288 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 Lagrangian and Tracer Evolution in the Vicinity of an Unstable Jet EMMANUEL BOSS AND LUANNE THOMPSON School of Oceanography, University of Washington, Seattle, Washington (Manuscript received 18 August 1997, in ®nal form 8 June 1998) ABSTRACT The dynamics of Lagrangian particles and tracers in the vicinity of a baroclinically unstable zonal jet are investigated in a simple two-layer model with an initially quiescent lower layer. The presence of a growing wave induces a particle drift dominated by Stokes drift rather then the contribution of the wave to the mean Eulerian velocity. Stable and unstable waves have zonal Stokes drift with similar meridional structure while only unstable waves possess meridional drift, which is in the direction of increasing meridional wave displace- ment. Particle dispersion in the upper layer is maximum at critical lines, where the jet and phase speeds are equal. In the lower layer, dispersion is maximum where the wave amplitude is maximum. Zonal mean tracer evolution is formulated as an advection±diffusion equation with an order Rossby number advection and an order- one eddy diffusion. The latter is proportional to two-particle dispersion. Finite amplitude simulations of the ¯ow reveal that small amplitude theory has predictive value beyond the range for which it is strictly valid. Mixing (as opposed to stirring) is maximum near cat's-eye-like recirculation regions at the critical lines. In the lower layer the pattern of convergence and divergence of the ¯ow locally increases tracer gradients, resulting in stirring yet with a much slower mixing rate than in the upper layer. Meridional eddy diffusion (or particle dispersion) alone is not suf®cient for prediction of mixing intensity. Rotation, which is quanti®ed by the cross-correlation of meridional and zonal displacements, must also be present for mixing. These results are consistent with observations of tracer and ¯oats in the vicinity of the Gulf Stream. 1. Introduction role of a barrier to mixing across the jet (Bower and Lozier 1994), while mixing has been associated with Floats and tracers in the ocean have been used by critical (or steering) lines where the phase speed of the physical oceanographers to infer both the Eulerian struc- meanders matches the zonal jet speed (e.g., Owens ture of the ¯ow ®eld and the mixing geometry associated 1984; Bower 1991; Pratt et al. 1995). Likewise, the with the ¯ow. The primary goal of this paper is to study presence of a critical level1 has been invoked to explain tracer mixing and Lagrangian particle behavior in a bar- the enhanced particle dispersion (Lozier and Bercovici oclinically unstable jet. In addition, we wish to study 1992) below the thermocline. Tracer gradients have been the link between the kinematics of ¯oats and tracer and found to be smaller at depth (Bower et al. 1985), con- the underlying ¯ow ®eld. The geophysical motivation sistent with potentially enhanced mixing there (Bower for this work comes from the Gulf Stream, a region in 1991). the ocean in which ¯oats have been used intensively. Theoretical studies of stirring and mixing2 with ap- Lagrangian ¯oats and tracer studies (e.g., Bower et al. plication to the Gulf Stream have recently focused on 1985; Bower and Rossby 1989; Bower and Lozier 1994; barotropic and equivalent-barotropic quasigeostrophic Song et al. 1995) in the Gulf Stream region have high- (QG) jets with stable Rossby wave perturbations. These lighted two main characteristics: on the one hand, the studies have emphasized the role of critical levels and/ Gulf Stream seems to provide a barrier that separates slope water from Saragasso Sea water; while on the other hand, it is an intense area of mixing (Song et al. 1995). The sharp potential vorticity (PV) gradient ob- 1 Critical level denotes the position, in the vertical, where down- served in the vicinity of the jet has been assigned the stream mean ¯ow speed equals phase speed (u0 5 c). Critical line denotes the horizontal position in a layered ¯ow where u0 5 c. In continuously strati®ed ¯ows with variations in the horizontal, u0 5 c de®nes a two-dimensional critical (or steering) surface (Pratt et al. 1995). Corresponding author address: Dr. Emmanuel Boss, COAS, 2 Stirring refers to dispersion of parcels from each other (diver- Oregon State University, 104 Ocean Admin. Bldg, Corvallis, OR gence) and the deformation of material lines in an inviscid model. 97331-5503. Mixing implies changes in properties of parcels due to interaction E-mail: [email protected] with surrounding ¯uid. q 1999 American Meteorological Society FEBRUARY 1999 BOSS AND THOMPSON 289 or lines as primary locations for mixing. Using several following the theoretical work of Rhines and coworkers analytical techniques derived from the theory of dy- (Rhines 1977; Rhines and Holland 1979) and Andrews namical systems, tracer mixing or stirring at these lo- and McIntyre [1978, see also McIntyre (1980) and a cations was quanti®ed (Samelson 1992; del-Castillo-Ne- review in Andrews et al. (1987), herein AHL]. The evo- grete and Morisson 1993; Rogerson et al. 1996, man- lution of zonal-mean or time-mean quantities is the fo- uscript submitted to J. Phys. Oceanogr.) or qualitatively cus of these approaches and has been used in analytical assessed (Pratt et al. 1995; Lozier et al. 1997). These studies (e.g., Rhines and Holland 1979; Uyru 1979; Mat- treatments have taken advantage of the two-dimension- suno 1980; Shepherd 1983), as well as in the analysis ality of the QG barotropic ¯ow (i.e., the ¯ow can be of numerical model results of baroclinically unstable derived from a streamfunction) and that the wave am- ¯ows (Dunkerton et al. 1981; Plumb and Malmann plitudes have no, or slow, evolution. 1987). Gulf Stream observations associate growing meander The Gulf Stream exhibits neither periodic nor steady events with strong subsurface ¯ows, indicating the pres- spatially growing meanders, two simple cases where the ence of baroclinic instability (Watts et al. 1995). In ad- above wave±mean ¯ow interaction theories are directly dition, the Gulf Stream has a fairly high Rossby number applicable. However, much of our theoretical under- [U/Ïg9H and z/f ; 0.5 (Kim and Watts 1994), where standing of the Gulf Stream is based on such simpli®ed U is the speed of the stream, g9 its reduced gravity, H assumptions that capture many of its observed features the thermocline depth, z the relative vorticity, and f the [e.g., meander growth and propagation speeds (Kill- Coriolis parameter]. This suggests that the effects of worth et al. 1984) and eddy forcing of the time mean baroclinicity, transient waves and mean ¯ows, and high ¯ow (Cronin 1996)]. Rossby numbers should be investigated (Lozier and Ber- In section 2, the model and the theoretical background covici 1992; Pratt et al. 1995). are introduced. We solve for Lagrangian and tracer di- Lozier and Bercovici (1992) investigated a baroclin- agnostics of unstable ¯ows of varied Rossby number in ically unstable zonal jet in a continuously strati®ed QG section 3 and discuss their relation to those of stable model. The basic-state jet varied only in the vertical. waves. Numerical model results for a ¯ow with inter- Using small-amplitude analysis they found that maxi- mediate Rossby number and ®nite-amplitude meander mum meridional particle excursion coincided with the are presented and contrasted with the small-amplitude depth of maximum meridional two-particle dispersion theoretical results to evaluate their applicability. In sec- that occurs at the critical level. Here, we add to this tion 4, the large amplitude numerical model results are study by considering a basic state that also varies in the presented. In section 5, the results are summarized and horizontal and has a ®nite Rossby number. We further their applicability to the Gulf Stream is discussed. investigate mean-¯ow evolution, both in the small am- plitude quasi-linear limit and in the ®nite amplitude non- 2. Model ¯ow and theoretical background linear regime. Several techniques are used to analyze the ¯ow: a small-amplitude theory for the zonal mean a. Basic state characteristics of ¯oats and tracers, a direct comparison To evaluate how Lagrangian ¯oats and tracers behave of the small amplitude results to a numerical model run, in a baroclinically unstable jet, we use a simple model and ®nally, the behavior of ¯oats and tracers when the whose Eulerian characteristics are known (Boss et al. baroclinically unstable waves grow to ®nite amplitude. 1996, herein BPT; Boss and Thompson 1999, herein Comparison of stirring and mixing in our study high- BT). There are two pools of constant PV in the upper lights the usefulness and limitations of the small-am- layer overlaying a quiescent lower layer (Fig. 1). The plitude theory. ¯uid is assumed to be Boussinesq, hydrostatic, rigid- The ¯ow analyzed here consists of two layers of con- lid, and on an f plane. We refer to this set of approx- stant density with a basic state that has a meridionally imations as the shallow water approximation (SW). The con®ned jet in the upper layer and a quiescent lower zonal geostrophic basic-state (denoted by superscript layer. This model shares several features observed in zero) depth and velocity structure in the upper layer are the Gulf Stream such as a near-surface jet, a PV front, 1/2 and the presence of baroclinic instability. To analyze H12 the ¯oat and tracer behavior in this ¯ow we apply the H11 2 1 exp(y/Rd,1) 1 1 for y , 0 []11H11 2 2 0 formalism of generalized Lagrangian mean (GLM) the- h1 5 ory. This theory provides a framework to analyze and H 1/2 H 11 1 exp( y/R ) 1 for y 0 quatify Lagrangian behavior.