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 ϭ c). Critical line denotes the horizontal position in a layered ¯ow where u0 ϭ c. In continuously strati®ed ¯ows with variations in the horizontal, u0 ϭ 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.
᭧ 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/͙gЈH and /f ϳ 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, gЈ its reduced gravity, H assumptions that capture many of its observed features the thermocline depth, 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 Ϫ 1 exp(y/Rd,1) ϩ 1 for y Ͻ 0 []H11 0 formalism of generalized Lagrangian mean (GLM) the- h1 ϭ 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. Using a small amplitude 12 Ϫ Ϫ d,2 ϩ Ͼ []H12 expansion of the governing equations, the theory gives (1) predictions of zonal mean quantities such as the velocity and displacement of the center of gravity of Lagrangian and ¯oats. The theory connects these quantities to the un- gЈ dh exp(y/R ) for y Ͻ 0 derlying wave ®eld through the Stokes drift. The evo- 0 1 d,1 u10ϭϪ ϭU (2) lution of the zonal-mean tracer distribution is analyzed fdy Άexp(Ϫy/Rd,2) for y Ͼ 0, 290 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29
et al. (1984) and Wood (1988). The QG case is recovered in the limit ⑀ ϭ 0 (BPT). The above model in the limit 1 r → ϱ becomes the 12 -layer model studied for ⑀ ϭ ͙2 by Stommel (1965) and Paldor (1983), for variable ⑀ by Williams (1991), and for QG by Pratt and Stern (1986) and Pratt et al. (1995). In BPT and Boss (1996), we summarize the results found in those studies and interpret them in terms of the linearized waves found in the different models. In particular we ®nd that the PV-front model is baroclinically unstable for all values of ⑀ and ®nite r. The vortical (Rossby like) waves taking part in the instability are trapped to the PV gradient near y ϭ 0 and are found for all ⑀. They are well approxi- mated by the QG solution. The Eulerian mean evolution FIG. 1. Schematic of the (a) depth and (b) velocity ®elds of an upper-layer potential vorticity front. The upper-layer ¯ow is geo- of the baroclinic instability is explored at small and large 0 strophic and has piecewise constant PV, q1j ϭ f/H1j, where j denotes meander amplitude and for various Rossby number the side of the front (1 being south, y Ͻ 0), H1j denotes the depth of ¯ows in BT. At all values of meander amplitude the the upper layer at y → ϯϱ, respectively, and the lower layer is baroclinic evolution of the QG ¯ow was found to ap- quiescent. When H12 ϭ 0, the interface between the layers intersects the surface, resulting in an outcropping front. proximate well that of the SW ¯ows. Here we analyze this model for its tracer and La- grangian characteristics in both its QG limit (assuming where gЈ and f are the reduced gravity and the Coriolis ⑀ ϭ ͙2/2) and a SW model with the same Rossby parameter and number for small amplitude meander. The depth ratio (r ϭ H /H ) is taken from the Gulf Stream's parameters U ϭ (gЈH )1/2Ϫ (gЈH ) 1/2 and R ϵ (gЈH )/ 1/2 f (3) T 1 01112 d,j 1j in the region east of Cape Hatteras, where the total depth are the jet maximum speed and the (different) defor- HT ϳ 4000 m and the mean thermocline depth H1 ϳ mation radii on each side of the front. The basic state's 500 m yielding r ϳ 8. 0 potential vorticity is given by q1j ϵ f/H1j, where j de- notes the side of the PV front (1 being south, y Ͻ 0) → b. Quasi-linear formulation and H1j the ¯uid depth at y ϯϱ (Fig. 1). The ¯ow is con®ned to within a deformation radius of the po- To understand the ®rst-order effect of unstable waves 0 on the transport of ¯oats and tracers, we assume that tential vorticity front. The lower-layer depth is h2 ϭ HT 00the deviations from the basic-state variables (the waves, Ϫ hu12, where HT is the total ¯uid depth, and ϭ 0. In order to study this ¯ow as a function of Rossby denoted by primes) are small relative to the amplitude number, it is convenient to nondimensionalize the var- of the basic state (introduced above and denoted with iables by transforming superscript zero) and have meridionally varying ampli- tudes and normal mode structure, ϭ aRe{Ј(y)ei(kxϪt)}, tR → d → → where Ј denotes any perturbation (wave) variable and t ,(x, y) (x, y)Rd,(u, ) (u, )U0, U0 a denotes the amplitude of the perturbation. For unstable ¯ows, the frequency is complex and its imaginary → → 1/2 hiihH1 and iip (gЈH10) U , part, I, is the growth rate. The solution for Ј(y)is found by linearizing the momentum and continuity where equations about the basic zonal-mean ¯ow and solving for the normal modes [Phillips (1954) and, for our ¯ow, 1/2 H1 ϵ (H11 ϩ H12)/2, Rd ϵ (gЈH1) / f (4) BPT]. in which H1 is the y-averaged upper-layer depth and Rd The modi®cation of zonal mean quantities from the the radius of deformation based on H1. The Rossby basic state due to the waves [denoted by tilde, (Ä )] is number, ⑀ ϵ |U 0|/ fRd, is a measure of the strength and then found by substituting these perturbation solutions asymmetry of the initial geostrophic ¯ow (BPT). The into zonally averaged equations [Phillips (1954) and for ¯ow depends on only the Rossby number ⑀ and the depth our ¯ow, BT] where the zonal mean is de®ned by ( ) Ϫ1 ∫ ratio (r ϭ HT/H1). ϵ 0 () dx, ϵ 2/k. While the normal mode This simple ¯ow exhibits a baroclinic zonal transport perturbations, A(y)ei and B(y)ei , have no zonal mean and a most unstable wave growth rate that are similar structure, their correlations can have a zonal mean struc- to those observed for the Gulf Stream (see BT for a ture since more detailed comparison). Special cases of this PV- 1 front model have been considered in the past and applied Re{A(y)ei}Re{B(y)ei} ϭ Re{A(y)B(y)*}e2Im{}, 2 to the Gulf Stream; when ⑀ ϭ ͙2 (H12 ϭ 0) the model reduces to the outcropping front studied by Killworth where the asterisk denotes the complex conjugate and FEBRUARY 1999 BOSS AND THOMPSON 291
ϭ ik(x Ϫ t). The modi®cations of mean quantities fact that in shallow water, the horizontal velocities are 2 2It are thus O(a ) and are proportional toe . The am- uniform within layers. The displacement vector (Јn ) is -Јn ϭ O(a)]. Notice that the displace ´ ١] plitude expansion of the zonal mean of a variable there- nondivergent 0 22It 3 0 fore is ϭ (y) ϩϩa ˜(y)e O(a ). Notice that ments are nearly singular near critical lines where un Ϫ with this de®nition˜ is O(1). c approaches zero. L Underlying this amplitude expansion is the require- The GLM velocityun for small-amplitude meanders ment of small perturbation amplitude, aeIt K 1. Also, is (Andrews and McIntyre 1978) it is assumed that only a single wave is present in each Ј2 layer, the one with the highest growth rate of linear L 0 n 0 unϭ u nϩ uÄ nϩ Ј nikuЈϩ nЈ nuЈϩ nyu nyy, baroclinic instability. Using this decomposition, both 2 the ®rst-order Lagrangian mean velocities and the evo- L ϭ Ä ϩ ЈikЈϩЈЈ , lution of a passive tracer can be predicted. nnnnnny hЈwЈ wL ϭ wÄ ϩ ЈikwЈϩЈwЈϩ nn. (9) nnnnnny 0 c. The generalized Lagrangian mean hn
0 Complete Eulerian and Lagrangian descriptions of a Here, un ϩ uÄ n is the zonal (Eulerian) mean velocity to 2 0 ¯ow contain the same information. However, once av- O(a ), whereun is the basic-state ¯ow. In addition, the eraging is performed, some information is lost, and the GLM velocity (minus the basic ¯ow) is proportional to zonal-mean Eulerian and Lagrangian descriptions con- e2It; thus it grows in time for unstable waves and is tain different information. Here we describe the zonal constant for stable waves. For the normal-mode waves mean movement of Lagrangian particles using the gen- studied here (which is proportional to eik(xϪct)), the av- eralized Lagrangian mean formalism (Andrews and erage of the velocity of particles equally distributed over McIntyre 1978). The GLM theory gives a prediction for a line spanning a wavelength (the GLM velocity) is the velocity of the center of mass of a line of particles equal to the average of the velocity of a single particle initially parallel to the basic-¯ow streamlines (which we over the wave period (both are averages over the wave refer to as the ``centroid''). For small amplitude waves phase). The latter is familiar from the theory of surface this velocity is the sum of the Eulerian mean velocity gravity waves. plus the Stokes drift. The difference between the GLM velocity and the SD L 0 Within a layer n (ϭ1, 2), the Lagrangian particle dis- Eulerian mean velocity uuunnnϵϪϪuÄ n is the Stokes placement Xn is related to the velocity ®eld un by the drift, that is, the difference between the average velocity kinematic relation d Xn/dt ϭ un. For small amplitude of a drifting particle relative to the velocity at the av- waves, Andrews and McIntyre (1978) de®ned a mean- erage position of that particle. It can arise both from free Lagrangian parcel displacement, Јnnnnϵ (,Ј Ј ,hЈ ), basic-state curvature and/or a spatially varying wave associated with a normal-mode wave (perturbation) ve- displacement. locityuЈn , which to O(a)is There are several differences between meridional Eu- 0 lerian mean and Lagrangian meridional velocities; the dun D ЈϭuЈ , (5) Eulerian zonal-mean (EM) meridional velocity (Ä n ) and nn ndy the meridional mass transport ( nhn ) are O(⑀) because D ЈϭЈ, (6) there is no mean zonal pressure gradient (so there is no nn n mean geostrophic meridional ¯ow) and because the ver- dh0 tical displacements (hЈ ) are O(⑀) compared to the hor- DhЈϭϪwЈϪЈ 1 , (7) n 11 1 1dy izontal displacements (BT). The meridional displace- ment, however, is not constrained by geostrophy (Uyru dh0 DhЈϭwЈϪЈ 2 , (8) 1979; McIntyre 1980) and therefore both the meridional 22 2 1dy Stokes drift and meridional GLM velocity are O(1) in Rossby number. The Stokes drift is nondivergent where D ϵ ik(u0 Ϫ c) is the O(1) expansion of the n n ( ´ uSD 0) for stable ¯ows and often is divergent for n ϭ ١ total derivative. Equations (7) and (8) are the evolution equations for the interface thickness and could be unstable ¯ows (McIntyre 1980). In contrast, the mean meridional mass transport h is always nondivergent. thought of as the kinematic boundary conditions at the n n layer's interface, where the vertical perturbation veloc- 3 0 ity is wЈnϭϮ(ikuЈ nϩ Ј ny)/h n from continuity and the d. Zonal mean tracer evolution The zonal mean evolution of a passive tracer is gov-
erned by the tracer conservation equation Dn/Dt ϭ 0. 3 Vertical velocities in this section are evaluated at the interface z The zonal mean tracer concentration to second order in h0 and are de®ned to be positive upward. They decay to zero at ϭ 1 022It the top and bottom boundary, are equal at the interface, and are amplitude is given by n ϭ nn(y) ϩ a Ä (y)e , with included here for comparison with the results of both Shepherd (1983) the second term arising from the presence of an O(a) ik(xϪt) and continuously strati®ed models (AHL). tracer perturbation, Re{aЈn (y)e }. To O(a) the trac- 292 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 er conservation equation (after division by ei(kxϪt))is particle dispersion) is indicative of stirring that occurs given by only within a layer (along isopycnals) and is due to eddy processes since the model is inviscid. In the numerical D Ј ϭ ik(u00Ϫ c)Ј ϩϭЈ 0. (10) n nnnnny model, small momentum and tracer diffusivities may Together with (6) it can be rewritten as cause irreversible mixing, but once again only along isopycnals. When the ¯ow is linear, steady and adiabatic D (Ј ϩ Ј 0 ) ϭ 0. (11) n nnny (under nonacceleration conditions, AHL), the tracer This implies that, if the perturbation of the tracer is transport velocity is equal to the mass transport velocity 0 initially zero [i.e., the tracer is zonally uniform (Rhines ( nnh /hn ). In general and in our model, both vanish for and Holland 1979)], then such stable (linear) waves, and no net meridional trans- port occurs (this result is referred to as the nontransport Ј ϭϪЈ 0 , (12) nnny theorem, AHL). which links the tracer perturbation with the zonal-mean Mixing requires both stretching and folding (e.g., Ot- tracer distribution. tino 1989). The Lagrangian diffusivity is indicative of
The mean tracer modi®cationÄ n(y) arises from the stretching but not of folding, which requires rotation. presence of waves and is found from the O(a 2) expan- As a measure of rotation (vorticity) we use another dif- sion of the zonally averaged tracer conservation equa- fusion-like term proportional to the cross-correlation of tion: the zonal and meridional displacement:
2It 0 xy DnInnynnynn/Dt ϭ e (2Ä ϩ Ä ϩ ЈЈϩuЈikЈ) ϭ 0. nnnnnϭ ЈuЈ ϭ IЈЈ. (13) While this term does not appear in the zonal-mean La- Substituting (12) in (13) and recasting (13) as an ad- grangian and tracer equations (due to zonal averaging, vection diffusion equation gives although it does appear in the time mean equations), we will demonstrate below that it is an important term to Ä ϩϭT 00( ЈЈ) ϵ ()yy 0 , (14) xy nt n ny n n nyy n ny y use to infer mixing from stirring. In QGn equals half where the angular momentum of the waves (Rhines and Hol- land 1979). When time mean equations are derived, T ϵ Ä ϩ wЈЈ /h , xy 11111 n appears in the off-diagonal terms of the diffusivity tensor (Rhines and Holland 1979). Tyyϵ Ä Ϫ wЈЈ/h , and ϵ ЈЈϭЈ2 , (15) 22222 nnnIn Several connections can be made between the mean are the tracer transport velocities in each layer, and tracer formulation and the GLM. First, the Lagrangian yy n is the Lagrangian diffusivity (Rhines and Holland velocity is the velocity of the center of mass of a patch yy 1979). Heren is proportional to the cross-stream two- of tracer (Plumb and Mahlman 1987). In addition (e.g., 2 particle dispersionЈn with the growth rate as the pro- AHL), portionality constant (AHL). Both are nonzero only for LTyyϭϩ, (17) transient waves and are strictly positive for growing nnny unstable waves. which for a small Rossby number ¯ow (17) implies that T L SD yy Similar to the EM meridional velocity, n is O(⑀). nnϳϳ ny; that is, the Lagrangian velocity is dom- yy Heren is O(1) and is spatially inhomogeneous in a inated by the Stokes drift, which is approximated by the divergent Stokes drift ®eld [such as for an unstable ¯ow, gradient of the Lagrangian diffusivity. Plumb and Mahlman (1987)]. Thus the dominant bal- ance in (14) for small Rossby number ¯ows is between e. Solution method the time rate of change of the tracer perturbation and the eddy diffusion term. In this section we outline the procedure for computing Another interesting result is derived by multiplying the Stokes drift, the Lagrangian and tracer velocities,
(12) byЈn and zonally averaging: and the Lagrangian diffusivity. For the analytical quasi- linear calculations we use the most unstable wave so- ЈЈ ϭϪyy 0 . (16) nn n ny lutions derived in BPT to compute the Stokes drift. Sum- This implies that for a growing (unstable) wave the ming it with the Eulerian mean velocities (solved in BT) meridional ``eddy'' ¯ux of tracer (as well as PV) is down provides the Lagrangian velocities (9). Similarly the the mean tracer gradient (Rhines and Holland 1979). tracer velocity and diffusivity are computed using (15). Substituting q for , (16) shows a fundamental differ- For the numerical model (the appendix), we compute ence between stable and unstable waves. Unstable the Lagrangian velocity from its de®nition, the ensemble waves ¯ux PV across mean PV contours, whereas stable averaged velocity of particles initially equally spaced waves do not (Rhines and Holland 1979; Lozier and and aligned parallel to the jet axis. The time rate of Bercovici 1992). change of the zonal-mean tracer distribution is calcu-
For the physical situation that we study below in the lated from its de®nition, nt. small amplitude limit, the Lagrangian diffusivity (two- To facilitate comparison of numerical results and the- FEBRUARY 1999 BOSS AND THOMPSON 293
1 FIG. 2. The (a) upper and (b) lower 12 -layer models that are used to interpret the instabilities present in the two-layer model of Fig. 1. The jet structure of the upper-layer model is identical 1 to that of the two-layer model. Both 12 -layer models are stable to normal-mode perturbation, while the two-layer model is unstable. ory, all quantities are normalized by the square pertur- and phase of the unstable wave were found to be very 2 bation amplitude a , where a ϭ [(21Ј y ϭ 0)]/ is the similar to those of the resonating stable waves computed ratio of the maximum horizontal displacement of the in two models, each with one layer identical to the two- front [1Ј (y ϭ 0)] to half the (dimensional) meander layer model plus a second in®nite layer (equivalent bar- 1 wavelength . The horizontal displacement of the front otropic or 12 -layer models, Fig. 2). Here, the same de- (dimensional) will be given by a/2 and since the most composition is used to understand the connection be- unstable wave in our model has a typical length of tween wave displacements and the Stokes drift for the 10.5Rd,(1Ј y ϭ 0) ϭ a ´ 5.25Rd. We compute the quasi- frontally trapped waves. linear GLM and mean tracer variables for two different The wave trajectories relative to the mean ¯ow and ¯ows: a QG ¯ow with ⑀ ϭ ͙2/2 (with most unstable the displacement ellipses (Fig. 3) display clearly how a wavenumber k ϭ 0.58) and a SW ¯ow with the same Stokes drift comes about. As the particle moves merid- ⑀ (k ϭ 0.6). ionally, the zonal wave displacement changes. Averaged over the phase of the wave, the zonal drift is in the same 3. Small amplitude results direction as the particle zonal velocity in the latitude where the zonal displacement is largest. For example, a. Application of the theory to the PV-front model in the lower layer for y ϽϪ2, the ellipses are cyclonic To develop some intuition about the Stokes drift and (clockwise) and the amplitude of the zonal displacement GLM velocities in the unstable ¯ow, we ®rst analyze increases with y resulting in a positive zonal drift. For the GLM velocities of stable waves (e.g., Matsuno the upper-layer model there is also a drift due to the 1 20 1980). In BPT we found that analyzing an instability as mean ¯ow curvature, 211u yy (9) which is everywhere a resonance between stable waves, one from each layer, positive for the model's jet, yet about 20 times smaller provides physical insight into the instability. Structure in magnitude than the wave±wave terms.
FIG. 3. Particle trajectory relative to the basic-state ¯ow (left side of each panel) and the displacement ellipses (right side of each panel) for 1 the ®rst vortical modes of the upper and lower 12 -layer QG models (k ϭ 0.6). Particles were initially at x ϭ 0 and drifted for two periods. All 1 0 displacements are normalized byЈn (y ϭ 0). Broken lines denote the positions of critical lines of the upper 12 -layer model (where u1 ϭ c). 294 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29
FIG. 4. Stokes drifts of both stable (left panels) and unstable (center and right panels) ¯ows. Results are shown for QG PV front and an intermediate ⑀ PV front. Solid (dashed) lines represent upper (lower) layer variables and broken lines denote both the position of the critical 2 lines in the upper layer (where u 1 ϭ Re{c}) and the position of the outcropping front. The Stokes drifts are normalized by a where a ϵ
[2max(n(0))]/ for the stable waves (modes of Fig. 2). For the unstable waves (modes of Fig. 1), a ϵ [2max(1(0))]/.
L SD The meridional structure of the zonal Stokes drift of The GLM meridional velocity (nnnϭϩÄ ) and the unstable mode of the two-layer ¯ow is similar in Lagrangian zonal acceleration4 are dominated by the structure and magnitude to that of the stable mode in Stokes drift in most of the domain except close to the 1 the two 12 -layer models (compare left and center panels upper-layer jet, where the EM and Stokes drift are of of Fig. 4). Differences are found in the upper layer in similar magnitude (Fig. 5 and Fig. 3 in BT). The GLM the vicinity of the critical lines, where the stable wave meridional velocity in the upper layer converges toward 0 Stokes drifts are singular [where Dn ϭ ik(u1 Ϫ c) ϭ 0]. the critical lines, being dominated by the Stokes drift. Also, while the amplitude of the Stokes drifts of stable Divergence of the GLM meridional velocity occurs near waves is constant in time, it grows likee2It for unstable the jet's center in the upper layer. Although dif®cult to waves. The amplitude of the zonal Stokes drift is sub- see in Fig. 5, in the lower layer there is a convergence stantial; a perturbation with a ϵ 21Ј (0)/ ഠ 0.2 results below the jet whose center shifts south with ⑀. in a 20% difference between the Eulerian and Lagrang- The zonal mean tracer velocity (15) is in the same ian mean velocities. direction as the zonal mean meridional mass ¯ux (Fig. No meridional Stokes drift is present for stable waves 6: Fig. 3 in BT), in the direction that reduces the po- SD with meridionally varying amplitudes ( ny ϭ 0; this is tential energy of the mean ¯ow. Unlike the mass ¯ux, similar to the absence of vertical drift of surface gravity it is divergent and is unequal and opposite in the layers. yy waves). The pattern of meridional displacement result- The upper-layer Lagrangian diffusivity (1 , Fig. 6) is ing from stable waves, however, is indicative of the maximum near the critical lines in the upper layer, being direction of meridional Stokes drift of the unstable proportional to |u 0 Ϫ c|Ϫ2 [(6), (15)]. A similar result waves. Since the maximum meridional displacement oc- was found in a critical level by Lozier and Bercovici cur at the critical lines in the upper-layer model and near (1992). In the lower layer the diffusivity is maximum y ϭ 0 in the lower layer (Fig. 3), where the perturbation where the perturbation amplitude is maximum. The amplitude is maximum (Fig. 6 in BPT), we expect the maximum is under the jet axis in QG and migrates south drift to the critical lines (Fig. 3). Indeed, the structure with increasing ⑀. yy of the meridional Stokes drift of unstable waves (Fig. Unlike the meridional diffusivity (n ) that is of the xy 4) is consistent with the insight derived from the struc- same magnitude in both layers,n is two orders of ture of the stable waves. magnitude larger in the upper layer (with largest value The QG solution for the Stokes drift is found to com- near the critical lines) than in the lower layer. Since both yy xy pare well with the SW one except for a discontinuity stretching (nn ) and rotation ( ) are needed for mixing in the zonal Stokes drift at y ϭ 0 in the SW model 0 owing to the discontinuity ofu1yy there. The loss of symmetry around y ϭ 0 in the SW model arises as a result of the loss of symmetry of the basic-state ¯ow as 4 We compute the zonal Lagrangian acceleration because it is pro- portional to the wave induced O(a2) part of the Lagrangian velocity the Rossby number is increased. In the lower layer the L SD uuntϭ 2I(uÄ n ϩ n ) and for comparison with ¯oats trajectory data agreement is better than in the upper layer, due to the from the numerical model. To recover the GLM zonal velocity, use L 0 L smaller relative layer depth change across the front. uuunnntϭϩ/(2I). FEBRUARY 1999 BOSS AND THOMPSON 295
FIG. 5. The upper-layer (solid line) and lower-layer (dotted line) meridional GLM velocity LL ( nnt , left panels) and zonal acceleration (u : right panels), for a QG PV front (with ⑀ ϭ 0.71) and SW PV front with the same ⑀. Each was normalized as in Fig. 4. Broken lines denote the positions 0 of critical lines (where u1 ϭ Re{c}) and the position of the outcropping front. to take place, mixing is predicted to be signi®cantly which we initialize with the basic ¯ow corresponding stronger in the upper layer, as will be seen with the to ⑀ ϭ ͙2/2. Details on the numerical procedures are numerical model in the next section. For cyclonic (an- found in the appendix. xy ticyclonic) rotation,n is negative (positive) and is of The GLM meridional velocities computed from the different sign at the two critical lines (Fig. 6). Stirring model ¯oats agree quantitatively well with the theoret- xy yy (as indicated by the amplitude of both11 and ) is ical prediction (Fig. 7). The largest difference is due to signi®cantly more intense in the northern critical line the difference in location of the critical lines in the in the SW model. This is an ageostrophic effect asso- numerical model due to the inevitable (yet small) dif- ciated with an increase in the horizontal displacements ferences in mean ¯ow between a free surface and a rigid- amplitude on the northern (shallow) side of the front. lid model. The numerical grid is such that the maximum By continuity, for a given perturbation in layer depth, jet velocity in the model occurs half a grid point relative the associated horizontal response will be larger on the to the jet center (0.12R d), and thus the biggest differ- shallower side. ences in u occur for the ¯oat released there. In addition, the presence of diffusion (numerical and explicit) affects both the mean ¯ow and the instability growth rate, and b. Numerical model result with a small-amplitude thus the position of the critical lines. Agreement be- meander tween the theoretical prediction with the modeled GLM In this section we compare the inviscid quasi-linear zonal acceleration is not as good as for the meridional results against results from a viscous numerical model velocities, especially close to the critical lines in the with small (but ®nite) meander amplitude. We use a free upper layer, which at ®nite amplitudes develop into a surface isopycnal model developed by Hallberg (1995), recirculation region with ®nite meridional extent (see
FIG. 6. The upper-layer (solid line) and lower-layer (dotted line) Eulerian mean meridional tracer velocity Tyy xy (nn , left panels), eddy diffusivity ( , center panels), and horizontal displacement cross-correlation ( n : left panel) for a QG PV front (with ⑀ ϭ 0.71) and SW PV front with the same ⑀. All are normalized as 0 in Fig. 4. Broken lines denote the positions of critical lines (where u1 ϭ Re{c}) and the position of the outcropping front. 296 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29
processes (neither included in our theoretical analysis nor in the way we compute the Lagrangian acceleration with the model results) are acting on the zonal mean jet and are affecting the O(a 2) zonal acceleration because 0 u1yy has a discontinuity at the front in the initial con- dition. The smaller value of the acceleration in both layers is caused by both reduction in the growth rate due to viscosity and nonlinearities as the perturbation grows (Pedlosky 1987) and the smearing of the signal when averaged over a meandering jet. The presence of an Eulerian barotropic mean ¯ow in the model was dis- cussed in BT. Here it causes the zonal GLM velocity to be nonzero at large y. Because there is no O(1) zonal- mean meridional velocity and because it is small close to the jet center where the meander amplitude is greatest, such a contamination of the O(a 2) by the O(1) dynamics is not observed in the meridional-mean Lagrangian ve- locities (Fig. 7). A basic-state tracer ®eld with a linear gradient was chosen to minimize numerical diffusion, and is given by
nnnny(t ϭ 0) ϭ (t ϭ 0) ϭ (y ϭ 0) ϩ y . (18)
n With n(y ϭ 0) ϭ 0.5 and ny ϭ (Ϫ1) 0.032, the con- centration varies from 0 to 1 over the model domain and is in opposite directions in each layer. The numerical
results of nt/ ny are in good quantitative agreement 0 with the quasi-linear calculations ofÄ nt / ny (Fig. 8). Differences between model and theory are biggest near the southern critical line. 0 yy 0 For our choice of mean tracer ®eldnnyny , ഠ SD 0 nny(17) is the forcing on the rhs of (14), which L explains the similarity between n in Fig. 7 and nt/ ny yy T ϭϪny n in Fig. 8. Results of a run with an explicit FIG. 7. Meridional GLM velocity and zonal acceleration in the 2 Ϫ1 upper and lower layers in theory (thin lines) and in the numerical tracer diffusivity ϭ 10 m s were quantitatively model (thick lines), both with intermediate ⑀ ϭ 0.71. All are nor- similar. malized with a2 where a ϭ 0.15 Ϯ 0.02 in the model. Dashed lines denote the position of the theoretical critical lines in layer 1. 4. Large-amplitude numerical model results below). Changing our interpolation scheme (from bilin- While the small-amplitude theory is useful where it ear to nonlinear) and increasing the number of grid is applicable, for meander amplitudes on the order of a points (from 1282 to 2002) did not change the results wavelength no similar theory is available. Here we in- signi®cantly. Lack of quantitative agreement in the up- vestigate the ®nite amplitude Lagrangian and tracer evo- per-layer zonal GLM accelerations is due to existence lution and inquire, in particular, whether the features of a strong O(1) sheared zonal mean ¯ow there. Viscous which have their seeds in the small-amplitude theory
FIG. 8. Time rate of change of zonal mean tracer concentration divided by the zonal mean gradient as predicted by theory (thin line) and as calculated from the numerical model results 2 (thick line) in both layers. All are normalized by a , where a ϭ [2 max(1(0))]/; a ϭ 0.05 Ϯ 0.01 for the model run with ϭ 0. FEBRUARY 1999 BOSS AND THOMPSON 297
to the migration of the critical lines toward y ϭ 0as the mean ¯ow weakens (the phase speed also weakens but, apparently, slower). Also, for ®nite amplitude waves, critical lines bifurcate, creating recirculation re- gions. For stable waves the meridional extent of the recirculation regions was found to be proportional to the square root of the amplitude and inversely propor- tional to the mean ¯ow shear (Pratt et al. 1995). Zonal averaging over these evolving regions is also respon- sible for the observed migration. Beyond equilibration (at about t ϳ 200) the convergence pattern is more erratic, especially on the northern side, and multiple zonal jets are present (BT). In the lower layer, conver- gence toward the front continues beyond t ഠ 200. As the lower layer jet is accelerated, the centroids spread into two main convergence zones (t ϳ 250). Centroids intersect prior to the instability equilibra- tion (Fig. 9), at which time the functional mapping from the initial position of the centroid to its ®nal position is nonunique. Since Lagrangian means are meaningful only relative to the parcel initialization, a reinitialization procedure may be warranted at such a time (McIntyre 1980; Dunkerton et al. 1981). However, it is interesting that even long after the centroid positions cross, the meridional Lagrangian velocity is in the same direction as predicted by the small-amplitude theory. By the end of the simulation (t ϳ 375), the spread of the centroids is approximately equal in both layers with slightly more centroids north of y ϭ 0 than south in both layers. In the upper layer this pattern is consis- tent with the increase in Lagrangian diffusivity to the north (Fig. 6). In the lower layer a similar pattern of Lagrangian diffusivity is expected once a substantial mean jet has accelerated there (BT). FIG. 9. Mean Lagrangian meridional position of the center of grav- ity of 21 lines of ¯oats, initially spaced evenly around y ϭ 0 (upper We emphasize that individual ¯oats in the upper layer two panels). In the initial stage of the instability, notice the conver- do not cross the local position of the PV front prior to gence toward critical lines in the upper layer and toward the center eddy detachment at t ϳ 290. Individual ¯oats and cen- in the lower layer. The evolution of total energy (solid), zonal mean troids of lines of ¯oats cross the mean position of the energy (dashed), perturbation energy (dotted), and zonal mean kinetic energy (dot-dashed) as a function of time is shown in the lowest front and jet (Fig. 9). Individual lower-layer ¯oats do panel. The solid line, parallel to the perturbation energy curve, rep- cross below the local position of the upper-layer PV resents the theoretical growth rate of the instability, exp(2It). front since the lower-layer perturbation is phase lagged relative to the upper layer's meander. This phase lag is characteristic of baroclinic instability. appear when the meanders reach ®nite amplitude. The Coincident with the ¯oat centroid convergence to the energetics evolution as a function of time (bottom of critical lines with increased meander amplitudes is the Fig. 9) is used to trace the instability evolution from its formation of recirculation regions around the location linear phase where the perturbation energy grows ex- of the critical lines. The recirculation regions are clearly ponentially to the equilibration phase over which the seen in the tracer distribution ®eld (Fig. 10). Two phe- long-term energetic growth is algebraic. nomena, pooling and recirculation, govern the tracer As predicted from the small-amplitude theory, the distribution. Pooling refers to the process of conver- ¯oat centroids initially converge toward the critical lines gence/divergence of ¯uid due to differential advection, in the upper layer and slightly south of y ϭ 0 in the and recirculation refers to the process whereby ¯uid lower layer due to migration of the maximum in per- parcels are folded against each other and requires the turbation amplitude southward (BPT, BT). In the upper presence of vorticity. Since the perturbation amplitude layer, with time, the ¯oat centroids continue to drift past decays meridionally away from the jet core, ¯uid is the positions where critical lines were initially found pooled between successive troughs and crests of the (Fig. 9). This does not indicate a change in the ¯oat meander near the front, with increases of tracer gradients response to mean-¯ow characteristics, but rather is due away from the front in both layers (Fig. 10). Recircu- 298 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29
FIG. 10. Tracer ®eld (left panels, with 10 contours of values equally spaced between 0 and 1), zonal mean meridional tracer gradients (center panels), and position of Lagrangian ¯oats (right panels). Floats were initially aligned parallel to the PV front at a distance of 1.2 (⅜), 2.4 (ϫ), and 3.9 (ϩ) deformation radii on both the north and south sides of the front. Solid line denotes the position of the PV front. The tracer ®elds were calculated at t ϭ 188 with tracer diffusivity ϭ 10 m2 sϪ1 in the upper (upper panels) and lower (lower panels) layers. Notice the ¯uid convergence pattern in the lower layer (lower left and right panels) and both pooling and recirculation in the upper layer (upper left and right panels) in both tracer ®eld and Lagrangian particles' position. lation in the pools in the upper layer causes mixing amplitude of the meander increases, the recirculation through shear dispersion, with increased gradients on regions overlap in their meridional position (although the edges of the recirculation regions, especially on the at different zonal locations), resulting in t ϳ 188 in the side facing the jet core. Pooling with no recirculation observed zonal mean tracer gradient. In the upper layer, occurs in the lower layer, resulting in a very different the tracer is entrained into the growing recirculation tracer ®eld evolution. Pooling occurs where the eddy regions where enhanced mixing takes place. As the diffusivity of the quasi-linear theory is maximum (Fig. growth of the recirculation regions slows and during 6), while recirculation occurs where the displacement eddy detachment, mixing within the recirculation re- cross-correlation is maximum. Note that the recircula- gions resembles mixing within closed streamlines; the tion direction is the same as predicted by the sign of tracer homogenizes within recirculation regions and re- xy n and that it is intensi®ed on the northern side, as sults in the migration of the tracer gradient toward the predicted by the previous section's results. edges of these regions, the jet axis on one side and the The structure of the zonal-mean meridional tracer gra- far ®eld on the other side. dients y is similar in both layers, yet it is an order of In order to quantify tracer mixing, as opposed to stir- magnitude stronger in the upper layer at t ϳ 188 (Fig. ring, we conducted experiments with varying explicit 10); the gradient is weak near the mean front position tracer diffusivities: ϭ 0, 10, 100, and 1000 m2 sϪ1 and strong on the edges of the meander, particularly on and respective Peclet numbers Pe ϭ U 0 R d/ ϭϱ, 6000, the northern side. This pattern is consistent with the 600, and 60. The momentum ®eld and initial basic tracer Lagrangian diffusivity structure in the lower layer (Fig. ®eld were the same for all the runs. We quanti®ed the 6). In the upper layer one may expect a gradient max- mixing by binning both the PV and tracer concentrations imum at the front (there exists a minimum in eddy dif- at each model grid point into six different bins. Assum- fusion there, Fig. 6). Indeed, for small amplitude (not ing is either tracer concentration or PV, the bin bound- shown) the upper-layer mean tracer gradient matches aries are min() ϩ (0, 0.1, 0.3, 0.5, 0.7, 0.9, that predicted from Fig. 6, with two regions of minimum 1.0) ´ [max() Ϫ min()]. At each time step we compute gradients on either side of the front. However, as the the ¯uid mass within each bin (i.e., tracer mass). The FEBRUARY 1999 BOSS AND THOMPSON 299
FIG. 11. Time evolution of the volume of PV and tracer bins. The volume in each bin is represented by the distance between two consecutive lines, from lower values up, with the bold line denoting the total cumulative volume in all the bins, which is constant. For example, the volume in the fourth bin (0.5 Ͻ Ͻ 0.7) is given by the distance between the fourth and third lines from the bottom. Change in the distance between two lines denotes conversion of water and hence net mixing. Tracer volumes are given for Pe ϭ 60 and 6000. conversion of ¯uid from one bin to another can only numerical model results. The Lagrangian zonal accel- occur if mixing is present and represents the net amount eration did not compare as well near the jet center. We of ¯uid converted and is a lower bound to the actual believe this is due to contamination of the mean by volume of water changing its tracer value. This analysis modi®cation owing to viscous effects on the jet, which is similar to Nakamura's (1996) formulation of tracer are not included in the theory. mixing in area coordinates. Comparison of QG and SW reveals that the QG model Tracer mass conversion intensi®es at t ϭ 200, about does well in predicting the magnitude and direction of when the primary instability equilibrates, and continues Lagrangian and tracer quantities. Differences in the re- at a rapid rate (almost exponentially) until t ϭ 290 when sults arise from the differences in the basic state between eddy detachment occurs (Fig. 11). The tracer is observed the two cases. This result is consistent with BT's result to mix sooner in the upper layer due to the presence of that the baroclinic evolution of the front is weakly de- the recirculation regions. For PV, less mixing occurs pendent on the Rossby number. initially in the upper layer than in the lower layer since Both the tracer and Lagrangian analysis support the initially there is no PV gradient in the vicinity of the following mixing scenario for a baroclinically unstable critical lines in the upper layer (the only PV gradient jet: in the initial phase of the instability, recirculation is at the front). Until t ϳ 290 mixing is weakly depen- regions located at the critical lines in the upper layer, dent on Pe, except for the bins of concentration adjacent grow exponentially, and entrain new ¯uid. At this stage, to the wall (where a no ¯ux condition is applied, forcing in the upper layer, there is little dependence of mixing the tracer values to change). After t Ͼ 290 mixing is on Peclet number (Fig. 11). As the baroclinic instability Pe dependent. equilibrates, ¯uid is found to mix vigorously within the recirculation regions with continuous (yet slower) en- trainment of new ¯uid into these regions. Mixing causes 5. Discussion and conclusions tracer gradients to migrate toward the edge of the re- We conducted a quantitative comparison of the La- circulation regions and a strong property gradient is grangian mean and zonal-mean tracer-related quantities maintained at the meandering jet and at the outer edge derived from theory of inviscid ¯ows with results from of the recirculation regions. The zonal mean gradients, a numerical model. For small-amplitude meanders we however, are reduced when the recirculation regions ®nd good quantitative agreement between theory and from both sides of the jet meridionally overlap (and yet 300 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 they continue to be zonally separated, Fig. 10). The of modes associated with critical lines. The kinematics recirculation regions continue to grow algebraically un- of tracers and Lagrangian particles are therefore not til barotropic shear destroys the coherent ¯ow; eddies expected to change qualitatively but possibly quanti- detach, and a cascade of secondary instabilities ensues tatively. While it has been shown that, for a neutral wave (BT). After eddy detachment occurs, mixing is found to exist, a critical line must coincide with a region of to depend more strongly on the value of the Peclet num- constant PV, unstable waves can have critical lines in ber (Fig. 11). regions of PV gradient (Lozier and Bercovici 1992; Pratt In the lower layer there are initially no recirculation et al. 1995). regions and consequently very little mixing occurs. In- Despite differences in their dynamics, we ®nd simi- tense lower-layer mixing does not take place until after larities between a snapshot of tracer distribution prior the lower-layer jet accelerates. to eddy detachment in our model (Fig. 10) and tracer Until eddy detachment, the mixing pattern is consis- distributions in barotropic and equivalent barotropic jet tent with that predicted from two eddy diffusivity terms. models that are either stable or equilibrated (e.g., Ro- yy xy At the critical lines, wherenn and are present, there gerson et al. 1996, manuscript submitted to J. Phys. is stretching and folding causing enhanced mixing, Oceanogr.). In all cases ®nite-amplitude waves create which is weakly dependent on the value of the tracer recirculation regions around which the critical lines lie diffusivity. Where only stretching is present, as in the and within which the tracer recirculates. In order for lower layer, mixing is signi®cantly smaller. mixing to occur within stable barotropic and equivalent Shepherd (1983) found that QG models with different barotropic jets, a second wave is needed that causes a basic states were similar in their Eulerian structure, yet recirculation region to exchange ¯uid with its surround- quite different (in direction and magnitude) in their ings. In our case, temporal growth of the unstable mode GLM velocity and attributed the difference to the chang- supplies the recirculation regions with new ¯uid. Mixing es in the structure of the basic-state shear. Similarly we within the recirculation regions in all models will de- ®nd that changes in the shear between the basic states pend, among other factors, on the tracer diffusivity and of QG and SW result in a bigger difference between on the rate of supply of new ¯uid into the recirculation the mean Lagrangian velocities of the two models than regions. between the Eulerian mean velocities (compare Fig. 4 For a single stable wave the meridional Stokes drift here with Fig. 2 in BT). vanishes. As one reviewer remarked, when more than As the instability equilibrates and growth becomes one wave is present, correlations between the waves algebraic, the ¯uid within the recirculation regions mix- may result in a Stokes drift. However, since the waves es in a process similar to, but faster than, tracer ho- have different phase speed, two critical lines will exist, mogenization within closed streamlines, which was changing the geometry of the problem. Still, it is con- found to be initially proportional to Pe2/3 (Rhines and ceivable that GLM theory may be generalized to cases Young 1983). When the recirculation regions detach to where stirring occurs when more than one stable wave form eddies, the theory of tracer homogenization within is present. It would be interesting to see how predictions closed streamlines should apply within them, and in- of such a calculation compare with those derived from deed, we notice a dependence of the mixing rate on the the theory of dynamical systems. Peclet number (Fig. 11). The theory and the model both point at the different Plumb and Mahlman (1987) found the zonally av- structure of the Lagrangian mean and Eulerian mean jet. eraged eddy diffusivity tensor to be a good predictor We ®nd the GLM zonal velocity to be faster than the for the rate of mixing observed in most regions of a Eulerian mean zonal velocity on the north side of the LL three-dimensional atmospheric GCM. As in our case, front and slower on the south side (Fig. 5, uuntϰϪ n xy 0 their diffusivity tensor does not include the term . un) for an intermediate O(1) Rossby number jet. While The disagreement, we believe, stems from the ¯ows such an analysis has not been attempted with Gulf under study. While we analyze the tracer evolution in Stream ¯oats, there is an indication that downstream a free, localized unstable ¯ow, Plumb and Mahlman velocity averaged over ¯oats differs from Eulerian mea- (1987) analyzed tracer evolution in an equilibrated, surements, even though the overall structure of the jet forced dissipative system. in streamline coordinates changes slowly downstream The PV structure of our model is such that there are of Cape Hatteras. For instance, a suggestion of the dif- no dynamical critical lines. The PV equation is regular ference between Eulerian and Lagrangian structure of at the upper layer's critical lines because the PV there the stream may be found in the analysis by Song et al. has no gradient. Inclusion of such a gradient may affect (1995), who calculated the downstream ¯oat velocities the results since additional unstable modes may be pres- of 64 RAFOS ¯oats adjacent to the 14.5Њ isotherm. ent. However, studies of similar baroclinic jets on a  Downstream velocity maps averaged over ¯oats present plane suggest that the dynamics are similar (Ikeda 1981; in either meander crests or troughs (their Fig. 10) show Wood 1988). Also, if a gradient in PV occurred there, an in¯ection point in the downstream on the anticyclonic one would expect it to be reduced by the same processes shear side of the jet (north), which was more pronounced that cause the tracer to mix due, possibly, to breaking for ¯oats at crests. Although this average over ¯oats is FEBRUARY 1999 BOSS AND THOMPSON 301 not the same as the averaging used in GLM, it is based occurring at the critical lines on both sides, ``expelling'' on ¯oat velocities, as opposed to Eulerian measure- the tracer gradient to the vicinity of the jet's axis. ments. Eulerian velocity measurements of the down- There are several ways in which the dynamical frame- stream velocity above the 12ЊC isotherm in the Gulf work used here could be applied to analyze ¯oat data Stream [e.g., Fig. 9 of Johns et al. (1995)] suggest that in the vicinity of jets. First, all the components of the the shear of the downstream current on either side of diffusivity tensor could be computed in the framework the jet maximum is single signed along both isotherms of a jet to establish the stirring geometry [Freeland et and isobars. Here we suggest that the difference between al. (1975) computed the diagonal terms of the diffusivity the mean Lagrangian and Eulerian observations of the tensor from ¯oat data collected in the mode region]. Gulf Stream is due to the inherent differences between Extending the analysis to spatially growing waves, using them, as formulated by the Stokes drift in the small- temporal rather than spatial averaging [e.g., Cronin amplitude theory. (1996) for the Eulerian mean], may be of more appli- We ®nd convergence of the centers of gravity of ¯oats cability to the Gulf Stream. For instance, analysis of to the critical lines (where the displacement are maxi- ¯oats released at a single location relative to the stream, mum) in the upper layer and to the location of the max- but on different phases of the meander, may be used to imum in displacement in the lower layer. Freeland et al. compute the Stokes drift. (1975) observed a migration of the centers of gravity The simple model introduced here lacks topography, of ¯oats toward areas of higher displacement (as quan- the  effect, continuous strati®cation, wind conver- ti®ed by increased eddy kinetic energy) in the MODE gence, and spatially growing dynamics (as opposed to region. a reentrant channel), all of which may modify our results Analysis of ¯oat trajectories, released within the Gulf with regard to a speci®c application such as the Gulf Stream, inferred that the PV front (Bower and Lozier Stream. However, the model's simplicity provides in- 1994) or the jet axis (Song et al. 1995) provided a barrier creased dynamical insight into the processes that affect to ¯oat crossing. The two sites cannot be distinguished geophysical ¯ows and ¯oats and tracers within them. with the available data (A. S. Bower 1997, personal communication). While in our model no individual La- Acknowledgments. We would like to acknowledge the grangian parcels crossed the PV front prior to eddy de- useful discussions with M. Kawase, K. K. Tung, P. B. tachment, we ®nd that, averaged over lines of ¯oats, it Rhines, and D. Swift. Assistance in running and de- is the location of the minimum in eddy diffusivity bugging the numerical model was kindly provided by (slightly south of the PV front in our SW model's upper D. Darr. Editorial comments by L. Landrum, D. LeBel, layer) that separates centroids migrating north and south F. Straneo, A. Weiss, and comments from two anony- to the maxima in diffusivity at the critical lines. The mous reviewer greatly improved this manuscript. E. B. role of the critical lines as the location where enhanced is supported by a National Science Foundation Young stirring takes place and where ¯uid and Lagrangian par- Investigator Award to L. T., and L. T. is supported by ticles are entrained seems robust in observations (Lozier an Of®ce of Naval Research Young Investigator Award. et al. 1997) as well as in previous theoretical work. Besides being sites of enhanced two-particle dispersion APPENDIX these regions exhibit vorticity, necessary for mixing. The role of the critical surface in the distribution of Numerical Model Parameters tracers in the vicinity of the Gulf Stream is suggested The numerical model used here was developed by by the distribution of O ; strong gradients are observed 2 Hallberg (1995) and initialized with the basic state of at the surface close to the PV front while weaker gra- . The model grid (128 by 128 grid points) dients are found on both sides of the stream. At depth ⑀ ϭ ͙2/2 contains three wavelengths of the most (linearly) un- (27.8 Ͼ Ͼ 27.0), both the PV and O2 fronts are weaker (Bower et al. 1985). This distribution is con- stable wave. We use a grid with ⌬x ϭ⌬y ϭ 0.247R d, 3 sistent with the presence of a critical (or steering) sur- a time step, ⌬t ϭ 0.015Rd/U 0, and R ϭ U 0 R d/ ϭ 5800, face within the jet where one would expect enhanced where is the biharmonic momentum diffusion coef- stirring (Bower 1991; Pratt et al. 1995). On the north ®cient. side of the stream the location of such a surface may Lagrangian ¯oats within the model are advected ac- be nearly parallel to isopycnals; on the south side this cording to surface is expected to be close to being normal to the d(x, y) isopycnals, potentially contributing to enhanced dia- ϭ (u, ), (A1) dt pycnal mixing. The oxygen and PV sections of Bower et al. (1985) indeed display a more homogeneous dis- solved using a fourth-order Runga±Kutta scheme. The tribution on the southern side. However, surface effects, velocities were estimated within a grid box using a four- such as deep winter mixing on the south side of the point bilinear interpolation from the box's corners. We stream, may confound this hypothesis. The strong prop- also used a nine-point nonlinear interpolation for com- erty front within the stream is consistent with mixing parison and found very little difference. We initialized 302 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 29 the model with 21 zonal lines of ¯oats symmetrically , and M. S. Lozier, 1994: A closer look at particle exchange in distributed about y ϭ 0, and separated by 0.3R from the Gulf Stream. J. Phys. Oceanogr., 24, 1399±1418. d , H. T. Rossby, and J. L. Lillibridge, 1985: The Gulf StreamÐ each other. Each line of ¯oats contained forty equally Barrier or blender? J. Phys. Oceanogr., 15, 24±32. spaced ¯oats. Cronin, M., 1996: Eddy±mean ¯ow interaction in the Gulf Stream at To compute the mean tracer evolution we added to 68ЊW. Part II: Eddy forcing on the time-mean ¯ow. J. Phys. Hallberg's (1995) model a tracer () advection±diffu- Oceanogr., 26, 2132±2151. sion equation solver for D/Dt ϭ ٌ2, with a constant del-Castillo-Negrete, D., and P. J. Morisson, 1993: Chaotic transport by Rossby waves in shear ¯ow. Phys. Fluids A, 5, 948±965. diffusivity , using an operator splitting method (Press Dunkerton, T. J., C.-P. F. Hsu, and M. E. McIntyre, 1981: Some et al. 1992). Advection is performed using a positive Eulerian and Lagrangian diagnostics for a model of stratospheric de®nite tracer advection scheme (Easter 1993) and dif- warming. J. Atmos. Sci, 38, 819±843. fusion by a ®nite difference explicit scheme. No-¯ux Easter, R. C., 1993: Two modi®ed versions of Bott's positive-de®nite numerical advection scheme. Mon. Wea. Rev., 121, 297±304. boundary conditions were applied at the channel's walls. Freeland, H. J., P. B. Rhines, and H. T. Rossby, 1975: Statistical The explicit tracer diffusivity was varied from 0 to observations of the trajectories of neutrally buoyant ¯oats in the 1000 m2 sϪ1 to assess its effect on mixing. north Atlantic. J. Mar. Res., 33, 383±404. In addition to the explicit tracer diffusivity (), there Hallberg, R., 1995: Some aspects of the circulation in ocean basins is an implicit numerical diffusion in the numerical mod- with isopycnals intersecting sloping boundaries. Ph.D. thesis, University of Washington, 244 pp. el (num). This diffusion can be either positive or neg- Ikeda, M., 1981: Meanders and detached eddies of a strong eastward- ative, varies with both time and space, and is strongest ¯owing jet using a two-layer quasi-geostrophic model. J. Phys. where the advective velocity is perpendicular to the trac- Oceanogr., 11, 526±540. er gradient. Dimensional and numerical analysis suggest Johns, W. E., T. J. Shay, J. M. Bane, and D. R. Watts, 1995: Gulf that Stream structure, transport and recirculation near 68ЊW. J. Geo- phys. Res., 100, 817±838. ١|(⌬x)2 Killworth, P. D., N. Paldor, and M. E. Stern, 1984: Wave propagation ´ u| and growth on a surface front in a two-layer geostrophic current. num ϰ , (A2) o J. Mar. Res., 42, 761±785. Kim, H. S., and D. R. Watts, 1994: An observational streamfunction where o is a representative tracer concentration. The in the Gulf Stream. J. Phys. Oceanogr., 24, 2639±2657. theoretical maximum of the numerical diffusivity Lozier, M. S., and D. Bercovici, 1992: Particle exchange in an un- max(u)´⌬x occurs if a strong O( ) discontinuity in stable jet. J. Phys. Oceanogr., 22, 1506±1516. o , L. J. Pratt, and A. M. Rogerson, 1997: Exchange geometry tracer concentration exists over a grid point perpendic- revealed by ¯oat trajectories in the Gulf Stream. J. Phys. Ocean- ular to the maximum in jet velocity (the gradient is ogr., 27, 2327±2341. parallel to the ¯ow). Here, we initialize the model with Matsuno, T., 1980: Lagrangian motion of air parcels in the strato- a concentration gradient perpendicular to the advection sphere in the presence of planetary waves. Pure Appl. Geophys., 118, 189±216. velocity, with a gradient less than 0.01 tracer units per McIntyre, M. E., 1980: Towards a Lagrangian-mean description of grid point. The tracer is advected by the modeled ¯ow stratospheric circulations and chemical transports. Philos. Trans. velocities, and its gradient is expected to stay largely Roy. Soc. London A, 296, 129±148. perpendicular to the direction of the ¯ow. We therefore Nakamura, N., 1996: Two-dimensional mixing, edge formation and expect that K 10 m2 sϪ1 (e.g., Fig. 10). The small permeability diagnosed in an area coordinate. J. Atmos. Sci., 53, num 1524±1537. sensitivity of the tracer ®elds to variation in ⌬x (not Ottino, J. M., 1989: The Kinematics of Mixing: Stretching, Chaos, shown) af®rmed this point. and Transport. 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