0.8

–0.6

–0.4

–0.2

0

–7

–6

–5

–4

–3

–2

–1

0

0 0.5 1.0 1.5 2.0 2.5 3.0

0 0.5 1.0 1.5 2.0 2.5 3.0

0 0.5 1.0 1.5 2.0 2.5 3.0

0 0.5 1.0 1.5 2.0 2.5 3.0

RankineParabolic

Lamb–OseenConical

RankineParabolic

Lamb–OseenConical

RankineParabolic

Lamb–OseenConical

RankineParabolic

Lamb–OseenConical

(a) (b)

(c) (d)

FIGURE 3. Circular averages of (a,b) velocity and (c,d) vorticity profiles are plotted (opencircles) for two different experiments: (a,c) a geostrophic anticyclone from experiment 2at t ' 1.2T0, and (b,d) a cyclogeostrophic anticyclone from experiment 10 at t ' 0.4T0.The velocity and vorticity profiles of idealized Rankine (dashed line), parabolic (solidline), conical (dot-dashed line), and Lamb–Oseen (heavy solid line) eddies are plotted forcomparison.

profile, ζ(r)/f = 3Ro[1− (2/3)(r/rmax)2] (see again figure 3). Note that the exact radial

profile is not very important. As we saw in part 1 (Lazar et al. 2013), the stabilitycriterion of stratified and viscid eddies is insensitive to the vorticity profile, if one usesthe vortex Rossby number as the parameter for vortex intensity. However, the ratiobetween the core vorticity and the Rossby number,

Γ ≡(ζ0

f

)1

Ro, (3.1)

is very sensitive to the vortex profile (where ζ0 = ζ(r = 0) is the extremum corevorticity). We saw that it is Γ = 2, 3 and 4, for idealized Rankine, parabolic andconical vorticity profiles, respectively. For a Lamb–Oseen vortex, Γ = 3.51 (seeappendix B for details). If we plot the relative core vorticity as a function of the vortexRossby number for many vortices in their initial stage (figure 4), we get a line with a

494 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

8

6

4

2

0

–2

–4

–6

–8–2.0 –1.5 –1.0 –0.5 0 0.5 1.0 1.5 2.0

Anticyclone

Cyclone

FIGURE 4. Relative core vorticity, ζ0/f , of cyclones (black squares) and anticyclones (opencircles) as a function of the vortex Rossby number, Ro = Vmax/(f rmax), at the initial time ofmany different experiments. The slope is Γ0 = 3.4.

slope Γ0 ' 3.4 (where the subscript 0 denotes that it is for the initial time). Hence, atfirst order, these experimental vortices should be correctly approximated by parabolicor Lamb–Oseen vorticity profiles. Note that even for a low vortex Rossby number(Ro ' 0.35) the relative core vorticity may reach here a finite value (ζ/f < −1)corresponding to negative absolute vorticity (f + ζ0 < 0).

4. Low Rossby number vorticesAs a geostrophic reference case, we first compare the evolution of a cyclonic and

anticyclonic eddy corresponding to low Ro (as an example we take experiment 2in tables 1 and 2). According to figure 5, at the initial stage of formationthe characteristic velocity, Vmax, is very close to the cylinder speed, Vc, and thecharacteristic radius is almost equal to the cylinder radius, rmax ' D/2. In this case,the values of the vortex Rossby number, |Ro| = 0.31–0.35, and the cylinder Rossbynumber, Roc = Vc/(Ω0D) ' 0.27, are close. This is unlike previous experiments byTeinturier et al. (2010), where Vmax was significantly smaller than Vc (Ro ' Roc/2).In that case, however, the dissipation was stronger (1/Ek ' 200) and the upper layerstratification was weaker. Hence, it seems that the combined effect of weak dissipationand strong stratification tends to increase the intensity of von Karman wake vortices.

For low Rossby numbers, both cyclonic and anticyclonic vortices are expected tofollow the same quasi-geostrophic evolution. To perform a quantitative comparisonbetween vortices generated at different times, we fix the time origin, t = 0, of eacheddy as the time when the vortex centre is located at the distance D/2 behind thecylinder. Evidently, the cyclonic and the anticyclonic vortices have almost identicalvelocity profiles during their respective decay (see figure 5), and the vortex amplitude,|Vmax|, decreases while the corresponding radius, rmax, stays almost constant. Hence,

Inertial instability of intense stratified anticyclones. Part 2 495

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

AnticycloneCyclone

(a) (b)

FIGURE 5. The azimuthal averaged velocity profile of a cyclone (black squares) and ananticyclone (open circles) in experiment 2, at (a) t = 3.2T0 and (b) t = 6.5T0. The azimuthalvelocity is rescaled by the cylinder speed Vc, and the radius by the cylinder diameter D.The initial vortex Rossby number of the anticyclone was Ro = −0.35, and for the cycloneRo= 0.31. The values of the other dynamical parameters are shown in tables 1 and 2.

the decay of opposite sign vortices is strictly identical, even for a vortex Rossbynumber that is not very small (Ro = 0.31–0.35 at t = 0), and we may assume here alinear viscous dissipation for these wake vortices.

To perform systematic studies and quantitative inter-comparisons between the decayof various vortices we use the evolution of the two characteristic parameters (ζ0/fand Ro), and the ratio between them, Γ . According to Teinturier et al. (2010),the core vorticity, ζ0, and the maximum velocity, Vmax, evolve differently whenanticyclonic vortices experience inertial destabilization. In the present case there isno cyclone–anticyclone asymmetry in the temporal evolution of these parameters (seefigure 6). For these stable and coherent structures, the relative core vorticity decaysslightly faster than the vortex Rossby number, leading to a slight decrease of the ratioΓ = (ζ0/f )/Ro from 3.7 at t = 2To to 3 at t = 8To for both vortices. Indeed, thisexample lies deep in the stable side of the stability diagram (figure 12, which wediscuss in more detail in § 6).

5. High Rossby number vorticesAccording to linear stability analysis, the unstable inertial perturbations are initially

located in the unstable annulus where χ(r) < 0. The growth of the unstable inertialmodes induces small three-dimensional perturbations (Afanasyev & Peltier 1998;Afanasyev 2002; Potylitsin & Peltier 1998; Kloosterziel et al. 2007; Lazar et al. 2013),which may contaminate the anticyclonic core (Kloosterziel et al. 2007; Teinturieret al. 2010) and lead to a complete vortex breakdown (Kloosterziel & van Heijst1991; Orlandi & Carnevale 1999; Carnevale et al. 2011) for the most unstablecases. However, the combined effect of stratification and dissipation (viscosity)strongly reduces the strength of inertial perturbations and can prevent such vortexbreakdown, or even stabilize the vortex completely. Although small-scale and three-dimensional perturbations are easily detected by dye tracer visualization (Afanasyev

496 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

2.0

1.5

1.0

0.5

0 1 2 3 4 5 6 7 8

0.5

0.4

0.3

0.1

0.2

0 1 2 3 4 5 6 7 8

6

5

4

1

2

3

0 1 2 3 4 5 6 7 8

CycloneAnticyclone

(a)

(b)

(c)

FIGURE 6. Evolution of (a) the relative core vorticity, (b) the vortex Rossby number, and(c) their ratio Γ , for the cyclone (black squares) and the anticyclone (open circles) shown infigure 5 (experiment 2). The parameters of the experiment are shown in tables 1 and 2.

2002; Teinturier et al. 2010), their signature on the mean horizontal velocity fieldmeasured by standard PIV is less visible, and a careful analysis of the vortex evolution(again using the parameters ζ0/f , Ro and Γ ) is needed in order to determine whether avortex is stable.

Inertial instability of intense stratified anticyclones. Part 2 497

80 120 160 200 240mm

80 120 160 200 240mm

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FIGURE 7. The surface vorticity field of experiment 10 at: (a) t = 0.35To, (b) t = 1.4To,and (c) t = 2.3To. The colour panel of cyclonic (red) and anticyclonic (blue) vorticity isnormalized by the Coriolis parameter f = 2Ω0. The parameters of the experiment are shownin tables 1 and 2.

5.1. Unstable vorticesIn figure 7 we plot the evolution of the surface vorticity field for a high Rossbynumber wake generated by a small cylinder (D = 25 cm) towed at high speed(Vc = 4 cm s−1). The cylinder Rossby number corresponding to this forcing isRoc = Vc/(Ω0D) ' −2.3. This set of parameters produces one of the most intensevortices in all of our experiments (experiment 10 in tables 1 and 2), with relativecore vorticity of up to |ζ0/f | ' 6.5 and vortex Rossby numbers of Ro ' −1.8. Suchintense anticyclones in rotating shallow water experiments are very hard to create byother methods. These vortices are too intense to be in a geostrophic balance, and wecan assume that the mean flow is in a cyclogeostrophic balance instead. Accordingto the generalized Rayleigh criterion (Kloosterziel & van Heijst 1991; Mutabazi et al.1992; Sipp et al. 2000), these strong anticyclones are well above this inviscid stabilitylimit. But surprisingly, these anticyclones do not break down and they remain circularand coherent for several rotation periods, and there is no signature of small-scaleperturbations in the vorticity field, as shown in the numerical simulations of Dong,McWilliams & Shchepetkin (2007). We do, however, detect a cyclone–anticycloneasymmetry. The radius of the core of these intense anticyclones shrinks much fasterthan that of the cyclones. Note that the azimuthal ribs evident on both cyclonesand anticyclones are fixed to the eddies and do not propagate around as you wouldexpect from a wave structure. It is a well-known artifact of PIV processing calledpeak-locking, which is a result of the discrete correlation values used for the curvefitting used to obtain the sub-pixel part of the displacement, and occurs generally whenthe particle image size is smaller than optimal, i.e. 2–4 pixels.

To quantify this asymmetry more precisely we focus on the evolution of two suchcyclogeostrophic eddies, the cyclone C2 and the anticyclone A2 defined in figure 7.Like geostrophic eddies, these too have almost identical velocity profiles at their initialstage of formation, and the characteristic velocities and radii are both very close to

498 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

1.0

0.8

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0.8

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0 0.25 0.50 0.75 1.00 1.25 1.50

CycloneAnticyclone

(a) (b)

FIGURE 8. The azimuthal averaged velocity profile of the cyclone C2 (black squares) andanticyclone A2 (open circles) indicated in figure 7, at: (a) t = 0.4T0, and (b) t = 1.9T0.The azimuthal velocity is rescaled by the cylinder speed Vc, and the radius by the cylinderdiameter D. Between the two dashed lines in both panels the Rayleigh discriminant of theanticyclones is negative.

the cylinder speed (Vc) and radius (rmax ' D/2), respectively (see figure 8a). However,in this case the maximal azimuthal velocity of the anticyclone decays much fasterthan its cyclonic counterpart (figure 8b). This anomalous decay of the mean azimuthalvelocity profile is a signature of the growth of inertial perturbations, or in other words,the three-dimensional recirculating cells initially localized in the unstable annulus atthe vortex edge, where χ(r) < 0. Similar results were found in numerical simulations(Kloosterziel et al. 2007) and previous laboratory experiments (Teinturier et al. 2010).The evolution of ζ/f , Ro, and the ratio between them are plotted in figure 9. Thecyclone–anticyclone asymmetry is apparent for the vortex Rossby number, caused bythe perturbations at the vortex edge, but not for the relative core vorticity, which is leftalmost unaffected.

Thus, the ratio Γ = (ζ0/f )/Ro is a relevant parameter to detect the signature ofinertial instability from the mean horizontal velocity field, as it essentially measuresthis change of the vorticity profile with time. According to figure 9, Γ increaseswith time for the unstable anticyclone A2 and may reach high values, up to 5, muchlarger than the initial values Γ0 = 3.4 (figure 4), while for the stable cyclogeostrophiccyclone C2 this ratio slightly decreases down to 3, similar to the geostrophic eddieswith small Rossby numbers. Thus we conclude that this vortex is indeed affected bythe growth of localized centrifugal–inertial perturbations, and is unstable, even thoughit did not reach complete breakdown. Although this vortex is well beyond the Rayleighcriterion it is only slightly unstable, which is in agreement with it being very close tothe marginal stability limit developed in part 1 (Lazar et al. 2013), in figure 12 (whichwe will elaborate on in § 6).

5.2. Stable vorticesAccording to the evolution of the ratio Γ , we find stable vortices with high vortexRossby numbers. For instance, experiment 7 seems to be stable, although it has astronger vortex Rossby number than the unstable experiment 5. It has an almost

Inertial instability of intense stratified anticyclones. Part 2 499

CycloneAnticyclone

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(a)

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(c)

FIGURE 9. Evolution of (a) the relative core vorticity, (b) the vortex Rossby number, and (c)their ratio Γ , for the cyclone C2 (black squares) and the anticyclone A2 (open circles) shownin figure 7.

constant Γ for more than 4 rotation periods for both the cyclone and the anticyclone(see figure 10a). The ratio Γ is also approximately constant for experiment 4 (seefigure 10b). Note that the relative core vorticity of both anticyclones is smallerthan the Coriolis parameter ζ0/f < −1, but we do not detect any signature ofinertial instability over a period of several rotations (t ∼ 4T0). Thus, the negative

500 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

6

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0 1 2 3 41 2 3 54

(a) (b)

FIGURE 10. Evolution of Γ for a cyclone (black squares) and an anticyclone (open circles)in (a) experiment 7, and (b) experiment 5. The parameters of the experiments are shown intables 1 and 2.

absolute vorticity criterion (i.e. ζ + f < 0, or the generalized Rayleigh criterion,χ(r) = (ζ(r)+ f ) (2V(r)/r + f ) < 0) does not apply here. This is in agreement withthe stability analysis in part 1, which showed that the combined effect of dissipation(here, molecular viscosity) and the vertical stratification significantly stabilizes theseintense vortices. The anticyclone of experiment 7 may be more intense than the one inexperiment 5, but its Burger number is much higher, and experiment 4 is quite close to(but on the stable side of) the viscous marginal stability curve found in part 1 (whichwe discuss in detail in § 6).

5.3. End-state of an unstable anticycloneAn additional objective of this study was to estimate the end result of thedestabilization process. Previous numerical studies (Kloosterziel et al. 2007; Carnevaleet al. 2011) at high Reynolds numbers have shown that the nonlinear evolutionof three-dimensional inertial perturbations induce a redistribution of the angularmomentum, and lead to a solid body rotation opposite to the rotation of the tank,inside the anticyclonic vortex core (Vθ(r) = −fr/2), and a sharp transition of theazimuthal velocity to the original velocity profile outside the core. For the non-stratified (N = 0) or weakly stratified (N/f ' 1) cases this equilibration process canresult in anticyclones that are susceptible to horizontal shear instabilities even if theywere not initially susceptible (Carnevale et al. 2011). Therefore, a small-scale andrapid inertial instability may trigger a slower barotropic instability at larger scale.However, the combined effect of dissipation and stratification may strongly affect thisnonlinear equilibration process. Indeed, according to the two-dimensional simulationsof Kloosterziel et al. (2007), when the stratification becomes substantial, N/f = 5–6(h/rmax = 1, Ro = −1 and Re = 104), the redistribution of angular momentum is weakand barely affects the velocity profile even after ten rotation periods.

To estimate the typical end result of an unstable submesoscale (i.e. large Burgernumber) anticyclone we again study the evolution of one of the most intense vorticesin our data set (experiment 10) for a few rotation periods. This case correspondsto a very negative core vorticity ζ0/f ' −5.8, a strong stratification N/f = 10 and

Inertial instability of intense stratified anticyclones. Part 2 501

a moderate aspect ratio h/rmax = 0.53. Unfortunately, due to memory limitation, werestricted the camera acquisition to a few thousand images and were only able tofollow the vortex evolution during 3–4 rotation periods (i.e. 1t = 300–350 s). In orderto reduce the experimental noise on the velocity field and its derivatives, we fit acircular velocity profile 〈Vθ(r)〉 with an analytical function, and get a smooth velocityderivative ∂rVθ . We assume a parabolic vorticity profile at the initial time (t = 0.5To)and a fifth-order polynomial in later stages.

The evolution of the anticyclonic velocity profile (see figure 11a) differssignificantly in this stratified experiment, from the weakly stratified simulations ofKloosterziel et al. (2007), shown in their figure 5, for instance. The small-scale mixinginduced by inertial perturbations smooths the velocity profile around the peak velocity,but does not change the vortex core (r 6 0.5rmax), nor does it intensify the velocitygradient at the vortex edge. The rapid decay of the negative part of the Rayleighdiscriminant χ(r) = (ζ(r)+ f ) (2V(r)/r + f ) (figure 11b) indicates a rapid evolutiontowards a more stable state. In approximately one rotation period, both the averagedvelocity profile and the corresponding Rayleigh discriminant reduce their amplitudesignificantly, after which the anticyclone seems to reach a new equilibrium state thatevolves slowly. The vorticity profile of this new state is more peaked towards thecentre, and there is rapid diffusion around the unstable annulus while the relative corevorticity remains high (ζ0/f ' −4.5 at t = 3.4To). It is as if the combined effect ofdissipation and stratification restricts the radial extension of the unstable perturbations.This seems to induce localized mixing and dissipation only around the peak velocity,where the generalized Rayleigh discriminant was, at the initial state, negative. Hence,the relative core vorticity of intense submesoscale anticyclones can retain a largenegative value for several rotation periods. Unfortunately, we did not measure thelong-term evolution of such unstable anticyclones. Perhaps another final end-state isreached after tens of rotation periods. However, in such long periods of time theimpact of viscous dissipation should become significant.

Another interesting issue is the evolution and the redistribution of angularmomentum by the three-dimensional inertial perturbations. According to axisymmetriccylindrical (Kloosterziel et al. 2007) and three-dimensional (Carnevale et al. 2011)numerical simulations, the mean absolute angular momentum L(r) = r 〈Vθ(r)〉 + r2fapproaches zero from the eddy centre to the edge of the unstable region. The evolutionof the averaged angular momentum at the surface for the same intense and stratifiedanticyclone is plotted in figure 11(c). We do indeed measure an attenuation of thenegative values of the angular momentum, due to the reduction of the peak velocity,but there is no global redistribution of the angular momentum with a decay of thepositive values at the outer edge as shown in figure 11 of Kloosterziel et al. (2007).The full three-dimensional redistribution and the viscous dissipation may invalidatehere the assumption of a full mixing between negative and positive angular momentumin the unstable region.

6. Stability diagram and comparison with linear stability analysisIn order to quantify more precisely the stability of anticyclones, we studied the

temporal evolution of the ratio Γ = (ζ0/f )/Ro. This ratio is initially around Γ0 ' 3.4,for the vortices created in our experiments. When it increases with time and reachesvalues larger than Γ = 4.5 within a few rotation periods, as in figure 9(c), we considerthe anticyclone unstable to inertial perturbations. Conversely, if this ratio remainsconstant or slightly decays with time, as in figures 6(c), 10(a) and 10(b), we consider

502 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

–1.0

–0.8

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(a)

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FIGURE 11. (a) The velocity profile, (b) the Rayleigh discriminant, and (c) the angularmomentum, for an unstable anticyclone from experiment 10, at times t = 0.5To (dashed line),t = 1.5To (solid line), and t = 3.4To (dotted line). The parameters of the experiment are givenin tables 1 and 2.

the anticyclones stable. We plot in figure 12 the locations of all the anticyclonesfrom the experiments given in tables 1 and 2 on the (Rd/rmax =

√Bu,Ro) parameter

space. Vortices that were found to be stable, by the above criterion, are indicated

Inertial instability of intense stratified anticyclones. Part 2 503

10

0.2 0.4 0.6 0.8 1.0 3.0

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500

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Stable viscous

Stable inviscid

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FIGURE 12. The initial location of stable (open circles) and unstable (black circles)anticyclones on the (Rd/rmax =

√Bu,Ro) parameter space. The experiment numbers

according to tables 1 and 2 are indicated. The thick solid line corresponds to Fd = Ro/√

Bu=1, the transition from subcritical to supercritical flow. The vertical dashed line corresponds tothe inviscid stability limit predicted by the generalized Rayleigh criterion for a parabolicvortex. The curves are the marginal stability limits for a parabolic vortex (dot-dashed),with 1/Ek = 500 and 1/Ek = 1000 (indicated), and the corresponding analytical marginalstability curves (solid), according to part 1 (Lazar et al. 2013). Grey trajectory lines in theinset show the time evolution of the unstable experiments up to approximately four tankrotation periods. The area of the parameter space shown in the inset is indicated by a thindashed line.

by open circles, and vortices found to be unstable are solid circles (the experimentnumber is indicated). As expected, unstable vortices are found for sufficiently largeabsolute values of the vortex Rossby number. However, for the same value of Ro(or even higher), we also find stable anticyclones. The time evolution of the unstableexperiments (shown in the inset of figure 12) describes the trajectory of the unstableanticyclones on the parameter space, reaching a stable configuration within a fewrotation periods.

The thick solid line in figure 12 is the Froude limit (Fd = 1) which corresponds tothe transition from subcritical to supercritical flow. As far as we know, supercriticalvortices cannot exist without a permanent external forcing such as the sink eddy.In our experiment the vortices are not forced. If we were to tow the cylinderat a sufficiently high speed, the flow around the cylinder would become locallysupercritical and high amplitude lee waves would be generated. An example of asupercritical wake is shown in Teinturier et al. (2010). A V-shaped lee wave pattern

504 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

attached to the moving obstacle is visible without any vortex formation in the nearwake. Further down (a few diameters behind the obstacle, in the far wake) the wakeflow is destabilized and a weak vortex street emerges, with much smaller relativevorticity than would be expected from the towing speed. Hence, we do not expect tofind any free vortices below the Froude limit.

One of the main goals of this study was to perform a quantitative laboratoryevaluation of the linear stability calculations for inviscid and viscous (i.e. dissipative)vortices, given in part 1. The stability analysis of circular barotropic vortices withina linearly stratified shallow layer (Lazar et al. 2013) provides an asymptotic marginalstability limit for viscous eddies, which is a function of the Ro,Bu and Ek numbersin the hydrostatic approximation (the Burger number combines both the stratification,N/f , and aspect ratio, h/rmax). It was developed for a Rankine vortex, and was shownto be insensitive to the vortex profile. This marginal stability limit, written in the(Rd/rmax,Ro) parameter space, is

Rd

rmax=√Bu=

[3

8 |a0|]3/2 1√

Ek

(|2Ro+ 1|)7/4|Ro| , (6.1)

where Ro is the vortex Rossby number defined in (2.1) (Ro < 0 for anticyclones),Ek is the Ekman number and a0 = −2.3381 is the first zero of the Airy function.In figure 12 we superimpose this marginal stability curve for two values of theEkman number (1/Ek = 500, 1000) with a solid line. The eigenvalue problem for themarginal stability of parabolic vorticity profiles was calculated numerically, since thereis no direct analytical solution, and the resulting stability curves for the same Ekmannumbers are indicated by dot-dashed lines. These two sets of lines are indeed veryclose, as the inertial instability, within a dissipative and a strongly stratified layer, isnot sensitive to the vorticity profile, as we showed in part 1. The dashed vertical linecorresponds to the inviscid marginal stability limit for a parabolic vortex, according tothe inviscid Rayleigh criterion (Kloosterziel & van Heijst 1991; Mutabazi et al. 1992),which translates simply to a negative absolute core (ζ0/f =−1 or Ro=−1/3).

We find agreement between the experimental data and the theoretical prediction.We confirm with these laboratory experiments that the combined effect of dissipation(molecular viscosity) and the vertical stratification induces a significant stabilization ofintense anticyclonic vortices. Even for moderate dissipation (Re = 1000–7000, 1/Ek >500) we found several cases of stable anticyclones that should have been unstableaccording to the Rayleigh criterion. Except for experiments 1 and 2, which may havebeen predicted to be stable according to the Rayleigh criterion (depending on the exactvorticity profile), the other experiments (e.g. 6, 4 and 9) are far into the unstablearea, and experiment 7 has an even higher Rossby number and core vorticity than theunstable experiment 5.

The three unstable vortices are close to the marginal stability curve, and indeedthey are unstable but only slightly, and do not completely break down. We did notmanage to create vortices deep in the unstable region, as our method limits theintensity of the vortices created. Increasing the towing speed over a certain speed willcreate a supercritical flow around the cylinder, which will generate a strong wavewake, and as a result, will reduce the vortex street intensity, as was observed byTeinturier et al. (2010). Moreover, even when not supercritical, some of the energycould dissipate in the near wake (due to inertial instability within the boundarylayer or elliptical instability within the elliptical vortices) before creating a regularwake of circular vortices. We note, however, that the vortices are slightly above the

Inertial instability of intense stratified anticyclones. Part 2 505

marginal stability curve 1/Ek = 500 while their Ekman numbers are estimated around1/Ek ' 500–630. Thus, the two-dimensional barotropic vortices used for the theoreticalanalysis might be oversimplified in comparison with the three-dimensional experimentafter all. However, this slight underestimation of the unstable region might stem froma different, more simple, reason. The quantitative estimation of the Ekman number,which is most crucial, as we can see here, depends on the square of the eddy thicknessh. Unfortunately, we do not have direct measurements of h, and we estimate its valueas close to the height hc = 5–7 cm of the cylinder immersed into the upper layer.However, if we use for h the typical value of the upper layer stratification hs, we geta higher characteristic thickness closer to 8–9 cm. This leads to 1/Ek ' 1000, a betteragreement of the marginal stability curve for the three unstable anticyclones. From thefact that experiment 4 is stable, we see that the Ekman number cannot be much lower.This quantitative comparison confirms that the simplified model used for the stabilityanalysis provides, at first order, a very good estimate of the unstable region in the(Ro,Bu,Ek) parameter space.

7. Summary and discussionWe studied the stability of intense and stratified vortices, with absolute values

of core vorticity larger than f , by means of large-scale laboratory experimentsperformed on the Coriolis platform. We generated several von Karman wakes toobtain cyclonic and anticyclonic vortices of various sizes and strengths within a thinupper stratified layer. Particle image velocimetry was used to quantify accurately thevelocity and the vorticity profiles of the vortices at the surface of the upper layer.The dynamical evolution of both eddies was compared to detect the signature of theselective destabilization of anticyclonic eddies by inertial–centrifugal perturbations.Unlike previous numerical studies (Dong et al. 2007; Kloosterziel et al. 2007),we did not observe a rapid or complete breakdown of anticyclones with a coreof negative absolute vorticity (negative potential vorticity). However, in a numberof intense anticyclones, we detected an anomalous decay of the mean azimuthalvelocity profile in the unstable annulus defined by the generalized Rayleigh criterionχ(r) = (ζ(r)+ f ) (2V(r)/r + f ) < 0. Such localized dissipation induces a rapid decayof the maximal azimuthal velocity Vmax of the unstable anticyclones, while their corevorticity ζ0 remains almost constant.

Hence, the evolution of the dimensionless parameter Γ = (ζ0/f )/Ro = (rmaxζ0)/Vmax

is a signature for possible inertial instability. In unstable anticyclones this parameterΓ increases rapidly, while for stable eddies this parameter remains almost constantor weakly decays due to viscous dissipation. The location of the stable and unstableanticyclones in the (Rd/rmax,Ro) parameter space is in agreement with the stabilityanalysis performed in our first paper (Lazar et al. 2013). The unstable eddies were allfound close to the marginal stability limit predicted by the analytical equation (6.1),which explains why they did not completely break down. Moreover, six intenseanticyclones with negative relative core vorticity, down to ζ0/f ' −3, were foundto be stable. Such cases show that the combined effect of strong stratification andmoderate dissipation strongly stabilizes the intense anticyclones. This confirms that theinviscid Rayleigh criterion is not relevant to stratified viscous eddies, even for largehorizontal Reynolds numbers (2000–7000), while the newly proposed stability criterionis relevant.

This mechanism may be of special interest in the ocean, where not only arestratification and dissipation important, but the submesoscale plays an important role in

506 A. Lazar, A. Stegner, R. Caldeira, C. Dong, H. Didelle and S. Viboud

the transition of energy in a direct cascade towards small-scale dissipation and mixing.As we mentioned in § 1, this was the motivation to study a setup resembling oceanicconditions. A von Karman wake was used for two main reasons: first, because it isa comfortable way of producing non-isolated (i.e. not sensitive to shear instabilities)cyclones and anticyclones of the same type and strength, and second, because islandwakes are the most likely places to find submesoscale anticyclones in the ocean.However, using this analysis on real oceanic eddies is difficult, first and foremostbecause the Ekman number is very difficult to estimate, unlike in our experimentswhere it is safe to assume the molecular viscosity, and second, in the relevant vortexdimensions, the submesoscale, we usually only have a snapshot of the dynamicalparameters or the flow field, if anything, but no time evolution. However, in Chavanne,Flament & Gurgel (2010) an accurate temporal evolution of the velocity structure ofan intense anticyclonic eddy in the lee of Hawaii is provided by a high frequencyradar (HFR) current meter with a spatial resolution of 1–2 km. When the vortex wasfirst detected as fully developed (26 October 2012) the vortex Rossby number wasRo ' −0.45. With HFR the local vorticity values can also be directly quantified, andthe core vorticity of this anticyclone was ζ0 = −1.5f . Note that the vortex seems likea non-isolated vortex throughout its evolution (see figure 6b of Chavanne et al. 2010),and that the ratio between the relative core vorticity and the vortex Rossby number(ζ0/f )/Ro∼ 3–4 is in good agreement with parabolic or conical vorticity profiles. Thisanticyclone is predicted to be unstable according to the standard Rayleigh criterion.However, as the authors note, it is stable, and indeed it can be seen from figure5(a,b) of Chavanne et al. (2010) that the time evolution of the ratio Γ shows adecay over a period of a few days. In order to predict its stability with the stabilityanalysis of part 1 (Lazar et al. 2013), the Burger number and the Ekman numbersshould be estimated. We estimate the mean Brunt–Vaisala frequency, N, measuredat a nearby station, and the typical depth of eddy h, with the measurements froman ADCP at the same station (figure 9 of Chavanne et al. 2010), and together withthe radius of the eddy get a Burger number of Bu ' 3. The Ekman number is evenmore difficult to estimate. Assuming that this was formed by strong shear flows orsignificant wind stress curl, we will use, as a first guess, a large diapycnal diffusivityof κz ∼ ν ∼ 10−4 m2 s−1, keeping in mind that it may vary by one order of magnitude,and we get 1/Ek ∼ 5000. These parameters correspond to a stable configuration withthe new criterion, even if the dissipation is weaker by an order of magnitude. Similarto the experiment, stratification together with the dissipation stabilizes this intensevortex, and the new criterion predicts this effect correctly. More data for intenseanticyclones of a similar quality are needed in order to evaluate this criterion in realoceanographic conditions, but from this case it seems promising.

Acknowledgements

These experiments were part of the TIRIS (Three Dimensional Vortex StreetInstability at High Reynolds Number around IslandS) project, funded by the6th European Commission (EC) Framework Program Hydralab III, within theTransnational Access Activities, Contract 022441. A.L. would like to thank the IsraeliMinistry of Science and Technology (MOST 3-6490) for her funding. A.L. and A.S.acknowledge the sustainable development chair of the Ecole Polytechnique for hissupport as well. C.D. is supported by ONR grant N00014-10-1-0564. Travel expensesof R.C. were partially funded by CIIMAR (PesT-C/MAR/LAOO15/2011). The authors

Inertial instability of intense stratified anticyclones. Part 2 507

are very grateful to E. Heifetz for his enlightening input, and J. Sommeria andL. Gostiaux for all their help.

Supplementary materialSupplementary material are available at http://dx.doi.org/10.1017/jfm.2013.413.

Appendix A. Scaling of isopycnal displacementConsider a stationary axisymmetric vortex in a shallow layer. It should satisfy the

equation

V2/r + fV = g∂rη, (A 1)

where V is the azimuthal velocity of the vortex and η is the vertical displacementfrom the original height of the shallow layer, which is H. We define the fractionaldisplacement as λ ≡ η/H. If we scale the velocity by the maximal vortex velocity,Vmax, and the radial scale by the radius in which this velocity is reached, rmax, (A 1)gives us a scaling for the fractional displacement:

λ∼ Ro (Ro+ 1)Bu

. (A 2)

The Rossby number is as defined in (2.1), Ro = Vmax/frmax, and the Burger number isas defined in (2.5), Bu = (rmax/Rd)2, but with a Rossby radius of deformation definedfor shallow water as Rd = √gH/f . Hence the displacement is inversely proportionalto the Burger number, which is quite large (see table 2), and directly proportional tothe Rossby number, which in our experiments is moderate even for the most intensevortices. Also, notice the asymmetry between positive and negative Rossby numbers,where anticyclones with |Ro| < 1 will have a smaller displacement in absolute valuethan cyclones. In our experiments this results in a fractional isopycnal displacement ofless than 1 % for half of the experiments and of the order of 5 % in the rest.

Appendix B. Lamb–Oseen vortexA Lamb–Oseen vortex defined by the maximal tangential velocity Vmax (similar to

the definition in Fabre, Sipp & Jacquin 2006) is

V(x)= VmaxC

x

(1− e−αx2

), (B 1)

where x= r/rmax, and the parameters α = 1.25643 and C = 1+1/2α are chosen so thatthe maximal velocity, V = Vmax, will occur at r = rmax (or xmax = 1). The correspondingvorticity is ζ(x)= 2αCVmax/rmax e−αx2

, of which the maximal absolute value is obtainedin the core (x = 0). The normalized core vorticity is ζ0/f = 2αCVmax/(rmaxf ). This canbe written in terms of the Rossby number Ro = Vmax/(rmaxf ) as ζ0/f = 2αCRo, whichgives a ratio between normalized vorticity and Rossby number of Γ = 2αC.

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