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    Results from an Earthlike run of the two-level model: (a) zonally averaged zonal winds and midlevel potential temperatures, (b) eddy fluxes of potential vorticity on the upper (F) and lower (B) levels, and (c) upper-level (250 hPa) eddy momentum flux and midlevel (500 hPa) eddy heat flux.

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    Total and marginal effects of eddies on the circulation of the two-level model (see text): (a) effects on upper-level zonal winds and (b) effects on midlevel temperatures.

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    (a) Zonally averaged zonal winds and (b) eddy fluxes of potential vorticity for model runs with ΔT = 24 K (dashed curves) and ΔT = 24.6 K (solid curves).

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    Maximum lower-level zonal wind (dashed and open circles) and maximum lower-level poleward eddy flux of potential vorticity (solid and filled squares) as functions of ΔT

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    Snapshots of upper-level potential vorticity for runs with (a) ΔT = 60 K, (b) ΔT = 24 K, and (c) ΔT = 24.6 K. The contour interval is 3 × 10−5 s−1, and the outer latitude for the polar stereographic projections is 20°N.

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    Midlevel zonal winds (thick curves) and wavenumber-7 phase speeds (thin curves) for model runs with ΔT = 24 K (dashed) and ΔT = 24.6 K (solid).

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    Latitude–time plots for the Southern Hemisphere of a model experiment in which ΔT is increased from 24 to 24.6 K over time: (a) upper-level zonal winds, where the contour interval is 1 m s−1; (b) lower-level zonal winds, where the contour interval is 0.2 m s−1; (c) B (lower-level eddy flux of potential vorticity), where the contour interval is 5 × 10−7 m s−2; and (d) −F (minus the upper-level eddy flux of potential vorticity), where the contour interval is 5 × 10−7 m s−2.

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    Results from a model experiment in which a westward barotropic body force is applied to the model Tropics, with ΔT = 24 K: (a) upper-level zonal winds for the forced experiment (thick solid curve), the forced experiment with no eddy feedback (thin solid curve), and the unforced experiment (dashed curve); (b) eddy fluxes of potential vorticity for the forced experiment.

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    Feedbacks that govern the transition to a self-maintaining jet.

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    Model results for ΔT = 40, 80, and 120 K: (a) marginal influence of the eddies on midlevel potential temperature, (b) eddy fluxes of potential vorticity on the upper level (F), and (c) latitudinal derivative of an upper-level tracer, relaxed to a sin(ϕ) profile with a relaxation time of 20 days.

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On the Self-Maintenance of Midlatitude Jets

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  • 1 Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois
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Abstract

In this paper an atmospheric jet is considered self-maintaining if the overall effect of baroclinic eddies is to preserve or enhance its westerly shear with height. Observations suggest that the wintertime jets in Earth’s atmosphere are self-maintaining. This has implications for the intrinsic variability of these jets—the annular modes—and for how the extratropics respond to tropical warming.

The theory of quasigeostrophic eddy–zonal flow interactions is employed to determine how a jet can be self-maintaining. Whether or not a jet is self-maintaining is found to depend sensitively on the meridional distribution of the absorption of wave activity.

The eddy driving of the jet in a simple two-level model of the global circulation is examined. It is found that, with approximately wintertime settings of parameters (a radiative equilibrium equator–pole temperature contrast of 60 K), the midlatitude jets in this model are self-maintaining. The jet is not self-maintaining, however, when the radiative equilibrium equator-to-pole temperature contrast is reduced below a critical value (∼24 K temperature contrast). Eddy amplitudes are also greatly reduced, in this case. The transition to a self-maintaining jet, as the radiative equilibrium temperature contrast is increased, suggests a set of feedback mechanisms that involve the strength of the baroclinicity in the jet center and where baroclinic eddies are absorbed in the subtropics. A barotropic eastward force applied to the model Tropics causes a poleward shift in the latitudes of greatest eddy absorption and induces a transition from a non-self-maintaining to a self-maintaining jet.

Self-maintaining behavior ultimately disappears, as the equator–pole thermal contrast, and thus the eddies, are strengthened. The flow is then highly disturbed and no longer dominated by wavelike baroclinic eddies.

Corresponding author address: Professor Walter A. Robinson, Dept. of Atmospheric Sciences, University of Illinois at Urbana–Champaign, 105 South Gregory Street, Urbana, IL 61801. Email: robinson@atmos.uiuc.edu

Abstract

In this paper an atmospheric jet is considered self-maintaining if the overall effect of baroclinic eddies is to preserve or enhance its westerly shear with height. Observations suggest that the wintertime jets in Earth’s atmosphere are self-maintaining. This has implications for the intrinsic variability of these jets—the annular modes—and for how the extratropics respond to tropical warming.

The theory of quasigeostrophic eddy–zonal flow interactions is employed to determine how a jet can be self-maintaining. Whether or not a jet is self-maintaining is found to depend sensitively on the meridional distribution of the absorption of wave activity.

The eddy driving of the jet in a simple two-level model of the global circulation is examined. It is found that, with approximately wintertime settings of parameters (a radiative equilibrium equator–pole temperature contrast of 60 K), the midlatitude jets in this model are self-maintaining. The jet is not self-maintaining, however, when the radiative equilibrium equator-to-pole temperature contrast is reduced below a critical value (∼24 K temperature contrast). Eddy amplitudes are also greatly reduced, in this case. The transition to a self-maintaining jet, as the radiative equilibrium temperature contrast is increased, suggests a set of feedback mechanisms that involve the strength of the baroclinicity in the jet center and where baroclinic eddies are absorbed in the subtropics. A barotropic eastward force applied to the model Tropics causes a poleward shift in the latitudes of greatest eddy absorption and induces a transition from a non-self-maintaining to a self-maintaining jet.

Self-maintaining behavior ultimately disappears, as the equator–pole thermal contrast, and thus the eddies, are strengthened. The flow is then highly disturbed and no longer dominated by wavelike baroclinic eddies.

Corresponding author address: Professor Walter A. Robinson, Dept. of Atmospheric Sciences, University of Illinois at Urbana–Champaign, 105 South Gregory Street, Urbana, IL 61801. Email: robinson@atmos.uiuc.edu

1. Introduction

It has been known for half a century that in the time and zonal mean eddies in earth’s atmosphere transport angular momentum poleward and to a lesser extent equatorward into the midlatitude jets (cf. Lorenz 1967; Starr 1968). What has been less clear is the overall effects of the eddies on the jet. These include the direct effects of eddy transports of heat and momentum and the effects of secondary circulations induced by these eddy transports. Transient eddies in the midlatitude atmosphere arise from the baroclinic instability of the jet. If the overall effect of these eddies is to enhance, or at least preserve, the baroclinicity of the jet where the baroclinic generation of eddy activity is most robust, the jet can be considered self-maintaining.

The definition of self-maintaining employed here is not the only possible one. Panetta and Held (1988) and Panetta (1993) obtained persistent localized jets in two-layer quasigeostrophic models, in which the background baroclinicity was uniform over a wide region, or, in doubly periodic models, everywhere. In the former study the model was severely truncated to a single zonal wavenumber and the zonal mean flow, whereas in the latter a full spectrum of eddies was resolved. The persistent jets that emerge in such models necessarily maintain themselves. In a quasigeostrophic model, in the presence of mechanical friction, eddies carry momentum into the westerly jets. This requires an export of Eliassen–Palm eddy activity from the jet, so that, in the time mean, the generation of wave activity within the jets exceeds its dissipation, while the converse is true between the jets. Summing over the two model layers, there is a poleward eddy flux of potential vorticity in the westerly jets, and an equatorward flux between them. In many of the cases discussed by Panetta and Held (1988) and Panetta (1993), it is principally the poleward flux of potential vorticity in the lower layer that varies strongly with latitude, while the dissipation of eddy activity in the upper layer is nearly uniform. As will be seen in the next section, this leads to jets that are nearly barotropic. If there is no thermal damping, as for many cases considered by Panetta, then, in the time mean, convergent or divergent eddy fluxes of potential vorticity aloft are either absent or are balanced by diffusive fluxes. In contrast, in the experiments described in sections 3 and 4, there are strong latitudinal variations in the equatorward flux of potential vorticity aloft, thermal damping is important, and baroclinic shear is concentrated in the jets.

Whether jets in Earth’s atmosphere are self-maintaining, in the baroclinic sense defined above, is not readily determined from observations. Diagnoses of the general circulation in the transformed Eulerian mean (TEM) framework (Edmon et al. 1980; Karoly 1982), however, suggest that the wintertime midlatitude jets are self-maintaining. The evidence is a region of the extratropics, across which, in the poleward direction, the vertical component of the TEM circulation changes from downward to upward through most of the depth of the troposphere (e.g., Edmon et al. 1980, their Fig. 6a). This “U” in the TEM streamfunction occurs at the same latitudes where the generation of baroclinic eddies is strongest, as indicated by the upward Eliassen–Palm (EP) fluxes from the lower troposphere (cf. Edmon et al. 1980, their Fig. 1b). Recent analyses, using modern observations yield essentially the same result (cf. Karoly et al. 1997; Tanaka et al. 2004). To the extent that the TEM circulation is eddy driven, the resulting adiabatic warming in the subtropics and adiabatic cooling further poleward implies that the thermal gradient is intensified by the overall influence of the eddies. The TEM circulation ultimately closes clockwise (in the Northern Hemisphere), so that hemispherewide, the eddies warm high latitudes and cool the Tropics. This must be the case if the eddies are to extract available potential energy from zonal mean flow. No energetic constraint, however, prevents the eddies from locally strengthening the thermal gradient.

Whether or not a jet is self-maintaining is of more than academic interest. Within the broad constraints of planetary geometry and the distribution of radiatively imposed thermal gradients, a self-maintaining jet can exist in a quasi-steady state over a range of latitudes. This was proposed by Robinson (2000) as an explanation for the prominence of annular modes (Thompson and Wallace 2000) among the patterns of intraseasonal variability. Panetta’s (1993) results suggest that the slow latitudinal wondering of a self-maintaining jet in a broad or uniform baroclinic zone is a plausible model for annular variability in the atmosphere. In a more realistic context, an idealized multilevel model on the sphere, Fyfe and Lorenz (2005) recently showed that the meridional wandering of a jet with a nearly fixed meridional structure offers a more parsimonious model for annular variability than a set of empirical orthogonal functions (EOFs) dominated by a latitudinal dipole. Their analysis of observations suggests this may be similarly true in the atmosphere.

That wintertime jets are self-maintaining also helps to explain some features of the zonally averaged response to tropical warming such as occurs during El Niño. In a series of papers in the 1990s (Chang 1995; Hou 1993; Hou and Molod 1995; Hou 1998) it was found in models that a strengthening of the Hadley cell, say by shifting the center of heating away from the equator, led to oppositely signed thermal changes within and immediately outside of the Tropics, suggesting that the enhanced Hadley cell caused a change in the overall heat transport by the eddies, including that due to eddy-induced secondary circulations. A closer examination of these results (Chang 1995) revealed that eddy heat transports and eddy momentum transports had opposing influences on the temperatures, but that the momentum fluxes “won.” When a change in the Hadley circulation is induced in the Tropics, the changes in the eddy momentum fluxes induce changes in the mean meridional circulation that in turn produce changes in midlatitude temperatures. These are larger than the temperature changes produced, directly and indirectly, by changes in the eddy heat fluxes.

This problem was revisited in a two-level model by Robinson (2002) and then in analyses of observations and general circulation model (GCM) output by Seager et al. (2003, 2005). The picture that emerges is as follows. A warming of the Tropics intensifies the eddy-driven jet and storm track. Because the eddy-driven jet is self-maintaining, this results in a band of cooling in middle latitudes, caused by a strengthening of the eddy-pumped rising motion at these latitudes. This rising motion causes rates of precipitation to increase (Seager et al. 2005).1 The jet and storm track also shift equatorward when the Tropics are warmed. In the model results of Robinson (2002), poleward eddy heat fluxes weaken slightly between 55° and 60°N. In the observed response to tropical warming associated with El Niño (Seager et al. 2003), the equatorward shift is more pronounced relative to the strengthening of the storm track, so that the accompanying shift in the band of rising motion, contributes to a band of anomalous cooling centered between 35° and 40°N.

Here we seek to explain why the jets are self-maintaining. It is not necessary that they are so—one can readily imagine a dynamically valid atmospheric circulation in which the meridional temperature gradient across the jet is significantly weaker than it would be in the absence of eddies. To address this question, the dynamics of jets are considered from the perspective of the eddy–zonal flow dynamics of a two-level quasigeostrophic model (section 2). It is found that whether or not a jet is self-maintaining depends sensitively on the meridional distribution of sinks of eddy activity, as measured by the convergence of the Eliassen–Palm flux (or, equivalently, the equatorward eddy flux of potential vorticity). These EP diagnostics are then applied to a two-level primitive equation model on the sphere (section 3). This model, in an Earthlike wintertime state, supports a self-maintaining jet. To address how such a jet arises, a non-self-maintaining jet is found by reducing the equator–pole temperature contrast beneath a threshold value. As this threshold is crossed, a self-maintaining jet emerges abruptly. Comparing these two model states (section 4) reveals the dynamical mechanisms at work in self-maintaining jets. These involve the wavelike nature of baroclinic eddies, specifically their narrow spectra and the resulting concentration of their absorption in a narrow band of critical latitudes. The positive feedbacks that cause the abrupt transition from a non-self-maintaining to a self-maintaining jet involve the shifts of these critical latitudes when the zonal flow is altered, and the responses of the zonal flow to resulting modifications in the distribution of wave driving. The transition to a self-maintaining jet can be brought about by applying a westward barotropic force to the Tropics, which causes a poleward shift in the latitudes of greatest eddy absorption. The paper concludes with a summary and discussion (section 5), including additional results that show the loss of self-maintaining jets for very strong forcing, and a comparison of the present results with those obtained from quasigeostrophic models on the beta plane.

2. Theory

We consider the steady and zonally averaged balance between the advection of potential vorticity by eddies and its destruction and creation by thermal damping and by surface drag. Equations (5) and (6) below were obtained by Pavan and Held (1996); the derivation is given here so as to make this paper self-contained. We employ quasigeostrophic dynamics, in a two-level model on the beta plane (cf. Holton 2004). The starting point is the potential vorticity equation,
i1520-0469-63-8-2109-e1
where q is the potential vorticity, ψ is the geostrophic streamfunction, and the subscripts, i = 1, 2, indicate the level, with 1 being the upper level and 2 the lower. The potential vorticity is given by
i1520-0469-63-8-2109-e2
where λ is the reciprocal of the internal Rossby radius of deformation, and the notation is otherwise standard. Sources of potential vorticity, S in (1), are associated with thermal damping and surface drag
i1520-0469-63-8-2109-e3
where N is the rate of Newtonian thermal damping, and E is the rate of Ekman drag in the lower level.
The zonally averaged steady-state balances of potential vorticity, between sources and sinks, on the left-hand side, and eddy transport, on the right, are
i1520-0469-63-8-2109-e4
where [·] indicates zonal averages, * denotes deviations from zonal averages, and F and B denote the zonally averaged eddy fluxes of potential vorticity on the upper and lower levels. In the quasigeostrophic system, these are equal to the divergences of the Eliassen–Palm fluxes on both levels (Edmon et al. 1980), so that B, which is typically positive in the time average, may be considered the source of Eliassen–Palm eddy activity, while −F, may be considered its dissipative sink aloft. Taking the sum of the two equations in (4) and integrating with respect to y yields
i1520-0469-63-8-2109-e5
assuming that the constant of integration vanishes. This is equivalent to requiring that the integration extends over all latitudes of eddy forcing, and that the zonal wind is forced only by the eddies. Differentiating the first line of (4) with respect to y and substituting from (5) gives an expression for the upper-level zonal wind in terms of the eddy forcing,
i1520-0469-63-8-2109-e6
The baroclinic part of the zonal flow, the second term on the right-hand side of (6), depends on the eddy forcing only on the upper level, and thermal wind balance then implies that only the upper-level eddy forcing determines the temperatures. This is a two-level model manifestation of the principle of downward control (Haynes et al. 1991). Thinking in the TEM framework: in the time average, the balance of zonal forces on the upper level is between the zonal Coriolis force due to the TEM meridional flow and the eddy deceleration of the zonal wind; that is, the convergence of the EP flux, or −F in the present notation. By continuity, TEM descent, and therefore the temperature, is proportional to (minus) the y derivative of the horizontal motion and thus to F′. The second y derivative in (6) then comes from the thermal wind relationship between the meridional derivative of temperature and the vertical shear in the zonal wind.

Equations (5) and (6) describe only the eddy-driven part of the zonal flow. In general this is superposed on a zonal flow that is in thermal wind balance with radiative equilibrium temperatures. In the absence of any external mechanical forcing, the zonal flow at radiative equilibrium vanishes on the lower level.

Consider the usual situation in which the radiative equilibrium imposes decreasing temperatures with latitude, and therefore westerly shear between the lower and upper levels. If this shear is sufficiently strong, the zonally averaged potential vorticity will increase poleward on the upper level and decrease poleward on the lower level, satisfying the Charney–Stern (1962) criterion for baroclinic instability. If the eddies arise spontaneously, as opposed to from forcing, and if dissipative processes deplete eddy potential enstrophy, than F is necessarily negative and B is necessarily positive. In addition as shown by Bretherton (1966), the global integral of the sum of F and B must vanish.

From (6) it is seen that a self-maintaining jet, for which eddies locally enhance the baroclinicity, is one for which F″ is positive in the jet center. This implies the presence of a local maximum of F in the jet, or a local minimum in the absorption of eddy activity. If it is assumed that the generation of eddy activity on the lower level, B, is greatest in the jet center, this implies that eddy activity propagates meridionally outward from the jet center before it is absorbed aloft. Such splitting of the region of eddy activity absorption, or EP flux convergence, near the tropopause, is found for the observed transient eddies in the Northern Hemisphere winter (e.g., Fig. 1b of Edmon et al. 1980), and, as is seen in the next section, in the output of a two-level numerical model.

3. Model experiments—Diagnoses of an Earthlike control run

The numerical model is a dry, global, spectral two-level primitive equation model, which has a long history of application to the study of atmospheric variability (Hendon and Hartmann 1985; Robinson 1991). The present version is run at a rhomboidal 15 spectral truncation, with a time step of one-half hour. The model levels are at 250 and 750 hPa. Linear drag, with a two-day damping time, is applied to winds on the lower level. Biharmonic diffusion is applied to all fields, with a diffusion coefficient of 4 × 1016 m4 s−1. Temperatures are linearly restored to a radiative equilibrium profile on a 15-day relaxation time scale.

There is no topography, and the radiative equilibrium temperatures are zonally symmetric. The midlevel (500 hPa) potential temperature in radiative equilibrium is given by
i1520-0469-63-8-2109-e7
where ϕ is the latitude, and ΔT is the equator–pole potential temperature contrast in radiative equilibrium. The radiative-equilibrium difference in potential temperatures between the upper and lower levels is 30 K everywhere.

A climate with reasonable similarity to that observed in midwinter is obtained with ΔT = 60 K. Figure 1 shows the time-averaged results from this run. Here and throughout the paper time averages are averages over the second 1000 days of 2000-day model runs. Each experiment is started from rest, with uniform temperatures on each level, except for a small random perturbation to the midlevel temperatures. Figure 1a shows the zonally averaged midlevel potential temperature and the zonally averaged zonal winds on the upper and lower levels. These are unremarkable, though it should be noted that in this dry model, without any added tropical heating, the Hadley circulation is weak (the strongest zonally averaged meridional winds are about 0.5 m s−1), so there is only a hint of a subtropical jet between 15° and 20°N. The dominant jet, at 40° to 45°N, is eddy driven (Lee and Kim 2003; Son and Lee 2005).

The second panel shows the eddy driving of this zonal flow—the zonally averaged eddy fluxes of potential vorticity on the upper (F) and lower (B) levels. A nearly quasigeostrophic version of the potential vorticity is used to compute these fluxes:
i1520-0469-63-8-2109-e8
where f0 = 10−4 s−1 and δθeq = 15 K. This expression differs from the usual quasigeostrophic potential vorticity in that the full, not the geostrophic, relative vorticity is used, and the Coriolis parameter is the actual value at each latitude, rather than obtained from a beta plane approximation. There is, in fact, no truly conserved potential vorticity for this model, and the results shown here are insensitive to precisely how the potential vorticity is computed.
The eddy fluxes of potential vorticity depicted in Fig. 1b are given by
i1520-0469-63-8-2109-e9
These are, as expected, poleward on the lower level and equatorward aloft. The meridional distribution of the poleward flux is smooth with a single maximum, just poleward of the center of the jet. The equatorward flux aloft spreads more widely over latitudes, which indicates that eddy activity generated baroclinically in the jet center propagates equatorward and poleward. The upper-level fluxes also feature a bump centered around 50°N. This region, with F″ < 0, is, as discussed above, necessary for the jet to be self-maintaining.
A direct, as opposed to diagnostic, approach for determining the full influence of eddies on the circulation is to compute time-averaged eddy fluxes of heat and momentum and then apply these as forcing to a zonally symmetric version of the model. In practice, it is simpler, and computational errors are avoided, if the eddy forcing is calculated as the time tendency that results when the zonally averaged model dynamics operate on the zonally and time-averaged model state. If the zonally averaged state of the model is denoted χ, then the zonally averaged dynamics of the model can be written symbolically as
i1520-0469-63-8-2109-e10
where NL is the nonlinear operator that describes all of the zonally symmetric model dynamics, including advection and dissipation, and Eddies is the eddy forcing of the zonal flow. The time average of (10) is approximated by
i1520-0469-63-8-2109-e11
where overbars indicate time averages. The appropriateness of approximating the time average of the nonlinear operator as the nonlinear operator acting on the time-averaged state of the model is confirmed by computing the eddy forcing using this approximation, and then applying this eddy forcing to a zonally symmetric version of the time-dependent model. The results of this calculation evolve to a steady state that reproduces the time-averaged state of the full model almost exactly.
The total effect of the eddies on the zonally averaged circulation can be determined by taking the difference between the results of the zonally symmetric model with and without eddy forcing applied; that is,
i1520-0469-63-8-2109-e12
This does not, however, produce good agreement with quasigeostrophic theory. In this model, with the chosen distribution of radiative equilibrium temperatures, the Hadley circulation is primarily eddy driven. In fact, the model without eddies has no Hadley circulation, no lower-level flow, and a zonal wind distribution aloft that is nearly in gradient thermal wind balance with the radiative equilibrium temperatures, with a slight modification due to horizontal diffusion. While the onset of an eddy-induced Hadley circulation is an effect of the eddies, it does not match the predictions of quasigeostrophic theory, which cannot capture the influence of the Hadley circulation. A more appropriate measure of the influence of eddies, for comparison with quasigeostrophic theory, is to calculate the response to a small enhancement of the eddy forcing. By so doing we compare the quantitative effects of eddies on circulations that are qualitatively similar, and obtain results relevant to how the circulation responds to perturbations, such as those associated with tropical forcing or the annular modes. The marginal response of the circulation to eddies, Δχm, is given by
i1520-0469-63-8-2109-e13
where Δχm, is scaled for comparison with the total influence, Δχt. Figure 2 shows the total and marginal influences of the eddies on the zonally averaged upper-level zonal winds and on the zonally averaged midlevel potential temperatures, from the same model run as before. The inclusion of eddies weakens the upper-level winds and the equator-pole temperature contrast. In middle latitudes, however, the eddies strengthen the westerlies aloft, and, across a narrow range of latitudes, the marginal influence of eddies is to strengthen the poleward decrease of temperature. This strengthening of the baroclinicity occurs at the same latitude as the bump in F (Fig. 1b), and so is in qualitative agreement with quasigeostrophic theory. This agreement is only qualitative, because quasigeostrophic theory neglects two effects that are important for wave driving that spans the hemisphere: the increase in Coriolis parameter with latitude, such that a given eddy driving yields greater latitudinal flow at lower latitudes, and the convergence of the meridians. Moreover, quasigeostrophic theory excludes thermal advection by the mean meridional circulation, which is included in the primitive equation model.

Assuming that the generation of baroclinic eddies increases with the baroclinicity of the zonal flow, the enhancement or preservation of baroclinicity by the eddies across some latitude is a positive feedback for eddy generation, or at least a local suppression of negative feedback that operates at other latitudes. As is seen in the next section this feedback makes the baroclinic eddies much stronger in a self-maintaining jet than in one with very similar parameters that is not self-maintaining.

4. The transition from a non-self-maintaining to a self-maintaining jet

To understand better the dynamics governing self-maintaining jets, it is helpful to have a non-self-maintaining jet for comparison. In the present two-level model, self-maintaining jets are robust features of the circulation, occurring over wide ranges of the parameters that govern thermal forcing and dissipation. When, however, the radiative equilibrium equator–pole temperature contrast, ΔT in (7), is reduced below a critical value, the jet abruptly ceases to be self-maintaining. A gradual loss of self-maintenance is also found for large ΔT; we return to this in section 5. The low ΔT transition is explored in this section. First, the abruptness of the transition, in terms of the parameter, ΔT, is demonstrated. The dynamics of the jets close to, but on opposite sides of, this transition are examined. A time-dependent transition from a non-self-maintaining jet is described. Finally, an experiment in which a transition from a non-self-maintaining to a self-maintaining jet is induced by a barotropic tropical zonal force provides support for the role of critical latitudes in the dynamics of self-maintaining jets.

It is found that for values of ΔT at or below 24 K, and with other parameters as before, the jets in the model are not self-maintaining. Figure 3a shows the time-averaged upper- and lower-level zonal winds above (ΔT = 24.6 K) and below (ΔT = 24.0 K) this threshold. Below the threshold, the upper-level jet is subtropical, and the lower-level winds are very weak, while above the threshold, the circulation is qualitatively similar to that shown in Fig. 1, albeit weaker. The eddy driving for these two model states is shown in Fig. 3b. Below and above the threshold, the latitudes of peak baroclinic generation of eddy activity (B) are similar, but this generation is more than 5 times stronger above the threshold. There is a similar difference in the magnitudes of the dissipation of eddy activity aloft (−F), but the distributions differ markedly. Below the threshold there are two peaks in the dissipation of eddy activity: one at the same latitude as its generation and one in the Tropics. From (6), it is expected that maximum of eddy activity dissipation aloft at the latitudes of its greatest generation acts to decrease the baroclinicity at these latitudes. This is a negative feedback on the baroclinic eddies. Above the threshold, there is sharp maximum in the dissipation of eddy activity in the subtropics, a local minimum equatorward of its latitude of greatest generation, and a weak maximum just poleward. Again, from (6), this distribution is consistent with a self-maintaining jet.

This transition is abrupt, with respect to changes in the controlling parameter, ΔT, as is shown in Fig. 4. The maximum values of the lower-level zonal wind and of B display a clear jump as ΔT is increased from 24.4 to 24.5 K. No other jump is found for values of ΔT at which the model supports eddies. There is another jump (not shown) at the threshold for baroclinic instability, ΔT ∼ 20 K, below which the model does not support eddies.

The model flows in these two cases are, in comparison with the Earthlike case depicted in Fig. 1, uncomplicated. This is indicated qualitatively by Fig. 5, which shows snapshots of the upper-level potential vorticity from the Earthlike model run (ΔT = 60 K) and the runs below and above the threshold (ΔT = 24.0, 24.6 K). In the Earthlike case, the dominant zonal wavenumber is around seven, while in the latter two cases it is exactly seven. The flow in the Earthlike case is irregular, while in the latter two cases it is extremely regular. The zonal flows do not change with time, and Hövmöller diagrams (not shown) indicate that the eddies propagate regularly with constant phase speeds and amplitudes. This phase speed is approximately twice as great for the case above the threshold as for that below it.

This regularity makes it possible to locate unambiguously the critical latitudes for the eddies. These are depicted in Fig. 6, which shows the midlevel (500 hPa) zonally averaged zonal winds and the zonal phase speeds of the wavenumber-7 eddies as functions of latitude. For both cases, the equatorward maximum in the absorption of eddy activity aloft, the minima in F visible in Fig. 4b, coincide with the critical latitudes, which are indicated in Fig. 6 by intersections between the phase-speed and zonal-wind curves. The poleward shift in the critical latitude as the threshold is crossed results from the greater phase speed of the eddies on the self-maintaining jet, as well as the modification of the zonal-wind profile near the critical latitude.

The much broader minima in F in Fig. 1b presumably come about because there is a broad distribution of eddy phase speeds. Also, this Earthlike case supports annular variability, which is manifested as variability, over time, of the latitudes of the jet and its associated eddy structures vary over time, smearing out time averages (cf. Fyfe and Lorenz 2005). A second minimum in F, in all three cases (Figs. 1b and 4b), occurs in a region of strongly negative meridional shear. These two regions of eddy absorption, for the self-maintaining jet, straddle the latitude of strongest eddy generation, giving rise to negative meridional curvature in F at that latitude, and thereby enhancing the baroclinicity.

Insight can be gained into the processes that maintain a circulation by examining its time-dependent establishment. Here we induce a time-dependent transition from a non-self-maintaining to a self-maintaining jet by increasing ΔT over time. Specifically,
i1520-0469-63-8-2109-e14
A transition to a self-maintaining jet takes place, but, within the length of this integration (5000 days), only in one model hemisphere: the notational Southern Hemisphere. This transition, which occurs between day 3000 and 3800 is shown in Fig. 7, in a set of time–latitude plots of zonally averaged quantities. As ΔT increases, the upper-level zonal wind strengthens (Fig. 7a), but the jet remains at the same latitude until around day 3150, when it begins to shift poleward. The poleward shift of the jet aloft is accompanied by the strengthening, in a series of pulses, of the zonal winds on the lower level (Fig. 7b) in middle latitudes, nearly simultaneous with easterly pulses in the Tropics. These pulses coincide with pulses of generation of eddy activity, or B (Fig. 7c). Figure 7d shows the absorption of eddy activity aloft, −F. Initially pulses in B coincide with pulses in −F, each with two maxima, one at the latitude of eddy generation and one in the Tropics. This is essentially a pulsing amplification of the distribution of F for ΔT = 24 K, shown in Fig. 3b. These transient tropical enhancements in −F decelerate the zonal winds in the Tropics, causing a poleward advance of the critical latitude. Starting around day 3400, this process becomes self-reinforcing, and the region of strong −F and accompanying deceleration of the upper-level zonal winds shifts continuously to its new equilibrium location in the subtropics. At the same time, Hövmöller diagrams (not shown) indicate that the changes in the zonal wind, especially the acceleration of midlatitude westerlies on the lower level, increase the phase speed of the baroclinic waves. This process, of eddies strengthening and inducing a poleward migration of the critical latitude, resembles the poleward migration of the critical latitude during some stratospheric sudden warmings (cf. Andrews et al. 1987), with the important distinction that, in this case, the strength and phase speed of the eddies is determined entirely by internal dynamics rather than by boundary conditions. Once the subtropical critical latitude captures most of the eddy activity generated in middle latitudes, there is a minimum in the absorption of eddy activity in the jet center, where the meridional shear is weak, and a weaker maximum in eddy absorption appears poleward of the jet.
The absorption of eddy activity is concentrated at the critical latitude, and the transition to a self-maintaining jet involves a poleward shift in this band of eddy absorption; this suggests that an artificially induced shift in the critical latitude could induce a transition to a self-maintaining jet. Additional experiments are performed in a westward zonal force is applied to the barotropic zonal wind in the Tropics. This is accomplished with equal and opposite zonally symmetric forcings applied to the barotropic vorticity in the subtropics of both hemispheres,
i1520-0469-63-8-2109-e15
where Z is the amplitude of the vorticity forcing, ϕ0 is the central latitude in both hemispheres at which is it applied, and Δϕ is its meridional width. A number of different cases were tried, many of which yielded similar results. Results are shown in Fig. 8 for a case in which ΔT = 24 K, Z = 5 × 10−13 s−2, ϕ0 = 20°, and Δϕ = 10°. The dashed curve in Fig. 8a shows the upper-level zonal wind for ΔT = 24 K (identical to the upper dashed curve in Fig. 3a). The thin solid curve shows the response to the forcing defined in (15), in the absence of any eddy feedback. These values are obtained by applying the vorticity forcing, (15), to the zonally symmetric version of the model with fixed eddy forcing calculated from the model climatology for ΔT = 24 K. The thick solid curve shows the upper-level zonal winds in the full model with the vorticity forcing applied. Applying a westward body force in the Tropics causes, in middle latitudes, a poleward shift of the jet, similar to that produced by increasing the value of ΔT across the threshold for a self-maintaining jet. The eddy forcing of the zonal flow, when the forcing defined in (15) is applied, is shown in Fig. 8b. These curves are very similar to those for ΔT = 24.6 K (i.e., a case above the threshold) shown in Fig. 3b and confirm that the application of a barotropic body force in the Tropics induces a transition to a self-maintaining jet.

5. Summary and discussion

The dynamics of self-maintaining jets are explored by examining the transition between jets that are and are not self-maintaining. In comparison with a non-self-maintaining jet, the self-maintaining jet features the export of eddy activity to the jet wings, especially on the equatorward flank, much stronger eddies, and stronger baroclinic and barotropic westerlies at the latitudes of the most robust eddy generation.

The transition to a self-maintaining jet, induced by a small increase in the strength of the radiative equilibrium equator-to-pole temperature contrast is abrupt. Such abruptness suggests that positive feedbacks operate to produce the transition. These positive feedback loops are shown schematically in Fig. 9. There are two primary feedback loops: one involves the strength of the eddies and thus the eddy driving of the zonal flow, while the second involves the meridional distribution of the eddy driving. If eddy activity is exported meridionally from its source latitude, then stronger eddies produce stronger eddy driving of the zonal flow, which makes the zonal flow more baroclinic in the jet center, which makes the eddies stronger. For stronger eddies to have this effect, however, their eddy activity must be deposited aloft in the jet wings. This occurs when the eddy critical latitudes are appropriately located, which, in turn, depends on the zonal flow profile, which sets the phase speeds for the eddies as well as determining where critical latitudes lie for eddies of a given phase speed. The highly regular eddies have well-defined critical latitudes. There is, however, observational support for similarly wavelike behavior in the dynamics of baroclinic eddies in the real atmosphere (Randel and Held 1991). Another possible positive feedback for the eddy strength is the reduction in barotropic shear in the self-maintaining jet, across the region of strongest baroclinic eddy generation (the maxima in B in Fig. 3b), since barotropic shear reduces the growth of baroclinic eddies (James 1987).

The results presented here differ in some important ways from previous work on the jets that emerge in the two-level quasigeostrophic model. In most cases presented by Panetta and Held (1988), Panetta (1993), and Pavan and Held (1996), westerly jets are associated with maxima in the poleward flux of potential vorticity in the lower level (B in present notation), while variations in the equatorward upper-level flux (F) are either weak or absent. Thus, according to Eqs. (5) and (6), the resulting jets are barotropic, and the baroclinicity in the jet core is that of the background. An exception is Panetta’s (1993) run 3, which combines a relatively high value for β, and thus relatively weak baroclinic instability, with strong mechanical and thermal dissipation. This run is among the least energetic of those he described. The westerly jets are, as expected, baroclinic, but do not display clear maxima in B, suggesting that even in this case, the jets are not self-maintaining in the sense of the present paper. Pavan and Held (1996) considered cases with a background baroclinic zone of finite width. They found that for a higher value of β, and thus weaker baroclinic instability, the equilibrated upper-level zonal winds are stronger than in radiative equilibrium, while in a more baroclinically unstable case, they are weaker. For wide baroclinic zones multiple baroclinic jets emerge for weaker, but not for stronger, baroclinicity.

These results suggest that the self-maintenance of jets may disappear as ΔT is increased and the background state becomes more unstable. Such experiments are carried out, running the model with values of ΔT up to 180 K, and it is found that self-maintaining jets survive over a wide range of values of ΔT, but disappear for a sufficiently large value. Figure 10 shows results from three experiments, with ΔT = 40, 80, and 120 K. Figure 10a shows the marginal influence of eddies on the midlevel zonally and time averaged potential temperature (as in Fig. 2b), Fig. 10b shows the distribution of F, and Fig. 10c shows the latitudinal gradient of the concentration of passive tracer on the upper level. The tracer is initialized with an initial concentration equal to the sine of latitude, and is relaxed back to its initial distribution with a 20-day time scale. For ΔT = 40 K (smaller than the nominally Earthlike value of 60 K used to obtain Figs. 1 and 2) the eddies enhance the baroclinicity of the jet. When ΔT is increased to 80 K, the eddies reduce the overall baroclinicity across middle latitudes, but less so in the jet, while for ΔT = 120 K, there is a reduction of baroclinicity at all latitudes, and no shelf is evident. Consistent with these results, the bump in F disappears for the largest value of ΔT. Similarly, the tracer distribution (Fig. 10c) shows enhanced tracer gradients, consistent with mixing barriers, in the jet center for the smaller values of ΔT, but not for its largest values. When ΔT = 120 K the tracer gradient is increased, relative to its background distribution, only at the poleward boundary of the model’s Hadley cell.

These more strongly forced experiments should be interpreted with caution, given the coarse resolution of the model, which is inadequate for highly turbulent flows. The overall picture, however, is that the wave–mean flow mechanisms that operate in self-maintaining jets eventually fail as the flow becomes more turbulent. Visual examination of the streamfunction and potential vorticity for ΔT = 120 K reveals that the eddy components are no longer dominated by eastward propagating baroclinic waves and that larger-scale quasi-stationary disturbances are prominent. That the self-maintenance of jets in the present model, on the sphere, appears to be more robust than for the quasigeostrophic model on the beta plane could result, in part, from the influence of spherical geometry, which, even in the absence of latitudinal shear, causes eddy activity to preferentially propagate toward lower latitudes and then dissipate away from its latitude of generation. A meaningful comparison with the quasigeostrophic results will require spherical model experiments at resolutions comparable to those used by Panetta (1993), a spectral truncation of ∼R40 or greater. Another question of interest is whether these highly turbulent states, which do not support self-maintaining jets, occur in Earth’s atmosphere. Recently Schneider (2004) argued that the circulation, through the influence of eddies on the height of the tropopause, organizes itself into a state with weak nonlinear interactions, and perhaps, therefore, with selfmaintaining jet. Exploring this issue will require the analysis of model experiments carried out with sufficient resolution in both the horizontal and the vertical.

The work most relevant to that presented here is a recent paper by Son and Lee (2005). In a multilevel model, they found an abrupt transition between an intermediate jet and a subtropical single jet as the strength of imposed equatorial heating is increased. The zonal flow of their intermediate jet state comprises a deep eddy-driven jet at ∼50°N with a more baroclinic subtropical shoulder. While, in their case, it appears that both the subtropical jet and the extratropical portion of the intermediate jet support strong eddy activity, there are similarities with the present results. Their enhanced tropical heating drives a stronger Hadley circulation, which in turn strengthens upper-tropospheric subtropical westerlies. As a result, the critical latitudes for baroclinic eddies and the resulting absorption of eddy activity shifts into the Tropics, causing the jet to shift equatorward as well. The westerly acceleration in the subtropics produced by the stronger Hadley circulation in these experiments is analogous, with opposite sign, to the effects, described in section 4, of an imposed tropical westward force.

In the present two-level model, self-maintaining jets emerge when some critical threshold in the radiative equilibrium baroclinicity is exceeded. This threshold is distinct from that at which baroclinic eddies first appear. The implication is that weakly nonlinear theories based on small baroclinic supercriticality are unlikely to be applicable to self-maintaining jets. More generally, even in a very simple context, in which the eddies are entirely regular, there is as yet no theory that allows one to predict a priori from the externally imposed parameters the characteristics of the self-maintaining jet or the values of parameters at which it will emerge. While the structures of baroclinic normal modes are useful for predicting the subsequent evolutions of baroclinic lifecycles, including abrupt transitions (Lee and Kim 2003), it appears these modes are less useful for quantitative predictions of equilibrated flows (Son and Lee 2005). Thus, while the results presented here point to mechanisms that may be important for sustaining jets in the atmosphere, we remain far from a predictive theory of the atmospheric general circulation or even of its representation in very simple models.

Acknowledgments

The author thanks Prof. Peter Haynes and Dr. Isaac Held for insightful comments on an earlier version of this paper, and Dr. Held for his suggestion that jets cease to be self-maintaining for strong thermal forcing. This work was supported by the Climate and Large-scale Atmospheric Dynamics Program of the National Science Foundation under Grant ATM-0237304.

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Fig. 1.
Fig. 1.

Results from an Earthlike run of the two-level model: (a) zonally averaged zonal winds and midlevel potential temperatures, (b) eddy fluxes of potential vorticity on the upper (F) and lower (B) levels, and (c) upper-level (250 hPa) eddy momentum flux and midlevel (500 hPa) eddy heat flux.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 2.
Fig. 2.

Total and marginal effects of eddies on the circulation of the two-level model (see text): (a) effects on upper-level zonal winds and (b) effects on midlevel temperatures.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 3.
Fig. 3.

(a) Zonally averaged zonal winds and (b) eddy fluxes of potential vorticity for model runs with ΔT = 24 K (dashed curves) and ΔT = 24.6 K (solid curves).

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 4.
Fig. 4.

Maximum lower-level zonal wind (dashed and open circles) and maximum lower-level poleward eddy flux of potential vorticity (solid and filled squares) as functions of ΔT

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 5.
Fig. 5.

Snapshots of upper-level potential vorticity for runs with (a) ΔT = 60 K, (b) ΔT = 24 K, and (c) ΔT = 24.6 K. The contour interval is 3 × 10−5 s−1, and the outer latitude for the polar stereographic projections is 20°N.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 6.
Fig. 6.

Midlevel zonal winds (thick curves) and wavenumber-7 phase speeds (thin curves) for model runs with ΔT = 24 K (dashed) and ΔT = 24.6 K (solid).

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 7.
Fig. 7.

Latitude–time plots for the Southern Hemisphere of a model experiment in which ΔT is increased from 24 to 24.6 K over time: (a) upper-level zonal winds, where the contour interval is 1 m s−1; (b) lower-level zonal winds, where the contour interval is 0.2 m s−1; (c) B (lower-level eddy flux of potential vorticity), where the contour interval is 5 × 10−7 m s−2; and (d) −F (minus the upper-level eddy flux of potential vorticity), where the contour interval is 5 × 10−7 m s−2.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 8.
Fig. 8.

Results from a model experiment in which a westward barotropic body force is applied to the model Tropics, with ΔT = 24 K: (a) upper-level zonal winds for the forced experiment (thick solid curve), the forced experiment with no eddy feedback (thin solid curve), and the unforced experiment (dashed curve); (b) eddy fluxes of potential vorticity for the forced experiment.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 9.
Fig. 9.

Feedbacks that govern the transition to a self-maintaining jet.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

Fig. 10.
Fig. 10.

Model results for ΔT = 40, 80, and 120 K: (a) marginal influence of the eddies on midlevel potential temperature, (b) eddy fluxes of potential vorticity on the upper level (F), and (c) latitudinal derivative of an upper-level tracer, relaxed to a sin(ϕ) profile with a relaxation time of 20 days.

Citation: Journal of the Atmospheric Sciences 63, 8; 10.1175/JAS3732.1

1

The eddy-modulated responses to tropical warming are reviewed in Robinson (2006).

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