1. Introduction
The time-varying circulations associated with geostrophic eddies in the atmosphere and ocean have a strong effect on tracer distributions and the background climatic state. In the atmosphere, the eddy circulation provides a poleward transport of heat in midlatitudes in the troposphere and a transport of ozone from the summer to winter hemisphere in the stratosphere (Andrews and McIntyre 1976; Plumb and Mahlman 1987; Andrews et al. 1987). In the ocean, the eddy circulations lead to extensive regions of nearly uniform potential vorticity along poorly ventilated density surfaces within the main thermocline (McDowell et al. 1982), as well as enabling the meridional transfer of water masses across the Antarctic Circumpolar Current (Danabasoglu et al. 1994; Marshall 1997; Speer et al. 2000; Karsten and Marshall 2002).
Given the importance of the eddy circulations, it is desirable to understand their effects on the time-mean tracer distribution, both to gain insight and improve how eddies are parameterized within coarse-resolution climate models. For the ocean, there is an ongoing debate as to how the eddy transfer should be represented. The eddy transfer is usually interpreted to act in a downgradient manner, which for a thickness closure leads to a flattening of isopycnals (Gent and McWilliams 1990) and for a potential vorticity closure leads to homogenization of conserved tracers within closed geostrophic contours (Rhines and Young 1982a, b). However, model tests of downgradient eddy closures in terms of thickness or potential vorticity only show limited skill when the parameterized fluxes are correlated locally with the eddy fluxes from eddy-resolving models (Roberts and Marshall 2000; Drijfhout and Hazeleger 2001). Indeed, support for downgradient closures of eddy transfer is only obtained if there are special symmetries in the flow, such as the spreading of convective waters from a cylindrical chimney (Visbeck et al. 1997) or the zonal-mean spreading of tracers in a zonal channel (Lee et al. 1997; Marshall et al. 1999). Our aim is to further explore the mechanism that controls why eddy tracer fluxes are sometimes directed downgradient, sometimes upgradient, and sometimes normal to the mean gradient, as demonstrated in our previous basin study (Wilson and Williams 2004). We make an analogy with how atmospheric storms have a characteristic life cycle involving initial growth and eventual decay. There is a downgradient, poleward heat transfer during the growth phase and an upgradient, horizontal momentum transfer during the decay phase. The development of storms is preferentially located in nearly zonal storm tracks over the western side of the ocean basins (Hoskins and Valdes 1990), which leads to the downgradient heat transfer occurring within the core of the storm track and the upgradient momentum transfer occurring downstream of the growth region. In analogy with the atmospheric storms, we propose that localized regions of down- or upgradient eddy transfer of tracers are associated with a characteristic life cycle of ocean eddies.
In particular, we examine how the direction of the eddy tracer flux relative to the mean tracer gradient is controlled by the Lagrangian evolution of tracer variance. We extend our previous study of tracer variance in an ocean basin (Wilson and Williams 2004) by considering the eddy fluxes and life cycles in an isopycnic, reentrant eddy-resolving channel model, which extends previous tracer variance studies by Rhines and Holland (1979), Holland and Rhines (1980), Marshall and Shutts (1981), Illari and Marshall (1983), and Marshall (1984). This domain includes an almost-zonal jet and topographical interactions and should be viewed as a highly idealized representation of a limited sector of the Southern Ocean. We examine different states of equilibration and develop an intuitive picture of a typical life cycle of an ocean eddy in terms of Lagrangian growth and decay of tracer variance. In section 2, a theoretical context is provided to understand the evolution of tracer variance and the necessary conditions for the eddy tracer flux to be directed downgradient. In section 3, the setup of the numerical experiments is described. In section 4, the mean tracer fields are examined, together with their associated variance and eddy flux components. In section 5, the tracer variance dynamics and their control of eddy tracer fluxes is assessed for limits of negligible tracer dissipation and for several diffusive sinks; Lagrangian diagnostics are exploited to provide a life cycle framework for understanding when eddy fluxes are directed downgradient in the former case. In section 6, the implications of the study are summarized.
2. Theoretical background
- (i) Dominance of Lagrangian increase in tracer variance. For a spinup, tracer variance is initially zero and increases with time, so for a closed volume and time mean,Therefore, the eddy tracer fluxes are downgradient on average over this closed volume and time mean. An example of this case occurs for potential vorticity (PV) in shielded layers within a wind-driven gyre: eddy enstrophy (PV variance) increases and, on average, PV contours are expelled from the region of initial eddy enstrophy growth. This homogenization of PV and expelling of mean PV contours is in accord with that expected from a downgradient eddy PV closure, as advocated by Rhines and Young (1982a, b), despite the local eddy fluxes being directed both up- and downgradient (Rhines and Holland 1979; Holland and Rhines 1980; Roberts and Marshall 2000; Wilson and Williams 2004)
In a Lagrangian frame, fluid may also experience localized changes in tracer deformation and variance (Figs. 1a,b). These Lagrangian changes might be associated with the characteristic life cycles of eddies deforming the tracer contours, as seen for atmospheric storms passing along a storm track. The tracer variance is largest where these tracer contours deviate most from their time-mean position (Fig. 1b). Assuming that tracer forcing and dissipation are small, there are associated Lagrangian increases and decreases in tracer variance. The increase in tracer variance following the flow is, on average, concomitant with a downgradient eddy tracer flux, implied by (2). Conversely, a decrease in tracer variance following the flow is associated with an upgradient eddy tracer flux.
(ii) Dissipation dominance. The tracer fluxes may also be downgradient when
D ′ must be comparable with those of C′ so that a significant correlation may be formed. This correlation is likely to depend on the particular tracer and its dissipation. Thus, one might expect that the direction of the eddy fluxes of a tracer with a strong sink, such as heat in the mixed layer or nitrate in the euphotic zone, may be different to that of a more conserved tracer, such as salinity. We will investigate the validity of these scenarios and how the dynamics might control them in the following numerical experiments.
3. Eddy-resolving isopycnic channel model
a. Model formulation
The model domain is a zonally cyclic channel in the Southern Hemisphere with a maximum depth of 5000 m with a partial topographical barrier and submerged ridge, which should be viewed as a highly idealized representation of the Pacific Ocean sector of the Southern Ocean (Fig. 2b, contours). The domain extends from 48.1° to 41.3°S in latitude and spans 45° in longitude. There are vertical sidewalls with a no-slip boundary condition. Initial intermediate interface depths are 1700 and 3500 m. The internal Rossby deformation radius,
The experiments are forced with a zonal wind stress of the form τx = τ0 sin(πy/L), where τ0 = 0.2 N m−2 and L is the meridional basin extent. There is no surface buoyancy forcing and no outcropping. The model is integrated for 50 years and unless otherwise stated, the time means shown correspond to years 40–50 of the uppermost layer after the model has equilibrated.
b. Time-mean background state
c. Relationship between Eady growth rate and eddy kinetic energy
Within the jet, there are noticeable alongstream variations in the Eady growth period that do not always locally coincide with the alongstream variation in eddy kinetic energy (Figs. 2a,b). The precise phase relationship between these variables is difficult to understand at a steady state. During the spinup, the Eady growth rate first develops (day 180) along the gap in the partial barrier (35°E) and along the downstream meander of the jet (37°–43°E) (Fig. 2c). At the same time, maxima in eddy kinetic energy (Fig. 2d) develop downstream of the Eady growth rate maximum (eastward of 38° to 5°E).
Subsequently, by day 720, the Eady growth rate increases all along the jet although there is still a local maximum rate located close to the gap in the topography (Fig. 2e). The eddy kinetic energy develops by an order of magnitude all along the jet with maxima centered along the jet at 40°–10°E and 20°–30°E (Fig. 2f). Thus, in our view, the maxima in eddy kinetic energy initially develop downstream of the maximum Eady growth rate, but this simple connection is obscured in the time-mean state by the persistence of background eddy kinetic energy. This background or fossil eddy kinetic energy remains since the dissipation of eddy kinetic energy is small in the model.
4. Mean, variance, and eddy fluxes of tracer
The mechanisms controlling the direction of eddy tracer fluxes are examined for two tracers: a passive tracer C, and a dynamic tracer, the PV. The passive tracer is initialized with a linear northward gradient from zero to unity, advected with no explicit diffusion, and has a source value of unity on the northern boundary and a sink value of zero on the southern boundary (with a relaxation time scale of 2.5 days, increasing linearly to 15 days in five grid cells toward the interior of the domain). The PV is conserved on the large scale, with only small explicit dissipation near the grid scale, because of the biharmonic momentum and thickness diffusion. We will focus on the idealized tracer, as it complements the previous gyre study of PV in Wilson and Williams (2004) and provides a simple parallel for fluxes of nutrients and carbon, which do not depend explicitly on velocity derivatives.
a. Mean tracer variations and total eddy fluxes
In the time mean for the upper layer, there is a front in both tracers following the time-mean jet, which acts as a partial dynamical barrier separating both higher C and PV (less negative) to the north and lower C and PV to the south (Figs. 3a,b, contours). The tracers are homogenized to different values to the north and south of the front. Maxima in the variance for both tracers are concentrated along the time-mean jet and associated tracer front (Figs. 3a,b, shaded) with broadly similar patterns along the jet. Away from the jet there are differences, with the passive tracer having higher variance toward the southern boundary. For both tracers, there are several anticyclonic eddies between the submerged ridge (10°E) and the partial barrier (35°E) (Figs. 3c,d), leading to a reversing pattern of up- and downgradient eddy fluxes (Figs. 3e,f) with tracer fluxes circulating anticlockwise around maxima in the corresponding tracer variance (Figs. 3a,b, shaded). The pattern of eddy fluxes for C and PV is similar in many places along the jet.
b. Spinup of tracer and its variance
Despite the simplicity of the model domain, there is a rich structure in the tracer variance and associated pattern of eddy fluxes. To reveal how these patterns emerge, maps of instantaneous tracer and tracer variance, 
c. Rotational and divergent eddy flux components and implied diffusivities
Although (2) provides a link between tracer variance and eddy flux direction, it is the total eddy flux that is considered rather than a divergent component. Since tracer evolution is affected by the divergence of the eddy tracer flux, it is useful to separate the eddy tracer flux into rotational and divergent components (e.g., Marshall and Shutts 1981; Roberts and Marshall 2000; Peterson and Greatbatch 2001).
Following Peterson and Greatbatch (2001), we decompose the eddy flux through solution of Poisson equations for each component with boundary conditions chosen for zero normal flux at the boundary. The rotational component is typically 5 times larger than the divergent component for the passive tracer (Figs. 5a,b) and is similar for PV. The rotational component dominates the total flux and circulates with up- and downgradient fluxes, whereas the divergent component is downgradient over most of the jet. This pattern is reflected in the eddy diffusivities, κ, for the total and the divergent components from the closure of the form
The divergent eddy tracer flux is the most relevant component for developing eddy parameterizations. As discussed in Wilson and Williams (2004), many commonly used downgradient eddy flux parameterizations (Gent et al. 1995; Visbeck et al. 1997; Treguier et al. 1997; Greatbatch 1998) might capture aspects of the large-scale effect of the eddies, even though they fail to mimic the local pattern of the eddy fluxes. However, in this study, we are primarily interested in the connection between tracer variance and eddy life cycles, so subsequently concentrate on the total eddy flux component.
5. Tracer variance budget
a. Negligible tracer variance dissipation
In summary, this analysis suggests that time-mean increases in tracer variance following the flow are accompanied by the eddy tracer flux being directed downgradient.
b. A Lagrangian eddy life cycle framework
The previous Eulerian time-mean eddy flux patterns consist of ensembles of many eddy events, including growth and decay of finite amplitude disturbances. For example, atmospheric eddies leave an Eulerian time-mean imprint in the varied distribution of eddy fluxes along storm tracks, while at the same time are understood in terms of eddy life cycles. We now examine an equivalent idea for the ocean, looking at the characteristic scales and regions of eddy fluxes and their direction throughout the life cycle.
To illustrate typical eddy life cycle behavior in the model, we consider local Lagrangian variation of Eady growth rate, eddy kinetic energy, tracer variance, and the direction of the eddy fluxes. The Lagrangian diagnostics are conducted following floats advected with the instantaneous velocity field for the case of negligible tracer variance dissipation in the top layer. The floats are advected with the 6-hourly velocity field, using a fourth-order Runge–Kutta scheme with a 6-h time step and bilinear interpolation of the velocity and Eulerian mean fields.
(i) Ensemble of trajectories. An ensemble of nine floats, arranged in a 3 by 3 uniform grid of spacing 1/6° centered at 43.6°S, 37.3°E is advected by the instantaneous flow starting at year 40 where the starting position (white square), trajectories, and end points (numbered) for each of the floats over a 2-month period are indicated in Fig. 7a. The ensemble of floats all experience an initial high Eady growth rate over the first 0.2 month, then there is a subsequent increase in eddy kinetic energy to a maximum at 0.75 month, then subsequent decay (Figs. 7b,c; the mean plus and minus a standard deviation, is shown). Likewise, there are two peaks of rate of increase of tracer variance (at 0.2 and 0.5 month) separated by a rate of decrease, which are associated with down- and upgradient eddy tracer fluxes, respectively (Figs. 7d,e). After 2 months, the trajectories become much more complex and there is a divergence in the individual responses of the floats (Fig. 7f), so two of the individual float trajectories from the ensemble are discussed in more detail.
(ii) Single trajectories. The life cycles for two single float trajectories are followed now for 7 months (Figs. 8 and 9), starting at almost identical initial positions (trajectories 9 and 4 of Fig. 7f) downstream of the gap in the topographic barrier at year 40. One of these floats loops round in a circuit, almost returning to its initial position just after 3 months and just after 5 months (Fig. 8a), while the other float follows the jet and zonally traverses the whole domain (Fig. 9a) in less than 5 months.
In summary, this Lagrangian view illustrates that there are characteristic eddy life cycles whose local behavior depends both on the stability of the local flow and on integrated effects such as downstream development of eddy kinetic energy and tracer variance. Over a period of several months, locally reversing changes in tracer variance and eddy tracer flux direction can accumulate to have a coherent signal of a particular sign.
c. Diffusive sink for tracer variance
Incorporating the additional tracer sink leads to the eddy flux being consistently directed downgradient for the Newtonian case (Fig. 10a), where the variance has been reduced by around 40% from that of the standard, no-diffusion case throughout the domain. For the case with Newtonian diffusion, the increased importance of the dissipation term leads to the correlation between −
d. Implied diffusivity from zonal-mean eddy tracer fluxes
There are also latitude ranges, such as between 46.5° and 47.5°S in Fig. 11a, where κ is much larger, of order 15 000 m2 s−1. This signal is an artifact of zonal averaging when the flow deviates from latitude circles and is due to the combination of significant zonal-mean fluxes (Fig. 5b) with weak zonal-mean tracer gradient (Fig. 3a).
The positive sign for the zonal-mean κ, implying downgradient transfer, is broadly unchanged for the different choices in tracer dissipation (Fig. 11), although the Newtonian case has a lower κ in the top layer and a higher κ in the middle and bottom layers.
While the eddy tracer flux has a complicated pattern in the time mean being directed both up- and downgradient, the eddy tracer flux is consistently directed downgradient in the zonal mean. This simpler result is due to the zonal mean together with the cyclic boundary condition automatically removing the rotational component of the eddy tracer flux, which usually dominates the remaining divergent component. However, the zonal-mean diagnostics of κ only have a limited usefulness for the ocean because of the presence of topography.
6. Summary
The direction of eddy tracer fluxes has been considered, exploiting the tracer variance budget. In the limit of small variance forcing, eddy tracer fluxes are directed downgradient when there is a Lagrangian increase in tracer variance or a dissipation of tracer variance. This response can be understood in terms of a Lagrangian view of ocean eddy life cycles, which is analogous to the life cycle view of atmospheric storms explained in terms of an ensemble of wave packets (Chang et al. 2002).
During spinup, there is an overall increase in the area-averaged tracer variance leading to an area-averaged, downgradient eddy tracer flux. While, locally, the eddy tracer fluxes can still be upgradient, this overall downgradient transfer is also reflected in the systematic smoothing of tracer gradients in unforced layers.
The steady-state Eulerian mean patterns may be considered to be a composite of many individual eddy life cycles, which are each affected by local flow stability and nonlocal boundary conditions. Again, an obvious analogy is that of the atmospheric storm tracks, whose Eulerian-mean eddy fields are made by the passage of many different individual eddies organized into latitudinal bands of high activity. For the ocean, we see that the rate of change of tracer variance and change in eddy flux direction locally reverse and often compensate in a Lagrangian sense, where the rotational eddy flux component plays a dominant role. However, there are also regions where these local reversals do not cancel when integrated over several eddy lifetimes; for example, there is a systematic downgradient eddy transfer over most of the jet associated with the divergent eddy flux component. Also, there are times during a typical eddy life cycle for which the tracer variance remains constant and there is no eddy tracer flux across mean contours.
The direction of the eddy tracer fluxes is sensitive to the tracer sink, where increasing the dissipation of tracer variance leads to a stronger downgradient flux. Consequently, if the tracer dissipation is concentrated in a surface mixed layer, such as for heat, then the eddy heat flux will be directed downgradient close to the outcrop of density surfaces and instead will have a reversing local pattern of up- and downgradient fluxes in shielded layers. Likewise, for a biogeochemical tracer, such as nitrate, which has a strong sink in the euphotic zone, the fluxes will be directed downgradient on density surfaces intersecting the euphotic zone.
Acknowledgments
We thank Dr. Vassil Roussenov for supplying a version of the float advection code. The work was supported by the AMT program supported by the UK NERC Consortium grant (NER/O/S/2001/01245). We also thank two anonymous reviewers for their constructive comments.
REFERENCES
Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen–Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33 , 2031–2048.
Andrews, D. G., J. R. Holton, and C. B. Leovy, 1987: Middle Atmosphere Dynamics. Academic Press, 489 pp.
Bleck, R., and L. T. Smith, 1990: A wind-driven isopycnic coordinate model of the north and equatorial Atlantic Ocean. 1. Model development and supporting experiments. J. Geophys. Res., 95 , C3,. 3273–3285.
Boudra, D. B., and E. P. Chassignet, 1988: Dynamics of Agulhas retroflection and ring formation in a numerical model. Part I: The vorticity balance. J. Phys. Oceanogr., 18 , 280–303.
Chang, E. K. M., S. Lee, and K. L. Swanson, 2002: Storm track dynamics. J. Climate, 15 , 2163–2183.
Danabasoglu, G., J. C. McWilliams, and P. R. Gent, 1994: The role of mesoscale tracer transport in the global ocean circulation. Science, 264 , 1123–1126.
Drijfhout, S. S., and W. Hazeleger, 2001: Eddy mixing of potential vorticity versus thickness in an isopycnic ocean model. J. Phys. Oceanogr., 31 , 481–505.
Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1 , 33–52.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnic mixing in ocean circulation models. J. Phys. Oceanogr., 20 , 150–155.
Gent, P. R., J. Willebrand, T. J. McDougal, and J. C. McWilliams, 1995: Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr., 25 , 463–474.
Gill, A. E., J. S. A. Green, and A. J. Simmons, 1974: Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Res., 21 , 499–528.
Greatbatch, R. J., 1998: Exploring the relationship between eddy-induced transport velocity, vertical momentum transfer, and the isopycnal flux of potential vorticity. J. Phys. Oceanogr., 28 , 422–432.
Holland, W. R., and P. B. Rhines, 1980: An example of eddy-induced ocean circulation. J. Phys. Oceanogr., 10 , 1010–1031.
Hoskins, B. J., and P. J. Valdes, 1990: On the existence of storm tracks. J. Atmos. Sci., 47 , 1854–1864.
Illari, L., and J. C. Marshall, 1983: On the interpretation of eddy fluxes during a blocking episode. J. Atmos. Sci., 40 , 2232–2242.
Karsten, R. H., and J. C. Marshall, 2002: Constructing the residual circulation of the ACC from observations. J. Phys. Oceanogr., 32 , 3315–3327.
Lee, M-M., D. P. Marshall, and R. G. Williams, 1997: On the eddy transfer of tracers: Diffusive or advective? J. Mar. Res., 55 , 1–24.
Lindzen, R. S., and B. Farrell, 1980: A simple approximate result for the maximum growth rate of baroclinic instabilities. J. Atmos. Sci., 37 , 1648–1654.
Marshall, D. P., 1997: Subduction of water masses in an eddying ocean. J. Mar. Res., 55 , 201–222.
Marshall, D. P., R. G. Williams, and M-M. Lee, 1999: The relation between eddy-induced transport and isopycnic gradients of potential vorticity. J. Phys. Oceanogr., 29 , 1571–1578.
Marshall, J. C., 1984: Eddy-mean-flow interaction in a barotropic ocean model. Quart. J. Roy. Meteor. Soc., 110 , 573–590.
Marshall, J. C., and G. J. Shutts, 1981: A note on rotational and divergent eddy fluxes. J. Phys. Oceanogr., 11 , 1677–1680.
McDowell, S., P. B. Rhines, and T. Keffer, 1982: North Atlantic potential vorticity and its relation to the general circulation. J. Phys. Oceanogr., 12 , 1417–1436.
Panetta, R. L., 1993: Zonal jets in wide baroclinically unstable regions—Persistence and scale selection. J. Atmos. Sci., 50 , 2073–2106.
Peterson, K. A., and R. J. Greatbatch, 2001: Vorticity fluxes in shallow water ocean models. Atmos.–Ocean, 39 , 1–14.
Plumb, R. A., and J. D. Mahlman, 1987: The zonally averaged transport characteristics of the GFDL general circulation/transport model. J. Atmos. Sci., 44 , 298–327.
Rhines, P. B., and W. R. Holland, 1979: A theoretical discussion of eddy-driven mean flows. Dyn. Atmos. Oceans, 3 , 289–325.
Rhines, P. B., and W. R. Young, 1982a: Homogenization of potential vorticity in planetary gyres. J. Fluid Mech., 122 , 347–368.
Rhines, P. B., and W. R. Young, 1982b: A theory of the wind-driven circulation. I. Mid-ocean gyres. J. Mar. Res., 40 , (Suppl.),. 559–596.
Roberts, M. J., and D. M. Marshall, 2000: On the validity of downgradient eddy closures in ocean models. J. Geophys. Res., 105 , 28613–28627.
Speer, K., S. R. Rintoul, and B. Sloyan, 2000: The diabatic Deacon cell. J. Phys. Oceanogr., 30 , 3212–3222.
Treguier, A. M., I. M. Held, and V. D. Larichev, 1997: Parameterization of quasigeostrophic eddies in primitive equation ocean models. J. Phys. Oceanogr., 27 , 567–580.
Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: Specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27 , 381–402.
Wilson, C., and R. G. Williams, 2004: Why are eddy fluxes of potential vorticity difficult to parameterize? J. Phys. Oceanogr., 34 , 142–155.
A schematic depicting (a) two tracer contours, C, with localized variations in concentration and (b) corresponding regions of high variance, C′2, in the tracer (shaded). Following a streamline (thick contour) there is a positive and negative advection of tracer variance, respectively, upstream and downstream of the maxima in tracer variance. Since a Lagrangian increase in tracer variance implies a downgradient eddy flux, there should be a down- and upgradient eddy flux, respectively, up- and downstream of the maxima in tracer variance.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
A schematic depicting (a) two tracer contours, C, with localized variations in concentration and (b) corresponding regions of high variance, C′2, in the tracer (shaded). Following a streamline (thick contour) there is a positive and negative advection of tracer variance, respectively, upstream and downstream of the maxima in tracer variance. Since a Lagrangian increase in tracer variance implies a downgradient eddy flux, there should be a down- and upgradient eddy flux, respectively, up- and downstream of the maxima in tracer variance.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
A schematic depicting (a) two tracer contours, C, with localized variations in concentration and (b) corresponding regions of high variance, C′2, in the tracer (shaded). Following a streamline (thick contour) there is a positive and negative advection of tracer variance, respectively, upstream and downstream of the maxima in tracer variance. Since a Lagrangian increase in tracer variance implies a downgradient eddy flux, there should be a down- and upgradient eddy flux, respectively, up- and downstream of the maxima in tracer variance.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Steady state and spinup: Time-mean maps from years 40–50 for (a) eddy kinetic energy (10−4 m2 s−2, shaded) and depth-integrated volume transport streamfunction (Sv, contours) in the top layer and (b) Eady growth period (days, shaded), evaluated at the lower interface of the top layer, and depth (dashed contours every 1 km, maximum 5 km). Note that, while there is both enhanced eddy kinetic energy and low Eady growth period along the eastward jet, the minima in Eady growth period occur upstream of the maxima in eddy kinetic energy. Instantaneous maps of Eady growth period (days) at (c) day 180 and (e) day 720 and eddy kinetic energy (10−4 m2 s−2) at (d) day 180 and (f) day 720, for the top layer, evaluated using a 300-day boxcar mean.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Steady state and spinup: Time-mean maps from years 40–50 for (a) eddy kinetic energy (10−4 m2 s−2, shaded) and depth-integrated volume transport streamfunction (Sv, contours) in the top layer and (b) Eady growth period (days, shaded), evaluated at the lower interface of the top layer, and depth (dashed contours every 1 km, maximum 5 km). Note that, while there is both enhanced eddy kinetic energy and low Eady growth period along the eastward jet, the minima in Eady growth period occur upstream of the maxima in eddy kinetic energy. Instantaneous maps of Eady growth period (days) at (c) day 180 and (e) day 720 and eddy kinetic energy (10−4 m2 s−2) at (d) day 180 and (f) day 720, for the top layer, evaluated using a 300-day boxcar mean.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Steady state and spinup: Time-mean maps from years 40–50 for (a) eddy kinetic energy (10−4 m2 s−2, shaded) and depth-integrated volume transport streamfunction (Sv, contours) in the top layer and (b) Eady growth period (days, shaded), evaluated at the lower interface of the top layer, and depth (dashed contours every 1 km, maximum 5 km). Note that, while there is both enhanced eddy kinetic energy and low Eady growth period along the eastward jet, the minima in Eady growth period occur upstream of the maxima in eddy kinetic energy. Instantaneous maps of Eady growth period (days) at (c) day 180 and (e) day 720 and eddy kinetic energy (10−4 m2 s−2) at (d) day 180 and (f) day 720, for the top layer, evaluated using a 300-day boxcar mean.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the upper model layer: (a) passive tracer concentration, C, (contours) and its variance (shaded, 10−2), where C is relaxed to 1 and 0 on the northern and southern boundaries, respectively; (b) dynamic tracer, the potential vorticity, Q (contours, 10−8 m−1 s−1), and its variance (shaded, 10−17 m−2 s−2); (c) and (d) the time mean of the eddy tracer and PV fluxes,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the upper model layer: (a) passive tracer concentration, C, (contours) and its variance (shaded, 10−2), where C is relaxed to 1 and 0 on the northern and southern boundaries, respectively; (b) dynamic tracer, the potential vorticity, Q (contours, 10−8 m−1 s−1), and its variance (shaded, 10−17 m−2 s−2); (c) and (d) the time mean of the eddy tracer and PV fluxes,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the upper model layer: (a) passive tracer concentration, C, (contours) and its variance (shaded, 10−2), where C is relaxed to 1 and 0 on the northern and southern boundaries, respectively; (b) dynamic tracer, the potential vorticity, Q (contours, 10−8 m−1 s−1), and its variance (shaded, 10−17 m−2 s−2); (c) and (d) the time mean of the eddy tracer and PV fluxes,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Instantaneous maps of tracer concentration (contours) and tracer variance (10−3, shaded) evaluated over a 300-day boxcar mean centered at (a) day 180 and (b) day 720 for the top layer and (c) day 180 and (d) day 720 for the middle layer. Over the spinup, the meridional tracer gradient changes from a constant to being concentrated as a front along the jet maintained by wind forcing in the top layer, but the front in the middle layer dissipates by day 720. In the top layer, the tracer variance first becomes significant downstream of the gap in the topography at 35°E, then develops with localized patches of high variance along the jet.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Instantaneous maps of tracer concentration (contours) and tracer variance (10−3, shaded) evaluated over a 300-day boxcar mean centered at (a) day 180 and (b) day 720 for the top layer and (c) day 180 and (d) day 720 for the middle layer. Over the spinup, the meridional tracer gradient changes from a constant to being concentrated as a front along the jet maintained by wind forcing in the top layer, but the front in the middle layer dissipates by day 720. In the top layer, the tracer variance first becomes significant downstream of the gap in the topography at 35°E, then develops with localized patches of high variance along the jet.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Instantaneous maps of tracer concentration (contours) and tracer variance (10−3, shaded) evaluated over a 300-day boxcar mean centered at (a) day 180 and (b) day 720 for the top layer and (c) day 180 and (d) day 720 for the middle layer. Over the spinup, the meridional tracer gradient changes from a constant to being concentrated as a front along the jet maintained by wind forcing in the top layer, but the front in the middle layer dissipates by day 720. In the top layer, the tracer variance first becomes significant downstream of the gap in the topography at 35°E, then develops with localized patches of high variance along the jet.
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the top layer: (a) rotational component of the eddy passive tracer flux, 
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the top layer: (a) rotational component of the eddy passive tracer flux,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Time-mean maps from years 40–50 for the top layer: (a) rotational component of the eddy passive tracer flux,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Mean advection of tracer variance (10−9 s−1) for a statistically steady state in the top layer: (a) by the time-mean flow, 
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Mean advection of tracer variance (10−9 s−1) for a statistically steady state in the top layer: (a) by the time-mean flow,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Mean advection of tracer variance (10−9 s−1) for a statistically steady state in the top layer: (a) by the time-mean flow,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an ensemble of nine floats released downstream of the gap in the topography (including the trajectories for Figs. 8 and 9): (a) mean tracer concentration (shaded) and each trajectory starting in the white square and lasting 2 months (with the number identifying the end point of each float); Lagrangian property evolution for the mean of the ensemble (thick line) and plus and minus one standard deviation (thin lines) for (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance, 
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an ensemble of nine floats released downstream of the gap in the topography (including the trajectories for Figs. 8 and 9): (a) mean tracer concentration (shaded) and each trajectory starting in the white square and lasting 2 months (with the number identifying the end point of each float); Lagrangian property evolution for the mean of the ensemble (thick line) and plus and minus one standard deviation (thin lines) for (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an ensemble of nine floats released downstream of the gap in the topography (including the trajectories for Figs. 8 and 9): (a) mean tracer concentration (shaded) and each trajectory starting in the white square and lasting 2 months (with the number identifying the end point of each float); Lagrangian property evolution for the mean of the ensemble (thick line) and plus and minus one standard deviation (thin lines) for (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 9 of the ensemble) released downstream of the partial barrier in the top layer where the float circulates from the jet into a quiescent part of the flow: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 9 of the ensemble) released downstream of the partial barrier in the top layer where the float circulates from the jet into a quiescent part of the flow: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 9 of the ensemble) released downstream of the partial barrier in the top layer where the float circulates from the jet into a quiescent part of the flow: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 4 of the ensemble) released downstream of the partial barrier in the top layer where the float stays within the jet as it circulates the full zonal extent of the domain: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 4 of the ensemble) released downstream of the partial barrier in the top layer where the float stays within the jet as it circulates the full zonal extent of the domain: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Lagrangian evolution following an individual float (member 4 of the ensemble) released downstream of the partial barrier in the top layer where the float stays within the jet as it circulates the full zonal extent of the domain: (a) mean tracer concentration (shaded) and trajectory starting in the white square and lasting 7 months (with month number marked), (b) Eady growth rate (day−1), (c) eddy kinetic energy (m2 s−2), (d) Lagrangian evolution of tracer variance,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Maps denoting whether the eddy tracer flux is downgradient (dark gray) or upgradient (light gray) based upon the negative scalar product of the eddy tracer flux and the mean tracer gradient, −
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Maps denoting whether the eddy tracer flux is downgradient (dark gray) or upgradient (light gray) based upon the negative scalar product of the eddy tracer flux and the mean tracer gradient, −
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Maps denoting whether the eddy tracer flux is downgradient (dark gray) or upgradient (light gray) based upon the negative scalar product of the eddy tracer flux and the mean tracer gradient, −
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Diagnostics of the eddy diffusivity (m2 s−1), κ, for the tracer, assuming both a zonal- and time-mean closure, 
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Diagnostics of the eddy diffusivity (m2 s−1), κ, for the tracer, assuming both a zonal- and time-mean closure,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Diagnostics of the eddy diffusivity (m2 s−1), κ, for the tracer, assuming both a zonal- and time-mean closure,
Citation: Journal of Physical Oceanography 36, 2; 10.1175/JPO2841.1
Correlation of flux direction term with components of advection of tracer variance for layer 1 for integrations with the idealized tracer and PV as well as with different forms of tracer diffusivity for the idealized tracer. Values in parentheses represent correlations for longitude 0°–30°E, which differ from the values for the whole domain because they omit large-amplitude small-scale signals near the gap in the topographical barrier and are more representative of the large-scale interior pattern. Correlation is significant at the 95% level unless italicized.
