An idealized atmospheric general circulation model is used to investigate the factors controlling the time scale of intraseasonal (10–100 day) variability of the extratropical atmosphere. Persistence on these time scales is found in patterns of variability that characterize meridional vacillations of the extratropical jet. Depending on the degree of asymmetry in the model forcing, patterns take on similar properties to the zonal index, annular modes, and North Atlantic Oscillation. It is found that the time scale of jet meandering is distinct from the obvious internal model time scales, suggesting that interaction between synoptic eddies and the large-scale flow establish a separate, intraseasonal time scale. A mechanism is presented by which eddy heat and momentum transport couple to retard motion of the jet, slowing its meridional variation and thereby extending the persistence of zonal index and annular mode anomalies. The feedback is strong and quite sensitive to model parameters when the model forcing is zonally uniform. However, the time scale of jet variation drops and nearly all sensitivity to parameters is lost when zonal asymmetries, in the form of topography and thermal perturbations that approximate land–sea contrast, are introduced. A diagnostic on the zonal structure of the zonal index provides intuition on the physical nature of the index and annular modes and hints at why zonal asymmetries limit the eddy–mean flow interactions.
In this paper we investigate processes that determine the persistence of the zonal index and North Atlantic Oscillation (NAO), focusing on intraseasonal time scales. By intraseasonal, we mean time scales of 10–100 days, longer than those associated with synoptic variability, but not so long as to allow for significant changes in the boundary conditions or forcing, such as sea surface temperature. On these shorter periods, then, the focus is on internal processes in the atmosphere that extend the persistence of these large scale patterns beyond synoptic time scales. Variability in this range may account for all, or at least a large fraction, of the variability of the NAO and zonal index on interannual time scales (e.g., Feldstein 2000a, b).
The zonal index as a measure of the strength of the zonally averaged circulation dates back at least to Rossby (1939), though it was not until somewhat later that it was understood that the index more accurately describes latitudinal variations of the jet (e.g., Namias 1950). It has become conventional to define the index as the principal component time series associated with the first empirical orthogonal function (EOF) of the zonally averaged zonal wind (e.g., Feldstein 2000a; Lorenz and Hartmann 2001). It is closely related to the so-called annular mode index, associated with the first EOF of sea level pressure (Thompson et al. 2000), and the NAO index, which is typically determined from rotated EOF analysis of pressure or geopotential surfaces (e.g., Barnston and Livezey 1987). To leading order, the zonal and annular mode indices characterize meridional shifts of the extratropical jet, with the implication of hemispheric scale variation, while the NAO suggests a more limited zonal scale of jet variation, with focus restricted to the North Atlantic region. Variability in this sector dominates the zonal average, however, so the NAO index highly correlated with the more zonal measures. All these indices essentially describe slow meanderings of the extratropical jet, and it is the time scale of these variations that is the subject of this study. We investigate the dynamics controlling variability of the zonally average jet and the influence of large-scale zonal asymmetries on the zonally averaged flow.
The zonal index and NAO exhibit considerable energy on intraseasonal time scales. Anomalies of the jet persist longer than typical baroclinic eddies; during the Northern Hemisphere winter, the e-folding time scales of the zonal index, annular mode, and NAO are 18, 10.6, and 9.5 days, respectively (Feldstein 2000a, b). They show no pronounced spectral peak at any frequency, however, exhibiting a relatively white spectrum at low frequencies. The stochastically stirred barotropic model of Vallis et al. (2004) provides perhaps the simplest explanation for both the persistence and flat spectrum of the patterns. There it is shown that annular mode and NAO-like patterns may simply be driven by random variations in the eddy vorticity fluxes. As the integral of eddy fluxes, the zonal flow exhibits greater power on lower frequencies than the eddies. If one assumes that the eddy fluxes are white in time (at least on time scales longer than 10 days) and that the damping of the large-scale anomalies is linear, the NAO and annular modes have a red spectrum as characterized by an autoregressive process of first order (AR-1) or, as it is also known, an Ornstein–Uhlenbeck process.
There is evidence, however, that the zonal index persists longer than would be expected from this simple reddening argument due to interactions between the eddies and the large-scale flow. Lorenz and Hartmann (2001) developed a technique to estimate the linear damping time scale of the zonal index absent any interaction between the eddies and large-scale flow. Their analyses suggest that the observed e-folding time scale of the zonal index in the Southern Hemisphere (∼13 days) is a little longer than the natural damping time scale of anomalies (8.9 days), providing some evidence of a feedback between the eddies and the large-scale flow. They propose a feedback based on changes in the index of refraction associated with shifts in the extratropical jet, which allow the jet to shape the higher frequency eddies. Independently, Kidson and Watterson (1999) and Watterson (2002) consider the behavior of the zonal index in the Southern Hemisphere of an atmospheric GCM coupled to a mixed layer ocean. While they found that a simple AR-1 model could replicate the basic behavior of the zonal index, a positive feedback between the high-frequency eddies and the index was necessary to explain the correlation of the index at long lags.
Mechanisms for enhanced persistence have been discussed previously in other forms by several authors, (e.g., Branstator 1992, 1995; Feldstein and Lee 1998; Robinson 1994, 1996, 2006). It is reasonably well established that the zonal index and NAO are forced by eddy momentum fluxes. The potential for low frequency anomalies, which are largely barotropic, to in turn influence the baroclinic eddies, thereby completing a feedback loop, has been harder to establish. Branstator (1995) used a linear model to predict the synoptic response to the intraseasonal anomalies. The resulting change in the storm track was such as to reinforce the anomaly. Robinson (2006) argues that the eddies increase the baroclinicity in the jet core relative to the flanks, so helping to maintain the existence of the jet. This self-maintenance mechanism reinforces the jet at the latitude of peak winds, and so could extend persistence of jet anomalies as well.
In this paper, we use a dry primitive equation model of the atmosphere to focus on the interactions between eddies and the large-scale flow. It has been demonstrated that atmosphere-only models can well capture the essential dynamics of both the NAO and annular modes (Limpasuvan and Hartmann 2000; Cash et al. 2002). Simplifications to the forcing of the GCM provide a realistic climatology governed by a few well-defined parameters, so that the internal damping time scales are known. The simplifications also allow us to explore the variability around a wide range of climatologies. We demonstrate that the time scale on which the jet varies is distinct from any of the prescribed time scales in the model, providing evidence of eddy–mean flow feedback. The feedback, however, is extremely sensitive to zonal asymmetries in the large-scale forcing. The destruction of the feedback appears to be related to zonal asymmetry in the eddy forcing, and is currently being investigated by the authors in a continuation of this study.
Our experiments and analysis procedures are discussed in section 2. We then survey the model over parameter space in section 3. We find that a number of aspects of the model’s behavior respond similarly to changes in three parameters—the equilibrium equator-to-pole temperature gradient and the temperature and momentum damping coefficients. Absent zonal asymmetry in the forcing, the time scale of the model’s annular mode is quite sensitive to the parameters. This sensitivity, however, is largely eliminated in the presence of zonally asymmetric forcing. Section 4 presents a diagnostic of the model’s zonal index to establish a more intuitive understanding of the physical pattern. A mechanism for a feedback between eddies and the mean flow is described in section 5. Section 6 summarizes our results and conclusions.
2. Experimental design
We use a spectral, dry primitive equation model with 20 evenly spaced σ = p/ps vertical levels, where ps is the surface pressure. The model forcing is idealized, as described in Held and Suarez (1994, hereafter HS), with a steady Newtonian relaxation to a prescribed temperature field. Thus, the thermodynamic equation is
where κ = R/cp, R is the ideal gas constant, and cp the specific heat of dry air. In the standard HS scheme, the damping, ka, is almost everywhere set to 40−1 day−1, the exception being in the low-level Tropics, where it is increased to 4−1 day−1 at the surface to establish a more realistic Hadley circulation. The equilibrium profile Teq is determined by a small number of parameters, the most significant for our purposes being the equator-to-pole temperature difference, ΔTeq. Momentum is removed from the model near the surface by a Rayleigh drag of strength kf , which decays linearly in σ to the 0.7 level, where it vanishes. A ∇8 damping is included to remove enstrophy at small scales, the coefficient set so that damping time on the smallest resolved scale is one-tenth of a day. For reference, the key parameters and their default values in the control run are listed in Table 1.
We report on eight sets of experiments, as outlined in Table 2. With the exception of set II, all simulations were completed with triangular truncation at total wavenumber 42, or T42. In exploring the model behavior, we recognized a troubling sensitivity to vertical resolution. At T42 horizontal resolution, the model’s zonal index becomes very persistent when the spacing of vertical levels is made particularly fine near the upper boundary of the atmosphere, as discussed in appendix A. Given this deficiency, we focus on trends in the time scales of variability as opposed to the absolute numbers themselves. Care was taken to verify key trends at resolution T63, where the behavior is less sensitive to vertical resolution.
Experiment sets I–IV explore the sensitivity of the variability to model parameters. The forcing is the same in both hemispheres and independent of longitude, as in the standard HS scheme. Since the two hemispheres were statistically independent, we sampled from both to double our dataset. In experiment III, where ka is varied, the strong damping in the low-level Tropics is not changed to minimize change to the tropical circulation. The sensitivity of the variability to zonally asymmetric forcing is determined in experiments V–VIII. Asymmetry is introduced through large scale perturbations to the equilibrium temperature profile that approximate land–sea contrast and idealized topography of comparable scale to the Rocky Mountains. A single zonally localized storm track is established, providing a rough approximation to the wintertime Northern Hemisphere, and producing NAO-like variability. The details of the perturbations are described in the appendix B, and their influence on the spatial structure of synoptic and intraseasonal variability is the focus of current research. Perturbations were placed only in the model’s Northern Hemisphere.
As is conventional, we define the zonal index as the principal component associated with the first EOF of the daily zonally averaged zonal winds on the σ = 0.975 level, the closest level to the surface. The bias in EOF computations that can result from the decrease in the earth’s radius with latitude was avoided by weighting fields with a factor of cos1/2θ (North et al. 1982). The annular mode is defined as the first EOF of the 10-day-averaged sea-level pressure (SLP), and the annular mode index is the time series of this pattern, found by projecting the daily SLP field onto the EOF. This first EOF (not shown) becomes NAO-like in structure in the zonally asymmetric runs. The correlation coefficients between the zonal index and annular mode index are greater than 0.9, even in simulations with significant zonal asymmetry. Thus the model’s zonal index, annular mode, and NAO (in asymmetric runs) describe much the same pattern of variability. The time scale of the zonal index/annular mode was quantified by the best exponential fit to the autocorrelation function of the index for all lags until the autocorrelation dropped below 0.3. The autocorrelation functions were not purely exponential, but tended to exhibit steeper decay on short lags with a broad shoulder at longer lags. As they are roughly self-similar, however, the e-folding time scale provides a good comparative measure.
The mean circulation and variability respond in a quite similar fashion to changes in the equilibrium temperature gradient ΔTeq, thermal damping rate, ka, or momentum damping rate kf . Figure 1 highlights the near equivalence of the climatological response. Figures 1a,b show the time and zonal mean zonal wind for two simulations with differing values of ΔTeq. With a doubling of the temperature gradient, the extratropical jet shifts poleward by approximately 9° and the time average surface winds double. As seen in Fig. 1c, the difference in the mean winds between these two simulations, the increase in ΔTeq leads to a mild strengthening of the subtropical jet near 20°, a more substantial increase in the extratropical jet near 50°, and a slight weakening of the winds in between. Figures 1d,e illustrate the difference between simulations when the values of ka and kf are varied, respectively. The response to all three parameters is similar both in structure and amplitude: the same fractional change in ΔTeq, ka, or kf leads to similar amplitude response. The sign of the response, however, has reversed for the surface friction; the poleward shift of the jet and increase in the surface westerlies observed with an increase in the thermal forcing occurs with a decrease in kf .
Perhaps most striking is the fact that the parameters have a similar effect on the time scale of intraseasonal variability. Figure 2 summarizes the results of the 14 simulations in experiment sets I–IV, showing the e-folding time scale of the zonal index as a function of the three parameters. In Fig. 2a, we focus on the similarity in the sensitivity to ΔTeq and ka. The solid and dot–dashed black curves show the time scale as a function of ΔTeq for two different resolutions, and correspond with the units on the lower x axis. The sensitivity to variations in ka is shown by the dashed gray curve, with values of the coefficient specified on the upper x axis. Increasing ΔTeq or ka leads to a drop in the time scale of variability; ΔTeq was varied over a wider range from the control value, 60 K, in experiment I. The sensitivity appears to saturate for high values above 80 K, and break down when the forcing becomes too weak, at 30 K. At T63 resolution, the model is significantly less sensitive to ΔTeq, but the sensitivity is still of the same sign. Figure 2b compares the sensitivity to ΔTeq with that to kf; the sensitivity to kf is similar, but again of opposite sign. In summary, the time scale of the zonal index is reduced by 1) an increase of the equilibrium temperature gradient, 2) a strengthening of the thermal damping, or 3) a weakening of the momentum damping.
Given the similar sensitivity of the model to each of the three parameters, we use only one, the equilibrium temperature gradient, to probe the sensitivity of intraseasonal variability to zonally asymmetric forcing. Here, ΔTeq is perhaps the most physical parameter, and can be linked to changes in forcing during the seasonal cycle, or changes to the climate, for example, a reduction of the ice–albedo effect due to global warming may reduce the equator-to-pole temperature contrast. Figure 3 summarizes the results from experiment sets I and V–VIII. For each experiment we compare the e-folding time scale of variability in three simulations in which the equilibrium temperature gradient is varied from 40 to 60 to 80 K. The black curve on top illustrates the dramatic sensitivity of the intraseasonal variability to ΔTeq absent any zonal asymmetries, as seen from Fig. 2. With increasing zonally asymmetric forcing, however, this sensitivity is weakened, and finally eliminated, as observed by the flattening of the sensitivity profiles.
When the equilibrium temperature gradient is varied from 40 to 80 K in the zonally symmetric model (experiment set I), the e-folding time scale of the annular mode decreases over fourfold, from 93 to 21 days. When the gradient is varied to the same extent in simulation set VIII, where we have introduced both topography and a diabatic heating anomaly to model land–sea contrast, the e-folding time scales are substantially shorter and nearly independent of the equilibrium temperature gradient, varying from 10–13 days. This is consistent with the time scale observed of the northern annular mode. The time scale of the southern annular mode, as analyzed in a more sophisticated GCM by Watterson (2007), exhibits a 50% variation within the seasonal cycle. This suggests that the sensitivity of the e-folding time scale to ΔTeq in experiments VI and VII is not unreasonable. These are perhaps the most relevant simulations to the Southern Hemisphere, which does contain substantial large scale asymmetry, such as the Andes Mountains.
To compare time scales of intraseasonal and synoptic variability, we compute an eddy turnover time scale L/K1/2, where K is the eddy kinetic energy, averaged over the depth of the troposphere ±15° from the jet core. Also, L is an eddy length scale determined from the zonal Fourier spectrum of the meridional velocity, υ, at the latitude and height of the peak eddy kinetic energy, θKmax and pKmax:
where r0 is the radius of the earth and the average wavenumber kave was computed by
where υ̃k(t) is the kth Fourier coefficient of υ(t, ϕ, θKmax, pKmax), and brackets refer to the time mean. The eddy turnover time scale varies only slightly with parameters. In simulation sets I and II, K increases linearly with the equilibrium temperature gradient, so that the eddy turnover time scale varies with ΔT −1/2eq, as shown in Fig. 4. The eddy kinetic energy did not change as much in experiments III and IV. There was a slight decrease in K with a decrease in the thermal or momentum damping, ka or kf , leading to rise in the eddy turnover time in both cases. The time scale of the zonal index increases with decreasing ka, but decreases when kf is reduced. Hence there is not a consistent relationship between the time scale of individual eddies and the jet as a whole. In addition, the relationship between the parameters and eddy statistics is not affected when large-scale asymmetries are introduced despite the dramatic changes in the jet variability.
4. The zonal structure of the zonal index and annular mode
What sets the time scale at which the extratropical jet varies in our model? In the zonally symmetric simulations, the parameters ΔTeq, ka, kf can be viewed as knobs that allow one to vary time scales in the model. Parameters ka and kf set damping time scales, while ΔTeq largely controls the eddy turnover time scale. The similar sensitivity of the time scale of the zonal index to all three knobs thus allows one to change it independently of any two of the three internal time scales. This suggests that the zonal index varies on a distinct time scale and is not simply determined by any one of the model’s internal time scales. When zonal asymmetries are included, the time scale of intraseasonal variability decreases, and becomes largely independent of one of the knobs, the equilibrium temperature gradient. The zonally averaged eddy kinetic energy and other eddy statistics, however, remain as sensitive to the gradient as they were in the zonally symmetric simulations. The chief difference is in the zonal structure of the eddy statistics, since eddies have become organized into a zonally localized storm track.
To seek out common features in simulations with the same degree of persistence, we develop a diagnostic of the annular mode as a function of both time and longitude [motivated in part by Feldstein (2000a), who considered the contribution of the zonal wind at each longitude to the zonal index to assess the coherence of the flow]. We compute the first EOF, Z(θ), of the daily zonally averaged surface zonal wind, us, and the associated principal component, ZI(t), which is the model’s zonal index. The EOF patterns are shown in relation to the mean surface winds for two different simulations in Fig. 5. The EOFs shift with the jet, remaining out of phase with the mean surface winds and so always characterizing a shift in the latitude of the jet. We then project the flow at all times and longitudes onto Z(θ), so generating a local dipole index that is a function of both time and longitude; that is, we compute
where * refers to deviations from the time mean and the cosine accounts for the latitude bias. The normalization constant α is chosen so that the zonal average of the dipole index has unit variance. Thus, ∫2π0 DI(t, λ) dλ = ZI(t), namely the original zonal index. In our model, the annular mode index captures the same variability as the zonal index; the EOFs associated with the two indices are in geostrophic balance and both characterize a meridional shift in the jet. The correlation between the two indices is above 0.9 in all simulations. Hence DI(t, λ) also allows us to see the zonal structure of the annular mode.
Figure 6 illustrates the evolution of the local dipole index for two simulations from experiment I with equilibrium temperature gradient ΔTeq = 40 and 80 K, respectively. The curves on the right of each map show the evolution of the zonal index. Both diagrams are dominated by short lived, eastward propagating features of limited zonal scale, perhaps wavenumber 5 or 6. Kushner and Lee (2007) found similar eastward propagating features. Even in this short time slice, one can see greater persistence of the zonally averaged signal in the simulation with ΔTeq = 40 K than in the simulation with ΔTeq = 80 K, but locally the slowly evolving signal is masked by the shorter-lived features. The amplitude of DI anomalies is larger in the ΔTeq = 80 K simulation. This is related to the scaling of the index, which is chosen so that the zonal average of the local index gives the traditional zonal index. Weaker coherence in the zonal flow will then lead to a higher variance of the dipole index.
We next compute the autocorrelation of the dipole index as a function of time lag and longitude. Given the statistical zonal symmetry of these simulations, the lag–longitude autocorrelation can be averaged over all longitudes to improve the statistics, producing one overall map. The maps in Figs. 7 and 9 are arbitrarily based from a reference longitude at 180°, but characterize the flow at all longitudes. Figure 7 focuses on the dominant feature of the lag–longitude correlation map, which captures the scale and slow eastward propagation of the local structures in Fig. 6. The features extend over a range of 50°–60° and persist for 2–4 days in both simulations. They propagate eastward at approximately 7° day−1 (8 m s−1 at the latitude of the jet maximum) in the simulation with ΔTeq = 40 K and 16° day−1 (14 m s−1) in the simulation with ΔTeq = 80 K. Such spatial and temporal scales suggest that these features are the projection of a single eddy, or single wave-breaking event, onto the meridional structure of the dipole index, as depicted in the diagram in Fig. 8. Indeed, composites of such features in Kushner and Lee (2007) had a baroclinic structure in the vertical, suggesting a composite of baroclinic eddies that have projected onto the zonal index.
Figure 9 illustrates the same lag–longitude autocorrelation maps, but the grayscale has been saturated at correlation 0.4 and the range widened. A wave train about the reference point provides further evidence that synoptic eddies are projecting onto the zonal index (or annular mode) pattern. The wave packet in the simulation with ΔTeq = 40 K is characterized by wavenumber 6 and propagates 23° day−1. When ΔTeq = 80 K, the packet is of wavenumber 5, and propagates 38° day−1. (The jet is located further poleward here, so the length scale has not increased significantly.) Based on the latitude of the jet maximum, the group velocities are 24 and 33 m s−1, respectively, consistent with the jet speeds in the two simulations, 27 and 37 m s−1.
The wave train in the simulation with ΔTeq = 40 K, however, is superimposed on a higher background correlation than with ΔTeq = 80 K. At lags beyond 10 days the correlation becomes independent of longitude, indicating zonally coherent behavior at longer time scales. We can quantify the significance of the hemispheric coherence at zero lag by estimating the strength of the uniform correlation beneath the projection of the individual eddies and wave trains on the local index. When ΔTeq = 80 K, the zonally uniform motion is at best 0.15 correlated with the local flow, while in the persistent simulation with ΔTeq = 40 K, this signal is more than doubled in strength. This zonally coherent signal represents a truly hemispheric pattern of variability, not necessarily a mode in the linear sense, but a slow vacillation of the entire eddy-driven jet with a time scale of its own.
This analysis suggests why it may be difficult to see an annular pattern in a synoptic map, as also highlighted by Fig. 7 of Cash et al. (2002). The zonal index is generally dominated by local features seen in the Hövmöller diagrams in Fig. 6. In the simulation with ΔTeq = 80 K, the zonally uniform signal is dominated by local events to the extent that it makes little difference to the zonal index. The annular mode in this case is largely a statistical feature, as suggested by the minimal model in Gerber and Vallis (2005). In the simulation with ΔTeq = 40 K, rather, there is a significant hemispheric signal that is masked, to some extent, by local events riding on top of the uniform movement of the jet. Synoptic variability hides the hemispheric signal unless it is filtered in space or time.
In simulations with zonally asymmetric forcing, analysis of the local dipole index suggests that the dynamics have changed qualitatively from the zonally symmetric case, consistent with the change in persistence. Consider a simulation from experiment VIII, where both land–sea contrast and topography have been included, with the standard value of ΔTeq = 60 K. Lag–longitude autocorrelation maps of the local dipole index are shown for two longitudes in Fig. 10. In Fig. 10a, the base longitude is in the peak baroclinic zone at the entrance to the storm track. Even with the grayscale saturated at correlation 0.45, the map is dominated by the local phase behavior, which indicates features propagating eastward 16° day−1. Excepting the region immediately downstream the base point, the correlation with other longitudes is below 0.05 for all lags. The base longitude chosen for Fig. 10b is located in the region of peak intraseasonal variability, as characterized by the first EOF of sea level pressure, but this pattern is characteristic of all latitudes outside the storm track. The map indicates slower phase propagation and more extended correlations in time and space. There is a distinct drop in the autocorrelation, however, with points in the baroclinic region between 80° and 120°. This barrier is observed for all base points chosen outside the storm track.
Similar structure is observed in other simulations with storm tracks. Zonal asymmetries, particularly topography, inhibit the uniform correlation observed in zonally symmetric simulations. Even when the large-scale forcing is favorable to strong persistence, with ΔTeq = 40 K, the storm track forms a barrier to zonally coherent motions. The addition of zonal asymmetries thus has the same effect as increasing the temperature gradient in the zonally uniform simulations. In all these simulations, the zonal index and annular mode do not reflect coherent movement of the jet. Rather, localized eddies, wave packets, and wave-breaking events are driving the zonally averaged flow about.
It is only in the more persistent simulations, with weak ΔTeq or ka, or with strong surface friction, kf , that the zonal index and annular mode describe a truly hemispheric variation. Individual eddies are less significant in determining shifts of the jet in these simulations. These uniform shifts of the extratropical jet may allow for, or rather result from a feedback between the eddies and the zonal mean flow.
5. Interaction between eddies and the large-scale flow
In this section we suggest a mechanism by which synoptic eddies interact with the mean flow to reinforce the extratropical jet at the latitude of maximum winds. This causes the zonal mean jet to become “sticky,” tending to meander slowly on a distinct time scale set by the strength of the eddy–mean flow interactions. We argue that synoptic eddies more efficiently reduce the shear at latitudes at which the upper-level flow is weak. Hence the eddies will increase the baroclinicity of the flow in the jet core relative to the baroclinicity on the flanks of the jet. While this property helps sharpen and reinforce the jet, as discussed in the context of a two-layer model by Robinson (2006), it also extends the persistence of jet anomalies.
The zonal index and annular mode anomalies are primarily barotropic, or at least equivalent barotropic, limiting their direct influence on the growth of baroclinic eddies themselves. [Barotropic shear can effect eddy growth, as James (1987) and Nakamura (1993) discussed. These effects will be discussed further below.] They indirectly affect the eddies, however, by changing the position of the critical latitudes of the flow, which influence wave breaking. Linear theory suggests that irreversible mixing—the damping of wave activity—occurs at critical layers where the phase speed of the wave, c, equals that of the mean flow u. Particle displacements, η, scale as ψ/(u − c), where ψ is the streamfunction perturbation, so that displacements will grow unbounded even for infinitesimal perturbations, leading to nonlinearity as the wave approaches the critical latitude (Randel and Held 1991).
As a thought experiment, consider a barotropic jet perturbation within a wider baroclinic region. The barotropic jet must be maintained against friction by an upgradient momentum flux. A divergence of wave activity generated by eddies growing within jet provides the momentum source (Edmon et al. 1980). Growth of eddies centered on the flanks of the jet yields a source of wave activity outside the jet core, and divergence of this flux will shift the latitude of the barotropic jet, thus changing the state of the zonal index or annular mode (Vallis et al. 2004). We contrast these two cases, eddy generation centered about the jet core versus on the flank, with a focus on their influence on the baroclinicity of the flow to determine how they feedback on future eddy generation.
Figures 11a–c schematically illustrate the influence of eddies generated in the jet core on the mean flow. Figure 11a shows the effect of heat fluxes generated by the growing eddy. Poleward heat fluxes are associated with the vertical propagation of wave activity (the Eliassen–Palm flux), which reduces the shear, drawing momentum from the upper to lower levels.1 Momentum is not extracted from the flow, however, until the wave activity dissipates. The zonal average flow u is strongest where the barotropic circulation augments the upper-level flow, allowing wave activity to propagate meridionally until it nears critical latitudes on the flanks of the jet. The resulting divergence of wave activity leads to an upgradient transfer of momentum at upper levels (Fig. 11b). The net effect of the eddy life cycle (Fig. 11c) is still a weakening of the upper flow in the jet core, but the weakening has been spread to the flanks of the jet as well. Thus the tendency of the eddy heat transport to reduce the baroclinicity has been partially offset by momentum transport in upper levels.
Now contrast this with an eddy that grows on either flank of the jet, as illustrated in Fig. 11d. In this case heat fluxes associated with eddy growth also lead to vertical propagation of Eliassen–Palm (EP) fluxes. But the weaker zonal velocity u at upper levels limits meridional propagation; weaker flow supports fewer waves, as only waves with phase speed less than u can exist. The waves are nearer their critical latitude, and damped more quickly. Thus the tendency of the heat fluxes to reduce the shear is not offset by a convergence of momentum. The initial eddy will begin to shift the barotropic circulation, but there is also a substantial change in the shear, making it less favorable for a second eddy to form at this latitude. This negative feedback limits the ability of the jet to move, thereby reinforcing the current position of the jet.
The barotropic jet associated with the zonal index and annular modes thus helps maintain the baroclinicity in the jet core, irrespective of the current latitude of the jet. We use maintain in a relative sense, however, as the eddies universally reduce the baroclinicity. The key is that the eddies are more efficient at reducing the baroclinicity if they form on the flanks of the jet. This suggests that the mechanism is perhaps best viewed as a negative feedback on movement of the jet. In terms of the persistence of the jet anomalies, however, a negative feedback on movement is the same as a positive feedback on staying.
Spherical geometry requires a small correction. Eddy growth is determined by the baroclinicity, which peak poleward of the region of maximum shear, and so poleward of the upper-level jet maximum. The offset arises primarily from the Coriolis parameter in the thermal wind equation,
There is thus a factor of sin−1θ shifting the maximum shear equatorward of the region of maximum temperature gradient. As eddy growth is preferred on the poleward flank of the maximum upper-level winds, equatorward propagation will be favored, and momentum is extracted from the subtropics preferentially over the polar region (Simmons and Hoskins 1978). This bias may lead to poleward propagation of zonal wind anomalies, as discussed by Feldstein (1998). If such propagation dominates, it may set the time scale of the zonal index.
We find evidence of the above mechanism in our model by analyzing the Eliassen–Palm fluxes (e.g., Edmon et al. 1980). The flux takes the form
where an overbar denotes the zonal average and primes deviations there from. Meridional divergence of the first term captures the direct momentum forcing by the eddies, while vertical divergence of the latter term characterizes the impact of eddy heat fluxes on the zonal momentum. We use the 2–10 bandpass-filtered EP fluxes to focus on the behavior of synoptic eddies. By averaging the net divergence of the flux over two halves of the atmosphere, above and below 500 hPa, we can simplify analysis of our GCM to that of a two-layer model and make a connection with the analysis of Robinson (2006).
We first consider the role of the eddies in the neutral state, the average over all times when |ZI(t)| < 1. The contribution of the eddy momentum and heat fluxes, divergence of the first and second terms in (5.2), to the zonal momentum forcing budget in the control simulation are shown in Fig. 12. Focus on the upper curves, which show the impact of the eddies on the flow above 500 hPa. The dot–dashed contours show acceleration of the mean flow due to eddy heat transport, negative (easterly) for all latitudes in the upper half of the atmosphere. The dashed curve, the impact of eddy momentum fluxes, indicates transport of momentum into the jet core. While this is not enough to offset the momentum lost to the lower layers, it does produce a relative minimum in the net torque, shown by the solid curves. This minimum is the self-maintaining, or self-sharpening property of the eddy fluxes described by Robinson (2006): the eddies, on average, tend to reduce the shear in the jet core slightly less than on the flanks. The key lies in the fact that easterly torque due to heat fluxes is broader than the westerly torque due to momentum fluxes. Sharpening of the baroclinicity is also dependent on surface friction to cancel the westerly torque in the lower layer. Otherwise the shear is reduced by an increase in the surface winds.
Composites associated with positive and negative zonal index events for the standard HS control run are shown in Fig. 13. The dashed curves on the top show the easterly acceleration of the zonal average zonal wind by 2–10 bandpass eddy fluxes in the upper half of the atmosphere. The minimum in the easterly acceleration on the upper-level flow remains coincident with the zonal mean jet, shown by the solid curve: eddies reinforce the jet anomalies in both cases. The dot–dashed curves show the westerly acceleration of the lower half of the model’s atmosphere. This is almost entirely due to heat fluxes; the poleward offset from the jet maximum is consistent with the discussion of (5.1). It can also be seen that the mechanism is best viewed as a negative feedback on movement of the jet. The eddies are everywhere reducing the baroclinicity; they are just more effective on the flanks of the jet than in the core.
6. Summary and conclusions
Using a simplified GCM, we have explored the way in which the intraseasonal variability depends on various parameters in order to understand the potential for interaction between synoptic eddies and the large-scale flow. Power on 10–100-day time scales was found in broad meridional dipole patterns similar in structure to the observed zonal index and annular modes. Absent zonally asymmetric forcing, the time scale of the model’s zonal index was found to be sensitive to three model parameters: the equilibrium equator-to-pole temperature gradient, ΔTeq, the thermal damping, ka, and the momentum damping, kf . The first parameter regulates the eddy heat transport of the model, and thus the eddy kinetic energy, while the latter two set internal damping time scales. A decrease in ΔTeq or ka, or an increase in kf , led to an equatorward shift in the mean jet and increase in the e-folding time scale of the zonal index. Since there exist three independent parameters that can be used change the time scale of the index, we conclude that it is distinct from any one of them individually; rather, the index time scale is being generated by an internal dynamical process, an eddy–mean flow interaction, although the index time scale must presumably depend, and in some complicated way, on a combination of the parameter time scales.
The time scales of the model changed quite dramatically when topography and large-scale thermal forcing were introduced to create a zonally localized storm track. Zonal asymmetries reduced the time scale of the intraseasonal variability and its sensitivity to model parameters. In the most realistic simulations, the time scale in fact became largely independent of the model parameters and quite consistent with time scales from observations.
We developed a simple diagnostic, a local dipole index, DI(t, λ), to explore the local, zonal structure of the zonal index and annular modes. The aim was seek out common features from simulations with the same degree of persistence and come to a more intuitive understanding on the nature of the zonal index in our model. On shorter time scales, the zonal index does not reflect zonally coherent motion. Locally, the dynamics are dominated by the projection of synoptic features onto the zonal index dipole. On longer time scales, however, we found that there is a true annular signal in the simulations with weak thermal forcing or strong surface friction—those simulations with more persistent zonal index behavior. This indicates the presence of a slow, coherent shift of the extratropical jet. Whereas the annular mode in these simulations is not a mode in the linear sense, it does have physical significance in that it characterizes a hemispheric shift in the extratropical jet where strong nonlocal interactions between the mean flow and eddies may be active. As seen in Fig. 9, after 10–15 days, the correlation of the local dipole projection becomes largely independent of longitude. This weaker, zonally coherent signal is masked, to some extent by synoptic variability that dominates the projection at shorter time scales.
In simulations with stronger thermal forcing, weaker surface drag, or zonally asymmetric forcing, where the zonal index was not as persistent, there was very little underlying zonally coherent behavior. The zonal index (and annular mode) are still the dominant patterns of low frequency variability in these simulations as determined by an EOF analysis, but their dominance is better understood from the statistical perspective of Gerber and Vallis (2005). Most of the power comes from smaller scale, local projections of individual eddies and wave-breaking events, not a hemispherically coherent movement of the jet. Zonally asymmetric forcing, topography in particular, creates a barrier to zonal coherence. In the storm track, the zonal index is dominated by swiftly moving synoptic eddies. The flow does become more zonal and persistent outside the storm track, but the zonally coherent signal is completely lost.
To explain the enhanced persistence in zonally symmetric simulations, we formulated a mechanism for coupling between eddies and the barotropic flow, inspired in part by the self-maintaining jet theory of Robinson (2006). We argue that the barotropic flow shields the baroclinicity of the jet, making eddies less efficient at reducing the shear in regions where the upper-level flow is strong. This causes the zonal mean jet to become “sticky,” tending to meander slowly on a distinct time scale set by the strength of the eddy–mean flow interactions. The mechanism depends on the balance between eddy heat transport in the lower troposphere (the production of wave activity) with eddy momentum transport above (the propagation and dissipation of wave activity). The surface westerlies and the barotropic jet may be considered to be maintained against surface friction by the westerly torque associated with the divergence of wave activity near the surface. The jet is thus tied to the latitude of peak eddy generation; a shift in the eddies leads to a shift in the jet on the relatively short time scale of the surface drag. If the eddies evolved independently of the mean flow, one would expect jet variations to be characterized by a red noise process in time, shouldering on a time scale set by the frictional drag and eddy processes that communicate surface friction to the upper-level flow. The barotropic flow, however, can influence eddy generation by shaping the meridional propagation and dissipation of waves in the upper atmosphere, making a feedback loop possible.
Linear theory suggests that waves are damped near critical latitudes, where the phase speed of the wave, c equals that of the flow, u. As waves approach the critical latitude, large particle displacements lead to wave breaking and irreversible mixing. The barotropic flow in the jet core increases the zonal velocity, permitting the jet to support more wave activity and enabling waves to propagate farther before they near critical latitudes. Momentum is transported in the opposite direction of the wave activity flux, and so tends to converge into regions of stronger flow. This torque partially offsets the tendency of eddies generated in the jet core to reduce the shear. Wave activity generated by eddies on the flanks of the jet, rather, will be damped more closely to the latitude at which it was generated; waves are simply closer to their critical latitudes. Hence eddies can more efficiently reduce the shear where the upper-level flow is weaker, and there is a negative feedback against sustained eddy growth on the flanks of the jet. This acts to retard motion of the jet, inhibiting eddies from shifting the barotropic circulation.
Surface friction plays a role in the feedback, in a fashion similar to that discussed in a two-layer model by Robinson (1996). Eddy heat fluxes produce a large westerly acceleration of the lower layers, so acting to reduce the shear, as shown in Fig. 12. By comparison, the feedback mechanism is evidenced in the shallow minimum in the easterly acceleration of the upper-level flow. If surface friction does not damp the westerly torque on the lower layers, the shear will be reduced by the increase in the winds at the surface, and the small minimum aloft will be rather inconsequential. In other words, without strong surface momentum damping, the shear is determined more by the low-level circulation. Momentum redistribution in the upper levels will not be able to maintain the shear and the barotropic circulation is no longer able to shape the stirring. The sensitivity of the time scale of the zonal index to the momentum damping, kf , may follow from these arguments. The jet becomes more persistent when kf is increased, and the westerly acceleration of the flow by the heat fluxes is more strongly damped.
The time scale of jet variation, however, is equally sensitive to the other parameters, the equilibrium temperature gradient, ΔTeq, and the thermal damping, ka. These parameters control the thermal restoring force on the flow, and are not easily connected to changes in the surface friction. With all three parameters, weaker persistence is consistently associated with a stronger barotropic circulation (see Fig. 1). As discussed by James (1987), and established in a more theoretical framework by Nakamura (1993), barotropic shear can directly influence eddy growth. These influences may compete with the indirect impact of the barotropic flow discussed in section 5 and limit the eddy–mean flow interaction when there is strong barotropic shear. This may link the stronger barotropic jet to the weaker persistence, in which case the sensitivity to all parameters can be related to their impact on the mean flow, and remains an open question for further research.
The dramatic breakdown of the jet variation time scale in simulations with topographic and large-scale thermal forcing is associated with the loss of the zonally coherent motion, as suggested by the analysis in section 4. A pathway for the breakdown of the eddy–zonal mean feedback in the presence of zonally asymmetric forcing is the subject of current research. There is evidence that the key lies in the separation of eddy growth and decay in the storm track.
We thank Ian Watterson and two anonymous reviewers for insightful comments and suggestions on an earlier version of the manuscript. This work is partially supported by the NSF, ATM division. EPG also thanks the Fannie and John Hertz Foundation for support.
At T42 resolution, the time scale of the zonal index is sensitive to vertical resolution in the upper atmosphere. This sensitivity can be found either by adding more levels evenly through the atmosphere, or by keeping the same number of levels and adjusting the spacing as to increase resolution in the upper atmosphere at the expense of the lower and midtroposphere. This increase in time scale (which can be dramatic) is associated with the same shift in the mean state observed in the parameter sweep experiments described in section 3.
It was discovered, however, that this increase in time scale is not robust to changes in horizontal resolution. At T63, the model ceases to be sensitive to the vertical resolution; the time scales of simulations with both low and high vertical resolution at the upper boundary do not change significantly. Experiments with the model at T63 and T85 resolution indicate that the model’s behavior at T42 is more robust with coarser vertical resolution than with finer resolution. This is to say, increasing the vertical resolution at T42 appears to introduce a pathology into the model. Mindful of this weakness of the model, we verified key trends at higher resolution, where the model behavior is more robust. A more thorough discussion of the resolution issue can also be found in section 5.4 of Gerber (2005).
We experimented with mountains of varying shape and height and found the results to be relatively robust provided the orography was positioned as to block the extratropical jet. In the simulations described here, we use a Gaussian ridge of variable height, as described in Table 2, comparable in horizontal scale to the Rocky or Andes Mountains. The half-width of the ridge in longitude is 12.5°. The ridge is 40° long in latitude and centered at 40°N. It tapers off with a half-width of 12.5° at either end, so that it does not extend into the Southern Hemisphere.
A thermal perturbationB1 was introduced to approximate the affect of land–sea contrast between a cold continent and warm ocean in the winter hemisphere. We experimented with several variations, settling with a simple perturbation δTeq to the zonally uniform profile that produced the desired zonal asymmetry in the synoptic variability,
where λ, θ, and σ = p/ps are the longitudinal, latitudinal, and vertical coordinates, respectively. The horizontal profile was constructed from sine functions,
In the vertical, the profile is maximum at the surfaces, and decays by a cosine function in σ to zero near the tropopause,
The resulting heating rates were greater than those observed in the atmosphere, but within range of reasonable. The maximum heat flux into the atmosphere was 400 W m−2.
Corresponding author address: Edwin P. Gerber, 435 W. 119th Street, Apt. 9M, New York, NY 10027. Email: email@example.com
This article included in the Jets and Annular Structures in Geophysical Fluids (Jets) special collection.
Eddy heat fluxes reduce the shear by weakening the temperature gradient. In the transformed Eulerian mean framework this is characterized by the vertical propagation of wave activity. In the Eulerian framework, the mean meridional circulation brings momentum to the surface.
Note that in the implementation of the HS forcing, Teq is not allowed to drop below 200 K, the temperature of the model’s stratosphere. We have kept this constraint when applying the perturbation δTeq, thereby truncating the cooling anomaly in upper levels at high latitudes where the temperature would otherwise drop below 200 K.