Abstract

The authors investigate the dynamics of zonal jets in a semihemisphere zonally reentrant ocean model. The forcings imposed in the model are an idealized atmospheric wind stress and relaxation to a latitudinal temperature profile held constant in time. While there are striking similarities to the observed atmospheric annular modes, where the leading mode of variability is associated with the primary zonal jet’s meridional undulation, secondary (weaker) jets emerge and systematically migrate equatorward.

The model output suggests the following mechanism for the equatorward migration: while the eddy momentum fluxes sustain the jets, the eddy heat fluxes have a poleward bias causing an anomalous residual circulation with poleward (equatorward) flow on the poleward (equatorward) flanks. By conservation of mass, there must be a rising residual flow at the jet. From the thermodynamics equation, the greatest cooling occurs at the jet core, thus creating a tendency to reduce the baroclinicity on the poleward flank, while enhancing it on the equatorward flank. Consequently, the baroclinic zone shifts, perpetuating the jet migration.

1. Introduction

In this paper, we describe some characteristics of zonal jets in a model of a zonally reentrant ocean, bounded by zonal walls at the equator and 50°S, driven by a steady eastward wind stress that peaks in middle latitudes. The model simulation was run as a test bed for ideas on eddy transport, and its major, climatological, characteristics are described elsewhere (Cerovecki et al. 2007, manuscript submitted to J. Phys. Oceanogr., hereafter CPH). Here we focus on the time variability of the zonal jets in the model. At any instant in time, the mean zonal flow comprises a dominant jet, together with two or three secondary jets primarily on its poleward side. The main jet wobbles quasiperiodically, in a manner that appears similar to the “annular mode” behavior of atmospheric jets. At the same time, the secondary jets, poleward of the main jet, migrate systematically equatorward such that, once every period of the main jet’s fluctuation, one secondary jet merges with the main jet, while another appears at the poleward flank of the secondary jets.

Annular modes are well documented in the atmosphere (e.g., Thompson and Wallace 2000) as the dominant modes of nonseasonal atmospheric variability; they form deep dipole structures straddling the mean jet in each hemisphere, and describe north–south oscillations of the jets. Both analyses of observations and model studies show feedback between the mean flow and baroclinic eddies to be at the heart of the oscillations (e.g., Robinson 1994; Lorenz and Hartmann 2001; Kushner and Polvani 2004). Likewise, multiple jets have long been understood to occur in wide domains and demonstrated in models ranging in complexity from barotropic and shallow water through to two-level quasigeostrophic and multilevel primitive equation models (e.g., Williams 1978; Panetta 1993; Cho and Polvani 1996; Lee 2005). Such structures have been observed in jets in the ocean (e.g., Roden 2000) and in the atmospheres of Jupiter and Saturn (e.g., de Pater and Lissauer 2001). A recent discussion of the dynamics of the formation of multiple jet structures, and factors controlling their width, can be found in Dritschel et al. (2007). While most of these studies have not revealed any tendency for the jets to migrate, systematic equatorward migration of multiple jets, under some circumstances, has been described by Williams (2003) in a model of Jupiter’s atmosphere.

There are two related points of interest in the present paper. First, the correspondence between the migration of the secondary jets and the oscillation of the main jet appears to be a manifestation of annular mode behavior in a multiple jet environment, which, to our knowledge, has not been reported before. In fact, we shall show that most of the variance in the zonal flow is captured by two spatial structures, each of which projects onto both the main jet and the secondary jets and which, taken together, describe the simultaneous oscillation/migration pattern. Second, while the narrowness of the jets is maintained by eddy momentum fluxes (as has long been understood), their equatorward migration is a response to the eddy heat fluxes, which act to reduce the baroclinicity on the poleward flank of the jets and increase it on the equatorward flank. We speculate that this behavior is ultimately determined by the latitudinal gradient of the background static stability.

The structure of the paper is as follows. A cursory description of the model and its time-averaged features is presented in section 2, and an analysis of the spatial and temporal variability is given in section 3. Properties of the baroclinic eddies, their effect on the zonal mean flow, and, in particular, their relation to the migration of the secondary jets are discussed in section 4. Conclusions are summarized in section 5.

2. Model characteristics

The data in this study were generated by CPH from the Massachusetts Institute of Technology (MIT) general circulation model (cf. Marshall et al. 1997a, b). Readers are referred to CPH for a complete description of the model setup and the equilibrated states. Here, we just provide a cursory description.

This is a zonally reentrant, semihemispheric model ranging from 50.67° to 0.17°S and 0° to 10°E on a ⅙° × ⅙° latitude × longitude grid with 15 vertical levels. The model does not include salinity; density is simply a linear function of temperature. The model-imposed forcings are shown in Fig. 1. The wind stress is eastward everywhere with weak winds near the equator. The heat forcing is applied to the upper surface layer, with a relaxation time of 30 days. Both forcings are constant in time and functions only of latitude. With the flow reaching statistically steady state by year 500, the model was integrated for a total of 1285 years, of which the last 313 years were examined for this study.

Fig. 1.

The prescribed model forcings are an (a) atmospheric wind stress and (b) heat forcing with a relaxation time of order one month. Both are constant in time, only a function of latitude, and applied to the top surface layer (22 m). Adapted from CPH.

Fig. 1.

The prescribed model forcings are an (a) atmospheric wind stress and (b) heat forcing with a relaxation time of order one month. Both are constant in time, only a function of latitude, and applied to the top surface layer (22 m). Adapted from CPH.

a. Time-averaged state

Despite the deformation radius increasing monotonically from about 10 km at 40°S to about 100 km at 15°S, the length scale of the forcings, shown in Fig. 1, is more than one to two orders of magnitude larger than the Rossby radius of deformation. With such a broad forcing, it is not surprising that multiple jets emerge (Panetta 1993). Figure 2 shows the time-mean zonal flow. A strong eastward jet (hereinafter the “main jet”) is located at 17°S, while a weaker westward flow is on its equatorward side. Poleward of the main jet is another eastward jet at about 26°. We shall see in what follows that this is the time-averaged remnant of several, time-dependent secondary jets.

Fig. 2.

Time- and zonally averaged zonal flow. Contour interval is 0.1 m s−1; zero contour is omitted.

Fig. 2.

Time- and zonally averaged zonal flow. Contour interval is 0.1 m s−1; zero contour is omitted.

Figure 3 shows the time-averaged density and potential vorticity (PV) distributions. Poleward of 12°S, PV is homogenized along isopycnals in the near-adiabatic interior. This region of homogenized PV coincides with the region of baroclinic eddy activity (CPH). PV gradients do appear in the nonadiabatic region near the surface, as is evident in Fig. 3, but the contribution in the surface “PV sheet” (Bretherton 1966) associated with the surface temperature gradient is not visible on the figure. To illustrate this point in the presence of multiple migrating jets, we show in Fig. 4a a snapshot of the zonally averaged zonal flow, in Fig. 4b the quasigeostrophic PV gradients integrated through the top 165 m (including the surface PV sheet), and in Fig. 4c the quasigeostrophic PV gradient at a typical interior level. As shown, the surface PV gradient dominates. Consistent with the arguments of Dritschel et al. (2007), each jet is associated with a sharp eddy transport barrier. There are multiple PV “steps,” that is, the gradients are concentrated at each jet and weak everywhere else. However, in our case the PV steps are manifested not in the interior but rather in strong, localized gradients of surface temperature.

Fig. 3.

Time-averaged quantities of Ertel’s potential vorticity (thick line) and isopycnals (thin line) with a contour interval of 0.5 × 10−10 °C m2 s−1 kg−1 and 0.5 kg m−3, respectively. Note the top 400 m has been enlarged.

Fig. 3.

Time-averaged quantities of Ertel’s potential vorticity (thick line) and isopycnals (thin line) with a contour interval of 0.5 × 10−10 °C m2 s−1 kg−1 and 0.5 kg m−3, respectively. Note the top 400 m has been enlarged.

Fig. 4.

Snapshot for year 1052 of (a) the zonally averaged zonal flow near the surface and in the interior, (b) the vertically averaged (top 165 m) quasigeostrophic potential vorticity gradient, and (c) the interior quasigeostrophic potential vorticity gradient. The vertical scale of (b) and (c) is different. Note that the near-surface gradient shown in (b) includes the contribution from the surface temperature gradient in the surface “PV sheet” and, in fact, is dominated by that contribution.

Fig. 4.

Snapshot for year 1052 of (a) the zonally averaged zonal flow near the surface and in the interior, (b) the vertically averaged (top 165 m) quasigeostrophic potential vorticity gradient, and (c) the interior quasigeostrophic potential vorticity gradient. The vertical scale of (b) and (c) is different. Note that the near-surface gradient shown in (b) includes the contribution from the surface temperature gradient in the surface “PV sheet” and, in fact, is dominated by that contribution.

The static stability varies greatly in depth and in latitude, as indicated in Fig. 5. At any particular depth, the stability at all depths monotonically increases equatorward up to approximately 20°S. As for the whole domain, several extrema are present. Perhaps, the most obvious is located between −15° and −10° above a depth of 750 m, a region associated with mode water formation (Cerovecki and Marshall 2007).

Fig. 5.

Time-averaged static stability (10−3 s−1).

Fig. 5.

Time-averaged static stability (10−3 s−1).

3. Description of variability

Figure 6 shows the annual-mean zonal flow, year by year over a 9-yr period. While the main jet oscillates meridionally, two to three secondary jets migrate equatorward and each eventually merges with the main jet. Figure 7 shows the vertically integrated, zonally averaged zonal flow anomalies as a function of latitude and time. The equatorward migration is clearly seen poleward of the primary jet, in particular, between 20° and 35°S. From the emergence of the secondary jet around 30°S to the time it takes to reach the primary jet varies from 8 to 12 yr and is a robust feature of this model.

Fig. 6.

Time- series of the annually averaged zonally averaged zonal flow. Time stamps (yr) are located on bottom left corner of each plot. Contour interval is 0.25 m s−1; zero contour is omitted.

Fig. 6.

Time- series of the annually averaged zonally averaged zonal flow. Time stamps (yr) are located on bottom left corner of each plot. Contour interval is 0.25 m s−1; zero contour is omitted.

Fig. 7.

Time series of the anomalous vertically integrated zonally averaged zonal flow. Positive contours start at 20 and increase in increments of 200. Negative contours (dashed lines) start at −200 and are also in increments of 200. Thick black line represents the time-averaged position of the main jet.

Fig. 7.

Time series of the anomalous vertically integrated zonally averaged zonal flow. Positive contours start at 20 and increase in increments of 200. Negative contours (dashed lines) start at −200 and are also in increments of 200. Thick black line represents the time-averaged position of the main jet.

Although less obvious, there is some evidence of zonal flow anomalies propagating poleward between −10° and −15°. Compared to the equatorward migration at higher latitudes, the poleward propagation occurs less systematically and over a shorter range. Because of its close proximity to the main jet, entangling its influence with the zonal flow anomalies itself becomes difficult. Consequently, our discussions of propagating zonal flow anomalies will mainly focus on the region of systematic equatorward propagation between −20° and −30°.

a. EOF analysis

The spatial and temporal variability of the entire time series can be best quantified by the use of empirical orthogonal functions (EOFs). The data were weighted to account for the decrease in area around latitude circles toward the pole but were not weighted to account for the varying layer depths. This will not be important as we are mostly interested in the horizontal variations of the zonal flow. Using the North et al. (1982) test, the first and second EOFs are well separated.

Figure 8 shows the horizontal structure of the leading two EOFs of the annually averaged surface temperature. With the model forcings independent of longitude, it comes as no surprise that there is little longitudinal variability. Thus, for the remainder of the paper, we consider zonal-mean budgets and explicitly examine the variability of the zonally averaged zonal flow.

Fig. 8.

The leading two EOFs of the annually averaged temperature at the surface layer. The percent variance is shown in the bottom left corner.

Fig. 8.

The leading two EOFs of the annually averaged temperature at the surface layer. The percent variance is shown in the bottom left corner.

As shown in Fig. 9, EOF1 displays an “equivalent barotropic” structure with maximum absolute anomalies at 19° and 14°S. The vertical black line represents the time-averaged location of the primary jet’s maximum value (17.2°S). By comparing the mode’s spatial structure and the mean location of the jet, EOF1 describes meridional fluctuations of the main jet or, in other words, it captures the jet “wobbling” in the north–south direction. This mode constitutes the largest amount (39.5%) of the total variability. In EOF2, the maximum anomalies are almost coincident with the mean location of the jet. Therefore, this mode indicates the intensifying and weakening of the main jet. Given the structure of the EOFs in Figs. 8 and 9, therefore, the spatial variability near the primary jet appears to be analogous to the atmospheric annular modes; the leading mode describes oscillations of the main jet, while the second mode captures its enhancement (e.g., Lorenz and Hartmann 2001) just as it does in the atmospheric eddy-driven jet.

Fig. 9.

The leading two EOFs of the annually averaged zonal-mean zonal flow. Solid (dashed) lines represent positive (negative) values. Note the nonuniform contour interval. Vertical black line indicates position of the time-averaged jet. The percent variance is shown in the bottom right corner.

Fig. 9.

The leading two EOFs of the annually averaged zonal-mean zonal flow. Solid (dashed) lines represent positive (negative) values. Note the nonuniform contour interval. Vertical black line indicates position of the time-averaged jet. The percent variance is shown in the bottom right corner.

Unlike the atmospheric case, there is an obvious nondipole structure in both modes; poleward of 20°S, three to four more additional extrema are present. These features capture the migrating secondary jets, with EOF2 comparable in magnitude with, and in quadrature with, EOF1 in this region. Thus EOF2 must account for more variance than is typical in the atmospheric case. Higher-order components are weaker with EOF3 capturing 11% and successive EOFs less than 4%; therefore, as Fig. 10 shows, the evolution of the zonal flow is well captured by the first two EOFs (cf. Fig. 6).

Fig. 10.

Time series of the reconstructed zonal flow using the leading two EOF modes and the time-averaged zonal flow.

Fig. 10.

Time series of the reconstructed zonal flow using the leading two EOF modes and the time-averaged zonal flow.

b. Description of EOF phases

The principal component captures the temporal variations associated with the spatial pattern of the EOFs. For instance, when the primary jet is displaced poleward (equatorward), the principal component associated with EOF1, PC1, will be positive (negative). Similarly, when the zonal flow at the primary jet is anomalously positive (negative), the principal component associated with EOF2, PC2, will be positive (negative).

Now that we have established how well the variability in the zonal average of u(y, z, t) is represented by the two EOFs, we define the following four phases (shown in Table 1) to capture both the temporal and spatial variability. For instance, we define Phase A as PC1 < 0 and PC2 < 0, Phase B as PC1 < 0, PC2 > 0, etc. A graphical representation of the four phases is shown in Fig. 11, as well as an example of a 21-yr time series in the PC space. Throughout the 313-yr time series, there is a sense of generally clockwise rotation, such that the following sequence occurs: Phase A → Phase B → Phase C → Phase D and then repeats back to Phase A. Since the secondary jets migrate, the principal components of both modes need to change sign to allow the secondary jets to advance equatorward and, hence, this sequence is ultimately dictated by the behavior of the secondary jets.

Table 1.

Physical characteristics of the primary jet in the four EOF phases.

Physical characteristics of the primary jet in the four EOF phases.
Physical characteristics of the primary jet in the four EOF phases.
Fig. 11.

The time series of the leading two principal components in PC space. The EOF phases are described in Table 1 and labeled in each quadrant. Each point is a 1-yr average.

Fig. 11.

The time series of the leading two principal components in PC space. The EOF phases are described in Table 1 and labeled in each quadrant. Each point is a 1-yr average.

Now that we have defined these four phases of the oscillation, we can thus describe the evolution of the structure of the zonal mean state as well as eddy fluxes by compositing all years associated with each phase. When analyzing migrating jets, a time average would “smooth” out its spatial structure. However, by defining these four phases, we will now be able to capture and examine the migrating jets. An example, which will be discussed in greater detail later, is shown in Fig. 17. The sequence shows that the primary jet wobbles, while the secondary jets migrate equatorward, precisely the behavior shown in Fig. 6. We can think of this sequence as a typical 8-yr cycle with each phase representing roughly 2 yr.

Fig. 17.

The total EP flux vectors for each EOF phase composite (labeled on bottom left corner) along with the zonally averaged zonal flow. The vectors are scaled the same for each plot.

Fig. 17.

The total EP flux vectors for each EOF phase composite (labeled on bottom left corner) along with the zonally averaged zonal flow. The vectors are scaled the same for each plot.

Table 2 shows how closely the sequence was followed. Each change in phase is followed by the correct phase at least 64% of the time; for example, the conditions prior to the onset of Phase A were correctly described to be in Phase D 64% of the time and incorrectly by Phase B or C 36% of the time. Similarly, Phase A described the PC space prior to Phase B 85% of the time. This shows that these preconditions are not symmetric, for example, there is a stronger relation between Phase A and Phase B than there is between Phase D and Phase A.

Table 2.

Statistical results on the conditions prior to the onset of each phase and zonal index.

Statistical results on the conditions prior to the onset of each phase and zonal index.
Statistical results on the conditions prior to the onset of each phase and zonal index.

Given the results in Table 2 and Fig. 17, there is a correlation between the strengthening of the main jet and the arrival of the migrating jets. When the secondary jet is closest (Phases B and C), there is also an intensification of the primary jet (see Table 1). The implication is that, as the migrating jet approaches, the main jet intensifies and subsequently displaces toward the direction (Phase C) of the approaching secondary jet. Conversely, in Phases A and D, where the strength of the main jet decreases, the closest secondary jet is farther away.

One last statistical result worth mentioning brings attention to the duration of each phase. For our 313-yr model study, Phase A constituted 95 yr in total, nearly 20% more than any other phase. Concurrent with anomalous weak eddy activity, the low zonal index (i.e., when PC1 is negative) lasting for longer durations appears consistent with the study done by Feldstein and Lee (1996), who examined the zonal index of the atmospheric jet in an aquaplanet.

4. The role of eddies in the flow evolution

The zonal flow anomalies shown in Fig. 7 are observed to persist on time scales that are larger than the frictional time scale associated with the bottom drag. Eddy–mean flow interaction is the only process capable of maintaining these anomalies.

a. Vertically integrated momentum budget

The zonal momentum equation for quasigeostrophic motion can be written as

 
formula

where the brackets represent zonal averages, primes denote deviations therefrom, and τ represents the applied forcing and the bottom friction. Integrating the entire column, we obtain

 
formula

where vertically integrated values are represented by angle brackets, the Coriolis force vanishes owing to mass conservation, and τsurf is the applied wind forcing shown in Fig. 1a; the bottom stress τbot is calculated in the model as

 
formula

where Az is the vertical viscosity; ubot and υbot the zonal and meridional velocity, respectively, at the bottom of the ocean; δbot the thickness of the bottom layer; and CD is the bottom drag coefficient. (For specific values, see CPH.)

The time-averaged momentum budget is discussed in CPH. The three terms on the right-hand side of (2) must balance. The wind stress is balanced, on the broad scale, by bottom drag as shown in Fig. 12. However, there are spatial variations in the latter on the jet scale that do not correspond to features in the wind stress; rather, these variations balance the convergence of eddy momentum fluxes (e.g., Held 1975; Ioannou and Lindzen 1986).

Fig. 12.

(a) Time-averaged quantities of the vertically integrated zonally averaged zonal momentum budget. Adapted from CPH. (b) Time average of the vertically integrated eddy momentum flux 〈[uυ′]〉.

Fig. 12.

(a) Time-averaged quantities of the vertically integrated zonally averaged zonal momentum budget. Adapted from CPH. (b) Time average of the vertically integrated eddy momentum flux 〈[uυ′]〉.

Since we are interested in the equatorward migration of the zonal flow anomalies, we will here examine departures from the time mean of (2). Since the wind forcing was held constant in time, there is only a three-way balance between the vertically integrated flow tendency, anomalous convergence of the eddy momentum flux and the anomalous bottom drag. After taking departures from the time-averaged state, Fig. 13 shows composites of each phase as described in the previous section. There is a quasi-steady balance between the bottom drag and the divergence of the Reynolds stress, the flow tendency being weak and virtually negligible compared to these two terms. The zonal flow anomalies are therefore sustained by the anomalous convergence of eddy momentum fluxes. Note that there is no obvious latitudinal bias in the momentum flux convergence. Thus, another approach is needed to explain why the jets are migrating equatorward.

Fig. 13.

Anomalous quantities of the vertically integrated zonally averaged zonal momentum budget for each EOF phase composite labeled at the bottom left corner of each plot. Solid line is the flow tendency; long dashed line is the eddy momentum flux convergence; vertically dashed line is the friction. Thick vertical dashed lines correspond to the maximum positive zonal flow anomalies.

Fig. 13.

Anomalous quantities of the vertically integrated zonally averaged zonal momentum budget for each EOF phase composite labeled at the bottom left corner of each plot. Solid line is the flow tendency; long dashed line is the eddy momentum flux convergence; vertically dashed line is the friction. Thick vertical dashed lines correspond to the maximum positive zonal flow anomalies.

b. Eddy–mean flow interaction

To understand fully the effect of the eddies on the mean flow, we use the transformed Eulerian mean (TEM) approach. This allows us to consider simultaneously the effects of the eddy momentum and heat fluxes (we shall in fact see that the heat budget plays a large role in the jet migration). The quasigeostrophic TEM equations are (Andrews and McIntyre 1976)

 
formula
 
formula

where Q represents diabatic effects,

 
formula

is the Eliassen–Palm (EP) flux vector (e.g., Edmon et al. 1980), and

 
formula
 
formula

are the residual circulations.

Once again, since we are interested in the departures from the time mean, (4) can be simplified. Away from the surface, we may neglect Q because there are no significant diabatic effects in the interior. Since the applied wind stress is constant in time, its anomalous value is always zero. Further, the bottom friction can be neglected when we focus our attention between the surface and the top 1600 m (where the activity is strongest). Thus, (4) and (5) can be rewritten as

 
formula
 
formula

where all of the above terms represent deviations from the time average.

Near the surface, we invoke Bretherton’s PV sheets (Bretherton 1966), exploiting the equivalence between an inhomogeneous boundary temperature distribution and a delta function PV anomaly just inside an isothermal boundary. Then both Fz and [*] vanish at the boundary, and the divergence of EP fluxes is concentrated in the surface PV sheet.

Accordingly, we first vertically integrate each of the terms in (9) through the top 1600 m for each phase of the evolution; the results of doing so are shown in Fig. 14. There are three important features to note. First, the flow tendency is essentially a residual; the local momentum budget for each phase is a quasi-steady balance between the anomalous divergence of the EP fluxes and the TEM Coriolis term. Second, the baroclinic component ∂Fz/∂z dominates over the barotropic component ∂Fy/∂y in the divergence of the EP flux. Third, the anomalous EP flux divergence is out of phase with the secondary jets; the strongest anomalous convergence lies on the poleward flank, while the strongest anomalous divergence occurs on the equatorward flank. The reason behind the spatial structure of · F will be discussed further in the next section.

Fig. 14.

Anomalous quantities of the vertically integrated TEM equation for each phase. The vertical integral was taken from the surface to a depth of 1600 m. Solid line is the divergence of the vertical component of the EP flux; long dashed line is the divergence of the horizontal component of the EP flux; dotted line is the flow tendency; and the dot-dashed line is the Coriolis force. Vertical dashed lines correspond to the maximum positive zonal flow anomalies.

Fig. 14.

Anomalous quantities of the vertically integrated TEM equation for each phase. The vertical integral was taken from the surface to a depth of 1600 m. Solid line is the divergence of the vertical component of the EP flux; long dashed line is the divergence of the horizontal component of the EP flux; dotted line is the flow tendency; and the dot-dashed line is the Coriolis force. Vertical dashed lines correspond to the maximum positive zonal flow anomalies.

The anomalous · F thus creates an anomalous residual circulation as shown in Fig. 15 (and schematically illustrated in Fig. 20b). On the poleward (equatorward) flanks of an anomalous eastward jet the EP flux is convergent (divergent), producing a poleward (equatorward) residual flow. By conservation of mass, near the jet cores the anomalous residual flow must rise and in between the jets must sink.

Fig. 15.

Anomalous residual circulation (vector) plotted over anomalous zonal flow (color) for each phase. The vectors are scaled the same for each plot, with the largest vector corresponding to 1.33 cm s−1.

Fig. 15.

Anomalous residual circulation (vector) plotted over anomalous zonal flow (color) for each phase. The vectors are scaled the same for each plot, with the largest vector corresponding to 1.33 cm s−1.

Fig. 20.

Schematic depiction of the interaction between baroclinic eddies and a localized jet. In (a) the baroclinic eddy activity is assumed to be symmetric about the jet; in (b) it is stronger on the poleward flanks of the jet, presumed to be a consequence of a general poleward decrease of static stability. CONV and DIV denote convergence and divergence, respectively, of the baroclinic EP fluxes; COOL and WARM indicate the local temperature tendencies attributable to the induced residual circulation, thus reducing (ℛ) and enhancing (ℰ) the local zonal mean baroclinicity. (Note that the Southern Hemisphere is depicted; the equator is to the right.) See text for discussion.

Fig. 20.

Schematic depiction of the interaction between baroclinic eddies and a localized jet. In (a) the baroclinic eddy activity is assumed to be symmetric about the jet; in (b) it is stronger on the poleward flanks of the jet, presumed to be a consequence of a general poleward decrease of static stability. CONV and DIV denote convergence and divergence, respectively, of the baroclinic EP fluxes; COOL and WARM indicate the local temperature tendencies attributable to the induced residual circulation, thus reducing (ℛ) and enhancing (ℰ) the local zonal mean baroclinicity. (Note that the Southern Hemisphere is depicted; the equator is to the right.) See text for discussion.

From (10), the rising residual circulation implies that the temperature tendency is a local minimum near the secondary jets (see Fig. 16). Therefore, the time tendency for the meridional temperature gradient is positive on the equatorward flank and negative on the poleward flank, thus encouraging equatorward migration of the baroclinic zone. From the thermal wind relation, vertical shear will increase (decrease) on the equatorward (poleward) side, furthering the equatorward jet migration. Thus, this pattern of strong/weak baroclinic jets moving equatorward is ultimately dictated by the pattern of the EP flux divergence.

Fig. 16.

Anomalous quantities of the transformed Eulerian mean thermodynamic equation (°C s−1) at z = −593 m. Solid line is the temperature tendency; long dashed line is the residual vertical advection of the vertical temperature gradient; dot-dashed line is the residual meridional advection of the meridional temperature gradient. Vertical dashed lines correspond to the maximum positive zonal flow anomalies.

Fig. 16.

Anomalous quantities of the transformed Eulerian mean thermodynamic equation (°C s−1) at z = −593 m. Solid line is the temperature tendency; long dashed line is the residual vertical advection of the vertical temperature gradient; dot-dashed line is the residual meridional advection of the meridional temperature gradient. Vertical dashed lines correspond to the maximum positive zonal flow anomalies.

c. EP fluxes

We have seen, therefore, that the equatorward migration of the secondary jets is a consequence of the structure of · F, in particular, the fact that the anomalous near-surface EP flux convergence is located on the poleward flank of each eastward jet rather than at the jet center, as one might naïvely expect. This finding, in turn, begs the question: Why are the fluxes organized in this way?

Before examining the anomalous EP fluxes, we show in Fig. 17 the total EP fluxes for each of the four phases described in section 3. In all cases, equatorward of 12°S, where the isopycnals are relatively flat, little to no eddy activity is observed. However, eddies are ubiquitous poleward of 12°S. In all phases, the flux is upward (i.e., eddy heat flux is poleward), consistent with baroclinic instability being the source of the eddies. Comparing the different phases, both the eddy heat flux and eddy momentum flux are more dominant in the high index phases, C and D, than in the low zonal index phases, A and B.

Since our interest is in the time-varying jets, we now focus on the anomalous EP flux vectors (i.e., departures from the time mean) shown in Fig. 18. In general, where there are positive zonal flow anomalies, the baroclinic component of the flux is enhanced (more upward). However, upon closer examination, the maximum of Fz is not coincident with the jet and is not symmetric: there is a poleward bias. From (6), the asymmetry must exist in the anomalous zonally averaged eddy heat flux or in the static stability. (We presume that variation in the Coriolis parameter across these narrow jets is too small to be of significance.) In fact, as Fig. 19 demonstrates, the composites of each phase indeed show that there is a robust poleward bias in the anomalous eddy heat flux.

Fig. 18.

Anomalous EP flux vectors for each EOF phase composite are plotted over the zonally averaged zonal flow anomalies. Each phase is labeled at the bottom left corner of plot. The vectors are scaled twice as large as Fig. 17 and are the same for each plot.

Fig. 18.

Anomalous EP flux vectors for each EOF phase composite are plotted over the zonally averaged zonal flow anomalies. Each phase is labeled at the bottom left corner of plot. The vectors are scaled twice as large as Fig. 17 and are the same for each plot.

Fig. 19.

Anomalous eddy heat flux (line) is plotted over zonal flow anomalies (in color). Each phase is labeled at the bottom left corner of plot.

Fig. 19.

Anomalous eddy heat flux (line) is plotted over zonal flow anomalies (in color). Each phase is labeled at the bottom left corner of plot.

Why are the largest poleward eddy heat fluxes located poleward of the eastward zonal flow anomaly? We speculate that the implied asymmetry in baroclinic eddy activity is consistent with a bias in the Eady growth rate (Eady 1949), defined to be

 
formula

We use σ as a local measure of instability and speculate that the upward EP fluxes of baroclinic eddies will maximize where σ is greatest. The Coriolis parameter varies only by a factor between 5% and 8% in the region of interest over the typical width (about 2°) of the jets. The vertical shear is essentially symmetric about the secondary jets. However, the buoyancy frequency N varies systematically with latitude in the region poleward of the main jet (see Fig. 5). In the migrating jet regime, N varies from 30% to over 55% over the length scale of the jets with smaller values on the poleward side. Thus, the absolute magnitude of the Eady growth rate is larger on the secondary jets’ poleward flank, consistent with the stronger poleward eddy heat fluxes there. Therefore, with N increasing monotonically equatorward in the migrating jet region, we speculate that, in the absence of any other asymmetries, the latitudinal variation in the background static stability may produce the asymmetry in the eddy heat fluxes.

d. Summary of the eddy–mean flow interaction

Drawing all of these results together, the nature of the interaction between the eddies and the evolving jets is summarized schematically in Fig. 20. In the absence of any latitudinal asymmetries in the background state, one might expect the situation depicted in Fig. 20a, the EP fluxes being symmetric about the jet, with divergence beneath the jet and convergence at and near the surface. Given our finding that the time derivative in (9) is negligible, the consequence is a pumping of the residual flow poleward at and near the surface and equatorward at the bottom, generating a circulation cell as shown. The residual upwelling and downwelling therefore produce cooling on the equatorward flank and warming on the poleward flank of the jet, thereby reducing the baroclinicity at the jet core and enhancing it at the flanks. This symmetric pattern would induce no tendency of the baroclinic zone, and the consequent jet, to migrate.

Consider now Fig. 20b in which we assume that the baroclinic eddy activity is displaced poleward of the jet as a consequence of basic-state static stability increasing equatorward in the background state. The residual circulation cell is, accordingly, displaced poleward with respect to the jet, thus producing tendencies of reducing baroclinicity on the poleward flank and enhancing it on the equatorward flank, leading to equatorward migration of the baroclinic zone and therefore of the jet itself.

5. Conclusions

As was shown in Fig. 6, multiple zonal jets emerge when extremely broad buoyancy and wind forcing are applied to the surface of the model ocean. Although both forcings are constant in time, there is significant variability in the zonally averaged zonal flow (see Fig. 7). Between 15° and 20°S, the main eastward jet oscillates meridionally, while between 20° and 35°S, secondary (weaker) jets systematically migrate equatorward. An EOF analysis describes the leading mode as an equivalent barotropic structure with the largest anomalies 3° north and south of the primary jet’s time-averaged position, thus describing north–south fluctuations of the main jet, in a manner qualitatively similar to atmospheric behavior (e.g., Thompson and Wallace 1998). However, unlike the atmospheric case, the two leading EOFs are in quadrature in the region of the secondary jets where, together, they capture the jets’ migration.

The secondary jets are maintained against bottom friction by convergence of the eddy momentum fluxes. However, it is the eddy heat fluxes (or equivalently, the vertical component of the EP fluxes) that control their equatorward migration. Although the anomalous convergence of the eddy momentum flux is nearly symmetric along the jet axis, convergence of the baroclinic (vertical) component of the EP flux is not. The vertical EP flux dominates locally and is stronger on the poleward flank of the jets (see Fig. 14), leading to the jets’ equatorward migration. We speculate that the asymmetry in the vertical EP flux is consistent with larger Eady growth rates associated with smaller values of static stability on the poleward side of the jets.

Recall from section 3 that equatorward of the main jet, there is some evidence (cf. Fig. 7) of poleward migration of zonal flow anomalies, although this migration is not as systematic as the equatorward migration farther poleward. As Fig. 5 makes clear, the latitudinal structure of the static stability changes character equatorward of the main jet. Below the depth of about 1000 m, its gradient changes sign, at about 14°S (though the gradient is weak equatorward thereof); at about 1000 m the structure is more complex. Therefore it is difficult to relate the behavior of the flow anomalies to the static stability structure in this region.

We might therefore expect equatorward (or poleward) migration of baroclinic jets whenever the static stability or, more generally, N/f increases (or decreases) equatorward throughout the vertical column. In fact, the multiple jets described by Williams (2003), either migrated equatorward or remained steady, depending on the background state. In the nonmigrating cases at altitudes of strongest baroclinic activity, there were little to no latitudinal variations of the vertical temperature gradient (Williams, Fig. 6a). However, in the migrating jet cases, the inferred static stability increased monotonically equatorward (Williams, Fig. 6b), consistent with the mechanism described above. Further, the patterns of EP flux divergence, in relation to the jets, showed similar characteristics to those described here.

Although observational records depict poleward migrating behavior in the observed atmosphere (Feldstein 1998), the mechanism discussed in this study may still be relevant. Other theories have been proposed in describing this behavior (e.g., James et al. 1994; Lee et al. 2007). Here, we note that above the boundary layer, at all heights, the static stability increases poleward (Peixoto and Oort 1992). Assuming no other asymmetries, larger Eady growth rates would be larger on the equatorward side and a similar argument discussed above would hold. While this may be a coincidence, the potential for waves to develop more strongly in a region of lower static stability leading to the poleward propagation of zonal flow anomalies warrants further study.

While there is a distinct migration in the atmosphere, whether this behavior arises in the observed ocean is uncertain. Perhaps limited by the length of observational records with sufficient spatial resolution encompassing a large enough domain, we are unaware of any documented observations reporting migrating jets. However, in the Antarctic Circumpolar Current (ACC), there are three important ingredients that would make this region a possibility. First, multiple jets have been reported (Nowlin and Klink 1986), and, second, as a zonally reentrant channel, this is similar to our idealized model. Finally, away from the mixed layer, the static stability monotonically increases equatorward from 0 s−1 to about 3 × 10−3 s−1 (Levitus et al. 1994). Therefore, migrating behavior in the ACC cannot be ruled out and merits further research.

Acknowledgments

We wish to thank three anonymous reviewers for their beneficial comments on this manuscript. This research was supported by the National Science Foundation under Grants OCE-0426307 and ATM-0314094.

REFERENCES

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
.
Bretherton
,
F. P.
,
1966
:
Critical layer instability in baroclinic flows.
Quart. J. Roy. Meteor. Soc.
,
92
,
325
334
.
Cerovecki
,
I.
, and
J. C.
Marshall
,
2007
:
Eddy modulation of air–sea interaction and convection.
J. Phys. Oceanogr.
,
in press
.
Cho
,
J. Y-K.
, and
L. M.
Polvani
,
1996
:
The emergence of jets and vorticies in freely evolving, shallow-water turbulence on a sphere.
Phys. Fluids
,
8
,
246
262
.
de Pater
,
I.
, and
J.
Lissauer
,
2001
:
Planetary Sciences.
Cambridge University Press, 544 pp
.
Dritschel
,
D. G.
,
P. H.
Haynes
, and
M. E.
McIntyre
,
2007
:
Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers.
J. Atmos. Sci.
,
in press
.
Eady
,
E. T.
,
1949
:
Long waves and cyclone waves.
Tellus
,
1
,
35
52
.
Edmon
Jr.,
H. J.
,
B. J.
Hoskins
, and
M. E.
McIntyre
,
1980
:
Eliassen–Palm cross sections for the troposphere.
J. Atmos. Sci.
,
37
,
2600
2615
.
Feldstein
,
S.
,
1998
:
An observational study of the intraseasonal poleward propagation of zonal mean flow anomalies.
J. Atmos. Sci.
,
55
,
2516
2529
.
Feldstein
,
S.
, and
S.
Lee
,
1996
:
Mechanisms of zonal index variability in an aquaplanet GCM.
J. Atmos. Sci.
,
53
,
3541
3556
.
Held
,
I. M.
,
1975
:
Momentum transport by quasi-geostrophic eddies.
J. Atmos. Sci.
,
32
,
1494
1497
.
Ioannou
,
P.
, and
R. S.
Lindzen
,
1986
:
Baroclinic instability in the presence of barotropic jets.
J. Atmos. Sci.
,
43
,
2999
3014
.
James
,
P.
,
K.
Fraedrich
, and
I. N.
James
,
1994
:
Wave-zonal-flow interaction and ultralow-frequency variability in a simplified global general circulation model.
Quart. J. Roy. Meteor. Soc.
,
120
,
1045
1067
.
Kushner
,
P. J.
, and
L. M.
Polvani
,
2004
:
Stratosphere–troposphere coupling in a relatively simple AGCM: The role of eddies.
J. Climate
,
17
,
629
639
.
Lee
,
S.
,
2005
:
Baroclinic multiple zonal jets on the sphere.
J. Atmos. Sci.
,
62
,
2484
2498
.
Lee
,
S.
,
S-W.
Son
,
K.
Grise
, and
S. B.
Feldstein
,
2007
:
Mechanism of poleward propagation of zonal mean flow anomalies.
J. Atmos. Sci.
,
64
,
849
868
.
Levitus
,
S.
, and
T. P.
Boyer
,
1994
:
Temperature. Vol. 4, World Ocean Atlas 1994, NOAA Atlas NESDIS 4, 117 pp
.
Levitus
,
S.
,
R.
Burgett
, and
T.
Boyer
,
1994
:
Salinity. Vol. 3, World Ocean Atlas 1994, NOAA Atlas NESDIS 3, 99 pp
.
Lorenz
,
D. J.
, and
D. L.
Hartmann
,
2001
:
Eddy–zonal flow feedback in the Southern Hemisphere.
J. Atmos. Sci.
,
58
,
3312
3327
.
Marshall
,
J.
,
A.
Adcroft
,
C.
Hill
,
L.
Perelman
, and
C.
Heisey
,
1997a
:
A finite-volume, incompressible Navier Stokes model for studies of the ocean on parallel computers.
J. Geophys. Res.
,
102
,
5753
5766
.
Marshall
,
J.
,
C.
Hill
,
L.
Perelman
, and
A.
Adcroft
,
1997b
:
Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling.
J. Geophys. Res.
,
102
,
5733
5752
.
North
,
G. R.
,
T. L.
Bell
,
R. F.
Cahalan
, and
F. J.
Moeng
,
1982
:
Sampling errors in the estimation of empirical orthogonal functions.
Mon. Wea. Rev.
,
110
,
699
706
.
Nowlin
,
W. D.
, and
J. M.
Klink
,
1986
:
The physics of the Antarctic circumpolar current.
Rev. Geophys.
,
24
,
469
491
.
Panetta
,
R. L.
,
1993
:
Zonal jets in wide baroclinically unstable regions: Persistence and scale selection.
J. Atmos. Sci.
,
50
,
2073
2106
.
Peixoto
,
J. P.
, and
A. H.
Oort
,
1992
:
Physics of Climate.
Springer, 564 pp
.
Robinson
,
W. A.
,
1994
:
Eddy feedbacks on the zonal index and eddy–zonal flow interactions induced by zonal flow transience.
J. Atmos. Sci.
,
51
,
2553
2562
.
Roden
,
G.
,
2000
:
Flow and water property structures between the Bering Sea and Fiji in the summer of 1993.
J. Geophys. Res.
,
105
,
28595
28612
.
Thompson
,
D. W. J.
, and
J. M.
Wallace
,
1998
:
The Arctic Oscillation signature in the wintertime geopotential height fields.
Geophys. Res. Lett.
,
25
,
1297
1300
.
Thompson
,
D. W. J.
, and
J. M.
Wallace
,
2000
:
Annular modes in the extratropical circulation. Part I: Month-to-month variability.
J. Climate
,
13
,
1000
1016
.
Williams
,
G. P.
,
1978
:
Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence.
J. Atmos. Sci.
,
35
,
1399
1426
.
Williams
,
G. P.
,
2003
:
Jovian dynamics. Part III: Multiple, migrating, and equatorial jets.
J. Atmos. Sci.
,
60
,
1270
1296
.

Footnotes

Corresponding author address: Cegeon Chan, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Building 54, Room 1719, 77 Massachusetts Avenue, Cambridge, MA 02139. Email: cegeon@mit.edu

This article included in the Jets and Annular Structures in Geophysical Fluids (Jets) special collection.