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 5 2.4 6 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 # dc/R # 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- q1997 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.