## 1. Introduction

In the tropical stratosphere, the circulation is dominated by the quasi-biennial oscillation (QBO) of equatorial wind. In the tropical troposphere, it is characterized by the Hadley circulation, which is forced by convection and the release of latent heat. Historically, these two circulations have been regarded as independent.

In the classical theory of the QBO, the stratospheric circulation depends upon the tropospheric circulation only indirectly, through vertically propagating wave activity that is likewise forced by convection; see Baldwin et al. (2001) for a recent review. Their limited interaction in classical theory follows unsuccessful attempts to explain the stratospheric QBO from simultaneous variations in the troposphere (Andrews et al. 1987). The Hadley circulation, on the other hand, depends upon sea surface temperature (SST) through its regulation of organized convection.

In a global context, the circulations of the stratosphere and troposphere must be interdependent. They are coupled through transfers of mass associated with the residual mean circulation of the stratosphere, the Brewer–Dobson circulation. Air descending at high latitudes of the winter hemisphere enters the troposphere across the polar tropopause. Compensating it at lower latitudes is air ascending across the tropical tropopause, which returns air to the stratosphere at the same rate. As the compensating motion occurs at latitudes of tropospheric convection, to some degree, it must involve the Hadley circulation. Such influence has been found in GCM experiments, extending downward several kilometers beneath the tropopause (Thuburn and Craig 2000). In fact, observed changes of the Hadley circulation vary coherently with changes of the Brewer–Dobson circulation (Salby and Callaghan 2005).

Transfers of mass between the stratosphere and troposphere play a key role in year-to-year changes of dynamical structure, the associated circulation accounting for a significant fraction of the interannual variance (Fusco and Salby 1999; Newman et al. 2001; Hadjinicolaou et al. 2002; Hu and Tung 2002). Such changes operate coherently in the stratosphere and troposphere, sharing major features with the Arctic Oscillation (AO; Thompson and Wallace 1998; Baldwin and Dunkerton 1999, 2001; Salby and Callaghan 2004a). They are introduced through two major mechanisms that influence the residual mean circulation: 1) changes of Eliassen–Palm (EP) flux transmitted upward by planetary waves and 2) changes of equatorial wind associated with the QBO. The latter determines the critical line of planetary waves, which controls where the absorption of wave activity forces residual mean motion. Each mechanism introduces interannual changes over the winter hemisphere, compensated over the Tropics by coherent changes of opposite sign (Salby and Callaghan 2002, 2004a).

The involvement of the QBO makes possible another influence, one that varies cyclically. The 11-yr cycle of solar irradiance becomes important in the upper stratosphere, where ozone heating involves wavelengths shorter than 200 nm. At those wavelengths, UV irradiance varies significantly between solar minimum and solar maximum (WMO 1987; Lean 2000).

A signature of such changes is, in fact, visible in the QBO. The duration of equatorial westerlies and easterlies varies systematically between solar minimum and solar maximum (Quiroz 1981; Salby and Callaghan 2000; Soukharev and Hood 2001). Such changes represent a decadal modulation in the frequency of the QBO, which varies about its mean value by ∼20%. They are related to interannual changes of wintertime polar temperature (Labitzke and van Loon 1988; van Loon and Labitzke 1992; McCormack 2003; Salby and Callaghan 2004b, 2006).

The decadal modulation of the QBO inspires a reexamination of its interaction with the Hadley circulation, one that accounts for a cyclic variation in their relationship, which may cancel and, hence, be invisible in the long-term average. It is within this wider context that interaction between the QBO and the Hadley circulation is considered here. Following an overview of the analysis, section 3 presents evidence of interannual changes in the tropical troposphere that vary coherently with the QBO in the tropical stratosphere. Section 4 presents evidence of reciprocal changes in the tropical stratosphere, namely, those that vary coherently with the Hadley circulation in the tropical troposphere. For each, the relationship between changes in the tropical stratosphere and troposphere reverses on the time scale of a decade. The systematic swing in their relationship is interpreted in section 5, in light of the cyclic variation of UV irradiance.

## 2. Covariance analysis

Daily reanalyses from the National Centers for Environmental Prediction (NCEP) between 1955 and 2000 provide dynamical structure upward to 10 mb (Kalnay et al. 1996). Prior to 1979, analyses rely upon the radiosonde network, the same measurements in which the QBO was discovered (see, e.g., Baldwin et al. 2001). Afterward, NCEP reanalyses are bolstered by satellite measurements. Although they provide global coverage, nadir-viewing satellite measurements have deep weighting functions that limit vertical resolution (see, e.g., Huesmann and Hitchman 2003).

To focus on interannual variability, the daily record has been consolidated into annual records of wintertime-mean [December–February (DJF)] temperature, height, and motion. The climatological mean of individual properties is then removed, leaving the interannual anomaly. The resulting 3D record describes interannual changes of the wintertime circulation over four decades, plus additional years that can be used to buttress statistical confidence (section 2b).

### a. Relationship between field properties

To discriminate the QBO, field properties are high-pass filtered. Time series are first convolved with a Hamming window that spans five consecutive winters. The deviation from this running mean then constitutes a high-pass filter. Having a smoothly varying frequency response, the filter discriminates periods shorter than 5 yr (see, e.g., Båth 1976). In the tropical stratosphere, it captures most of the interannual variance, which is concentrated about periods of the QBO. In the tropical troposphere, it captures interannual variance associated with the so-called tropospheric biennial oscillation (TBO), which involves similar periods (Li et al. 2001).

Analogous operations define the running cross correlation between two field properties. The running Hamming window isolates fluctuations in those properties locally in time: they are nonzero only within a neighborhood of years about the window’s center. The correlation between the windowed properties then contains contributions from only those years. As the window shifts, so do contributions to the cross correlation; see Salby and Callaghan (2004b) for a more detailed development.

The running correlation measures the instantaneous relationship between fluctuations in those properties, specifically, the cosine of their relative phase. It also represents instantaneous contributions to the standard correlation between the properties, which reflects their average relationship. In fact, when the running correlation is integrated over time, it yields a time-mean correlation that is very close to the standard correlation.

A systematic variation in the relationship between two properties may be reflected in a cyclic swing of their running correlation (e.g., between positive and negative values). Such behavior can occur only if two criteria are satisfied: (i) the two properties are coherent instantaneously (e.g., locally in time, they involve similar periodicity), and (ii) a systematic variation is present in one or both of the properties. How systematically their relationship varies, in turn, is measured by its standard correlation to a deterministic reference signal. For example, if the relationship is strongly correlated to an oscillation of specified period, it must cycle deterministically at that period.

Although a cyclic swing in the relationship between two properties may reflect a systematic variation, it need not. For a record of fixed length, there is a small but finite probability that their relationship will vary cyclically simply through chance (see, e.g., Gershunov et al. 2001). Isolating variations that are truly systematic requires that we discriminate to those that will remain correlated to the deterministic reference signal as the record is extended.

### b. Statistical confidence

The reliability of observed variations is evaluated through Monte Carlo simulation. We define a “null hypothesis” as a cyclic variation in the relationship between two properties that actually occurred through chance. The probability that their relationship tracks the deterministic reference signal over four decades is then required to be small. Under those circumstances, the null hypothesis can be rejected with a high degree of confidence; see Salby and Callaghan (2004b) for a more detailed development.

The same operations that are performed on the observed record are performed on field properties that are generated randomly. The NCEP record is then replaced by a large stochastic ensemble of records. In it, we evaluate the probability that, over four decades, the relationship between two properties tracks the deterministic reference signal simply through chance. The associated probability measures the confidence with which the null hypothesis can be rejected.

The procedure is illustrated by the running correlation between a stochastic property and 10-mb equatorial wind, *u*^{10}_{EQ} (observed). From their running correlation, we calculate the standard correlation to a decadal oscillation, which serves as the deterministic reference signal. Of some 400 000 stochastic realizations, less than 10% achieve a standard correlation to the decadal oscillation that exceeds 0.40. A correlation of 0.40 is therefore significant at the 90% level.^{1}

Although high, 90% probability is not adequate to reject the null hypothesis with certainty. The confidence level is raised further by requiring additional criteria to be satisfied, namely, by records that are shifted. Applied to each extended record is a moving window, four decades long. Shifting the observation window successively yields records that are each four decades long, but which begin at successively later years.

If two properties have a relationship that is truly systematic (e.g., in the observed records), then shifting the window introduces new data at the tails of the records that maintain the existing relationship. The standard correlation of the relationship to the deterministic reference signal is therefore preserved. If, on the other hand, two properties have a relationship that occurred merely through chance (e.g., in the stochastic records), shifting the window introduces new data at the tails of the records that have only a small probability of maintaining the existing relationship. The standard correlation to the deterministic reference signal therefore deteriorates with increasing shift.

Requiring the standard correlation with the deterministic reference signal to be maintained simultaneously under *all shifts* of the four-decadal record (i.e., jointly in records over 1955–96, 1956–97, and so forth) sharply reduces the probability of a chance correlation to that reference signal; see the appendix of Salby and Callaghan (2005) for an analytical treatment. Of 400 000 realizations, each shifted successively by an integral number of years, less than 0.5% yield a correlation to the deterministic reference signal that exceeds 0.40 under all shifts of the four-decadal record. Observed records that satisfy this more restrictive criterion are therefore significant at the 99.5% level.

## 3. Interannual variability coherent with the QBO

Plotted in Fig. 1a is the running correlation between 10-mb equatorial wind, a proxy for the QBO, and wintertime temperature at 25°N and 70 mb, in the subtropical stratosphere (solid). The running correlation between *u*^{10}_{EQ} and *T*^{70}_{25N}, lagged by 1 yr, swings repeatedly from positive toward negative values. It does so near the beginning of each of the four decades, reflecting a cyclic reversal in the relationship between stratospheric temperature and *u*^{10}_{EQ}.

That reversal is visible even in the raw time series of the two properties, plotted in Fig. 1b (normalized). Here, *T*^{70}_{25N} (solid) is seen to fluctuate *in phase* with *u*^{10}_{EQ} (dashed) in the middle of the decades, for example, around 1956, 1965, 1974, and again around 1986. However, *T*^{70}_{25N} fluctuates *out of phase* with *u*^{10}_{EQ} near the beginning of each decade, around 1960, 1970, 1982, and again around 1991. Those are the years when the running correlation deviates toward negative values.

The cyclic swing of the relationship in Fig. 1a is nearly opposite to the variation of 10.7-cm solar flux *F _{s}*, lagged by 1 yr (dashed). It serves as a deterministic reference signal. The two curves in Fig. 1a achieve a standard correlation of −0.83, which is highly significant.

The cyclic form of the running correlation implies a cancellation among instantaneous contributions to the average correlation. The standard correlation should therefore be small. In fact, the standard correlation between *T*^{70}_{25N} and *u*^{10}_{EQ} is only 0.15—not even significant.^{2}

Figure 2 plots analogous information, but for temperature at 15°S and 400 mb, inside the tropical troposphere. It coincides with the descending branch of the Hadley circulation. Temperature at this site has a running correlation with *u*^{10}_{EQ} (Fig. 2a) that reproduces much of the cyclic behavior evident in the stratosphere (Fig. 1a), but out of phase. The running correlation between *T* ^{400}_{15S} and *u*^{10}_{EQ} swings repeatedly from negative toward positive values. The reversal in their relationship is visible even in the raw time series (Fig. 2b). Here, *T* ^{400}_{15S} and *u*^{10}_{EQ} fluctuate *in phase* near the beginning of the decades, but *out of phase* in the middle of the decades.

The swing in their running correlation (Fig. 2a) varies nearly in phase with *F _{s}*. Exceptional is the swing to positive

*F*during the early 1980s, which is mirrored only weakly in the running correlation. Those anomalous years coincide with a major El Niño, which disrupted the normal seasonality of tropospheric temperature. Occurring simultaneously was the major eruption of El Chichón. Aerosol ejected by it detectably altered diabatic heating and upwelling in the tropical stratosphere (WMO 1995). Contemporaneous with those disturbances was a disruption in the cyclic modulation of QBO frequency, from the more regular pattern apparent in other years (cf. Fig. 4 of Salby and Callaghan 2000). The deviation in Fig. 2 during the early 1980s limits the standard correlation with

_{s}*F*to ∼0.60. Although smaller than the correlation to

_{s}*F*in the stratosphere, it is still highly significant. Excluding the early 1980s yields a standard correlation with

_{s}*F*close to 0.80—as large as in the stratosphere.

_{s}As in the stratosphere, the cyclic form of the running correlation in Fig. 2 implies a cancellation among instantaneous contributions to the average correlation. In fact, the standard correlation between *T* ^{400}_{15S} and *u*^{10}_{EQ} is only 0.26—not significant. Ordinarily, this would imply instantaneous contributions to the standard correlation that vary randomly, indicating the absence of a systematic relationship between *T* ^{400}_{15S} and *u*^{10}_{EQ}. However, the cyclic variation in Fig. 2, supported by a similar variation in the stratosphere (Fig. 1), indicates that something else may be going on. The standard correlation between tropospheric temperature and the QBO is small—not through absence of a systematic relationship between *T* ^{400}_{15S} and *u*^{10}_{EQ}, but through a cyclic reversal in their relationship.^{3}

The structure of changes operating coherently with *u*^{10}_{EQ} and *F _{s}* has been composited by renormalizing the product correlation: the swing in running correlation between

*T*and

*u*

^{10}

_{EQ}times its standard correlation to

*F*, by the corresponding temperature variance. The associated covariance represents the instantaneous projection of temperature variance onto

_{s}*u*

^{10}

_{EQ}(

*t*) and then its overall projection onto

*F*(

_{s}*t*); see Salby and Callaghan (2004b) for a more detailed discussion. Plotted in Fig. 3 is anomalous wintertime temperature that varies coherently with

*u*

^{10}

_{EQ}and

*F*(contoured). Superimposed is the associated significance, at the 99.5% and 99.9% levels (shaded). Figure 3 represents wintertime temperature that fluctuates interannually with

_{s}*u*

^{10}

_{EQ}(open circle), but gradually drifts into and out of phase with

*u*

^{10}

_{EQ}per Figs. 1 and 2. Anomalous temperature during individual years follows by scaling values in Fig. 3 by the product of anomalous

*u*

^{10}

_{EQ}and

*F*during those years, each normalized by the respective standard deviation.

_{s}In the lower stratosphere, cold anomalies straddle the equator. They are strongly coherent with *u*^{10}_{EQ} and *F _{s}*. Those anomalies reflect the residual circulation of the equatorial QBO, which induces vertical motion to either side of the equator. Intensified in the winter hemisphere, they also reflect the QBO’s impact upon the Brewer–Dobson circulation (Salby and Callaghan 2006). The hemispherically symmetric structure resembles temperature structure that is recovered through direct correlation with the solar cycle (e.g., van Loon and Labitzke 1998, 1999; Crooks and Gray 2005). However, the anomalies in Fig. 3, which are defined directly from the QBO, are stronger and more coherent. The correlation with

*F*and its hemispheric symmetry likewise strengthen if the record is sampled around solstice and during extremal phases of the QBO (van Loon and Labitzke 2000; Labitzke 2003).

_{s}The cold anomalies in Fig. 3 are a signature of adiabatic cooling and anomalous upwelling. They extend downward into the extratropical troposphere, especially in the winter hemisphere. That signature is excluded below ∼200 mb and equatorward of 30°, where thermal structure is shaped by convection and the Hadley circulation. There, anomalous temperature reverses sign. When positive, it too is strongly coherent with *u*^{10}_{EQ} and *F _{s}*. The warm anomaly inside the tropical troposphere resembles structure that accompanies interannual changes of the Brewer–Dobson circulation (Salby and Callaghan 2005). The latter, like the tropospheric structure in Fig. 3, involves interannual variance that operates on periods shorter than 5 yr.

The warm anomaly in Fig. 3 and attending coherence with *u*^{10}_{EQ} and *F _{s}* reflect an intensification of the Hadley circulation: during those winters when upwelling in the subtropical stratosphere is intensified, so are upwelling and latent heating in the equatorial troposphere. They are accompanied in the subtropical troposphere by intensified downwelling and adiabatic warming. Both imply anomalously warm temperature in a neighborhood of the equator. According to Fig. 3, these features of the Hadley circulation vary coherently with the QBO but gradually drift into and out of phase with

*u*

^{10}

_{EQ}on the time scale of a decade.

The cyclic reversal in the relationship between *T* ^{400}_{15S} and *u*^{10}_{EQ} and its coherence with *F _{s}* reappear at lags between those properties of ±5–6 yr, ±10–12 yr, and so forth. Hence, the correlation also oscillates with lag, on the time scale of a decade. The cyclic dependence on lag reflects a decadal swing in the relationship between

*T*

^{400}

_{15S}and

*u*

^{10}

_{EQ}, as would result from a modulation of one or both of those properties.

Analogous behavior appears in properties that directly characterize the Hadley circulation. Plotted in Fig. 4 is the running correlation between *u*^{10}_{EQ} and cross-equatorial motion in the upper troposphere and lower troposphere, advanced by 6 yr. At 200 mb (Fig. 4a), cross-equatorial motion reflects the poleward branch of the Hadley circulation. It is driven by outflow from deep convection that is positioned south of the equator during northern winter. The running correlation between *υ*^{200}_{EQ} and *u*^{10}_{EQ} (solid) swings from negative toward positive values in each of the four decades. Implied is a cyclic reversal in their relationship. It varies nearly *in phase* with *F _{s}*, after the latter has been shifted by 6 yr (dashed). The relationship between

*υ*

^{200}

_{EQ}and

*u*

^{10}

_{EQ}then achieves a standard correlation with

*F*of 0.80. By contrast, the standard correlation between

_{s}*υ*

^{200}

_{EQ}and

*u*

^{10}

_{EQ}is only 0.02. It has been virtually erased by the cyclic reversal of instantaneous contributions.

At 925 mb (Fig. 4b), cross-equatorial motion reflects the equatorward branch of the Hadley circulation and the trade winds. The running correlation between *υ*^{925}_{EQ} and *u*^{10}_{EQ} likewise swings from positive toward negative values in each of the four decades. Implied is a similar reversal in their relationship. It varies nearly *out of phase* with *F _{s}*, after the latter has been shifted by 6 yr. The relationship between

*υ*

^{925}

_{EQ}and

*u*

^{10}

_{EQ}then achieves a standard correlation with

*F*of −0.86. By contrast, the standard correlation between

_{s}*υ*

^{925}

_{EQ}and

*u*

^{10}

_{EQ}is only −0.03.

Plotted in Fig. 5 is the structure of anomalous *υ* that varies coherently with *u*^{10}_{EQ} and *F _{s}*. Meridional motion is strongly coherent with

*u*

^{10}

_{EQ}and

*F*in the upper troposphere and again in the lower troposphere, where it has opposite sign. Positioned just south of the equator, the vertical dipole in

_{s}*υ*has the same structure as the Hadley circulation during northern winter. Analogous structure accompanies changes of the Brewer–Dobson circulation (Salby and Callaghan 2005). The associated changes reflect ∼15% of climatological-mean

*υ*in the upper troposphere, but as much as half of the interannual variance. Implied by Figs. 4 and 5 are interannual changes of the Hadley circulation that vary coherently with the QBO, but only over durations shorter than a decade. Over longer durations, their average relationship cancels through cyclic reversals that track the variation of

*F*and ozone heating in the upper stratosphere.

_{s}## 4. Interannual variability coherent with the Hadley circulation

Figure 6 plots the running correlation between cross-equatorial motion at 925 mb, a proxy for the Hadley circulation, and zonal wind at 2.5°N and 70 mb, lagged by 10 yr. The running correlation between *u*^{70}_{2.5N} and *υ*^{925}_{EQ} (solid) swings from negative toward positive values in each of the four decades. It closely tracks *F _{s},* after the latter has been shifted by half a solar cycle (dashed). The two then achieve a standard correlation of 0.88. The cyclic reversal in the running correlation erases the average correlation between

*u*

^{70}

_{2.5N}and

*υ*

^{925}

_{EQ}. Accordingly, their standard correlation is smaller than −0.01.

The corresponding structure of *u* that varies coherently with *υ*^{925}_{EQ} and *F _{s}* is plotted in Fig. 7. As before, the anomaly during individual years follows by scaling values in Fig. 7 by the product of anomalous

*υ*

^{925}

_{EQ}and

*F*each normalized by the respective standard deviation. Anomalous

_{s},*u*is strongly coherent in the tropical stratosphere. Nearly symmetric about the equator, it has the same form as the QBO of zonal wind. The magnitude of anomalous

*u*there approaches 4 m s

^{−1}, about 20% of the amplitude of the QBO. Flanking it in the winter subtropics is a weaker anomaly of opposite sign. Although concentrated in the stratosphere, anomalous

*u*operates coherently with cross-equatorial motion in the lower troposphere (open circle). Their relationship, however, reverses cyclically on the time scale of a decade (Fig. 6).

The cyclic variation also appears in the running correlation against cross-equatorial motion at 200 mb (not shown), likewise a proxy for the Hadley circulation. Figure 8 plots the corresponding structure of anomalous *u*, lagged by 12 yr: anomalous zonal wind that varies coherently with *υ*^{200}_{EQ} and *F _{s}*. Anomalous

*u*again maximizes in the tropical stratosphere, where it is strongly coherent with

*υ*

^{200}

_{EQ}and

*F*Symmetric about the equator, it too has the form of the QBO. There, the running correlation behaves similarly to that in Fig. 6, achieving a standard correlation with

_{s}.*F*of 0.79.

_{s}Cross-equatorial motion in the upper troposphere (*υ*^{200}_{EQ}) is forced by divergence, which reflects the outflow from organized convection. The latter, in turn, strongly influences the tropical tropopause. Plotted in Fig. 9 is the running correlation between equatorial divergence at 200 mb and equatorial temperature at 100 mb, a proxy for the tropical tropopause. Like other properties of the tropical circulation, the running correlation between *T*^{100}_{EQ} and (**∇** · ** υ**)

^{200}

_{EQ}swings between negative and positive values in each of the decades. It varies nearly in phase with

*F*but less coherently than does the running correlation with stratospheric properties. The two curves in Fig. 9 achieve a standard correlation of 0.70, still strongly significant.

_{s},Figure 10 presents the structure of anomalous temperature that varies coherently with (**∇** · ** υ**)

^{200}

_{EQ}and

*F*. Anomalous temperature is strongly coherent in a neighborhood of the tropical tropopause. Of order 1 K, it represents ∼20% of the rms interannual anomaly (Reid and Gage 1996; Salby et al. 2003). A secondary region of strong coherence appears at the highest levels, in the subtropics of the winter hemisphere.

_{s}Interannual changes of the tropical tropopause, like those in Fig. 10, can be induced directly by the equatorial QBO. Through changes of static stability, the QBO can influence thermal structure that defines the tropopause, as well as cold cloud associated with convective overshoots (Reid and Gage 1985; Gray et al. 1992; Collimore et al. 2003). Implied is a local interaction in the Tropics between the QBO and the Hadley circulation.^{4}

The QBO and the Hadley circulation can also influence one another through global interaction. The QBO influences downwelling over the winter pole (Holton and Tan 1980; Labitzke 1982; Tung and Yang 1994), which transfers mass to the troposphere across the polar tropopause. That downwelling must be compensated at lower latitudes by anomalous upwelling, which returns mass to the stratosphere across the tropical tropopause. However it is introduced, the coherence between the tropical stratosphere and tropical troposphere implies an interaction between the Hadley circulation and the QBO overhead, the latter being sensitive to the variation of UV irradiance.

## 5. Conclusions

The QBO’s relationship to the Hadley circulation reverses on the time scale of a decade, as does its relationship to the wintertime polar stratosphere. The cyclic reversal in the relationship between the QBO and the Hadley circulation has two important implications: 1) Interannual changes of one circulation operate coherently with changes of the other, reflecting their interaction. 2) At least one circulation is influenced by a decadal variation, perhaps related to the variation of UV irradiance. The latter need not influence both circulations. So long as interannual changes in the tropical stratosphere and troposphere involve the same periods, enabling them to be instantaneously coherent, it is sufficient that the stratospheric circulation alone be sensitive to the decadal variation of UV irradiance.

The absence of a standard correlation between the QBO in the tropical stratosphere and the Hadley circulation in the tropical troposphere follows, not from incoherent (e.g., random) changes between those circulations, but from a cyclic reversal in their relationship. The latter results in a cancellation among instantaneous contributions, erasing their average correlation. Behavior analogous to that presented here would follow, were data in the tropical stratosphere and troposphere grouped according to those periodic intervals when their instantaneous correlation is of one sign.

Monte Carlo simulations indicate that the cyclic reversal in the relationship between the QBO and the Hadley circulation is strongly significant. Varying on the time scale of a decade, their relationship tracks the cyclic variation of UV irradiance, which modifies ozone heating in the upper stratosphere. It is there that the phase of the QBO is set, through interaction with the seasonal cycle [e.g., with the semiannual oscillation (SAO)].

It should be underscored that the implied interaction between the tropical stratosphere and troposphere refers, not to leading order, but to changes about it. Involving periods shorter than 5 yr, their interaction includes the QBO in the stratosphere, as well as a biennial oscillation that accompanies it (Baldwin and Dunkerton 1998; Salby et al. 1997; Baldwin et al. 2001; Salby and Callaghan 2006). It also includes the TBO in the troposphere, which operates on similar periods. As both represent a significant fraction of the interannual variance in the upper troposphere, the results suggest that, at least over durations shorter than a decade, these forms of interannual variability are related.

The tropical stratosphere and troposphere can interact locally through the tropical tropopause. Its modulation by the equatorial QBO, which is sensitive to the variation of UV irradiance, is one mechanism through which the decadal variation of ozone heating can be conveyed to the tropical troposphere. Another involves global interaction between the stratosphere and troposphere, via transfers of mass. In that regard, the equatorial QBO also influences the Brewer–Dobson circulation. The latter transfers stratospheric air into the polar troposphere with a compensating transfer of tropospheric air into the tropical stratosphere.

Support for the latter mechanism comes from tropospheric structure that has been found here to vary coherently with the QBO, but whose relationship to it reverses with *F _{s}* (Fig. 3). It has the same form as tropospheric structure that varies coherently with changes of the Brewer–Dobson circulation (Salby and Callaghan 2005). In each, the covariance between the stratosphere and troposphere is concentrated at periods shorter than 5 yr. A decadal modulation of the Brewer–Dobson circulation, introduced through the QBO, would be accompanied by a modulation of the mass flux into the troposphere over the winter pole. That would require a compensating modulation of the mass flux returned to the stratosphere over the Tropics, which, in turn, would require the involvement of the Hadley circulation.

The authors are grateful for constructive remarks provided during review. Figures were produced by Jackie Gratrix. This work was supported by NSF Grant ATM-0127671.

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^{1}

If both field properties are stochastic, a correlation to the decadal oscillation of 0.40 is significant at the 99.99% level.

^{2}

The standard correlation between two properties reflects the projection of their running correlation onto a reference signal of zero frequency. The latter is orthogonal to the nonzero frequency that characterizes *F _{s}.* Consequently, as the running correlation between the properties becomes strongly coherent with

*F*, the standard correlation of those properties must approach zero.

_{s}^{3}

Analogous behavior operates in the Madden–Julian oscillation (MJO) of the tropical troposphere (Madden 1986). In the MJO, the relationship between zonal and cross-equatorial motion reverses during the annual cycle, leading to a cancellation in their average relationship. The standard correlation then implies no relationship between those properties. However, grouping data in individual seasons yields an instantaneous correlation that is strong, but that reverses systematically during the annual cycle.

^{4}

It is noteworthy that the relationship between cold cloud and the QBO appears to reverse on the time scale of a decade (Collimore et al. 2003). Derived independently from the record of outgoing longwave radiation (OLR), that behavior is broadly consistent with the reversal in Fig. 4.