524 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27
Interactions of Baroclinic Isolated Vortices: The Dominant Effect of Shielding
S. VALCKE AND J. VERRON CNRS, Laboratoire des Ecoulements GeÂophysiques et Industriels, Institut de MeÂcanique de Grenoble, Grenoble, France (Manuscript received 15 November 1995, in ®nal form 12 August 1996)
ABSTRACT The interactions of two quasigeostrophic isolated shielded vortices are considered in a two-layer model and in the reduced-gravity approximation. Each shielded vortex is de®ned by a realistic horizontal pro®le of relative vorticity in the upper layer. In the numerical experiments, the initial separation distance between the vortices, d, and the degree of the ambient baroclinicity (i.e., the density strati®cation) are varied. The results show that the interactions of shielded vortices are dictated by their horizontal structure of potential
vorticity, which depends on the baroclinicity of the system. Globally, the critical distance of merging is dc/R ϭ 2.4 Ϯ 0.3 (where R is the radius of the vortices). The results denote also a favoring effect of baroclinicity on the merging ef®ciency, which increases when the baroclinicity increases. Furthermore, a new mechanism of interaction inhibiting the merging is identi®ed. When two vortices char- acterized by an annulus of opposite-sign potential vorticity surrounding their core interact, the potential vorticity of the annuli is redistributed and forms two lateral poles. Under the action of these poles, the vortices move apart from one another and their merging is inhibited. It is therefore concluded that the merging of vortices possessing a shielded potential vorticity structure is very unlikely. To apply these results to the oceanic reality, a better knowledge of the horizontal structure of oceanic vortices remains essential.
1. Introduction rents such as the Gulf Stream, the Kuroshio, or the Aghulas Current rings. These vortices, capable of im- Since the 1970s, interactions between vortices have portant heat transport, probably have a signi®cant im- received considerable attention from the scienti®c com- pact on the ocean general circulation and on the climate. munity. In their laboratory experiments, Brown and Understanding their dynamics and, in particular, their Roshko (1974) and Winant and Browand (1974) re- interactions is therefore essential to understanding the vealed, in particular, one of the most fundamental vortex oceanic dynamics itself. interactions, the ``coalescence'' or ``merging'' of two The best documented example of the merging of oce- like-sign vortices. Later, numerical simulations of two- anic vortices came from Cresswell (1982) concerning dimensional turbulence showed that coherent structures two East Australian Current warm core rings. Yasuda of vorticity grow by successive mergings and tend to et al. (1992) also identi®ed the partial merging of two dominate the ¯ow (e.g., Basdevant et al. 1981; Mc- Kuroshio warm core rings. Some Geosat altimetry ob- Williams 1984). This phenomena has also been ob- servations also suggested that Aghulas rings may coa- served in the laboratory (e.g., Hop®nger et al. 1982; lesce (van Ballegooyen et al. 1994). Finally, Schultz- Couder and Basdevant 1986) and in numerical simu- Tokos et al. (1994) reported that two meddies identi®ed lations of geostrophic turbulence, which constitutes a in the Iberian Basin interacted and merged. useful idealization of many geophysical ¯ows such as An abundant literature exists concerning the merging the atmosphere or the ocean (McWilliams 1989; Cush- of ®nite-core Rankine vortices in the barotropic (un- man-Roisin and Tang 1990). strati®ed) case. A Rankine vortex is formed by a circular Relatively recently oceanographers began to study the anomaly of uniform vorticity of radius R. Many nu- ocean variability associated with the ``mesoscale ed- merical, theoretical, and experimental studies identi®ed dies,'' characterized by length scales of 50±200 km, the critical distance of merging d /R for such vortices, which are now thought of as containing most of the c oceanic kinetic energy. Among these, one ®nds coherent that is, the greatest center-to-center separation under vortices resulting from the meandering of strong cur- which the vortices merge. Depending on the studies, 3.2 Յ dc/R Յ 3.4 (e.g., Overman and Zabusky 1982; Grif- ®ths and Hop®nger 1987; Melander et al. 1988; Drit- schel and Legras 1991; Waugh 1992; Ritchie and Hol- Corresponding author address: Sophie Valcke, SEOS/CEOR, Uni- land 1993). versity of Victoria, P.O. Box 1700, Victoria, BC V8W 2Y2, Canada. Concerning baroclinic (i.e., evolving in a strati®ed E-mail: [email protected] ¯uid) Rankine vortices, most of the numerical and ex-
᭧1997 American Meteorological Society
Unauthenticated | Downloaded 10/01/21 05:48 PM UTC APRIL 1997 VALCKE AND VERRON 525 perimental investigations were realized in the two-layer size of the outer rings may affect their interactions and quasigeostrophic (QG) framework. In their laboratory their critical distance of merging dc/R. He found that experiments, Grif®ths and Hop®nger (1987) created the shielding generally reduces the value of dc/R and their vortices by suction or injection of ¯uid in the upper that, in some cases, an internal barotropic instability of layer of a two-equal-layer strati®ed ¯uid and found a each vortex may inhibit their merging. Depending on marked effect of the strati®cation on the vortex merging. the characteristics of the outer rings, the vortices were However, Polvani et al. (1989) showed, by numerical stable or unstable, and various outcomes were observed simulations based on the contour dynamics, that the depending also on the initial separation distance d. In merging of two vortices de®ned by circular anomalies some cases, merging of the cores or of the annuli (``in- of potential vorticity in the upper layer is almost in- verted merger'') occured. In other cases, the interaction sensitive to the strength of the strati®cation, when the destabilized each vortex, forming a tripole or a quad- layers are of equal depths. The problem was reconsid- rupole (weak instability) or breaking into two dipoles ered by Verron et al. (1990) and by Verron and Valcke (strong instability). (1994) who described, in particular, how the merging Major achievements were also gained in the meteo- of two vortices de®ned as circular anomalies of relative rological ®eld concerning the interactions of shielded vorticity in the upper layer depends strongly on the vortices, by studies devoted to tropical cyclones. In the background strati®cation. They showed that the inter- barotropic case, for vortices made up of concentric vor- actions of baroclinic vortices are, in fact, determined by tex patches (``compound'' vortices, Ritchie and Holland the structure of their potential vorticity. 1993) as well as for continuous vortices (Holland and The merging of isolated lenses, which may be re- Dietachmayer 1993), it was shown that during the in- garded as a front closed onto itself and therefore cannot teraction distortion of the weak outer vorticity ®eld be treated in the QG approximation, also raised consid- modi®es the advecting ¯ow over each vortex core; de- erable interest. Such isolated frontal eddies generate no pending on the shape of the vorticity ®elds, this leads velocity outside their front and have a well-de®ned en- to mutual approach or divergence of the two vortices. ergy. Unlike Rankine vortices, which by de®nition are Similarly, Pokhil and Polyakova (1994) found that, de- not isolated, these lenses have to come into contact with pending on the vortex outer structure and initial sepa- each other in order to interact. Gill and Grif®ths (1981) ration distance, interaction of the tangential wind of one pointed out that, if two such lenses were to totally merge vortex with the vorticity ®eld of the other vortex may into one, the ®nal lens would have more energy than decrease or increase the distance between them, thereby the combined energies of the two original lenses. In their determining their critical distance of merging. Also, laboratory experiments, Nof and Simon (1987) ob- Chan and Law (1995) concluded that whether two in- served that the merging is possible without any external teracting barotropic shielded vortices attract or repel source of energy and concluded that the potential vor- each other depends on the advection of their asymmetric ticity of the lenses must somehow be altered during the vorticity distribution, which is governed by their struc- process. Nof (1988) suggested that shock waves, alter- ture and the separation distance between them. Sup- ing the potential voricity, may form along narrow in- porting these ®ndings, Wang and Holland (1995), con- trusions developing during the merging process. Cush- sidering baroclinic vortices, identi®ed three fundamen- man-Roisin (1989) argued that the merging need not be tal modes of interaction separated by two critical sep- complete: the ®nal product consists of a central lens aration distances: the mutual approach separation containing almost all the original energy but only a frac- (MAS) under which the vortices approach each other tion of the mass, surrounded by ®laments holding the and above which they move on divergent orbits, and rest of the mass, almost no energy, but an important the mutual merger separation (MMS) below which the part of the initial angular momentum. Dewar and Kill- vortices rapidly merge together. All these works sup- worth (1990) also pointed out that the energy paradox ported the suggestion of Lander and Holland (1993) that can be resolved if mixing of potential vorticity outside the classical Fujiwhara (1921) model for tropical inter- the eddies is allowed. action, that is, orbit, mutual approach, and merger, More on the line of our work, some authors also should be modi®ed to consist of several quasi-stable considered, in the barotropic and in the baroclinic frame- states, that is, approach, orbit, and merger or sudden works, the interactions of vortices characterized by a release and escape. The work presented here, which core of relative vorticity of one sign surrounded by an focuses on oceanic vortices, shares a certain similarity annulus of opposite sign. This speci®c eddy structure with some aspects of the above studies on tropical cy- denoted by ``shielding'' is more realistic than the Ran- clones; whenever it is the case, the analogies will be kine structure, for oceanic vortices such as the Gulf detailed in the text. However, the reader is referred to Stream rings (Olson 1980), as well as for atmospheric the above authors for a more detailed description of the vortices such as tropical cyclones (Holland 1980). different sensitivity experiments presented in these In the barotropic case, Carton (1992) proceeded to works (presence of an earth-vorticity gradient and other contour surgery experiments on piecewise-constant external in¯uences, vortex structure, baroclinicity, dia- shielded vortices and investigated how the strength and batic heating, etc.).
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In the oceanographic context, Masina and Pinardi The layout of the paper is as follows. In section 2, (1991, 1993; MP91 and MP93 hereafter) studied by we present the two-layer quasigeostrophic model, also numerical simulations the QG merging of oceanic iso- used in the ``reduced-gravity'' approximation, and we lated shielded vortices on the  plane. Horizontally, their brie¯y describe the numerical code used. The shielded vortices had the realistic velocity pro®le proposed by vortices chosen in our numerical experiments are pre- Olson (1980) for Gulf Stream rings, also used in our sented in section 3. The stability of these vortices is simulations (see section 3). MP91 considered the baro- brie¯y investigated in section 4, while our main results tropic case. In particular, they identi®ed a critical dis- concerning their merging are reported in section 5. In tance of merging, dc/R ϭ 2.4 Ϯ 0.3, R being the radius section 6, we discuss these results and evaluate their of the cores. When d Ͻ dc, they observed that the merg- oceanic implications. Finally, in section 7, we sum- ing of the cores is accompanied by the development of marize the main ®ndings of this work. lateral arms, that may detach and form a dipolar struc- ture with opposite-sign ``near-®eld'' vortices, originat- 2. The quasigeostrophic model and the numerical ing from the annuli. On the base of environmental pa- code rameter sensitivity experiments, they concluded that the merging is a nonlinear process and that the  parameter In this study, we investigate how the particular hor- does not affect appreciably the merging per se, even if izontal structure of isolated shielded vortices affects it notably in¯uences the development of the arms and their baroclinic merging. The simple quasigeostrophic near-®eld vortices. MP93 used a six-layer model rep- framework is used here in the two-layer and in the 1½- resenting strati®cation of the Gulf Stream region, in- layer (reduced gravity) approximations on the f plane cluding the  effect. Vertically, the velocity pro®le of (i.e., neglecting the  effect). Analysis of the dynami- their vortices decreased linearly, becoming zero at a cally conserved quantity, that is, the potential vorticity, depth zmax below which initial rest was assumed. They will allow us to clearly identify some key processes proceeded to a detailed analysis of the local energy and determining or inhibiting the merging of such vortices. vorticity balances during the merging and identi®ed the same critical distance of merging as in the barotropic a. The quasigeostrophic model case, dc/R ϭ 2.4 Ϯ 0.3. In particular, they observed that the effect of increasing the baroclinicity of the system The two-layer QG equations on the f plane, without is to slow down the merging; this is what they called forcing but including dissipative terms Vi, can be written the ``halting effect of baroclinity'' on the merging. Com- pared to the barotropic case, the near-®eld vortices of dd H (Qϭٌ2ϩ 2Ϫ2(Ϫ)ϭV (1 the baroclinic cases were weaker or did not develop at dt11 dt H ϩ H 211 []12 all and the vorticity ®lamentation was clearly inhibited. Our previous work showed that the dynamics of baro- dd H 2 1 Ϫ2 (clinic vortices is governed by their horizontal and ver- Q22ϭٌϪ ( 212Ϫ)ϭV, (2 dt dt H ϩ H tical structure of potential vorticity (Verron and Valcke []12 1994). Here the key role of the horizontal potential vor- where ticity structure will also emerge. [Let us note that in ץץץץץ d meteorology the analysis of dynamical phenomena in ϭϪii ϩ yץ xץ xץ yץ tץ terms of potential vorticity is probably more widely used dt than in oceanography, e.g., Hoskins et al. (1985)]. In with the streamfunction in layer i of thickness H .In comparison with MP93, we will study here the inter- i i each layer, Qi is the potential vorticity anomaly made actions of baroclinic shielded vortices in a simpler 2 -up of the relative vorticity i ϭٌi and of the stretch framework, that is, a simple two-layer model without  ing term. The internal Rossby radius is a measure of effect in the quasigeostrophic formulation. This will al- the baroclinicity of the system, in the sense that a greater low us to relate the behavior of the vortices to the par- (smaller) Rossby radius corresponds to a weaker (stron- ticular horizontal structure of their potential vorticity ger) baroclinicity. It is written and, consequently, to identify and explain basic mech- anisms playing a possible role in the merging of bar- 1/2 gЈHH12 oclinic shielded vortices. Globally, we will ®nd the same ϭ , (3) f2(HϩH) critical distance of merging as MP91 and MP93, and []01 2 we will observe the halting effect of the baroclinicity where gЈϭg⌬/Å ϭg(2 Ϫ1)/[(2 ϩ 1)/2] is the on the speed of merging. In addition, we will describe reduced gravity, i being the density. Under the f-plane and explain a favoring effect of baroclinicity on the approximation, the Coriolis parameter f, which is a mea- merging ef®ciency, which we will then de®ne. Fur- sure of the local vertical component of the planetary thermore, we will describe a new mechanism of inter- vorticity, is supposed to be a constant f0 ϭ 2⍀p sin(0), action inhibiting, in some cases, the merging of shielded where ⍀p is the earth rotation rate and 0 a latitude of vortices. reference.
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The limitations of this model are the ones usually where rg, the internal Rossby radius, is now expressed associated with the QG approximation on the f plane. by
In de®ning Rp the earth radius (Rp ഠ 6400 km),  the 1/2 (gЈH1) local gradient of the planetary vorticity, L and D the rg ϭ . (5) horizontal and vertical length scales, U the horizontal f 0 velocity scale, and T the time scale, one can form the The reduced-gravity approximation neglects the bottom following dimensionless numbers: the ``sphericity'' layer dynamics, keeping however the interface dynam- number L/Rp, the aspect ratio ␦ ϭ D/L, the number * ics. Using this approximation, it is important to be aware
ϭ L/f0, the spatial Rossby number Ro ϭ U/f0L, the that the merging of vortices much greater than the in- temporal Rossby number RoT ϭ 1/f0T, the Burger num- ternal Rossby radius may be severely altered (less ®- 2 2 ber Bu ϭ gЈD/f0L , and the external Froude number F lamentation, less axisymmetrization, important decrease 2 ϭ (f0L) /gD. For the vortex dynamics, the relevant time in the critical distance of merging, etc.) with respect to scale T is given by the turnover period, which leads to the two-layer case where H2 is ®nite, as shown by Pol-
T ϭ L/U and therefore to Ro ϭ RoT. In addition to the vani et al. (1989). usual shallow-water restrictions (L/Rp K 1, ⌬/ K 1, ␦ K 1), the QG approximation also requires Ro ϭ RoT b. The numerical code K 1 and Ro/Bu K 1. These two additional constraints imply that the vortex vorticity is negligible compared The numerical simulations were realized with two- to the planetary vorticity and that the interface deviation layer and reduced-gravity versions of the Holland is small with respect to the layers thicknesses. Further- (1978) original QG code. This code solves (1) and (2) more, as we use the f-plane approximation, * should using classical ®nite-difference second-order schemes. be K 1. The Helmholtz equations resulting from the discreti- At the latitude of the Gulf Stream typical of the mid- zation in time are solved with a pseudospectral algo- rithm. latitude ocean, the rotation parameters are f0 ϭ 9.3 ϫ 10Ϫ5 sϪ1 and  ϭ 2 ϫ 10Ϫ11 mϪ1 sϪ1. For oceanic vortices The model considers a square box of ¯uid of side such as the Gulf Stream rings, the different scales are length Lb with, in our case, a resolution of 153 ϫ 153. U ϭ 1msϪ1,Lϭ50 km, D ϭ 1000 m (Olson 1991). As the radius of each vortex core R is chosen such that Following Olson (1980) and Olson et al. (1985), the R/Lb ϭ 1/20, a total of about 15 points discretizes each best ®t for a cold core and a warm core ring to a two- vortex diameter. layer model gives respectively gЈϭ0.016 m sϪ2 and gЈ Free-slip is chosen as the lateral boundary condition. Ϫ2 At the surface, the rigid-lid approximation is used, and ϭ 0.011 m s . This leads to L/Rp ϭ 0.008, ␦ ϭ 0.02, * ϭ 0.01, Ro ϭ Ro ϭ 0.2, Bu ϭ 0.7 or 0.5, and F we assume no bottom topography and no bottom fric- T tion. The terms V of (1) and (2) are needed to dissipate ϭ 0.002. We see that the usual shallow-water restric- i tions are well ful®lled. Also, as F K 1, the rigid-lid the enstrophy, which tends to accumulate at the small approximation will be justi®ed. However, the value of scales not resolved by the model. These terms take the form of a biharmonic operator V ϭϪAٌ4. As our * ϭ 0.01 indicates that the Coriolis parameter f may i 4 i aim is to simulate a nearly inviscid ¯ow, the coef®cient vary by 2% across the diameter of a ring (ϳ 100 km): A is chosen as small as possible without allowing spu- the f-plane approximation might not be totally valid. 4 rious numerical noise to develop at the grid scale of the Also, as the Ro, RoT, and the ratio Ro/Bu are not really 4 model. In all the simulations, we chose A4* ϭ A4/(M´R ) K1, the quasigeostrophic hypotheses are not totally re- Ϫ5 ϭ 2.0 ϫ 10 , where M is the value of the relative spected. With these considerations in mind, we will use vorticity in the core of the vortex. the f-plane quasigeostrophic framework anyhow, as it allows us to proceed to a ®rst study in a simpli®ed framework. Furthermore, we recall here that MP91 3. The shielded vortices showed that the  effect does not appreciably affect the The vortices considered in the simulations have, in merging per se. Also, as shown by Valcke and Verron the upper layer, the realistic velocity pro®le for cyclonic (1995), the quasigeostrophic approximation retains Gulf Stream rings proposed by Olson (1980). This pro- probably the essential features of the complete dynamics ®le was also used, in particular, by MP91 and MP93. for vortices having a Ro number as high as Ro ϭ 0.5. There is no radial ¯ow, and the tangential velocity in- In some numerical experiments, we use the reduced- creases linearly from zero at the center to a maximum, Ϫ1 gravity or ``1½-layer'' QG approximation, which is the V,atrϭR(for Gulf Stream rings, V ഠ 1.6, m s limiting case for which the thickness of the lower layer and R ഠ 60 km): becomes in®nitely deep (H → ϱ). Equations (1) and 2 V (2) then reduce to one equation for the upper layer: (r) ϭ r, r Ͻ R. (6) R dQ d 1 ϭ [ Ϫ Ϫ2 ]ϭ0, (4) Outside the core of radius R, it follows an exponential dt dt 1rg1 decay:
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(r) ϭ Vexp[3(1 Ϫ r/R)], r Ͼ R. (7) By de®nition, a vortex is ``isolated'' if its circulation,
C(r) ϭ 2r(r), approaches zero beyond a certain ra- dius (Hop®nger and van Heijst 1993). In the present case, C(r) → 0asr→ϱ. The shielded vortex de®ned by (6) and (7) is therefore isolated, which also means that the total surface integral of its relative vorticity is zero. In relative vorticity, this corresponds to a core in solid-body rotation:
ϭ M ϭ 2V/R, r Ͻ R, (8) surrounded by an annulus of opposite sign:
13 ϭVϪ exp[3(1 Ϫ r/R)], r Ͼ R. (9) rR Figure 1 shows the nondimensional pro®les of the 2 upper-layer streamfunction 1/MR , the relative vortic- ity 1/M, and the azimuthal velocity 1/M R. For nu- merical reasons, the actual vorticity pro®le 1/M has been slightly smoothed by a hyperbolic tangent in order to avoid discontinuities incompatible with the ®nite dis- cretization. a. The two-layer case In one series of numerical experiments, we will use the QG model in a two-layer version. Following Olson (1980) and Olson et al. (1985), the 10Њ and 15Њ isotherms give, respectively for cyclonic and anticyclonic rings, the best ®t to the interface of a two-layer model; the corresponding upper-layer thicknesses are then H1 ϭ 700 m and H1 ϭ 200 m. In the numerical experiments, we assumed an ocean of total depth HT ϭ 5000 m and, based on the observations, we gave to H1 the average value of H1 ϭ 500 m, corresponding therefore to H1/H2 ϭ 1/9.
In the upper layer, the streamfunction 10 correspond- ing to the relative vorticity ®eld of two vortices ( ϭٌ2 ϭ (10 10 10 kϭ1,2 is prescribed. The sum ⌺kϭ1,2 indicates the superposition, a distance d apart, of two ®elds given by (8) and (9). In this study, we suppose that the material pertur- bation associated with a vortex is con®ned to the upper layer only. The potential vorticity in the bottom layer is assumed to be conserved such that the anomaly Q20 ϭ 0 everywhere. This initialization differs substantially from an initialization assuming initial rest in the bottom layer. In this last case, Verron and Valcke (1994) showed that the potential vorticity anomaly created in the bottom layer for small Rossby deformation radius can have an important impact on the upper-layer dynamics and leads to what they called the ``heton interaction.'' In the pres- ent case, the bottom layer cannot have this direct dy- FIG. 1. The upper-layer initial pro®les of the isolated shielded vor- 2 namical effect and, therefore, we will be able to better tex chosen in the numerical experiments: (a) streamfunction 1/MR , (b) relative vorticity / , (c) azimuthal velocity / R. appreciate the effect of the baroclinicity on the upper- 1 M 1 M
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2 FIG. 2. The upper-layer initial pro®les of the shielded vortex. Two-layer case: (a) potential vorticity Q1/M, (b) stretching termf0 (20 Ϫ 2 10)/gЈH1M. Reduced-gravity case: (c) potential vorticity Q1/M, (d) stretching term Ϫf010/gЈH1M. The three curves correspond to /R or
rg/R ϭ 0.3 (A), 0.5 (B), ϱ (C).
layer dynamics. Following the de®nition of Q2 [see Eq. for 35 rings of the World Ocean; in terms of /R,he (2)], our initialization gives an initial streamfunction in ®nds that, in general, 0.24 K /RK 1.3. → the lower-layer 20 corresponding to Figure 2 shows that for /R ϱ, the stretching term is zero and the potential vorticity Q1M has exactly the ff22 200 same shielded pro®le as the relative vorticity 1/M. (In → (20Ϫ 20ϭϪ 10. (11ٌ gЈHg22ЈH what follows, we will refer to the case with /R ϱ as the ``barotropic'' case and, in that case, we will use This is what we will call the two-layer case. the term ``vorticity'' for the potential vorticity and for Figures 2a and 2b illustrate the nondimensional up- the relative vorticity as these two quantities are equal.) per-layer potential vorticity Q1/M and stretching term When the baroclinicity increases, corresponding to 2 f0(20 Ϫ 10)/gЈH1M, corresponding to /R ϭ 0.3, 0.5, smaller /R, the stretching term increases and acquires and ϱ for the two-layer case. Note that Olson (1991) a shielded structure. Consequently, the potential vortic- 2 2 2 evaluates the Burger number Bu ϭ gЈH1/f0R ഠ (/R) ity keeps, for all values of /R, a shielded structure,
Unauthenticated | Downloaded 10/01/21 05:48 PM UTC 530 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 even if the annulus becomes progressively less impor- unstable and forms a tripole; the relative vorticity ®eld tant compared to the core. We will see that this particular at t/T ϭ 28, where T ϭ 2/M is the turnover period, structure of the potential vorticity has a major impact is shown in Fig. 3b. When the baroclinicity increases, on the vortex interactions. corresponding to smaller /R or rg/R, the intensity of Based on available observations of Gulf Stream rings, the barotropic instability decreases. For /R Յ 0.5 or one cannot determine precisely if the rings effectively rg/R, Յ 0.5, both the two-layer and the reduced-gravity possess a shielded potential vorticity structure. So we vortices are almost perfectly stable: even if part of the looked for a second type of initialization for which, in annulus forms two weak lateral poles, the resulting tri- the highly baroclinic regimes corresponding to oceanic pole is still surrounded by a closed annulus. This is rings, the potential vorticity would not exhibit a shielded shown in Fig. 3c for the two-layer case with structure. /R ϭ 0.3. The reduced-gravity case with rg/R ϭ 0.3, for which the potential vorticity is not shielded at all, is even perfectly stable, as shown in Fig. 3d. b. The reduced-gravity case In conclusion, we can con®rm that the two-layer and In a second series of numerical experiments, the reduced-gravity vortices will not be subject to the strong shielded vortices de®ned in the upper layer by (8) and mode-2 instability that could interfere with the merging. (9) will be studied in the reduced-gravity approximation Our shielded vortex in the two-layer and reduced-grav- described in section 3 [Eqs. (4) and (5)]. In this ap- ity cases is, in general, weakly unstable and tends to proximation, the bottom layer is supposed to be in®- form a tripole, except in the reduced-gravity case with nitely deep and motionless: the bottom-layer stream- rg/R ϭ 0.3, which is perfectly stable. In section 6b, we function 2 is neglected. This is what we will call the will verify that this weak instability does not affect our reduced-gravity case. main conclusions. The resulting nondimensional upper-layer potential vorticity Q / and stretching term Ϫf2 /gЈH cor- 1 M 0 10 1 M 5. Merging and divergence of shielded vortices responding to rg/R ϭ 0.3, 0.5, and ϱ for the reduced- gravity case are illustrated at Figs. 2c and 2d. For rg/ In the numerical experiments, the nondimensional ini- → R ϱ, the reduced-gravity case is of course identical tial separation distance was varied between d/R ϭ 2.0 to the two-layer case: the stretching term is zero and and 3.0 for values of /R or rg/R ϭ 0.3, 0.5, 1.5, and the potential vorticity has a shielded pro®le. When the ϱ. The results are summarized graphically in Fig. 4 for Rossby radius rg/R is smaller, the stretching term is the two-layer cases (H1/H2 ϭ 1/9 and Q20 ϭ 0) and in stronger but is not shielded in this case; consequently, Fig. 5 for the reduced-gravity cases. On each ®gure, the the shielded structure of the potential vorticity is less critical distance of merging dc/R, which is the limit be- pronounced. For rg/R ϭ 0.3, the potential vorticity is tween the merging and non-merging regions, is shown. not at all shielded. This reduced-gravity case will there- Globally, this critical distance does not depend very fore also allow us to study the interactions between much on the baroclinicity of the system, measured by isolated vortices for which potential vorticity structure the value of /R or /R. In both cases, the critical is not shielded. rg distance of merging, dc/R is between d/R ഠ 2.1 and d/R ഠ 2.7, which agrees with MP93's results (dc/R ϭ 4. Stability of the shielded vortices 2.4 Ϯ 0.3). The symbols and numbers identifying the experiments are explained in the following paragraphs. We ®rst investigate the intrinsic stability properties of the isolated vorticity pro®les proposed by Olson (1980) [Eqs.(8) and (9)] in the two-layer and reduced- a. The merging cases gravity cases. Some recent studies on the stability of axisymmetric vortices show that the presence of an an- In the merging region, the vortices merge rapidly: In nulus of opposite-sign vorticity around a circular core all cases, we observed that the two initial cores have may lead to a mode 2 barotropic instability (e.g., Carton completely fused at t/T Ͻ 10 Ϯ 2. The numbers reported 1992; Carton and Legras 1994). When this instability in Figs. 4 and 5 for each experiment indicate the merging is weak, a tripole is generated; if the instability is strong, ef®ciency, E, de®ned as the percentage of potential vor- the vortex breaks into two dipoles. We want here to ticity initially in the cores of the two original vortices verify a priori that our vortices are not subject to this that stays in the core of the resulting vortex after the strong instability that may inhibit the merging (Carton merging: 1992). Therefore, we examine for various values of /R or QdA ͵1 f rg/R the evolution of a unique vortex slightly perturbed Af by a mode 2, in the two-layer and reduced-gravity cases. Eϭ . (12) Figure 3a illustrates the initial relative vorticity ®eld. QdA 1 i kϭ1,2 ͵ In the barotropic case (/R→ ϱ), the vortex is weakly Ai
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FIG. 3. Stability of the shielded vortices. (a) Initial relative vorticity ®eld slightly perturbed by a mode 2. Relative vorticity at t/T ϭ → 28: (b) barotropic case (/R ϱ), (c) two-layer case with /R ϭ 0.3, (d) reduced-gravity case with rg/R ϭ 0.3.
Initially, the core of each original vortex of area Ai when the parameters /R or rg/R are weak. To illustrate is easily identi®able, being the only region where Q1/ this phenomenon, the potential vorticity ®elds Q1 of two M is positive. After the merging, the core of the re- experiments with d/R ϭ 2.0 are shown in Figs. 6 and → sulting vortex of area Af is identi®ed by selecting the 7, respectively in the barotropic case /R ϱ (E ϭ central region where Q1/M is positive, without includ- 53%) and in the two-layer highly baroclinic case where ing ®laments. This is done at t/T ϭ 28, when the re- /R ϭ 0.3 (E ϭ 83%). sulting vortex has reached a stable form after the merg- In both ®gures, one sees that during the merging, the ing (except for the reduced-gravity case with d/R ϭ 2.75 potential vorticity of the annuli forms two ``lateral and rg/R ϭ 0.3, for which the calculation was done at poles'' (corresponding to a local maximum of vorticity t/T ϭ 120Ðsee section 5c). marked by an H, more visible in Fig. 7). These lateral As reported in Figs. 4 and 5, the ef®ciency E is greater poles diverge from one another following anticyclonic
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FIG. 4. Critical distance of merging dc/R as a function of /R for FIG. 5. As in Fig. 4 for shielded vortices in the reduced-gravity case. shielded vortices in the two-layer case. In the merging region, the merging ef®ciency in percent, E (expression 12), is given. The sym- bols of the non-merging region are explained in the text. alies are analogous to our lateral poles and their effect is, as in our experiments, to increase the ®lamentation trajectories, dragged along by the negative cores which and to decrease the ef®ciency of the merging. merge. Conversely, the lateral poles act on the cores by The merging evolution of two shielded vortices ini- tearing out some of their vorticity that winds around the tially close together, as shown on Figs. 6 and 7, shares poles. The lateral poles thereby increase the ®lamen- many characteristics with the instability process of one tation process inherent in the merging. Finally, a tripole single elliptical shielded vortex, as studied by Carton is formed. and Legras (1994). In fact, the con®guration of two There are, however, important differences between the shielded vortices superposed in such a way that their barotropic and strongly baroclinic cases. In the barotropic annuli overlap is analogous to the con®guration of one case, as in Fig. 6 where /R → ϱ, the intensity of the elliptical shielded vortex. So it is not surprising that, in opposite-sign potential vorticity annuli is relatively im- that case, the evolution of two circular shielded vortices portant (see Fig. 2a). Therefore, the lateral poles are rel- is analogous to the evolution of one ellipitical shielded atively strong, and a great part of the potential vorticity vortex, which may be subject to an instability leading is torn out from the cores and forms ®laments. The re- to the formation of one tripole. sulting tripole has a small central core surrounded by two As noted by MP93, the merging evolution is slower important poles: the ef®ciency (E ϭ 53% in this case) is when the baroclinicity is stronger (smaller /R); this is low. When the baroclinicity is stronger, as in Fig. 7 where what MP93 called the ``halting'' effect of baroclinicity, /R ϭ 0.3, the intensity of the annuli and the strength of considering the speed of merging. Between the barotropic the lateral poles are weaker. The ®lamentation process is case (Fig. 6) and the strongly baroclinic case (Fig. 7), we reduced, the merging cores retain more of their initial observed a difference in the time of merging of t/T ഠ 2.0, potential vorticity and the ef®ciency (E ϭ 83% in this which gives about 2±3 days for Gulf Stream rings and case) is high. The resulting tripole has an important central agrees with MP93. However, the fact that the ef®ciency core and two weak lateral poles. is greater when the baroclinicity is stronger is certainly The lateral poles correspond to the ``near-®eld'' vor- also a remarkable phenomenon: it indicates a ``favoring'' tices identi®ed in the relative vorticity ®elds by MP93. effect of the baroclinicity on the ef®ciency of the merging. The above description agrees with MP93's observations In any case, as the vortices are de®ned by their relative that, compared to the barotropic case, the intensity of vorticity, the effect of the baroclinicity is in fact related the near-®eld vortices and the ®lamentation process are to the impact that the baroclinicity has on the structure of clearly reduced in the baroclinic case. It is also coherent potential vorticity of the vorticesÐthat dictates the dy- with the description made by Wang and Holland (1995) namicsÐrather than on the merging dynamics itself. concerning merging experiments of two tropical cy- clones using a multilevel model. In particular, these au- b. The nonmerging cases and the critical distance of thors observe that the merging of the cyclones at the merging middle levels, where low PV anomalies concurrently develop from the wrap-up of an outer anticyclonic ring, Let us look now at the nonmerging two-layer and is accompanied by much stronger ®lamentation than at reduced-gravity cases. For the experiments identi®ed by the lower levels (see their Fig. 6). These low PV anom- the letters ID, the cores have an initial tendency to
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→ FIG. 6. Potential vorticity ®eld Q1 for the barotropic case (/R ϱ) with d/R ϭ 2.0 at t/T ϭ (a) 0.0, (b) 4.0, (c) 8.0, (d) 12.0 (merging region). The dashed and solid isocontours illustrate, respectively, the negative and positive potential vorticity of the cores and of the annuli. The strength of the lateral poles is high and the merging ef®ciency is low (E ϭ 53%). merge, but this tendency seems now totally inhibited by original cores (Fig. 8d). The orientation of the dipoles the action of the lateral poles. The two-layer experiment is such that they tend to travel in opposite directions with d/R ϭ 2.75 and /R ϭ 0.3, shown in Fig. 8, il- (Fig. 8e): this reseparates the cores (Fig. 8f) and inhibits lustrates this evolution. Initially, the cores of the vortices the merging. The letters ID stand for ``interaction of the are well separated (Fig. 8a). During the evolution, the coresÐdivergence.'' cores interact, exchange vorticity, and tend to merge; In the experiments identi®ed by the letter D, the initial however, as in the merging cases, the opposite-sign po- separation distance d/R ϭ 3.0 and the cores of the vor- tential vorticity of the annuli is redistributed and forms tices are too far apart to exchange vorticity. However, two lateral poles (Figs. 8b and 8c). Each one of these the interaction still leads to the redistribution of the lateral poles forms a dipolar structure with one of the potential vorticity of the annuli that forms two lateral
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FIG. 7. As in Fig. 6 for the two-layer strongly baroclinic case (/R ϭ 0.3). The lateral poles are weak and therefore the merging ef®ciency is high (E ϭ 83%). poles. As in the ID experiments, two dipoles are formed It is interesting to note here that the D-type of inter- and diverge from one another. The letter D stands for action presents a clear analogy with the interactions ``divergence.'' Hereafter, we will refer to this particular identi®ed by Ritchie and Holland (1993) between a interaction of shielded vortices, leading to the formation compound shielded vortex and a single vortex patch of two diverging dipoles, and inhibiting the merging, (their Figs. 10, 11c, and 11d). In their case, the presence as the ``D-type'' of interaction. of an outer ring having a vorticity of opposite sign com-
→
FIG. 8. Potential vorticity ®eld Q1 for the two-layer experiment with d/R ϭ 2.75 and /R ϭ 0.3 at t/T ϭ (a) 0.0, (b) 2.0, (c) 4.0, (d) 6.0, (e) 8.0, (f) 10.0 (non-merging region). The initial merging tendency of the cores is inhibited by the action of two lateral poles resulting from a redistribution of the annuli potential vorticity.
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Unauthenticated | Downloaded 10/01/21 05:48 PM UTC APRIL 1997 VALCKE AND VERRON 537 pared to the core results in a net divergence of the two ing tendency of the cores is totally counteracted by the vortex cores; however, the two dipolar structures spe- action of these lateral poles. These ones form, with the ci®c to the D-type of interaction are not formed, as only original cores, two opposite-traveling dipoles and the one of the two vortices is shielded. Also, in Pokhil and merging is inhibited. This is the case for all two-layer
Polyakova (1994) and in Chan and Law (1995), inter- and reduced-gravity experiments with d/R Ͼ dc/R except acting shielded vortices are sometimes observed to di- for the reduced-gravity experiment with rg/R ϭ 0.3, verge; the D-type of interaction is probably active there, discussed in more detail hereafter. although, as brie¯y presented in the introduction, their presentations of the phenomenon differ from ours. c. The reduced-gravity experiment with /R ϭ 0.3 Concerning the critical distance of merging, one can rg see in Figs. 4 and 5 that it is in any case smaller than The reduced-gravity experiment with rg/R ϭ 0.3 is the critical distance of merging for Rankine vortices of the only case where the vortices do not present an an-
3.2 X dc/R X 3.4 (see section 6c). We note here that nulus of opposite-sign potential vorticity around their Carton (1992) also ®nds that the shielding generally core, as shown in Fig. 2c. The D-type of interaction, reduces the critical distance of merging even if he does described in section 5b and based on the presence of not identify the D-type of interaction as such. Also, dc/R an opposite-sign potential vorticity annulus, may there- seems to increase slightly when /R or rg/R decrease. fore not occur. The reduced-gravity cases with rg/R ϭ This phenomenon is more pronounced in the reduced- 0.3 and d/R X 2.5 merge rapidly; in all cases, the two gravity case (Fig. 5) than in the two layer case (Fig. 4), initial cores have completely fused at t/T ϭ 10 Ϯ 2. even if, in both cases, the variations are small. Some For the experiment with d/R x 3.0, the vortices do not caution should be taken in interpreting these variations merge and almost do not interact at all. They just acquire as they are close to the numerical resolution of the mod- a very slow corotating motion (the letter R stands for el, corresponding to 0.17R. ``rotation''). The reduced-gravity case with rg/R ϭ 0.3 These variations can be simply related to the size of the and d/R ϭ 2.75 is more ambiguous: for a nondimen- Ϫ5 potential vorticity core of the vortices, which increases sional viscosity of A4* ϭ 2.0 ϫ 10 , merging happens slightly when /R or rg/R decrease, as can be seen in Fig. at t/T ϭ 40 Ϯ 6, which is a relatively long time. Fur- 2a for the two-layer case and more evidently in Fig. 2c thermore, with other experiments, we observed that the for the reduced-gravity case. As was shown by Verron and merging time is in¯uenced by the value of the numerical
Valcke (1994), two vortices having a core of potential dissipationA44*: the greaterA* , the smaller the merging vorticity more extended will interact and merge from a time. So in this reduced-gravity experiment (with rg/R greater distance. For example, let us consider the reduced- ϭ 0.3 and d/R ϭ 2.75), it is not clear if this merging → gravity cases with rg/R ϱ (Fig. 2c, curve C) and rg/ is a ``real'' convective one, or if it is a ``viscous'' merg- R ϭ 0.5 (Fig. 2c, curve B). In the ®rst case, 2.0 Ͻ dc/R ing in the sense that it occurs only because of the nu- Ͻ 2.25 and the radius of the core of positive potential merical dissipation. This is why in the reduced-gravity vorticity Q1/M, that we will call RPV, is equal to the radius case with rg/R ϭ 0.3 (but only in that case) the critical of the core of relative vorticity R. In the second case, 2.5 distance of merging cannot be precisely determined and
Ͻ dc/R Ͻ 2.75 and RPV ഠ 1.2R. If we rede®ne the critical the curve is dashed in Fig. 5 indicating an uncertainty. distance of merging as dc/RPV, this gives 2.0 Ͻ dc/RPV Ͻ 2.25 in the ®rst case and 2.1 Ͻ d /R Ͻ 2.3 in the second c PV 6. Discussion case: These differences in dc/RPV are not signi®cant as they are below the resolution of the model. Also, it is likely Before making additional remarks on the D-type of that variations in the intensity of the shielding could induce interaction and discussing oceanic implications, let us some variations in the critical distance of merging (Carton consider here two particular aspects of our simulations, 1992; Pokhil and Polyakova 1994). However, as in our that is, the role of the initialization and the role of the experiments the variations in dc/R are anyway very small, weak instability described in section 4 in the formation we cannot quantify this effect and its relative importance. of the lateral poles. In conclusion, the interaction between two vortices characterized by a shielded potential vorticity structure a. Role of the initialization in the formation of the always leads to a redistribution of the opposite-sign po- lateral poles tential vorticity of the annuli that forms two lateral poles. If the initial separation distance between two such We verify here that, in our experiments, the formation vortices is greater than a critical value, the initial merg- of lateral poles is not an arti®cial result of the initiali-
←
FIG. 9. Vorticity ®eld of the barotropic experiment (/R → ϱ) without any initial superposition for d/R ϭ 2.75 at t/T ϭ (a) 0.0, (b) 0.7, (c) 1.4, (d) 2.0, (e) 2.7, (f) 3.4. Even if the lateral poles have been initially ``®ltered out,'' the merging tendency of the cores is inhibited by the action of two lateral poles that are formed by a subsequent redistribution of the opposite-sign potential vorticity coming from the annuli.
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zation in which we superpose (at a distance d apart) the may also occur in this case. The experiment with /R0 → initial ®elds associated with each vortex. This super- ϱ and d/R0 ϭ 2.5 is illustrated in Fig. 10. It is clear position creates initially two small lateral poles on both here that the cores of the vortices initially interact and sides of the line joining the centers of the vortices (e.g., tend to merge; once again, this tendency is inhibited by see Fig. 7a). In reality, if two vortices get closer re- the action of lateral poles formed by the redistribution specting the principle of material conservation of the of the vorticity coming from the annuli, and the vortices potential vorticity, this quantity cannot be superposed. diverge from one another. We proceeded to additional experiments in which the In conclusion, the formation of lateral poles inhibiting initial ®eld was ``®ltered'' in order to eliminate the lat- the merging is a fundamental characteristic of the in- eral poles: in the right (left) half of the ®eld, the vorticity teraction between shielded vortices; it is related to the associated to the right (left) vortex only was initialized, asymmetric redistribution of the potential vorticity of thereby avoiding any superposition. Obviously, this the annuli, and it is not linked to the stability properties mode of initialization is not more realistic as the total of each initial vortex. potential vorticity of the annuli is now reduced. We should therefore consider these new experiments only as test cases that will allow us to determine if the for- c. The D-type of interaction mation of lateral poles is, or not, an artifact linked to the initialization. Let us ®rst compare the interactions of shielded vor- These new experiments were realized in the baro- tices with the interactions of Rankine vortices in the → tropic case (/R → ϱ) and in the two-layer case with barotropic case (/R0 ϱ). For barotropic Rankine /R ϭ 0.3, for 2.0 X d/R X 3.0. The temporal evolution vortices, 3.2 X dc/R X 3.4 (see section 1). As already of the experiment with /R → ϱ and d/R ϭ 2.75 is noted, the critical distance of merging of our barotropic shown in Fig. 9. Initially, there is no lateral pole any- shielded vortices (2.0 Ͻ dc/R Ͻ 2.25) is much smaller. more. During the interaction, there is an asymmetrical This is not surprising as the velocity pro®le beyond the redistribution of the opposite-sign potential vorticity core of the vortex decreases much faster for a shielded coming from the annuli: two lateral poles are subse- vortex [in exp[3(1 Ϫ r/R) in this case] than for a Rankine quently formed and inhibit the merging of the cores. In vortex (in 1/r). Furthermore, the effect of the shielding, these new experiments, the intensity of the lateral poles causing the D-type of interaction and inhibiting the is reduced, as the total quantity of potential vorticity in merging, can certainly not occur for Rankine vortices. the annuli is reduced; their action is, however, strong The D-type of interaction, which occurs also in the enough to induce the D-type of interaction for d/R Ͼ baroclinic cases when the vortices present a shielded 2.25 when /R → ϱ and d/R Ͼ 2.5 when /R ϭ 0.3. potential vorticity structure, is particularly interesting. In conclusion, in our standard two-layer and reduced- In the ocean, two interacting vortices are not suddenly gravity experiments, the D-type of interaction inhibiting generated very close together. More likely, they are the merging is not an artifact of the initialization, but is brought close together by some external forcing (cur- caused by a subsequent asymmetric redistribution of the rent, other vortices, etc.) and interact progressively. potential vorticity of the annuli that forms the lateral poles. Based on our results, we conclude that if two vortices actually possess an annulus of opposite-sign potential vorticity, the ®rst interaction that takes place involves b. Role of the instability in the formation of the a redistribution of the potential vorticity of the annuli; lateral poles the vortices then form dipolar structures, diverge from We verify here that the formation of lateral poles is one another, and the merging is inhibited. Also, as the not a phenomenon linked to the weak instability of our redistribution of potential vorticity occurs in the early shielded vortices, described in section 5. To do so, we stages of the interaction, the  effect would probably consider the merging of another type of vortex that is not alter the basis of this mechanism, even if it would shielded and perfectly stable. As shown by Carton and certainly affect the subsequent trajectories of the dipolar McWilliams (1989), the vortex de®ned by the following structures. relative vorticity 0 has these two properties: In conclusion, our experiments allowed us to quantify the critical distance of merging of shielded vortices. (r) ϭ (1 Ϫ r/R ) exp(Ϫ2r/R ). (13) 0 max 0 0 However, a more interesting result is certainly the iden- Additional experiments, realized with these perfectly ti®cation of the D-type of interaction, peculiar to shield- stable vortices, showed that the D-type of interaction ed vortices, that prevents their merging.
← → FIG. 10. Vorticity ®eld during the interaction of two shielded and perfectly stable vortices [Eq. (13)] for /R0 ϱ and d/R0 ϭ 2.5 at t/T ϭ (a) 0.0, (b) 3.6, (c) 7.1, (d) 10.7, (e) 14.3, (f) 74.9. The asymmetric redistribution of the vorticity of the annuli makes the vortices to diverge, thereby inhibiting their merging.
Unauthenticated | Downloaded 10/01/21 05:48 PM UTC 540 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 d. Oceanic implications shielded potential vorticity. In the baroclinic cases, the shielded structure of the relative vorticity is more or less To evaluate the potential role of the D-type of inter- masked by the stretching term, related to the interface action in the ocean, we have to determine if the oceanic deviation. Consequently, depending on the strength of vortices actually possess a shielded potential vorticity the baroclinicity of the system, the vortices present or structure. In the ocean, it is pretty clear that the vortices not a shielded potential vorticity. such as the Gulf Stream rings are isolated and therefore When two vortices characterized by a shielded potential present a shielded relative vorticity structure (e.g., Olson vorticity structure interact, lateral poles are formed by the 1980; Joyce 1984; Olson et al. 1985; Kunze 1986). redistribution of the potential vorticity coming from the However, the relative importance of the stretching term annuli. The action of these lateral poles is to compete with and, therefore, the structure of the potential vorticity are the merging tendency of the cores. When the vortices are not well quanti®ed. Furthermore, the very de®nition of initially close together, the merging tendency of the cores the potential vorticity, the materially conserved quan- is stronger and the action of the lateral poles is to reduce tity, depends on the theoretical framework considered. the merging ef®ciency. If the baroclinicity is stronger In fact, very few authors give maps of the potential (weaker), the opposite-sign potential vorticity annulus of vorticity for the vortices they observed in the ocean. the vortices is weaker (stronger). Consequently, the ef®- The shallow-water equations, applied to one adiabatic ciency of the merging is greater for a stronger baroclinicity. layer of constant and uniform density of thickness h, These results express a ``favoring'' effect of the baroclin- conserve the following potential vorticity ⌸ (if dissi- icity on vortex merging ef®ciency. When the vortices are pation is neglected): initially farther apart, the action of the lateral poles dom- ϩ f inates the interaction: the vortices form dipolar structures ⌸ϭ . (14) h that diverge from one another and the merging of the cores is totally inhibited. This is what we called the ``D-type'' Olson et al. (1985) and Joyce and Kennelly (1985) give of interaction. maps of the potential vorticity ⌸ associated to the warm It is interesting to note that many of our observations core ring 82B. It is calculated for an upper layer of agree with the work of Masina and Pinardi (1993), even thickness h1 (in a two-layer model) in which the ring is if all aspects of these two studies are not directly com- supposed to be entirely located. The map of this poten- parable as they differ substantially in their frameworks tial vorticity ⌸ shows only a central negative anomaly, (two-layer versus six-layer model, f plane versus  plane, following globally a 1/h1 pro®le without any shielding. etc.) and in the initialization of the vortices (potential vor- Olson (1980) shows that the materially conserved ticity anomaly con®ned to the upper layer versus relative quantity for a ring is more exactly expressed by ⌸E: vorticity con®ned to the upper layers and rest in the bottom / layers). In any case, the critical distance of merging is dcץ ⌸E ഠ ( ϩ f ). (15) R ϭ 2.4 Ϯ 0.3. In both studies, the lateral poles (or near- z eld vortices) are stronger and the ®lamentation is more® ץ Olson (1980) and Koshlyakov and Sazhina (1994) pres- important (the ef®ciency is lower) in the barotropic case ent a vertical section of the potential vorticity ⌸E for than in the baroclinic case. Masina and Pinardi (1993) two different cyclonic rings they observed. For Kosh- point out a ``halting'' effect of baroclinicity with regard lyakov and Sazhina (1994), the strong positive anomaly to the speed of merging which we also observe. In ad- associated to the cyclonic ring in the upper layers is dition, it is shown here that baroclinicity has a favoring surrounded by a weaker negative annulus. However, in effect on the ef®ciency of merging. Olson (1980) no such annulus is detectable. The most interesting result from our experiments is In summary, we cannot draw any ®rm conclusion on certainly the identi®cation of the D-type of interaction, the horizontal potential vorticity structure of oceanic peculiar to vortices having a shielded potential vorticity vortices: the observations presently available are too structure. We concluded that the ®rst interaction that scarce and certainly not precise enough. To evaluate the would occur if two such vortices were brought pro- role in the ocean of the D-type of interaction identi®ed gressively closer together would be dominated by the for shielded vortices in our experiments, we certainly action of the lateral poles inhibiting the merging. The need more detailed observations of the potential vor- merging of vortices characterized by a shielded potential ticity structure of the oceanic vortices. vorticity structure is therefore very unlikely. To evaluate the potential role of the D-type of interaction for oceanic vortices, we have to gain a better knowledge of their 7. Conclusions horizontal structure of potential vorticity. To do so, more The results presented here show that the interactions detailed in situ observations are essential. of isolated shielded vortices are in fact dictated by their horizontal structure of potential vorticity. In the baro- Acknowledgments. We would like to thank Y. Morel tropic case, the potential vorticity is equal to the relative for fruitful discussions and the reviewers for their de- vorticity and isolated vortices necessarily present a tailed and constructive comments. We are also thankful
Unauthenticated | Downloaded 10/01/21 05:48 PM UTC APRIL 1997 VALCKE AND VERRON 541 to M. Eby for his help in revising an earlier version of , and M. A. Kennelly, 1985: Upper-ocean velocity structure of this paper. Calculations were made using the numerical Gulf Stream warm-core ring 82B. J. Geophys. Res., 90 (C5), 8839±8844. facilities of the `Institut de DeÂveloppement et des Res- Koshlyakov, M. N., and T. G. Sazhina, 1994: Water circulation and sources en Informatique Scienti®que' in Palaiseau, a cyclonic ring in the Gulf Stream splitting region in May±June France. The ®rst author was supported by a Conseil de 1990. J. Geophys. Res., 99 (C7), 14 091±14 100. Recherches en Sciences Naturelles et en GeÂnie du Can- Kunze, E., 1986: The mean and near-inertial velocity ®elds in a warm- core ring J. Phys. Oceanogr., 16, 1444±1461. ada scholarship. Lander, M., and G. J. Holland, 1993: On the interaction of tropical- cyclone-scale vortices I: Observations. Quart. J. Roy. Meteor. 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