In simple GCMs, the time scale associated with the persistence of one particular phase of the model’s leading mode of variability can often be unrealistically large. In a particularly extreme example, the time scale in the Polvani–Kushner model is about an order of magnitude larger than the observed atmosphere. From the fluctuation–dissipation theorem, one implication of these simple models is that responses are exaggerated, since such setups are overly sensitive to any external forcing. Although the model’s equilibrium temperature is set up to represent perpetual Southern Hemisphere winter solstice, it is found that the tropospheric eddy-driven jet has a preference for two distinct regions: the subtropics and midlatitudes. Because of this bimodality, the jet persists in one region for thousands of days before “switching” to another. As a result, the time scale associated with the intrinsic variability is unrealistic. In this paper, the authors systematically vary the model’s tropospheric equilibrium temperature profile, one configuration being identical to that of Polvani and Kushner. Modest changes to the tropospheric state to either side of the parameter space removed the bimodality in the zonal-mean zonal jet’s spatial distribution and significantly reduced the time scale associated with the model’s internal mode. Consequently, the tropospheric response to the same stratospheric forcing is significantly weaker than in the Polvani and Kushner case.
Several previous studies (e.g., Song and Robinson 2004; Polvani and Kushner 2002, hereafter PK02; Kushner and Polvani 2004, hereafter KP04; Son and Lee 2006; Ring and Plumb 2007, 2008) have described the relationship between external forcings and the climatological response in model simulations. With extratropical forcings, the spatial structure of the response is dominated by the model’s “annular modes” (Thompson and Wallace 2000; Lorenz and Hartmann 2001)—that is, the leading patterns of variability in the model climatology. Ring and Plumb (2007, 2008) found linear relationships between the annular mode response to external forcings and the projection of the forcing onto the modes, with the constant of proportionality being approximately equal to the decorrelation time of the unforced modes, as suggested by the fluctuation–dissipation theorem (FDT).
The results presented by PK02 and KP04 are especially dramatic: they found a large poleward shift (of about 10° latitude) and intensification (by almost 5 m s−1) of the surface wind maximum in the southern (winterlike) hemisphere of a simplified GCM in response to imposed perturbations to the imposed equilibrium temperature in the stratosphere. However, as emphasized by Gerber and Polvani (2009) and as will be discussed further here, in the particular cases of PK02 and KP04 (hereafter, the PK cases), the decorrelation time of the model’s leading annular mode is extremely long (200–500 days) as compared to the time of 10–20 days that characterizes annular modes in the atmosphere (Feldstein 2000). Simplified GCMs without topography typically produce decorrelation times several times larger than those observed (Gerber and Vallis 2007; Gerber and Polvani 2008), but those in the PK cases are long even by the standards of such models. One consequence of unrealistically long decorrelation times is that the fluctuation–dissipation theorem predicts unrealistically strong responses to external forcings.
In this paper, we illustrate the anomalous nature of the sensitivity of the climatological state in PK02 and KP04 by investigating how the response to perturbed stratospheric equilibrium temperatures depends on the climatological state of the troposphere. Following PK02 and KP04, we define an equilibrium temperature (Te) distribution in the troposphere like that of Held and Suarez (1994), but vary a single parameter to shift the latitude of the peak Te between the equator and the summer subtropics. One such state is identical to that of PK02 and KP04. We show that the decorrelation time of the leading annular mode in the model’s winter hemisphere is particularly long for this state. Modest changes in either direction of this tropospheric parameter reduce the decorrelation time considerably. Consequently, the response to changes in stratospheric Te becomes much weaker than that found by PK02 and KP04. These results complement those of Gerber and Polvani (2009), who found the PK state and its annular mode decorrelation times to be very sensitive to the model’s lower boundary condition. The long decorrelation time of the PK cases is a reflection of the fact that their tropospheric climatology sits at a transition at which the eddy-driven jet separates from the subtropical jet.
The method of how the decorrelation time τ is calculated, how the model is set up, and details of each model experiment are described in section 2. The results of increasing the seasonality of the temperature distribution are shown in section 3. In cases with a more realistic τ, our results will show a dramatically reduced effect from a stratospheric perturbation. In the context of the fluctuation–dissipation theorem, a discussion will follow in section 4; finally, our conclusions can be found in section 5.
2. Method and model setup
There are many ways to measure the persistence of the model’s internal variability. Here, we adapt the method used in Gerber et al. (2008). First, we determine the principal component (PC) of the zonally averaged zonal winds poleward of 20°S. After that, we define the decorrelation time τ to be the time when the autocorrelation of the leading PC crosses 1/e. For reference, this value has been observed to be approximately between 10 and 20 days, depending on the season (Feldstein 2000; Baldwin et al. 2003).
The data in this paper were generated by the Geophysical Fluid Dynamics Laboratory (GFDL) atmospheric general circulation model. This is a dry hydrostatic primitive equation model in σ coordinates, in which equations are solved using spectral transforms in the horizontal and a Simmons and Burridge (1981) finite difference in the vertical. The results shown here use a T30 resolution with 3.75° in latitude and longitude grid points. In the vertical, to sufficiently resolve stratospheric dynamics, we use 40 vertical sigma levels, with about 15 levels in the troposphere, 18 in the stratosphere, and 7 above the stratopause. The key experiments were also run at T60 resolution; our main conclusions are unaffected by this change of resolution. A full model description is detailed in the appendix of PK02.
Both surface drag (parameterized as Rayleigh friction) and radiation (represented by Newtonian cooling toward an equilibrium tropospheric temperature profile) are similar to that used by Held and Suarez (1994) as shown:
where ϕ is the latitude, p is the pressure, and TT, T0, p0, δy, δz, and ε are constants. In (2), the first and third terms are symmetric about the equator. Thus, only the second term determines the strength of the seasonality. When ε = 0, the equilibrium temperature profile for both hemispheres is identical. As the magnitude of ε increases, so does the asymmetry between the summer and winter hemispheres. In the following experiments, multiples of 10 K ranging from 0 to −30 K will be used for ε. The different latitudinal surface equilibrium temperature structures are shown in Fig. 1. As the magnitude of ε is increased, the peak temperature is displaced away from the equator and into the Northern Hemisphere; this profile is broadly representative of a Northern Hemisphere summer and a Southern Hemisphere winter. Except for the time step and model resolution, the setup is identical to that in the PK cases when ε = −10 K.
Following PK02 and KP04, we employ the same stratospheric equilibrium temperature profiles. In Table 1, “no vortex” refers to a stratospheric equilibrium temperature independent of latitude at constant pressure. In these cases, easterlies or weak westerlies are prevalent in the stratosphere. For cases with a cold polar vortex, we use Eqs. (A1) and (A2) of PK02, in which the equilibrium temperatures within the vortex are controlled by the parameter γ. For γ = 2 (“weak vortex” cases), the modeled vortex climatological wind speeds are typically 50 m s−1, whereas for γ = 3 and γ = 4 (“moderate vortex” and “strong vortex” cases, respectively) they reach 70 and 90 m s−1 respectively.
The cooling rate is 1/40 day−1 above σ = 0.7 and increases linearly to 1/4 day−1 toward the surface. To reduce numerical diffusion, we have used a sixth-order hyperdiffusion damping the largest resolved wavenumber on a time scale of half a day. In the runs shown below, with no topography or any other longitudinally varying forcings, each experiment was integrated for at least 4000 days with a time step of at most 1800 s. To obtain robust statistics, as seen in the right column of Table 1, for experiments with a decorrelation time exceeding 100 days, experiments were run out to at least 8000 days. In all cases, we disregard the initial spinup period of 500 days and analyze the remaining period.
Table 1 lists the various experiments reported here, with their different imposed tropospheric and stratospheric equilibrium states, along with the decorrelation time for the leading mode of variability of zonal-mean zonal wind in the Southern Hemisphere. Runs 1a–d with ε = −10 K are identical to those of PK02 and KP04, and the weak vortex cases 1a and 1b are noteworthy for their very long decorrelation times. In runs 2 and 3, the tropospheric Te maximum is shifted poleward of that in the PK cases. For comparison with our other cases, the climatological zonal-mean zonal winds for runs 1a, 1b, and 1d are shown in Figs. 2a–c. The large impact reported by PK02 and KP04 of the strong vortex case 1d (with γ = 4) onto the mean zonal winds is highlighted in Fig. 2d. In particular, this illustrates the annular-mode nature of the tropospheric response, as explicitly demonstrated by PK02 and KP04.
Inspection of Figs. 2a–c reveals a change in the time-mean tropospheric jet from a deep single jet centered near 35°S to a double structure comprising a deep jet near 45° with an almost-separated subtropical jet in the upper troposphere near 25°. In fact, the time-mean picture in the weak vortex cases 1a and 1b is rather misleading. Figures 3e–h show a relative histogram of the latitudinal location of the maximum daily-averaged near-surface zonal-mean zonal winds for runs 1a–1d; the near-bimodal distribution in case 1a reveals that in reality the eddy-driven surface wind maximum is fluctuating between locations at 30° and 40°. Case 1b brings out this dual regime behavior more clearly (Fig. 3f). There is in fact a minimum at the location of the time-mean jet. Thus, as discussed in Gerber and Polvani (2008), in the PK cases with a stratospheric weak vortex, the eddy-driven jet in the troposphere is actually teetering between two states. In one state, the eddy-driven jet merges with the subtropical jet, whereas in the other the two are well separated.
The situation changes markedly when the tropospheric state is altered by changing the factor ε [see (2)]. Time-averaged zonal-mean zonal winds for the no-vortex runs 2a and 3a and the corresponding strong-vortex runs 2d and 3d are shown in Figs. 4a,b and 4d,e, respectively. In each case, the tropospheric eddy-driven and subtropical jets are well separated in the time mean; in each case, the “teetering” behavior of runs 1a and 1b is absent, and the decorrelation time of the leading fluctuating mode is considerably shorter (see Figs. 3i,j and 3m,n, respectively).
The contrasting behavior of the fluctuations in the near-surface tropospheric zonal flow is illustrated for the no-vortex runs 1a (ε = −10 K) and 3a (ε = −30 K) in Figs. 5a and 5c, respectively. The long persistence of each of the two regimes evident in the location of the eddy-driven jet in case 1a contrasts with the more rapid (note the different time scales on the two plots), weaker, and occasionally poleward-propagating fluctuations in case 3a, similar to the cases found in an observational study by Feldstein (1998).
In fact, case 1a also shows more rapid fluctuations like those of case 3a, but these are masked in Fig. 5a by the larger long-lived anomalies. This is shown in Fig. 5b, whose anomalies are calculated from individual periods of run 1a when the surface jet is in its poleward position. Within this regime, a comparison with Fig. 5c shows that the jet displays fluctuations similar in magnitude and time scale to those of run 3a. Thus, the persistent regimes seen in the PK case 1a do not replace the more rapid variations but rather coexist with them; from this viewpoint, they constitute an additional mode of variability.
The impact on the time-mean zonal winds of changing the specification of stratospheric Te “no vortex” to “strong vortex” for the cases with ε = −20 and −30 K is shown in Figs. 4c and 4f, respectively. Although in each case the changes to the stratospheric jet are comparable to the PK cases with ε = −10 K, the tropospheric impact is much weaker. This is made explicit in the comparison shown in Figs. 6b–d, showing the response of the surface zonal winds to the altered stratosphere for the three tropospheric states, and suggesting that the strong response found in PK02 and KP04 is anomalous.
Although both experiments 2a and 3a were performed with a stratospheric relaxation profile that is isothermal in pressure, Figs. 4a and 4d exhibit westerlies in the stratosphere. Further tests using T42 and T60 resolution exhibit the same property. We suggest the following as the reason for this behavior: As the magnitude of ε increases, the meridional temperature gradient increases, and hence the maximum tropospheric midlatitude winds increase as well, as shown by comparing Figs. 2a, 4a, and 4d. However, there is not enough compensating wave drag above the jet in experiments 2a and 3a (not shown) to reduce the increased vertical shear, preventing the background winds from forming easterlies in the stratosphere. In the extreme case of experiment 3a, with insufficient wave damping above the tropospheric jet and a meridional stratospheric equilibrium temperature gradient equal to zero, to a first-order approximation, the winds turn approximately barotropic throughout the lower two-thirds of the extratropical stratosphere. The reason for the insufficient wave drag above the jet appears to be related to the poleward jet position and an increase in meridional wave propagation (not shown). (Note that as |ε| increases, the jet shifts poleward, as shown in Fig. 6). As a result, the ratio between the wave drag above the jet and equatorward of the jet decreases. Although the lack of wave drag can explain the westerlies in the stratosphere, this process does not explain the localized maximum wind speed at about 5 hPa for experiment 3a. Because this particular characteristic does not affect our main results and appears to be resolution dependent, this property will not be discussed further. In any case, as the magnitude of ε increases, there does appear to be a robust propensity for the midlatitude winds to be more barotropic and hence the existence of stratospheric westerlies above the jet, but a more thorough explanation is beyond the scope of this study.
Other simple models have demonstrated that gradually changing a single parameter (e.g., increasing the diabatic heating in the tropics) can change the location of peak eddy activity from one latitude to another (e.g., Lee and Kim 2003). As discussed in Gerber and Polvani (2009), there is evidence that this system is actually “teetering” between these two states. This is clearly demonstrated in Fig. 3f by the bimodal distribution of the eddy-driven jet’s distribution in experiment 1b.
To demonstrate fully that ε = −10 K lies in between two regimes, we perform an additional experiment with ε = 0 K, thus placing the tropospheric Te maximum equatorward of that of the PK case. As shown in Figs. 3a, 3i, and 3m, the distribution of the eddy-driven jet’s location is unimodal and is unambiguously located in the subtropics in experiment 0a or in the midlatitudes in cases 2a and 3a. A comparison with Fig. 3e shows that the control case of the PK setup indeed sat in between these two states.
There are other examples in the atmosphere that exhibit “regime behavior.” For instance, using a primitive equation model, Akahori and Yoden (1997) showed that it was possible for there to be a bimodality in the frequency distribution of the eddy life cycle index. Using a weak surface drag value, the zonal-mean jet was located in high latitudes and the life cycle of baroclinic eddies was predominantly characterized by anticyclonic breaking. Conversely, using a high drag value, the tropospheric jet shifts to the low latitudes, with eddies being characterized by cyclonic breaking. However, for an intermediate surface drag value, there was a bimodality in the frequency distribution function of this eddy life cycle index. In an observational example, Christiansen (2009) noted the bimodality in the winter stratospheric circulation: either strong or weak polar vortex winds existed, with rarely anything in between.
We speculate that the physical reasons for this particular regime behavior and its associated long decorrelation have to do with this eddy-driven jet’s preference for the two distinctly separate locations. Using the dynamical core of the GFDL GCM, Lee and Kim (2003) have shown that the location of the eddy-driven jet could exist in either region depending on the location of the greatest instability. In one regime, imposing sufficiently strong equatorial diabatic heating, the subtropical jet intensifies to the point where the strongest hemispheric baroclinicity resides in this region and subsequently constrains the eddy-driven jet’s location within the subtropics. However, with weak diabatic heating, the greatest instability would then be associated with the model’s imposed equilibrium temperature, whose meridional gradient maximizes in midlatitudes, resulting in the eddies organizing their activity in this region. In the PK setup, we thus speculate that the instability in midlatitudes and the subtropics are comparable in magnitude, and, as a result, teetering between these two regimes.
a. Consistency with the fluctuation–dissipation theorem
Why does the tropospheric response differ so much between experiments? According to the fluctuation–dissipation theorem, the linear response to an imposed forcing is proportional to the projection of that forcing onto the system’s natural modes of variability and to the decorrelation time associated with that mode, according to an equation of the form
An obvious difference between the no-vortex cases (experiments 2a and 3a with 1a) is the location of the jet. The change in the stratospheric equilibrium temperature profile resulted in the tropospheric jet shifting to the midlatitudes. An important consideration is how that changed the stratospheric thermal forcing’s projection onto the leading mode of variability. Presumably, with the change of the jet location, the mode structure will change as well. There is then the possibility that the inner product of the two decreased noticeably and hence could explain the significantly weaker response. Thus, we first determine whether the weak response can be explained by the poor projection of the forcing onto the mode. To determine the mode of variability for these thermally forced cases, we perform an analysis similar to that of Ring and Plumb (2008). First, the covariance of the zonal wind and temperature anomalies is obtained. Then, through singular value decomposition (SVD), we determine the leading pattern of the temperature variability. Figure 7 shows a comparison between experiments 1 and 3 for the leading mode and the applied forcings. Note that the perturbation is the same for both cases. As shown in Table 2, the forcing does not project less onto the leading mode; instead, the projection is actually greater in the ε = −30 K case.
With the projection of the forcing onto the mode being larger, Eq. (3) suggests that a weaker response must be associated with a shorter decorrelation time. Indeed, as shown in the right column of Table 1, the decorrelation time of experiment 3a is nearly 10 times smaller than that of experiment 1a. Thus, in these experiments, the magnitude of the tropospheric response is controlled by the model’s time scale of internal variability.
The dependence on the decorrelation time provides an explanation of the nonlinearity of the surface wind response found by PK02, KP04, and our experiment 0 to changes in the parameter γ (their Fig. 2 and our Fig. 6a, respectively). As (3) makes clear, linearization of the response/forcing relationship about a basic state presumes that the characteristics of a mode—in particular, the modal structure and its decorrelation time—remain essentially unchanged by the perturbation. As Table 1 makes clear, in cases 2 and 3, the decorrelation time changes little under perturbations, but in cases 1a and 1b, it is sensitive to the parameter γ, being especially large near γ = 2 and γ = 3 in case 0c—when the system is “on the edge” of switching from one jet location to the other—in which case (3) predicts nonlinearity in the response with the greatest sensitivity near γ = 2, as PK02 and KP04 found.
In this paper, we have demonstrated the sensitivity to the results obtained by PK02. As Fig. 3 makes clear, a long decorrelation time is found to exist in the cases in which the latitudinal preference for the eddy-driven jet teetered between 1) coexisting with and 2) being well separated from the subtropical jet. Modest changes to the tropospheric state removed this behavior and significantly reduced the decorrelation time to a more realistic value (from 250 to 30 days). Consequently, the response to an identical forcing was much weaker than that found by PK02 and KP04 (cf. Figs. 2d and 4f). Consistent with the fluctuation–dissipation theorem (FDT), these experiments support a previous study by Gerber et al. (2008) that the decorrelation time associated with the internal variability is equally as important as how the forcing projects onto the mode.
Previous work has qualitatively shown that FDT is a simple and effective way to predict climatological changes when tropospheric forcings have been prescribed (e.g., Ring and Plumb 2007, 2008; Gerber et al. 2008). Using this framework, we have shown that FDT can also be used in a stratosphere–troposphere system in which stratospheric forcings can be used to predict the qualitative response. Unfortunately, similar to those previous studies, we find that the quantitative accuracy of the FDT is limited. One of the issues in applying the FDT is the appropriate definition of the “modes” of the unforced system. Ring and Plumb (2008) argued for the use of principal oscillation patterns (POPs) rather than EOFs to define the leading modes, but in their mostly tropospheric model they found little difference between the structures of the leading POPs and EOFs. A model that includes the stratosphere, however, is likely to be more problematic because the vertical structure of the leading EOFs is sensitive to how the covariances are weighted when calculating the EOFs. In this paper, we have followed Thompson and Wallace (2000) in defining how the vertical structure is weighted; we regard it as unlikely that our main conclusions are unduly sensitive to this choice.
We thank Gang Chen, Ed Gerber, and Mike Ring for helpful discussions and two anonymous reviewers for their beneficial comments on this manuscript. This research was supported by the National Science Foundation under Grants ATM-0314094 and ATM-0808831.
Corresponding author address: Cegeon Chan, Department of Earth and Atmospheric Science, MIT Building 54, Room 1719, 77 Massachusetts Avenue, Cambridge, MA 02139. Email: firstname.lastname@example.org