## 1. Introduction

The annular modes are the leading patterns of variability in the extratropics of each hemisphere (Thompson and Wallace 1998, 2000). The spatial structure is often described in terms of a dipole of pressure or geopotential, with a ring of anomalously high values in the midlatitudes, and a region of anomalously low values centered toward the pole. The patterns’ signature appears robustly in other meteorological fields, most notably zonal wind (Lorenz and Hartmann 2001, 2003), which contains a dipole indicating wobbles of the zonal jet about its time-mean position. These same spatial patterns also appear to dominate the dynamical response of the atmosphere to perturbations such as greenhouse warming (e.g., Kushner et al. 2001; Rind et al. 2002, 2005) and ozone depletion (e.g., Thompson and Solomon 2002; Gillett and Thompson 2003) and to be intermediaries of stratospheric influence on the troposphere (e.g., Baldwin and Dunkerton 1999, 2001; Kushner and Polvani 2004; Song and Robinson 2004; Thompson et al. 2005, 2006). Thus, the patterns appear both as unforced natural variability and as a forced response to perturbations of the climate system.

In a previous study (Ring and Plumb 2007, hereafter RP07), using a simple atmospheric general circulation model, we investigated the extent to which the annular modes constitute such a preferred response of the climate system to forcing by prescribed torques [modeled on those used by Song and Robinson (2004)] that were monopolar in each hemisphere. RP07 found that in most trials, the response of the model was dominated by the annular mode patterns and that, consistent with earlier work (Lorenz and Hartmann 2001, 2003), eddy–mean flow feedback was crucial in organizing and magnifying the response. The magnitude and sign of the annular mode response were found to depend on the location, especially in latitude, of the applied torque and in fact to be approximately linearly dependent on the projection of the torque onto the dominant EOF of zonal-mean zonal wind. These characteristics—the appearance of the modes of unforced variability dominating the forced response and the linear dependence of that response on the modal projection of the forcing—are reminiscent of the predictions of the fluctuation–dissipation theorem (Nyquist 1928; Leith 1975). In this paper, we shall explore the applicability of this theorem to this problem.

First, the methodology used in RP07 is extended to model trials which are forced thermally rather than mechanically. We will show in sections 3 and 4 below that for cases forced by imposed perturbations to the reference temperature *T*_{ref}, whose gradients are located in extratropical latitudes, the response is indeed “annular mode–like” in its structure, with eddy feedback again being the central factor controlling that structure. When the *T*_{ref} gradients extend into the tropics, however, the response is less cleanly annular mode–like, and Hadley cell dynamics become a significant factor. [Similar behavior was noted in RP07 for cases with applied torques in low latitudes of the upper troposphere, as well as by Son and Lee (2006).] For the cases with extratropical forcing, we (like RP07) find a linear relationship between the projections of response and forcing onto the annular modes.

The ad hoc procedure followed in RP07 and in section 3 here—using projections onto the modal structures defined by EOF or singular value decomposition (SVD) analysis of the unforced variability of zonal-mean wind and temperature—is difficult to justify rigorously. Although it is not unreasonable to assume that these patterns reflect the leading eigenvectors of the underlying dynamical operator, there is no a priori basis for assuming the system’s eigenvectors to be orthogonal. Accordingly, we take in section 5 a more rigorous approach to the problem, assuming a simple but nonlocal representation of eddy feedback, using principal oscillation pattern (POP) analysis (von Storch et al. 1988) to determine the leading eigenvectors of the underlying dynamical system operator. We find for these experiments that both left and right eigenvectors of the dominant mode in each hemisphere are very similar in structure to those of the corresponding EOFs, justifying a posteriori the approach taken by RP07.

To render the dynamics of the zonal-mean flow, in the presence of eddy feedback, as transparent as possible, we first formulate it in such a way that the perturbed zonal-mean flow is the only state variable. Eddy feedback is parameterized as a nonlocal but linear function of the zonal wind anomaly plus a stochastic term. It is assumed that the variability of the zonal flow in the absence of external forcings is the response to this stochastic forcing. In the presence of external forcings, there is an effective torque acting on the zonal flow, which includes not only any directly applied torque but also the impact of the mean meridional circulation induced by the imposed torques and thermal perturbations. The theory predicts that the slope of the relationship between the response in a given mode (to be obtained through the fluctuation–dissipation analysis) and the effective torque projected on that mode is just the decorrelation time of the mode in the unforced case, a manifestation of the fluctuation–dissipation theorem (Nyquist 1928; Leith 1975) applied to this problem.

We find that for both the mechanically forced trials and thermally forced trials, each considered separately, the relationship between the response in the given mode and the effective torque projected onto that mode is linear. However, the mechanical and thermal cases follow different slopes, with differences of roughly a factor of 2 between the computed slopes and the theoretical prediction. Although the model results are approximately consistent with the theoretical prediction, we find quantitative discrepancies that we are currently unable to explain.

## 2. Model setup and control climatology

This study uses the dynamical core of the Geophysical Fluid Dynamics Laboratory (GFDL) atmospheric general circulation model. This is a dry, hydrostatic, primitive equation model. For the results shown here, trials were conducted using T30 spectral resolution and 20 equally spaced sigma levels; selected trials performed at higher spectral resolutions produced similar results.

The control climatology is based on a run of 7000 model days, with data sampled once daily after the model had spun up from rest and reached a statistically steady state. All of the forced climatologies are based on model runs of at least 5000 days of data, sampled daily. Some of these runs include 7000 days of data.

The setup of the model is like that of Held and Suarez (1994), with linear friction and radiation schemes. The modifications made to the Held and Suarez (1994) setup are exactly those used in RP07, so the reader is referred there for details. Notably, we used a *T*_{ref} profile with a maximum value displaced slightly off the equator and different equator–pole *T*_{ref} gradients in each hemisphere, which results in the Southern Hemisphere being the winter hemisphere in our model. Additionally, we chose a value of Rayleigh drag which was twice that used by Held and Suarez (1994) because it ameliorated—but did not eliminate—the unrealistically long decorrelation time scales found for the annular modes, which is a problem for simple models with a zonally symmetric lower boundary (Gerber et al. 2007; Gerber and Vallis 2007).

The model, with no external forcing applied, produces a climatology which is relatively similar to that of the earth, with westerly jets in each hemisphere. Each hemisphere contains an annular mode, with a dipole of positive and negative anomalies easily identified as the leading patterns of variability in the surface pressure, geopotential, and zonal-mean zonal wind fields. We repeat the figure displaying the leading EOF of zonal-mean zonal wind anomalies in each hemisphere from RP07 as Fig. 1 and refer the reader to RP07 for more details on the control climatology.

The leading EOF, confined to the Southern Hemisphere, explains 53% of the variance globally. The second EOF, confined to the Northern Hemisphere, accounts for 17%. The third and fourth EOFs (not shown), which are analogous to the tripolar patterns found by Lorenz and Hartmann (2001, 2003) as the second leading EOF in each hemisphere, explain 10% and 4% of the variance, respectively. We shall focus on the dipolar EOFs here because they explain a much larger percentage of the variability as compared to the other patterns.

One feature of the control run relevant to our discussions of thermally forced trials not shown in RP07 is the variability of the temperature field of the control run. Here we show the leading patterns of the temperature anomalies, obtained through singular value decomposition of the covariance with the zonal wind anomalies, in Fig. 2. The leading spatial pattern of temperature is in the Southern Hemisphere, with the second pattern in the Northern Hemisphere; the two are shown together on the same plot. In each hemisphere, the dominant temperature anomaly appears in the midlatitudes, extending from the upper troposphere toward the surface. In the subpolar and polar regions, a weaker temperature anomaly of the opposite sign prevails. The zonal wind patterns obtained from the SVD analysis are virtually identical to the EOFs shown in Fig. 1.

## 3. Results of thermally forced trials

In this section, the climatologies of model runs that are forced with perturbations to the control *T*_{ref} are considered. This methodology is similar to that used by Son and Lee (2005, 2006). As for the mechanically forced trials shown in RP07, a long-time climatology is compiled for each forced trial here and compared to the climatology of the control run to determine whether the changes in climatology are annular mode–like.

The types of *T*_{ref} perturbations we examine may be split into two categories: trials with broad, hemisphere-scale changes to *T*_{ref} and trials with *T*_{ref} changes confined to the midlatitude and polar regions. Below we show examples of each.

Figure 3 displays the difference in *T*_{ref} and the meridional gradient of this difference for two of these trials with the broader forcing. In the trial featured on the upper row of the figure, the maximum *T*_{ref} anomaly of 5 K occurs just off the equator, at the warmest point in the control’s *T*_{ref} profile. Although the largest anomalies are found in the tropics, the anomalies extend well into the midlatitudes. Also notable is the broad latitudinal extent of the gradient of the change in *T*_{ref}, which is linked to the vertical gradient of wind through the balance assumption (i.e., the thermal wind shear). This change, as may be seen in Fig. 3, has large values in the subtropics.

The lower row of Fig. 3 displays the difference in *T*_{ref} and the gradient of that difference for a case with a 5-K cooling perturbation at the pole. As with the profile displayed on the upper row, both the *T*_{ref} perturbation and the gradient of that perturbation are latitudinally broad, with the gradient of the perturbation having maximum amplitude in the subtropics.

In addition to these two trials, we also conduct runs with perturbations equal in magnitude but opposite in sign to those in Fig. 3 (e.g., a cooling perturbation centered at the equator and a warming perturbation centered at the poles).

The results for these four trials are not as cleanly annular mode–like as for most of the mechanically forced trials discussed in RP07. Note from Fig. 1 that, in the EOF patterns, the equatorward lobe of each dipole is slightly stronger than the higher-latitude lobe. The change in zonal-mean zonal wind is shown in Fig. 4 for each of the four trials versus the control. Deviations from the annular mode pattern are especially notable for the two trials with the *T*_{ref} perturbation maximized in the tropics: although the response is somewhat dipolar, the equatorward lobes are much too weak and the nodal lines are shifted equatorward from those of the EOF patterns. The responses for the two cases with the *T*_{ref} perturbation maximized at the pole are more reminiscent of the annular mode profile, but even here some features of the response differ from the annular mode pattern. In the trial with a 5-K warming *T*_{ref} perturbation at the pole, for example, the Southern Hemisphere nodal line is at 30°, compared with 37° for EOF1 of the unforced run.

We consider the amount of variance in the zonal wind responses explained by the annular modes by comparing the total variance in each case to that of a projection of the wind response pattern onto the annular mode pattern. For the Southern Hemisphere response to the polar cooling, 83% of the variance in the response may be explained through the annular mode pattern. This is comparable to percentages found for the cases in RP07. For the responses in the other three Southern Hemisphere cases, and the responses for all cases in the Northern Hemisphere, the percentage explained by the annular modes is lower; in fact it is under 50% for the cases with 55° gradients in the Southern Hemisphere and for all cases in the Northern Hemisphere.

The streamfunction and Eliassen–Palm (E-P) flux divergence anomalies (normalized to units of zonal wind acceleration) for the warm equator trial are shown in Fig. 5. The streamfunction anomalies produced are broad, with significant amplitude extending deep into the tropics. This is in contrast to the responses found by RP07 when annular mode–like behavior was present, in which the streamfunction anomalies were instead much more concentrated, with sharper gradients in the subtropical region. The E-P flux divergence difference field is noteworthy for the lack of a strong surface dipole characteristic of the annular modes (cf. Fig. 6 of RP07); instead, the response in that field reflects a general increase in eddy generation at the surface, with no concomitant decrease of similar magnitude at another latitude. These results are similar to those of the cold equator trial, although the trials with *T*_{ref} anomalies centered at the poles did show differences more similar to the annular mode signal.

One factor which contributes to the poor annular mode–like responses to the thermal forcings shown here is the concentration of much of the forcing in the tropics. Son and Lee (2006), using more geographically concentrated forcings than those chosen here, found a less predictable zonal wind response to forcing placed deep in the tropics as compared to forcing placed deep in the polar region. Because they discerned a wind response for the former trials that was not completely extratropical in nature, the wind response to tropical forcings may not be as easily described in terms of the annular modes. An unexpectedly weak zonal wind response for some trials with a mechanical forcing placed on the equatorward flank of the jet, especially in the upper troposphere, was also found by RP07, who noted that the Hadley cell becomes more involved in such cases.

Additionally, the forcings chosen in the four trials above are extremely broad in their latitudinal extent, and their projection on the annular modes is relatively weak. Examining, for example, the latitudinal extent of the gradients of *T*_{ref} differences in Fig. 3, we note that it encompasses regions in which the leading EOF of zonal wind in each hemisphere is both positive and negative. The chosen forcing projects positively in some places but negatively in others.

With these points in mind, we conduct a second set of thermally forced trials. As above, the forcing is accomplished by altering the *T*_{ref} profile, but for these trials the change in *T*_{ref} is nonzero only poleward of 45° in each hemisphere. An example of the change in *T*_{ref}, together with the gradient of that change, is shown in Fig. 6. The *T*_{ref} anomaly and its gradient are more latitudinally confined than in the previous cases. All trials in this suite use this shape of forcing; however, the polarity and amplitude vary. We conduct trials using maximum anomalies of 2, 4, 6, and 10 K, performing a trial with polar warming and a trial with polar cooling for each magnitude.

The climatological changes in the zonal-mean zonal wind and temperature fields under the influence of the perturbed *T*_{ref} profiles, versus that of the control, are shown for the trials with 6-K maximum forcing in Fig. 7. The shapes of the responses were similar in other trials, with stronger responses at 10 K and weaker responses at 2 and 4 K. The strong, annular mode–like anomalies were produced in the zonal wind field, with vertically coherent westerly and easterly anomalies extending from the ground to the top of the model atmosphere. The anomalies are centered about the position of the unforced time-mean jet, and the relative amplitudes of positive and negative centers are much closer to 1/1 in each hemisphere as compared to the trials with broader forcing. The percentage of the response variance explained by the annular modes in these cases are all above 80% for the Southern Hemisphere and between 67% and 76% in the Northern Hemisphere, which are comparable to the mechanically forced cases in RP07.

Considering the temperature changes (as seen in Fig. 7) for each case, the largest temperature anomalies are found at the pole, reflecting the imparted difference in *T*_{ref} between each run and the control. However, in each case there is an opposite-signed temperature anomaly in the midlatitudes, stretching from near the surface to the upper troposphere. This feature was also visible in the leading SVD pattern of each hemisphere (Fig. 2).

We also examine other climatological fields and find responses there consistent with those found by RP07 for mechanically forced runs. As an example, the streamfunction and E-P flux divergence differences between the run with a −6 K anomaly and the control are shown in Fig. 8. The streamfunction anomalies are now much more concentrated, with a sharper gradient in the subtropics rather than diffuse cells extending more deeply toward the equator. The E-P flux divergence anomalies show the surface dipole, indicating a latitudinal movement of the largest region of eddy generation, and hint at a dipole at the tropopause level. The latter feature is much clearer when we consider solely the meridional component of the divergence (not shown) and suggests movement of the eddy momentum flux convergence along with the zonal wind.

As with the mechanically forced trials in RP07, we compare the strength of both the response and the forcing for each case. We do so below for the eight trials using thermal forcing concentrated poleward of 45°. For each case, we project the perturbed *T*_{ref} onto the (nondimensional) SVD pattern of temperature in each hemisphere. Likewise, we project the forced temperature response on the same SVD pattern. For both the Southern (Fig. 9) and Northern Hemispheres (Fig. 10), the pattern between response and forcing is fairly linear, as was true for the mechanically forced trials in RP07. The slopes of the best-fit lines are 44 days for the Southern Hemisphere and 20 days for the Northern Hemisphere. This encourages us to examine further the relationship between response and forcing using the fluctuation–dissipation theorem (see section 5).

In summary, the model runs with spatially concentrated reference temperature perturbations, maximized at the pole, produce annular mode–like patterns in response to the forcing. This suggests, as do the mechanically forced trials from RP07, that the annular modes are the preferred response of the model to the forcing.

However, as found by Son and Lee (2006) and RP07, the climatological response of the model is less simple when forcings with a significant tropical extent are considered. This highlights the extratropical nature of the annular mode dynamics, such as the eddy–mean flow feedback (which we discuss below), while processes occurring in the tropics may obscure or curtail the annular mode–like response.

## 4. Diagnoses using a zonally symmetric model

As in RP07 for cases which were mechanically forced, we consider results using a zonally symmetric model for the thermally forced trials presented in section 3 above. This allows us to consider the direct response to the forcing without changing the eddy fluxes compared to the response in trials for which both the direct forcing and the perturbed eddy fluxes have been prescribed.

The procedure is the same as that used by RP07. For each trial from the full model, two runs in the zonally symmetric model are conducted. In one case, the zonally symmetric model is run including the applied perturbation to *T*_{ref}, but keeping the same climatological eddy fluxes derived from the control run, to examine the direct effects of the applied thermal perturbation without eddy feedback changes. These eddy fluxes are calculated from the full model run and are prescribed to the zonally symmetric model, where they are not allowed to change with time. Then, the zonally symmetric model is run, including both the applied *T*_{ref} changes and the climatological eddy fluxes derived from the analogous full model run. In all cases, the forced results are compared to a run using the *T*_{ref} field and the climatological eddy fluxes calculated from the control run, which provides a good reproduction of the full model’s climate. The results shown below are averaged over the final 500 days of 2000-day model runs.

In Fig. 11, we consider the changes in zonal wind and temperature for a zonally symmetric model run including the directly applied forcing only and compare the results to a run including the eddy changes. These runs used *T*_{ref} forcing with a maximum amplitude of +4 K, but their results are representative of those from other trials.

As found by RP07 for mechanically forced cases, the direct response to the *T*_{ref} perturbations is clearly not annular mode–like, being monopolar in structure in each hemisphere and weaker than those found from the analogous full model runs. Including both the directly applied forcing and the changes in eddy flux divergences from the full model run results in a much better agreement. The zonally symmetric model is now able to capture the dipolar structure of the anomalies in each hemisphere, with a westerly anomaly and an easterly anomaly centered about the position of the old time-mean jet. Similarly, the direct forcing alone does not capture the temperature changes resulting from the full model. Although the polar change in temperature stemming from the forcing is reflected in the direct run, the dynamic warming or cooling in the midlatitudes is not. Again, eddy feedback is essential to reproduce the full model results.

## 5. Comparison of response and forcing strengths in a fluctuation–dissipation framework

As seen above, the strength of the response for trials with forcing confined to poleward of 45°, when considering the projection on the SVD pattern of temperature, appears fairly linear. The mechanically forced trials in RP07 likewise demonstrated a linear relationship between the projection of the response on the wind EOF and the projection of the forcing on the same pattern.

The linear forcing–response relationship suggests that the fluctuation–dissipation theorem is a fruitful avenue for further investigation of this problem. The theorem relates the forced response of a system to the unforced variability. Specifically, for systems in which the fluctuation–dissipation theorem holds, the response of the system, projected upon a mode obtained through the analysis, should scale linearly with the projection of the forcing, with the decorrelation time of the unforced mode indicating the slope of the relationship.

Introduced to the climate sciences by Leith (1975), the theorem had already been applied to problems in other disciplines of the physical sciences (e.g., Nyquist 1928; Kubo 1957; Kraichnan 1958). Since Leith (1975), a number of other authors have also made use of the theorem to study problems in the atmospheric sciences (e.g., Bell 1980; North et al. 1993; Dymnikov and Gritsun 2005).

As presented by Leith (1975), the fluctuation–dissipation theorem assumes linearity of the system and Gaussian statistics—conditions which Leith (1975) noted may not be satisfied in the real atmosphere. This analysis was recently expanded by Majda et al. (2005), who found a theoretical basis for what they termed a “quasi-Gaussian” approximation in which some of the assumptions could be relaxed.

We will pursue a fluctuation–dissipation analysis here, but our formulation of the problem will be somewhat different from that presented by Leith (1975); instead, we follow the work presented by Penland (1989). The basic assumption of Penland’s (1989) approach is that the variance in the control trial is stochastically forced, with the decorrelation time of the forcing being much less than that of the response.

Our mathematical formulation may be broken into three pieces. First, we reduce the system to one in which zonal wind is the sole variable. Second, we derive equivalent forcings for each case. Finally, we analyze the system by determining its eigenvectors and relate the projections of forcing and response.

**u**, representing the departure of the zonal mean flow from the basic state (defined as the time average of the control run), satisfies an equation of the type

Deviations from the basic state are small.

The zonal-mean flow satisfies gradient wind balance.

The terms involving the product of the basic state vertical velocity and the latitudinal gradient of perturbation static stability can be neglected. [Under quasigeostrophic scaling, these terms are

*O*(*R*) smaller than the others, where*R*is Rossby number; moreover, the term is found to be small for most locations in our trials, including the tropics.]The perturbations to the eddy flux divergences can be represented by terms of the form 𝗘

**u**, where the matrix 𝗘 is some function of the basic state, plus a stochastic term. [The stochastic term is then subsumed into the forcing term on the right of (1).] This assumption [which extends the approach of Lorenz and Hartmann (2001, 2003)] is perhaps the most restrictive of the assumptions because it presumes that(i) the eddy fluxes respond instantaneously to the mean flow changes (in view of the long characteristic time scales of the latter, this does not seem unreasonable) and

(ii) the eddy fluxes are insensitive to changes in mean static stability (or, alternatively, that such changes are too small to matter).

The forcing **f** in (1) is not simply the applied torque (expressed as acceleration). A localized torque such as we have been using affects **u** not only directly but also by driving a meridional circulation, which in turn affects **u** through angular momentum advection. A perturbed *T*_{ref} affects **u** in the same way. In fact, **f** actually represents what might be called the “Eliassen response” to the forcing. What is meant by this is the instantaneous acceleration induced by the forcing (thermal or mechanical) when **u** = 0—that is, the solution to (the nongeostrophic version of) the classic balanced vortex problem of Eliassen (1951). [This becomes obvious from (1): when **u** = 0, ∂**u**/∂*t* =** f**.] For the localized torques that we have used in RP07, we shall see that **f** turns out to be predominantly a vertically smoothed version of the torque.

The problem described by (1) is of course nothing more than the purely zonally symmetric problem discussed at length by Haynes et al. (1991) with the addition of eddy feedback. The operator in (1) is just 𝗕 = 𝗟 + 𝗘, where 𝗟 is the operator in the absence of eddy feedback. Lacking an analytical theory for the eddy feedback, 𝗘 is unknown, and therefore so is 𝗕. Therefore, we cannot attempt to solve (1) from first principles. Moreover, we have no basis a priori to assume that 𝗕 is symmetric.

**f**represents spatially and temporally stochastic forcing, a natural way to approach (1) is through the standard eigenvalue decomposition 𝗕 = 𝗩

**Λ**𝗪

^{T}, where

**Λ**is the diagonal matrix containing the eigenvalues and the matrices containing the left and right eigenvectors are biorthogonal (𝗩𝗪

^{T}= 𝗜). Then (e.g., von Storch et al. 1988; Penland 1989), if the lag covariance of

**u**is 𝗖

*= 〈*

_{τ}**u**(

*t*)

**u**

^{T}(

*t*−

*τ*)〉, where the angle brackets denote a time average, it follows that

*λ*of the unknown matrix 𝗕 from those of the known matrix 𝗚

*. Note that the eigenvalue*

_{τ}*λ*of each mode represents the inverse of the decorrelation time for that mode.

In principle, _{n}, *λ*_{n}, and _{n} may be complex; however, for the cases we study here, we obtain real values for these quantities.

Among other things, (5) is consistent with the key prediction of the fluctuation–dissipation theorem (Leith 1975), namely, that the slope of the relationship between the response in a given mode and the forcing of that mode is just the inverse of the mode’s decorrelation time in the stochastically forced case. In other words, the decorrelation time of the dominant mode is a measure of the sensitivity of the zonal flow to external perturbations. Because simplified GCMs of the kind used here and in many similar studies tend to exhibit decorrelation times that are much longer than those of the observed annular modes (Gerber and Vallis 2007; Gerber et al. 2007) it seems likely that such models exaggerate the response to external forcing.

The effective torque **f** is determined from a two-dimensional model that solves the elliptic equation (A11) in pressure coordinates for the anomalous streamfunction *χ _{E}*. This is then substituted into (A1);

*ω*= 0 is prescribed as the top boundary condition and

*υ*= 0 as the side boundary condition. The bottom boundary condition is

*ω*

_{p=ps}=

*dp*/

_{s}*dt*, where

*p*is surface pressure. This is linearized to provide a boundary condition at the fixed-pressure boundary of the 2D model, following Haynes and Shepherd (1989).

_{s}The effective torques for trials 4 and 5 from RP07, which are representative of the other cases, are shown in Fig. 12. They are plotted alongside the original torques. In both cases the effects of the overturning circulation have smoothed the forcing vertically. Rather than being concentrated in a bull’s eye, the effective torque is more barotropic. Although in each case the magnitude is still strongest near the level of the forcing, the strength there has been much reduced.

The effective torques for two thermally forced cases (10-K maximum warming and cooling) are shown in Fig. 13. They are shown alongside the buoyancy forcing in each case. This forcing is defined as *α _{b}* ∂

*b*/∂

_{e}*ϕ*, where

*b*is the applied perturbation buoyancy. This, of course, is related to the difference in reference temperature between the forced and control trials:

_{e}*b*=

_{e}*g*(

*T*

_{ref}−

*T*

_{ref,control})/

*T*

_{*}. Here,

*α*is a relaxation coefficient that is constant at (40 day)

_{b}^{−1}above 700 hPa but increases linearly to (4 day)

^{−1}at 1000 hPa. We choose 255 K for

*T*

_{*}.

In each case, the buoyancy forcing is concentrated at the ground (because of the increase in *α _{b}* closer to the surface) in the midlatitude and subpolar region. The effective torque resulting as a consequence of the applied buoyancy anomalies, however, has a much different shape. In both trials the main feature in both hemispheres is a broad region of effective torque in the midlatitudes of the free troposphere, with torques of opposite sign in smaller regions at the surface.

To find the leading POPs, we first use (2) to define 𝗚* _{τ}* from the lagged covariances of the control run and then determine its eigenvectors. The covariance matrices using the entire dataset are rather noisy, and we followed Penland (1989) and first reduced the number of degrees of freedom, while retaining most of the variance, by projecting

**u**onto a small number of EOFs. This procedure made the calculations both more efficient and more robust than using the full 960 × 960 covariance matrix.

Figure 14 shows the POPs, the leading eigenvectors of 𝗚* _{τ}* and of 𝗕. The eigenvectors are similar to the EOFs of the control run (Fig. 1). In each hemisphere, the leading POP describes a vertically coherent dipole centered about the position of the unforced time-mean jet. This offers an a posteriori justification for the projections of the responses and mechanical forcings onto the EOFs conducted by RP07. Although the EOFs themselves do not contain dynamical information, the spatial patterns describing the leading eigenvectors of the system’s dynamical operator strongly resemble the EOFs.

Nothing in the POP theory requires a specific choice of lag time *τ*, and indeed the patterns should be insensitive to this choice of lag. Accordingly, we calculated the POPs for several different choices of time lag and confirmed that our patterns are in fact insensitive to the choice of lag time when the covariance matrix is well posed.

The similarity between the leading POPs and the EOFs is promising, but there are two caveats of which we must make note. The first potential concern is a poorer separation of amplitude between the hemispheres for the POPs as compared to the EOFs. For the latter patterns, the features were always confined to one hemisphere, with virtually no residual amplitude in the opposite hemisphere. Here, however, there is a larger residual amplitude in the hemisphere that does not contain the primary pattern. This behavior was also noted for the eigenvectors calculated for other time lags, as well as for different choices of truncation for number of EOFs.

The second caveat is the ill-posedness of the larger covariance matrices of this system. The same problem was noted by Dymnikov and Gritsun (2005) in their fluctuation–dissipation analysis. For our system, the covariance matrix becomes ill-posed if it contains information on more than the first dozen EOFs, and the matrix also appears to be ill-posed for some time lags in the 8–12 EOF range. Fortunately these higher EOFs describe very little variance of the unforced run, so little information is lost by working on a reduced basis. In fact, the first four EOFs contain 85% of the variability in the zonal wind field, the first eight EOFs collectively describe 90% of the variability, and the first twelve EOFs contain 93% of the variability. Although it is mathematically necessary to reduce the field to a smaller set of EOFs, the retained EOFs still describe the vast majority of the full system’s variability.

Having obtained both the effective torques and the POPs, we are now able to project the wind responses as well as the effective torques onto 𝗪^{T}. These projections are shown for the Southern Hemisphere in Fig. 15 and for the Northern Hemisphere in Fig. 16. For each hemisphere, the behavior of the mechanically forced trials or the thermally forced trials, each considered separately, is qualitatively consistent with (5). In each case the relationship between forcing and response appears fairly linear. We emphasize that the strength of the response may be increased by one of two means—either by increasing the strength of the forcing or by improving the projection of the forcing on the POPs. The thermally forced trials selected here used forcings of similar shape, but the mechanically forced trials include runs with changes in the location of the forcing, as well as trials with changes to the amplitude of the forcing. The linearity of the result is consistent with similar studies (Gerber et al. 2007).

We also note that the time scales derived from the eigenvalues associated with the POPs compare favorably with the decorrelation time scales of the unforced annular mode patterns. The eigenvalues found through the POP analysis closely match the *e*-folding autocorrelation times of the patterns in each hemisphere—58 days in the Southern Hemisphere and 48 days in the Northern Hemisphere. This agreement is found for different choices of lag and truncation.

Although these aspects of the response-forcing comparison are consistent with the expectations of fluctuation–dissipation theory, as in Gerber et al. (2007), there are also aspects in which our results diverge from those expectations. Most notably, the mechanically and thermally forced trials fall about two separate lines. Although the relation between response and forcing projections is linear within each subset of trials, the mechanically forced trials appear about one line, whereas the thermally forced trials follow a second, distinct line. The difference in the forcing–response relationships is not expected from fluctuation–dissipation theory; we would expect a single sensitivity line from the biorthogonal basis of the GCM’s response operator.

We also note that the autocorrelation *e*-folding decay time scales (58 days in the Southern Hemisphere and 48 days in the Northern Hemisphere) are not well represented by the response–forcing curves. In both hemispheres, the mechanically forced trials follow a line with a shallower slope than the prediction from fluctuation–dissipation theory (in the Southern Hemisphere, 29 days; in the Northern Hemisphere, 16 days). In contrast, the best linear fit to the thermally forced trials is steeper than the fluctuation–dissipation prediction, with the thermally forced slope being 85 days in the Southern Hemisphere and 100 days in the Northern Hemisphere.

In summary, although the response of the system is qualitatively consistent with fluctuation–dissipation theory, significant quantitative differences are evident. In the final section of this paper, we will examine possible reasons for this discrepancy.

## 6. Discussion

We have conducted a series of thermally forced trials to determine further whether the annular modes are the preferred response of a simple model to generic forcing. Trials with thermal forcing confined poleward of 45° did feature annular mode–like responses. The responses were not as cleanly annular mode–like for trials with broader latitudinal forcing extending into the tropics, in which cases the dynamics of the Hadley cell appeared to become involved in the response.

We then examine the relationship between forcing and response in a fluctuation–dissipation context, using the thermally forced trials presented here and the mechanically forced trials from RP07 as our sample set. We project the zonal wind response and effective torque from each trial onto the principal oscillation patterns (POPs), the eigenvectors of the dynamical matrix of the system, rather than onto the statistical EOF or SVD patterns. Consistent with fluctuation–dissipation theory, the response increases linearly for either an increase in forcing strength or an improved projection of forcing on the POPs. But the results diverge quantitatively from the predictions of fluctuation–dissipation theory for reasons we do not fully understand.

First, we emphasize one important finding of this study—the similarity between the POPs and the EOF or SVD patterns of zonal wind. EOF analysis is statistical and not dynamical in nature, and thus although EOFs can tell us which pattern of variability is most likely to occur in a system, they cannot in and of themselves provide us with information on that system’s dynamics. In contrast, the POPs do provide this information because they are the leading eigenvectors of a dynamical operator.

The high degree of similarity between the leading POPs and EOFs offers an a posteriori explanation for the finding in RP07 and in section 3 here that response and forcing are linearly related when projected onto the EOFs. Given (5), this finding is then consistent with the theory.

For the thermally forced cases discussed in section 3, the linear response–forcing relationship could be seen solely as an indication of linearity in the perturbed dynamics because the different cases differ only in the amplitude of the imposed perturbations in *T*_{ref}. For the mechanically forced cases described in RP07, however, the different projections of the forcing onto the modes were in many cases a result of different locations of the applied torques, rather than of their magnitudes. In particular, a localized torque applied near the node of the leading EOFs (and hence of the leading POPs) produced a very weak response, again in accordance with (5) when one considers the vertically coherent nature of the effective torques and of the POPs.

In an attempt to understand the discrepancies between our results and the theory, we now consider the assumptions in the forcing–response relationship. Perhaps the broadest of these is the parameterization of the eddy feedback operators as linearly dependent on the instantaneous value of the perturbation zonal wind. Most significantly, this prevents variations in the static stability from affecting the eddy feedback processes. Thus, the mathematical framework would be insufficient if the static stability changes do materially affect feedback processes.

There is a difference in the changes in static stability computed for the mechanically and thermally forced trials. In the mechanically forced trials there is obviously no directly applied change to the static stability, although as temperature changes associated with the anomalous overturning circulation occur, the static stability may be altered through indirect processes. In the thermally forced trials, however, the directly imposed static stability anomalies are large, particularly near the tropopause, because the imposed *T*_{ref} anomaly is always zero in the stratosphere. The imposed temperature changes, although modified by the overturning circulation, are still clearly evident in the climatological temperature difference fields, so the thermally forced trials contain much larger changes in static stability than do the mechanically forced trials.

In our formulation, the anomalies in eddy divergences include a stochastic component assumed to have a much shorter time scale than the response; in the control run, this stochastic component constitutes the entire forcing. The long decorrelation time scale of the zonal wind response suggests that this assumption is appropriate. We have also assumed that the eddy feedback operators depend on the instantaneous, and not the lagged, values of zonal wind. Again, given the long decorrelation times of the model’s annular mode patterns, we believe this simplification is well justified.

Beyond the assumptions we made in the reduction of the dynamical system to a single variable, the fluctuation–dissipation analysis still requires that perturbations be small enough to allow for linear dynamics to hold. As noted by Leith (1975), a decrease of forcing strength in a model—the logical step if linearity of results is a concern—unfortunately works at cross-purposes with the desire to obtain a statistically robust result, which suggests stronger forcings. However, the results here do appear linear when considering mechanical forcings alone or thermal forcings alone.

The autocorrelation times of the annular modes in our setup (58 days in the Southern Hemisphere and 48 days in the Northern Hemisphere) are much longer than those found for the patterns on the earth (Feldstein 2000; Baldwin et al. 2003). Like Gerber et al. (2007), we note that the simulated patterns in a model setup with an unrealistically long autocorrelation time scale will be much more sensitive to the applied forcing than would the patterns in nature. While keeping these reservations in mind, the success of fluctuation–dissipation theory in other recent studies (e.g., Gritsun and Branstator 2007) indicates the theory is an attractive means by which to evaluate the relationship between the forcing and response of a climate model.

## Acknowledgments

We gratefully acknowledge Ed Gerber for his helpful discussions and two anonymous reviewers for their constructive comments. This work was supported by the National Science Foundation through Grant ATM-0314094.

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## APPENDIX

### Definition of the Effective Torque and Derivation of (1)

In this appendix, we present our reduction of the dynamical system to one whose state vector depends solely on the zonal wind. In doing so, we derive an effective forcing for each case, dependent on both the applied torque and the advection of angular momentum by an overturning response.

Here, *M* is the basic state absolute angular momentum per unit mass and *m* is the perturbation therefrom. The anomalous eddy momentum flux divergence is *d _{m}*; the anomalous eddy buoyancy flux divergence is

*d*. The torque per unit mass is

_{b}*h*. The dissipation coefficients for momentum and buoyancy are represented by

*α*and

_{m}*α*, respectively. The basic state buoyancy is represented as

_{b}*B*=

*g*(

*T*

_{basic}−

*T*

_{*}) /

*T*

_{*}, while the perturbation buoyancy is

*b*=

*g*(

*T*−

*T*

_{basic}) /

*T*

_{*}. The forcing of the buoyancy equation is the relaxation to the reference buoyancy; we write the reference buoyancy as

*B*=

_{e}*g*(

*T*

_{ref},

_{basic}−

*T*

_{*}) /

*T*

_{*}and the perturbation that constitutes the forcing as

*b*=

_{e}*g*(

*T*

_{ref}−

*T*

_{ref},

_{basic}) /

*T*

_{*}. We choose 255 K for

*T*

_{*}.

*χ*:

^{1}

*ϵ*.

*a*

^{−1}∂

*(A2) −*

_{ϕ}*ϵ*∂

*[*

_{p}*M*× (A1)], where the subscripts

*ϕ*and

*p*indicate partial differentiation with respect to that variable, and use (A5) to eliminate the terms containing time derivatives. Finally, substitution of

*χ*for

*υ*and

*ω*leaves a diagnostic equation for streamfunction:

*S*=

*κ*(

*B*+

*g*)/

*p*−

*B*.

_{p}*m*and

*b.*Those proportional to the buoyancy gradient

*b*can be expressed in terms of

_{ϕ}*m*through use of the balance equation (A5). As noted in the main text, we assume that the term

_{p}*a*

^{−1}Ω

*(*

_{ϕ}*b*−

_{p}*κb*/

*p*) is small; further, in our case

*α*is independent of

_{b}*ϕ*, and so the last term on the left of (A6) vanishes. We may then write (A6) in the form

*m*= 0) balanced response to the prescribed thermal and mechanical forcing in (A7). The inversion (A11) is subject to conditions of boundedness at the poles, and

*ω*= 0 at

*p*= 0; at the surface the condition

*ω*=

*dp*/

_{s}*dt,*where

*p*is surface pressure, is linearized and applied at a constant pressure surface, as in Haynes and Shepherd (1989).

_{s}*i*th grid point,

*χ*is the Eliassen streamfunction response to the imposed forcing, so 𝗳 as defined by (A15) and (A17) is the zonal acceleration in the Eliassen problem.

_{E}^{1}

As written here, *χ* does not have units of mass streamfunction, but the substitution *χ* → (*g*/2*π**a* cos*ϕ*)*ψ* returns this quantity.