Dynamics of the Ocean Rings on the Beta-Plane inQuasi-Geostrophic Two-Layer Model

Mike RudkoUniversity of Miami-RSMAS

1 Introduction

Quiet often we encounter very long-lived, coher-ent structures in nature. In the atmosphere, thelifecycle of tropical cyclones and hurricanes maybe up to one month, while so-called oceanic ringscan live up to several years. This study is pri-marily devoted to the oceanic features, howeverthe same philosophy may be applied to the at-mosphere as well. The process of formation andpropagation of such structures is a long-standingproblem in geophysical fluid dynamics. Typically,rings are formed due to the retroflection of thewestern boundary currents, such as Gulf Stream,North Brazil Current, Agulhas etc. (Castelaoet al., 2011; Olson, 1991). Numerous observa-tional studies (Castelao et al., 2011; Joyce et .al ,1992) suggest that the maximum swirl velocity ofthe rings ranges from 0.5 m/s to 1.3 m/s with ra-dius of maximum velocity from 100 km to 160 km.The total size of the rings is about 400 km.

The main goal of this study is to examinethe dynamics of the oceanic rings and their in-teraction with the environment. Fiorino et al.(1988) investigated the dynamics of tropical cy-clones (as it was said above, cyclones in the at-mosphere are dynamically similar to ocean rings)and found that the interaction with the environ-ment is a very complicated process. It turned outthat the size of the vortex (i.e. the radius of max-imum winds) significantly changes the trajectoryof the vortex, while, at the same time, the inten-sity doesn’t contribute much to the propagation.Unlike in the atmosphere, in the ocean the cen-tral role in the behavior of the vortex plays theβ-effect (meridional gradient of planetary vortic-ity). Reznik et al. (1994) showed that the β term

in the vorticity equation creates dipolar circula-tion (“β-gyres”), which, in turn, pushes the vor-tex along its trajectory. But because of the rota-tion of the vortex, the axis of dipolar circulationis also rotated. The direction of the axis rotationdepends on the sense of vortex-counterclockwiseand clockwise for cyclonic and anticyclonic vor-tices, respectively. Thus, the overall direction ofpropagation of the cyclone (anticyclone) is to thenorthwest (southwest). Another intriguing fact,which was pointed out is that for times larger thenO(βL)−1, where L is characteristic length scale ofthe vortex, the radiation of Rossby waves proba-bly cannot be neglected.

The fact that stratification may influence themotion of the vortex was considered in severalstudies. In particular, Flierl (1994) using two-layer quasi-geostrophic (QG) model (the simplestmodel of stratified ocean) gives the time scale forwhich the Rossby wave radiation becomes effi-cient of T = 1

δβL , where δ is the layer depth ratio.Reznik et al. (2007), using the same model, inves-tigated the dynamics of point vortices. The mainresults from their studies are as follows: the prop-agation of a mixed-mode vortex (initially it hasboth baroclinic and barotropic components) dif-fers from pure barotropic (only barotropic mode ispresent initially) and pure baroclinic (only baro-clinic mode is present initially) vortices. However,the influence of baroclinicity attenuates with thepassage of time, and the flow becomes essentiallybarotropic.

Given our current understanding, there areseveral questions which remain unanswered. Howdoes the vortex interact with the backgroundflow? How does stratification influence the trajec-tory of distributed vortices? What is a mechanism

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of interaction between a vortex and Rossby waves.To address these questions we investigate the dy-namics of the vortex under different regimes (withand without background flow). The evolution ofbeta-gyres is computed and analyzed.

In section 2 we describe the model and themethodology of the numerical experiments. Theresults are described in section 3. In section 4 wediscuss our results and make conclusions.

2 Model and Methodology

The numerical model is based on a traditional QGapproximation supplemented with lateral viscos-ity and bottom friction (Pedlosky , 1987). Themodel represented by two stack isopycnal layersallows us to represent the ocean stratification inthe simplest possible way (Berloff et al., 2009).The governing equations are the conservation ofpotential vorticity in each layer:

∂

∂t(∇

2ψ1 −1

R21

(ψ1 − ψ2)) + β∂ψ1

∂x+ J(ψ1,∇

2ψ1) =

ν∇4ψ1

∂

∂t(∇

2ψ2 −1

R22

(ψ2 − ψ1)) + β∂ψ2

∂x+ J(ψ2,∇

2ψ2) =

ν∇4ψ2 − γ∇2ψ2

where ψn is the streamfunction in layer n, n = 1,2(indices 1,2 hereafter refer to the top and bottomlayers, respectively), J(a, b) = ∂a

∂x∂b∂y −

∂a∂y

∂b∂x is the

Jacobian operator. The stratification parameters

Rn are given by Rn =1f

√δρρ gHn and Rossby de-

formation radius is defined as Rd = R1R2√R2

1+R22

. Here

g is a gravitational acceleration, ρ is a mean den-sity and δρ is a density difference between twolayers. Hn stand for the thicknesses of the layers.In this study we used two layers of equal thick-ness, H1 = H2 = 2 km. The value of the Rossbydeformation radius is 25 km and β = 2 ∗ 10−11

m−1s−1. For almost all experiments the domainsize is 2560 × 2560 km, with a resolution of 5 kmin both directions. To compare our results withbarotropic dynamics, we increased Rd to 1000, sothat vorticity equations are decoupled.

The model was initialized with a Gaussianprofile ψ1 = a exp(− r

2

b2 ) in the upper layer andψ2 = 0 in the lower layer. Note that in this casethe vortex contains both barotropic and baroclinicmodes. The parameters a and b are defined asa = VmaxRmaxexp(

12), b =

√

2Rmax, where Vmax isa maximum swirling velocity and Rmax is a radiusof maximum swirling velocity. For all runs, wechose the realistic oceanic values of Vmax = 1m/sand Rmax = 100 km. Vertical shear may be easilyintroduced if needed. However, it should be em-phasized, that the model artificially maintains thevertical shear throughout the integration. For thecase where horizontal shear (Ψ = Uy2) was used,

the model was initialized as ψ1 = a exp(− r2

b2 )+Uy2

,ψ2 = Uy2. The different values for U were chosenso that background velocity is equal to 6,10,−6or −10 cm/s in the center of the domain and 0 atthe left bottom corner.

To find the trajectory of the vortex we need toknow the location of the vortex at each time step.In this study the center of the vortex is defined asan extremum of the streamfunction at each timestep. Following the traditional approach we de-compose the streamfucntion into barotropic andbaroclinic components:

ψ1 = ψBT +H2

H1

ψBC

ψ2 = ψBT −H1

H2

ψBC

where ψBT is a barotropic streamfunction andψBC is a baroclinic streamfucntion. Each of themodal streamfunction may be decomposed intosymmetric and asymmetric parts:

ψBT = ψBTs + ψBTa

ψBC = ψBCs + ψBCa

It is the asymmetric part of the streamfunctionthat defines the evolution of “β-gyres”. Thus, instratified fluid we must consider two types of beta-gyres: barotropic and baroclinic.

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3 Results

3.1 Vortex Track, No Environmen-tal Influences

Figure 1 shows the vortex trajectory for a 70-daysimulation. For the first 20 days, the trajectory ofcyclonic vortex follows the anticipated trajectorytoward the northwest. However on the 21st daywe clearly observe a jump to the east. After this,the vortex again moves along a smooth trajectoryto the northwest, but on the 40th day we againsee an eastward jump which is even larger. Af-ter the 50th day, the vortex is moving perfectlywestward.

Figure 1: Vortex track with no environmental in-fluences, baroclinic case. After 50th day the vor-tex translates to the west.

The vortex track in the barotropic dynamics(see fig.2) is very similar to the baroclinic case forthe first 20 days. However after the 21th day, thevortex starts to move strictly westward, while forbaroclinic case it happens only after the 50th day.The associated wave timescale is 1

δβL = 5.8 days(it is the same for barotropic run, because δ = 1).Figure 3 shows the snapshots of the upper-layerstream function for the 1, 5, 21 and 70 days. Notethat for the 1st and 5th day the vortex core re-mains almost undisturbed. For longer times, theRossby waves come into play and the perturba-tions extract energy from the vortex.

Figure 2: Vortex track with no environmental in-fluences, barotropic case. After 21st day the vor-tex translates to the west.

Figure 3: The snapshots of upper-layer streamfunction at 1, 5, 21 and 70th day. No environ-mental influences.

One central finding of this study is that amixed-mode vortex experiences clearly observedjumps, while this does not happen in barotropicsystem. A somewhat similar result was reportedin the work Reznik et al. (2007) in which analyti-cal theory for barotropic dynamics was comparedto the numerical simulation (also barotropic) forGausian vortices. The analytical prediction ofposition of the vortex coincides with the modelresults for the first 9 days (the wave timescale).However, the theory overestimates displacementof the vortical structure, because there is no lossof amplitude of the mean-state vortex due to the

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wave activity. More importantly, their barotropicmodel does not exhibit the jump-like behavior ob-served in this study.

3.2 Vortex Track, Vertical andHorizontal Shear

The rings trajectories for different values of verti-cal shear are depicted in figure 4. With the weakeastward background flow (U = 4 and 6 cm/s inthe upper layer and 0 in the lower layer), theoceanic ring being slightly advected by the back-ground flow resists moving to the northwest. Nev-ertheless, if U = 10 cm/s the vortex changes its di-rection to the north-east. It is because the back-ground flow is strong enough, so that the advec-tion of the vortex is noticeable. Again, the vor-tex track exhibits jump-like behavior at certaintime steps. The westward-oriented backgroundflow merely advects the ring further to the west.For all runs with vertical shear the structure ofthe vortex core evolves approximately like for theruns without any background flow.

Figure 4: Vortex track with vertical shear U=4,6, 10 and -6 cm/s.

Another behavior of the coherent structurewas found for the simulations with imposed hori-zontal shear. Figure 5 shows the vortex locationswith U = 6,10,−6 and −10 cm/s (see section 2).On the one hand, the eastward jumps are stillpresent like in the simulation without backgroundflow. Besides, they are not smoothed like in thecase with vertical shear, which further indicates,

that the jumps are a baroclinic phenomenon. Onthe other hand, the vortex core is destroyed ear-lier if the shear is eastward, and later if the shearis westward (compare the snapshots of 21st dayon fig.3 and fig6.).

Figure 5: Vortex track with horizontal shear U=6,10, -6 and -10 cm/s.

Figure 6: The snapshots of upper-layer streamfunction at 1, 5, 21 and 50th day. Horizontal shearU = −10cm/s

3.3 Dynamics of the β-Gyres

To understand the dynamics of the north-westward propagation of cyclonic vortices the dy-namics of the β-gyres should be considered. Look-ing at figure 7 one may conclude the the ventilatedvector (the vector which coincides with the direc-

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tion of the flow between barotropic dipolar circu-lations) rotates to the north-west direction. Baro-clinic β-gyres evolve in different way: the venti-lated vector turns to the north-east (see fig.8). Al-though, one may think that the baroclinic β-gyrescan explain the jump-like behavior of the vortex,the magnitude of asymmetric baroclinic stream-function is small relatively to the barotropic one.The simulations for evolution of β-gyres for thecases with environmental influences are literallysimilar to the case with no background flow, butslightly stretched by the background flow.

Figure 7: Evolution of barotropic β-gyres with noenvironmental influences at 1, 5, 10, 15 and 20thday. On the 20th day the barotropic β-gyres areessentially destroyed.

Figure 8: Evolution of baroclinic β-gyres with noenvironmental influences at 1, 5, 10, 15 and 20thday. Note the stability of baroclinic β-gyres incomparison to barotropic.

4 Discussion and Conclusion

To summarize, the behavior of baroclinic sys-tem has both similar and opposite features to thebarotropic one. The overall direction of propaga-tion of cyclonic vortex is to the north-west. How-

ever, the trajectory of the baroclinic vortex expe-riences so-called jumps, which we did not find forbarotropic case. The underlaying physics of thisphenomena is largely remains unclear, and ourbelief that this may be due to the resonant in-teraction of Rossby waves and vortex itself. An-other issue what should be verified is the influ-ences of meridional walls. The environmental in-fluences are also different: vertical shear tends tosmooth the sharpest of the jumps, but practicallyhas no impact on the vortex core. The behaviorof the vortex in horizontal shear somewhat op-posite, namely the core of the vortex is gettingweaker earlier for the eastward shear and later forthe westward flow, but the jumps are very sharp.The reason of such behavior is intuitively clear:if the background flow has the same sign of vor-ticity as vortex itself, it tends to strengthen thevortex, and if the sign is opposite - to weakenit. With regard to the β-gyres, the barotropicdipolar circulation affect the the dynamics muchstronger then the baroclinic one, which is consis-tent with the results of previous studies. However,the fact that baroclinic β-gyres are rotated to thenorth-east requires further investigation. To lookat the dynamical balances of the vorticity equa-tion would be useful step toward the explanationof this fact.

Acknowledgement

The author would like to thank Dr. David Nolanfor opening this course and providing guidancethroughout the project. The author would alsolike to thank Dr. Pavel Berloff for providing thenumerical model. Helpful advice and suggestionsfrom Dr. Igor Kamenkovich were also greatly ap-preciated.

References

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Reznik, G.M. and Z. Kizner (2007), Two-layerquasi-geostrophic singular vortices embedded ina regular flow. Part 2. Steady and unsteady

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Reznik, G.M., and W.K., Dewar (1994), Ananalytical theory of distributed axisymmetricbarotropic vortices on the beta-plane, J. FluidMech., 269, 301–321.

Flierl, G.R., (1994), Rings: Semicoherent OceanicFeatures, Chaos, 355 (4), doi:10.1063/1.166013

Pedlosky, J., (1987), Geophysical Fluid Dynamics,Springer-Verlab.

Castelao, D.B, and W.E. Johns (2011), Sea sur-face structure of North Brazil current ringsderived from shipboards and moored acousticdoppler current, J. Geophys. Res, 116, C01010,doi:10.1029/2010JC006575

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