## 1. Introduction

The circulation of the stratosphere varies from one year to the next. The interannual anomaly (deviation from the climatological average) is closely related to changes of the residual mean circulation (Fusco and Salby 1999; Newman et al. 2001; Hu and Tung 2002). Forced by planetary waves, the residual circulation regulates wintertime temperature through downwelling and adiabatic warming. It also regulates wintertime ozone through poleward transport from its chemical source in the Tropics. Interannual changes of North Polar temperature *T*_{NP} and Northern Hemisphere ozone are, in fact, strongly coherent with anomalous forcing of the residual circulation (ibid.; Salby and Callaghan 2002; Hadjinicolaou et al. 2002). The latter is characterized by the change of upward Eliassen–Palm (EP) flux from the troposphere, which measures the anomalous momentum transmitted upward by planetary waves.

An analogous influence is exerted by the quasi-biennial oscillation (QBO) of equatorial wind *u*_{EQ} (Holton and Tan 1980; Labitzke 1982). It too operates coherently with interannual changes of *T*_{NP}. The QBO determines the position of the critical line (*u*

Jointly, changes of upward EP flux from the troposphere and of equatorial wind represent anomalous forcing of the residual circulation. These two influences account for much of the interannual variance of wintertime temperature and ozone (Salby and Callaghan 2002). In fact, the structure of anomalous temperature bears the imprint of the residual circulation: a signature of anomalous downwelling and adiabatic warming over the winter pole, compensated at lower latitudes by a signature of anomalous upwelling and adiabatic cooling.

The involvement of the QBO opens the door to an auxiliary influence. The 11-yr variation of solar irradiance assumes increased importance in the upper stratosphere, where ozone heating involves wavelengths shorter than 200 nm. Unlike longer wavelengths, UV irradiance at short wavelengths changes significantly between solar min and solar max (e.g., WMO 1987; Lean 2000). Implied are analogous changes of ozone heating, which shapes thermal structure and the global circulation. A signature of such changes is, in fact, inherent in wintertime polar temperature (Labitzke and van Loon 1988, 1996). An 11-yr variation emerges conspicuously when *T*_{NP} is stratified against the equatorial QBO—even though no systematic variation is evident in the raw record of polar temperature.

A clue to the origin of such changes comes from the equatorial QBO, which itself varies with solar activity (Quiroz 1981; Salby and Callaghan 2000; Soukharev and Hood 2001). Equatorial wind in the lower stratosphere undergoes a systematic modulation of frequency, one that tracks the 11-yr variation of UV irradiance. Near solar min, the downward migration of westerlies and easterlies, characteristic of the QBO, stalls. This prolongs the duration of westerlies below 30 mb and easterlies overhead, reducing the QBO's frequency. The systematic modulation of frequency influences how long *u*_{EQ} of one sign is maintained during winter, when planetary waves disturb the polar-night vorted and make it sensitive to equatorial wind.

During winter, midlatitude westerlies support planetary wave propagation, which couples the polar and equatorial stratosphere. Those months comprise the “disturbed season,” when the vortex is weakened by stratospheric warmings. Figure 1 plots, as a function of year, the number of winter months during which 30-mb *u*_{EQ} is westerly and easterly [adapted from the RAOB, analysis of Salby and Callaghan (2000)]. Near solar min (in the middle of the decades), *u*_{EQ} of one sign persists throughout the winter season: It occupies all four months of the disturbed season; *u*_{EQ} then persists with opposite sign throughout the next winter season. Wintertime-mean *u*_{EQ} thus alternates between opposite extremes—biennially. However, near solar max (near the beginning of each decade), equatorial wind changes sign during the winter season: *u*_{EQ} of one sign occupies only part of the disturbed season. Equatorial wind is thus both westerly and easterly during one winter season and likewise during the next. Wintertime-mean *u*_{EQ} is then weak and it changes only gradually between consecutive years.

The polar vortex during late winter is shaped by planetary waves and their interaction with equatorial wind throughout the preceding months of winter. The systematic variation of *u*_{EQ} between solar min and solar max should then produce an analogous variation of the polar-night vortex. According to Fig. 1, wintertime-mean *u*_{EQ} favors a biennial swing in the state of the vortex near solar min, but a more gradual variation near solar max.

Interannual changes of the polar-night vortex are, in fact, punctuated by a biennial oscillation (BO). Corresponding to a frequency of 0.50 cpy, the BO is manifested in *T*_{NP}, as well as in potential vorticity (PV) (Salby et al. 1997; Baldwin and Dunkerton 1998). Both reveal a component that alternates between consecutive winters. The BO, in fact, accounts for almost as much interannual variance of *T*_{NP} as the QBO. It is separated in frequency from the QBO (which has a mean frequency of 0.41 cpy) by 0.09 ≅ 11 yr^{−1}. The BO is closely related to signatures of the solar cycle. If it is filtered out, evidence of a systematic 11-yr variation disappears.

Below, we employ a long record of National Centers for Environmental Prediction (NCEP) reanalyses to isolate systematic variations of the stratospheric circulation and determine what relationship they have to the 11-yr variation of UV irradiance. Following a description of the data and various analyses, sections 3 and 4 identify systematic variations that operated coherently over the last four decades with solar activity. Section 5 then validates those variations against Monte Carlo simulations.

## 2. Data and analysis

### a. The atmospheric record

Daily reanalyses from NCEP between 1955 and 2000 provide records of the global circulation upward to 10 mb (Kalnay et al. 1996). Those records describe behavior over four complete decades, plus additional years that are used to bolster statistical confidence (section 5). They have been consolidated into monthly mean records of temperature, height, and wind. The resulting 3D distributions are used to isolate changes that varied systematically over four decades.

The analyzed fields derive from a wide range of operational measurements. They are based initially on ground-based RAOBs that are distributed nonuniformly over the earth. In later decades, they also include satellite measurements, which have complete horizontal coverage but limited vertical resolution. The observations are married with a forecast model, from which the analyzed 3D distributions are produced.

Analysis error (e.g., introduced by changes in the operational network and the introduction of satellites) limits the accuracy of the NCEP record. However, such error can only interfere with systematic variations. It cannot produce them. A definitive assessment of analysis error is made difficult by the heterogeneous nature of the data and analysis procedure. Instead, we appeal to the deterministic nature of UV irradiance, which varies predictably over the four decades. In tandem with Monte Carlo simulations, this feature is used to establish the reliability of systematic variations that operate coherently with solar flux *F*_{s}. The deterministic nature of UV irradiance enables the confidence in such variations to be increased dramatically (section 5).

### b. Analysis for systematic variations

A systematic variation can manifest itself at low frequency (LF), where it appears as a gradual drift of a field property *ψ*(*t*) that simply tracks *F*_{s}(*t*). This represents a direct or *linear response* to the 11-yr variation of UV irradiance. A systematic variation can also manifest itself at high frequency (HF). Involving changes between neighboring years, the high-frequency component will be treated synonymously with “interannual variability.” The dependence on *F*_{s} is then more complex, representing a generally *nonlinear response* to the 11-yr variation of UV irradiance (e.g., Dunkerton and Baldwin 1992). It can occur through interaction with the annual cycle, for example, through a mechanism that is restricted to winter months (Salby et al. 1997). At high frequency, a systematic variation can assume the form of an amplitude modulation and/or a frequency modulation of interannual variability (e.g., like the one in Fig. 1 manifested by the equatorial QBO).

*ψ*into low- and high-frequency components:

*ψ*

*t*

*ψ*

_{LF}

*t*

*ψ*

_{HF}

*t*

*ψ*(

*t*). It constitutes a low-pass filter that discriminates to periods longer than about 5 yr (see, e.g., Båth 1976), which defines

*ψ*

_{LF}

*t*

*ψ*

*t*

*ψ*

_{HF}

*t*

*ψ*

*t*

*ψ*

*t*

*ψ*

_{LF}and

*ψ*

_{HF}account for all of the variance. Consequently, they decompose

*ψ*into changes operating on long and short time scales.

*χ*of low- and high-frequency variance. A diagnostic of LF variance is provided directly by

*ψ*

_{LF}(

*t*). For HF variance, diagnostics are considered for both amplitude modulation and frequency modulation. Amplitude modulation is described by the 3-yr running deviationwhere Δ

*t*= 1 yr and extrema refer to values over the three neighboring years. Then ‖

*ψ*

_{HF}‖(

*t*) measures the instantaneous amplitude of interannual variability. Frequency modulation is described by the running cross-correlation between

*ψ*

_{HF}(

*t*) and a reference time series Γ(

*t*):

*c*[

*ψ*

_{HF}, Γ](

*t*). The reference time series is chosen to reveal how

*ψ*

_{HF}depends upon another field property, for example, how temperature depends upon equatorial wind, which influences the residual circulation. In this capacity, Γ(

*t*) serves as an interannual clock, against which the phase of

*ψ*

_{HF}(

*t*) is measured. Then

*c*[

*ψ*

_{HF}, Γ](

*t*) represents the cosine of the instantaneous phase between the two records. Like the amplitude of interannual variability, it varies over the four decades.

Collectively, the diagnostics *χ*(*t*) = *ψ*_{LF}(*t*), ‖*ψ*_{HF}‖(*t*), and *c*[*ψ*_{HF}, Γ](*t*) account for the major forms of systematic variation through which the 11-yr variation of UV irradiance can manifest itself: a linear response at low frequency, as well as a nonlinear response at high frequency. Each has been subjected to a variational analysis that isolates, in the 3D NCEP record, interannual changes which operate coherently over the four decades with solar activity. For an individual diagnostic *χ*(*t*), the analysis first evaluates the corresponding time series at each longitude, latitude, and pressure, and likewise over all possible seasons. It then calculates, over all possible lags, the correlation between *χ*(*t*) and the record of solar flux *F*_{s}(*t*). The analysis isolates those locations, seasons, and lags for which the diagnostic *χ*(*t*) is strongly correlated to *F*_{s}(*t*).

*F*

_{s}over four decades is composited by renormalizing the correlation between the diagnostic

*χ*and

*F*

_{s}:

*ψ̂*

**x**

*c*

*χ,*

*F*

_{s}

**x**

*ψ*

^{2}

^{1/2}

**x**

**x**refers to position and 〈

*ψ*

^{2}〉

^{1/2}to the corresponding variance of either

*ψ*

_{LF}or

*ψ*

_{HF}. In (3),

*c*[

*χ,*

*F*

_{s}] represents the projection of

*χ*(

*t*) onto

*F*

_{s}(

*t*). If

*χ*equals the running correlation

*c*[

*ψ*

_{HF}, Γ](

*t*), which measures the instantaneous phase of

*ψ*

_{HF}(

*t*) relative to the reference time series Γ(

*t*), then

*c*[

*χ,*

*F*

_{s}] represents a double projection of

*ψ*

_{HF}: first onto the reference time series Γ(

*t*) and then onto the solar time series

*F*

_{s}(

*t*).

The composite structure defined by (3) describes changes of *ψ* for which the diagnostic *χ*(*t*) = *ψ*_{LF}(*t*) or ‖*ψ*_{HF}‖(*t*) or *c*[*ψ*_{HF}, Γ](*t*) tracks the 11-yr variation of *F*_{s}. For LF variability, the composite structure describes the change of *ψ* that simply drifts with the systematic variation of *F*_{s}. For HF variability, it describes the change of *ψ* associated with an amplitude or a frequency modulation of interannual variability, one that likewise varies systematically with *F*_{s}.

A signature of the 11-yr variation of UV irradiance emerges prominently in the HF component. The LF component also evidences a systematic variation. However, it is comparatively weak, limited chiefly to the highest levels in the analyses. For this reason, the presentation here concentrates on interannual variability, represented in the HF component.

## 3. Frequency modulation of interannual variability

*T*

^{Feb–Sep}

*T*

^{Feb}

*T*

^{Sep}

*T*

^{Feb–Sep}is coupled directly to the residual circulation through adiabatic warming, which forces the tendency of temperature in the thermodynamic equation. The wintertime tendency, in turn, determines

*T*

_{NP}during late winter (often used as a proxy for the vortex during individual years).

### a. Relationship to equatorial wind

Figure 2 plots, as a function of year, the wintertime tendency of 70-mb temperature over the Arctic, Δ*T*^{Feb–Sep} (solid). The raw annual record exhibits no systematic drift or modulation of amplitude. However, its power spectrum (not shown) contains a peak at 0.50 cpy. It is analogous to the signature of the BO in the monthly record of temperature (Salby et al. 1997). Superposed in Fig. 2 is the annual record of 50-mb *u*_{EQ} (dashed), averaged over the same months. It alternates sign between neighboring years, reflecting the QBO. Like Δ*T*^{Feb–Sep}, *u*_{EQ} exhibits little evidence of a systematic long-term variation.

Consider now the instantaneous phase of Δ*T*^{Feb–Sep} relative to *u*_{EQ}, which then serves as the reference time series. Figure 3 plots the running correlation between the records in Fig. 2 (solid). Despite the apparent absence of long-term changes in the individual records, their instantaneous correlation varies systematically on the time scale of a decade: *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) swings from −1.0 in the middle of the decades (Δ*T*^{Feb–Sep} *out of phase* with *u*_{EQ}) towards +1.0 around solar max (Δ*T*^{Feb–Sep} *in phase* with *u*_{EQ}), in each of the 4 decades. In fact, *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) tracks the record of 10.7-cm flux (dashed). For a lag of 1 yr, it has a correlation to *F*_{s} of 0.84, which is highly significant.

Near solar min, the correlation between Δ*T*^{Feb–Sep} and *u*_{EQ} is negative. It reflects a polar-night vortex that is anomalously warm (Δ*T*^{Feb–Sep} > 0) during QBO easterlies (*u*_{EQ} < 0), but one that is anomalously cold (Δ*T*^{Feb–Sep} < 0) during QBO westerlies (*u*_{EQ} > 0). The relationship between Δ*T*^{Feb–Sep} and *u*_{EQ} during those years is *consistent* with the dependence of the vortex on the critical line. However, near solar max, the relationship reverses, as first reported by Labitzke and van Loon (1988). During those years, the correlation between Δ*T*^{Feb–Sep} and *u*_{EQ} is positive. It reflects a polar-night vortex that is anomalously warm (Δ*T*^{Feb–Sep} > 0) during QBO westerlies (*u*_{EQ} > 0), but one that is anomalously cold (Δ*T*^{Feb–Sep} < 0) during QBO easterlies (*u*_{EQ} < 0). The relationship between Δ*T*^{Feb–Sep} and *u*_{EQ} during those years is *inconsistent* with the dependence of the vortex on the critical line.

The origin of the systematic variation in Fig. 3 can be traced back to the raw time series (Fig. 2). In the middle of the decades, Δ*T*^{Feb–Sep} varies *out of phase* with *u*_{EQ}. However, in the late 1950s, it misses a beat. Then Δ*T*^{Feb–Sep} becomes temporarily *in phase* with *u*_{EQ}. A similar phase transition occurs in the late 1960s–early 1970s, in the late 1970s–early 1980s, and again in the late 1980s–early 1990s. Notice that, at certain times, Δ*T*^{Feb–Sep} alternates between consecutive years. It changes almost biennially in the middle 1960s, in the middle 1970s, and again in the middle to late 1980s. Those are the same years when the correlation between Δ*T*^{Feb–Sep} and *u*_{EQ} approaches −1.0 (i.e., near solar min).

The relationship between Δ*T*^{Feb–Sep} and *u*_{EQ} is consistent with the expected dependence on the critical line near solar min, when equatorial wind of one sign persists throughout the winter season (Fig. 1). Wintertime-mean *u*_{EQ} is then strong and positive during one year, but strong and negative during the next. The swing between extreme values produces a large quasi-biennial displacement of the critical line about its climatological-mean position.

On the other hand, the relationship between Δ*T*^{Feb–Sep} and *u*_{EQ} is inconsistent with the expected dependence near solar max, when the equatorial wind reverses during the winter season. Wintertime-mean *u*_{EQ} is then weak. In particular, it is shifted towards an equatorial wind of opposite sign: Wintertime-mean easterlies near solar max are thus *westerly* relative to conditions near solar min, when easterlies persist throughout the disturbed season. Accordingly, the wintertime-mean critical line is removed *equatorward* of its extreme poleward position, found during easterly winters near solar min. Analogous reasoning applies to wintertime-mean westerlies. Near solar max, the critical line is advanced poleward of its extreme equatorward position, found during westerly winters near solar min.

The systematic variation of *u*_{EQ} favors a vortex during easterly winters that is colder near solar max than near solar min. Likewise, it favors a vortex during westerly winters that is warmer near solar max than near solar min. The reversed dependence is implied by a weakening of wintertime-mean *u*_{EQ} near solar max when the equatorial wind changes sign during the disturbed season (Fig. 1). In fact, near solar max, wintertime-mean *u*_{EQ}, integrated upward over 30–10 mb, actually reverses: Column-integrated *u*_{EQ} then assumes sign opposite to *u*_{EQ} at 50 mb (see Salby and Callaghan 2000, Fig. 9). It is on *u*_{EQ} inside a deep overlying layer that the 70-mb Δ*T*^{Feb–Sep} depends (through the so-called downward control principle). Formally, this dependence also involves altitudes above 10 mb (e.g., Gray et al. 2001), which lie beyond the range of conventional measurements and NCEP reanalyses. Nonetheless, the reversal of *u*_{EQ} at 30–10 mb suggests that anomalous Δ*T*^{Feb–Sep} at 70 mb should also reverse near solar max: Δ*T*^{Feb–Sep} should then become anomalously cold when the 50-mb *u*_{EQ} is easterly and anomalously warm when the 50-mb *u*_{EQ} is westerly. Polar temperature, which changes *out of phase* with 50-mb *u*_{EQ} near solar min, should then change *in phase* with 50-mb *u*_{EQ} near solar max.

Behavior analogous to that in Fig. 3 is found at lower latitude. Figure 4 plots the instantaneous correlation between the wintertime tendency of the 100-mb temperature at 45°N and 50-mb *u*_{EQ} averaged over the same months. At this latitude, *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) also varies coherently with solar activity—but out of phase. It has a correlation to *F*_{s} of −0.74, likewise highly significant. As *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) over the Arctic is positively correlated to *F*_{s} (Fig. 3), the systematic variation of Δ*T*^{Feb–Sep} at high latitude is compensated at lower latitude by a systematic variation of opposite sign.

At both high and low latitudes, *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) undergoes a decadal swing between −1.0 and +1.0. This systematic variation leads to a cancellation in the overall correlation between temperature and equatorial wind. When calculated over all four decades, the correlation between Δ*T*^{Feb–Sep} and *u*_{EQ} is close to 0.

Figure 5 plots the structure of anomalous Δ*T*^{Feb–Sep} (contoured) operating coherently with 50-mb *u*_{EQ} and *F*_{s}, along with the corresponding significance (shaded). It has been composited according to (3) with *χ* = *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}]. Anomalous temperature during individual years then follows by scaling values in Fig. 5 by the product of anomalous *F*_{s} and anomalous *u*_{EQ} during those years (each normalized).

Anomalous temperature over the Arctic is highly coherent with *u*_{EQ} and *F*_{s} (significant at the 99.99% level). According to the above scaling, it is strong and positive during winters when *F*_{s} < 0 and *u*_{EQ} < 0 (which leaves the sign of Δ*T*^{Feb–Sep} in Fig. 5 unchanged). Then Δ*T*^{Feb–Sep} reflects a warmer vortex during easterly winters near solar min when *u*_{EQ} < 0 persists throughout the disturbed season (Fig. 1). Wintertime-mean easterlies are therefore strong. The critical region, which limits the size and strength of the vortex, is then advanced deep into the winter hemisphere. Implied is intensified downwelling and adiabatic warming at high latitudes, consistent with anomalous temperature.

The same behavior, however, is also implied during winters when *F*_{s} > 0 and *u*_{EQ} > 0 (which likewise leaves the sign of Δ*T*^{Feb–Sep} in Fig. 5 unchanged). Then Δ*T*^{Feb–Sep} reflects a warmer vortex during westerly winters near solar max, when *u*_{EQ} reverses during the disturbed season. Wintertime-mean westerlies are therefore weak, in fact, reversed at higher levels. The critical region is then advanced poleward of its position during westerly winters near solar min. Implied is intensified downwelling and adiabatic warming at high latitudes, likewise consistent with anomalous temperature.

At subpolar latitudes is anomalous temperature of opposite sign. Although weaker, it too is highly coherent with *u*_{EQ} and *F*_{s}. Anomalous temperature is negative from midlatitudes of the winter hemisphere, across the Tropics, and into the subtropics of the summer hemisphere. There, Δ*T*^{Feb–Sep} reflects intensified upwelling and adiabatic cooling during winters when *F*_{s} < 0 and *u*_{EQ} < 0. Corresponding to easterly winters near solar min, the intensified upwelling compensates intensified downwelling over the Arctic that is found during those winters. The same behavior is also implied during winters when *F*_{s} > 0 and *u*_{EQ} > 0. They correspond to westerly winters near solar max, when *u*_{EQ} reverses during the disturbed season.

Anomalous temperature in the Arctic stratosphere exceeds 6 K. This is comparable to the systematic variation of temperature that is recovered when *T*_{NP} is stratified against the QBO (Labitzke and van Loon 1988). Both are almost as large as the full rms variation of wintertime temperature. They can be understood from the systematic phase shift of interannual variability represented in Fig. 3. Interannual variability includes variance from the QBO. It undergoes a systematic modulation of phase that tracks the 11-yr variation of *F*_{s} (e.g., Fig. 1). When windowed to winter months, the QBO's phase then drifts with *F*_{s}: gradually passing from the westerly extremum to the easterly extremum. In this manner, the QBO can introduce a systematic variation in wintertime *T*_{NP} that is as large as the full rms variation—even though no systematic variation in amplitude or baseline is evident.

Anomalous Δ*T*^{Feb–Sep} in Fig. 5 has the same basic structure as that found to operate coherently with anomalous forcing of the residual circulation (see Salby and Callaghan 2002). For each, anomalous temperature is strong and positive over the Arctic, where it reflects intensified downwelling and adiabatic warming. Compensating it at subpolar latitudes is negative anomalous temperature, which reflects intensified upwelling and adiabatic cooling. Although weaker, that structure is consistent with the residual circulation and the wider area over which upwelling occurs. The resemblance between these anomalous structures suggests that the systematic variation of Δ*T*^{Feb–Sep}, which operates coherently with both *u*_{EQ} and *F*_{s}, enters through a modulation of the residual circulation.

Changes of thermal structure in Fig. 5 are, through thermal wind balance, related to changes of zonal wind. Undergoing a similar systematic variation is the wintertime tendency of zonal wind Δ*u*^{Feb–Sep}. The correlation *c*[Δ*u*^{Feb–Sep}, *u*_{EQ}](*t*) varies coherently with *F*_{s}, but out of phase at high latitude. This makes it out of phase with *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*) at high latitude. Hence, *c*[Δ*u*^{Feb–Sep}, *u*_{EQ}] ≅ +1.0 near solar min and ≅−1.0 near solar max, just opposite to the variation in Fig. 3.

Figure 6 plots the anomalous wintertime tendency of zonal wind operating coherently with the 50-mb *u*_{EQ} and *F*_{s}. As before, anomalous wind during individual years follows by scaling values by the product of anomalous *F*_{s} and *u*_{EQ} during those years. Anomalous wind at polar latitudes is highly coherent with *u*_{EQ} and *F*_{s} (significant at the 99.99% level). It is strong and negative during winters when *F*_{s} < 0 and *u*_{EQ} < 0. Then Δ*u*^{Feb–Sep} reflects a weakening of the polar-night jet during easterly winters near solar min when *u*_{EQ} < 0 persists throughout the disturbed season (Fig. 1). The behavior is consistent with warmer temperature over the Arctic during those winters (Fig. 5). The same behavior, however, is also implied during winters when *F*_{s} > 0 and *u*_{EQ} > 0. Then Δ*u*^{Feb–Sep} reflects a weakening of the jet during westerly winters near solar max when *u*_{EQ} reverses during the disturbed season. Wintertime-mean equatorial westerlies are therefore weak, in fact, reversed at higher levels. The critical region is then advanced poleward, implying intensified downwelling and adiabatic warming at high latitudes.

At subpolar latitudes, the wind anomaly is also highly coherent, but reversed. Positive anomalous wind reflects a weakening of easterlies that lie equatorward of the Aleutian high; it plays a key role in the dynamics of the polar-night vortex. Anomalous westerlies lie equatorward of the Aleutian high, whereas anomalous easterlies lie poleward of it. These opposing changes of zonal wind flatten its gradient at middle and high latitudes. They reflect a poleward shift of the critical region where the PV gradient is weak and planetary waves experience strong absorption, which in turn forces residual mean motion.

Figure 7 plots, as a function of longitude and latitude, the anomalous wintertime tendency of the 30-mb height operating coherently with the 50-mb *u*_{EQ} and *F*_{s}. The correlation *c*[Δ*Z*^{Feb–Sep}_{30}*u*_{EQ}](*t*) varies coherently and in phase with *F*_{s} at high latitude, making it in phase with *c*[Δ*T*^{Feb–Sep}, *u*_{EQ}](*t*). Hence, *c*[Δ*Z*^{Feb–Sep}_{30}*u*_{EQ}](*t*) ≅ −1.0 near solar min and ≅+1.0 near solar max, as in Fig. 3. As for the zonal-mean structure, anomalous height during individual years follows by scaling values in Fig. 7 by the product of anomalous *F*_{s} and *u*_{EQ} during those years.

Anomalous Δ*Z*^{Feb–Sep}_{30}*F*_{s} < 0 and *u*_{EQ} < 0. These conditions correspond to easterly winters near solar min when *u*_{EQ} < 0 persists throughout the disturbed season (Fig. 1). Wintertime-mean easterlies are then strong, producing a wintertime-mean critical region that is advanced deep into the winter hemisphere. Positive anomalous Δ*Z*^{Feb–Sep}_{30}*u*_{EQ} and *F*_{s} (van Loon and Labitzke 1994).

The same behavior, however, is also implied during winters when *F*_{s} > 0 and *u*_{EQ} > 0. These conditions correspond to westerly winters near solar max when *u*_{EQ} reverses during the disturbed season. Wintertime-mean westerlies are therefore weak, in fact, reversed at higher levels. The critical region is then advanced poleward, implying intensified downwelling and adiabatic warming at high latitudes.

Flanking the vortex over the date line is a negative wave anomaly. It opposes the wintertime amplification of the Aleutian high, at least at midlatitudes. However, north of the Arctic Circle, the Aleutian high is reinforced by positive Δ*Z*^{Feb–Sep}_{30}

Wavenumber 1 is shifted poleward during winters when *F*_{s} < 0 and *u*_{EQ} < 0. Found near solar min, those are winters when *u*_{EQ} < 0 persists throughout the disturbed season (Fig. 1). This produces wintertime-mean easterlies that are strong and a critical region that is advanced deep into the winter hemisphere. Nonlinear mixing then damps wavenumber 1 at midlatitudes. Conversely, the poleward advance of wave activity inside the critical region amplifies wavenumber 1 at high latitudes.

The same behavior is also implied during winters of *F*_{s} > 0 and *u*_{EQ} > 0. Found near solar max, those are winters when *u*_{EQ} reverses during the disturbed season (Fig. 1). They produce wintertime-mean westerlies that are weak, in fact, reversed at higher levels, and a critical region that is likewise advanced poleward. The Aleutian high should then also experience a poleward shift, with wavenumber 1 weakened at midlatitudes and amplified at high latitudes.

The wintertime reversal of *u*_{EQ} near solar max implies different behavior during early and late winter. Even though *u*_{EQ} during late winter is westerly, behavior during early winter should resemble that of easterly years. This is just what is found for winters grouped near solar max (van Loon and Labitzke 1994, their Fig. 10).

Surrounding the vortex in Fig. 7 is a collar of negative anomalous height. It modifies the meridional gradient, which characterizes the edge of the vortex. Bounding the vortex to its south is the critical region, where nonlinear mixing flattens the height gradient and limits the size of the vortex. According to Fig. 7, the meridional gradient is steepened near 30°N, where anomalous Δ*Z*^{Feb–Sep}_{30}*Z*^{Feb–Sep}_{30}*F*_{s} < 0 and *u*_{EQ} < 0, when equatorial easterlies persist throughout the disturbed season. The same behavior is implied during winters of *F*_{s} > 0 and *u*_{EQ} > 0, when equatorial wind reverses during the disturbed season. During both winters, the poleward shift of the critical region should be accompanied by a weakening of planetary waves at midlatitude and an amplification at high latitude.

Figure 8 compares the full tendency (mean + anomaly) of the 30-mb height during easterly winters at extrema of *F*_{s}. (Note: the wintertime tendency is negative throughout.) Years when *F*_{s} < 0 and *u*_{EQ} < 0 (Fig. 8a) correspond to easterly winters near solar min when *u*_{EQ} < 0 persists throughout the disturbed season. The vortex is then anomalously weak. It is displaced out of polar symmetry by an Aleutian high that invades the Arctic. Analogous structure is represented in PV (not shown), wherein the vortex is small, weak, and distorted. Straddling the pole in Fig. 8a are opposing height anomalies. Low and high values of Δ*Z*^{Feb–Sep}_{30}

Years when *F*_{s} > 0 and *u*_{EQ} < 0 (Fig. 8b) correspond to easterly winters near solar max when *u*_{EQ} reverses during the disturbed season. The vortex is then anomalously strong. It is left nearly in polar symmetry by an Aleutian high that, while amplified at midlatitudes, scarcely crosses the Arctic circle. Increased zonal symmetry over the Arctic represents a weakening of wavenumber 1 at high latitude, accompanied by an equatorward retreat of the critical region.

### b. Relationship to other field variables

The systematic modulation of interannual variability reappears at particular lags of the field property (e.g., Δ*T*^{Feb–Sep}) relative to the reference time series *u*_{EQ}: at lags of 5–6 yr, 10–12 yr, etc. Reflecting multiples of half a solar cycle, those lags also recover a systematic variation of phase. Like the variation in Fig. 3, it operates coherently with *F*_{s}. That phase variation is associated with dynamical structure very similar to the structure in Figs. 5–8.

Analogous behavior emerges if the phase of interannual variability is referenced against time series other than *u*_{EQ}. Figure 9 plots, for a lag of one and a half solar cycles, the running correlation between Δ*T*^{Feb–Sep} at 50 mb over the Arctic and North-Polar temperature at 50 mb, which then serves as the reference time series. Here, *c*[Δ*T*^{Feb–Sep}, *T*_{NP}] (solid) swings systematically from −1.0 towards +1.0 in each of the four decades. It operates coherently with *F*_{s} (dashed). The behavior in Fig. 9 does not explicitly involve *u*_{EQ}. Consequently, it represents a systematic variation within temperature itself. Analogous behavior is recovered by high-pass filtering *T*_{NP} to the QBO and BO, which likewise reveals an 11-yr modulation (Salby and Callaghan 2000).

Anomalous Δ*T*^{Feb–Sep} operating coherently with *T*_{NP} and *F*_{s} has the same basic structure (not shown) as that operating coherently with *u*_{EQ} and *F*_{s} (Fig. 5). In each, anomalous temperature is strong and positive over the Arctic during easterly winters near solar min, but also during westerly winters near solar max. Compensating it at subpolar latitudes is negative anomalous temperature, consistent with the structure of the residual circulation.

A systematic modulation emerges even if interannual variability is referenced against properties at the tropical tropopause. Figure 10 plots the running correlation between the wintertime tendency of the 20-mb height over the Arctic, lagged by half a solar cycle, and 100-mb height over the equator. Here *c*[Δ*Z*^{Feb–Sep}_{20}*Z*_{EQ}] (solid) swings systematically from −1.0 towards +1.0 in each of the Four decades. It has an overall correlation to *F*_{s} (dashed) of 0.80, which is highly significant.

Figure 11 plots the anomalous wintertime tendency of height operating coherently with *Z*_{EQ} and *F*_{s}. Anomalous height (contoured) is highly significant (shaded) and positive over the Arctic, where it reflects anomalous downwelling. Here Δ*Z*^{Feb–Sep} intensifies upward through the roof of the NCEP analyses, to levels where the variation of UV absorption between solar min and solar max becomes substantial. The upward intensification may also reflect the dependence on *u*_{EQ} in the upper stratosphere and mesosphere, which influences polar temperature at lower levels (Gray et al. 2001). Compensating positive Δ*Z*^{Feb–Sep} over the Arctic is negative anomalous height at subpolar latitudes. Although weaker, it too is highly significant. Reflecting anomalous upwelling, negative Δ*Z*^{Feb–Sep} coincides with the collar of negative anomalous height surrounding the vortex that was seen earlier (Fig. 7).

A systematic modulation of interannual variability is evident even if phase is referenced against the local field property itself. Setting the interannual clock Γ equal to *ψ*_{HF} references changes at each location against those at the same location, but lagged in time. The running cross-correlation is then replaced by the running autocorrelation. The latter measures the cosine of the phase at one time relative to the phase at another time. Figure 12 plots, for a lag of two solar cycles, the running autocorrelation of Δ*Z*^{Feb–Sep}_{30}*c*[Δ*Z*^{Feb–Sep}_{30}*Z*^{Feb–Sep}_{30}*F*_{s}, producing a correlation of −0.83.

The behavior in Fig. 12 describes a systematic variation of phase *within the height record itself.* The phase of interannual variability changes rapidly in some years, but slowly in others. This systematic variation reflects a frequency modulation, one that determines the phase of interannual variability during winter months. The corresponding structure of anomalous height (not shown) is similar to that presented earlier, when height is referenced against other field properties.

## 4. Amplitude modulation of interannual variability

The variational analysis also isolates a systematic modulation in the amplitude of interannual variability. However, relative to frequency modulation, the signature of amplitude modulation is weak. Figure 13 plots the instantaneous power of the 20-mb height tendency over the Arctic, ‖Δ*Z*^{Feb–Sep}_{20}^{2} (solid). It describes interannual variance, inclusive of the QBO and BO. Superposed is the record of *F*_{s}, lagged by half a solar cycle (dashed). Instantaneous height power in Fig. 13 has an overall correlation to *F*_{s} of 0.72. This is smaller than the correlation to *F*_{s} of frequency modulation, which tracks UV irradiance closely. Nonetheless, the implied variation of amplitude is still highly significant according to Monte Carlo tests (section 5).

With the lag accounted for, interannual power amplifies near solar min in each of the four decades. Those are the years when wintertime-mean *u*_{EQ} alternates between consecutive years—biennially (Fig. 1). During the same years, Arctic temperature changes almost biennially (Fig. 2). An analogous amplitude modulation results if *T*_{NP} is filtered about the QBO and BO (Salby and Callaghan 2000, their Fig. 6).

The corresponding structure of anomalous height is plotted in Fig. 14. It has been composited according to (3), now with *χ* = ‖*ψ*_{HF}‖. Anomalous Δ*Z*^{Feb–Sep} has much the same form as that associated with frequency modulation (Fig. 11). Anomalous height maximizes over the Arctic. It increases upward through the roof of the NCEP analyses, where the change of UV absorption between solar min and solar max becomes substantial.

## 5. Reliability of systematic variations

The variational analysis isolates systematic variations in a diagnostic *χ* that operate coherently over four decades with *F*_{s}. For a record of any fixed length, however, there is a small but finite probability that *χ* will track *F*_{s} simply through chance. Isolating systematic variations that are truly related to the variation of UV irradiance thus requires us to discriminate to those which will maintain their relationship to *F*_{s} as the atmospheric record is extended. The challenge then is to separate the wheat from the chaff.

We define a null hypothesis that the correspondence between the diagnostic *χ* and *F*_{s} (e.g., measured by their overall correlation) actually occurred through chance. The probability that *χ* track *F*_{s} over four decades is then required to be small. Under these circumstances, the null hypothesis can be rejected with a high level of confidence.

### a. Monte Carlo simulation

To evaluate the reliability of systematic variations, the same operations used to define *χ* and its relationship to *F*_{s} are now performed on field properties that are generated randomly. The individual NCEP record is then replaced by a large stochastic ensemble of such records. Each represents one realization of a stochastic process—unrelated to solar activity. The diagnostic *χ*(*t*) is calculated over the stochastic ensemble, producing many realizations of the corresponding time series. From them, we evaluate the probability that a systematic variation operating coherently with *F*_{s} over four decades occurs through chance.

The running correlation is based on a field property (e.g., *ψ*_{HF} = Δ*T*^{Feb–Sep}) and a reference time series (e.g., Γ = *u*_{EQ}). Each is now represented as a stochastic process. The process is defined from a Gaussian spectrum, with a correlation time of about 2 yr (representative of observed variability). The null hypothesis then holds that the correspondence between *χ* = *c*[*ψ*_{HF}, Γ] and *F*_{s} follows randomly via this red spectrum.

From the stochastic process, some 400 000 realizations are generated—each four decades long. They are used to produce the running correlation between the stochastic field property and reference time series. Of some 400 000 realizations, less than 10% yield an overall correlation to *F*_{s} exceeding 0.40. Realizations satisfying this criterion are collected in Fig. 15 into a histogram, as a function of lag between the field property and the reference time series (solid). Notice that realizations satisfying the above criterion are distributed over lag almost uniformly.^{1} A correlation to *F*_{s} of 0.40 is therefore exceeded *randomly with respect to lag.* This contrasts with behavior in the observed records, wherein the criterion is satisfied at *preferred lags:* A strong correlation to *F*_{s} reappears at lags of 5–6 yr, 10–12 yr, and so forth.

### b. Shifted records

The ensemble of records is randomized further by shifting each by an integral number of years. One set of 400 000 records begins in 1955, extending forward for four decades. Another set begins in 1956, and so forth. Systematic variations that are genuinely related to *F*_{s} (e.g., in the observed record) are then modified only through a phase shift because new data introduced at the tail of the record maintain a consistent relationship to *F*_{s}. On the other hand, systematic variations over four decades that occur through chance (e.g., in the stochastic records) quickly deteriorate with increasing shift because new data introduced have only a small probability of maintaining the existing relationship to *F*_{s}. Requiring the diagnostic *χ* to vary coherently with *F*_{s} under *all shifts* of the 4-decadal record: by 1, 2, 3, 4, and 5 yr (i.e., jointly in records over 1955–96, 1956–97, and so forth) reduces the chance correlation to *F*_{s} dramatically.

Superposed in Fig. 15 are realizations that yield an overall correlation to *F*_{s} exceeding 0.40 in *all* of the shifted records (dashed). The probability of satisfying the null hypothesis through chance is sharply reduced. Of 400 000 realizations, each shifted successively by an integral number of years, less than 0.001% yield an overall correlation to *F*_{s} exceeding 0.40 under all shifts of the 4-decadal record. Observed records that satisfy this more restrictive criterion are therefore significant at the 99.999% level.

### c. Referenced against observed u_{EQ}

The picture is unchanged even if the reference time series is not stochastic. If the observed record of *u*_{EQ} is employed, then the reference time series includes a frequency modulation that varies systematically with *F*_{s} (Fig. 1). Still generated stochastically, however, is the field property. For an unshifted 4-decadal record, realizations yielding an overall correlation to *F*_{s} greater than 0.40 still comprise less than 10% of the ensemble. The histogram of those realizations resembles the solid curve in Fig. 15. As when the field property and reference time series are both stochastic, those realizations are distributed over lag almost uniformly. Therefore, a correlation to *F*_{s} of 0.40 is still exceeded randomly with respect to lag, even though the reference time series now includes a systematic variation that favors preferred lags.

Requiring the same criterion to be satisfied jointly under shifts of the 4-decadal record again reduces the chance correlation to *F*_{s} dramatically. The correlation to *F*_{s} must now exceed 0.40 in *all* of the shifted records. The probability of satisfying the null hypothesis through chance is then sharply reduced. Of some 400 000 realizations, each one shifted successively by an integral number of years, less than 0.1% satisfy this condition. Observed records that satisfy the more restrictive criterion (shaded in preceding figures) are then significant at the 99.9% level.

## 6. Conclusions

The circulation of the wintertime stratosphere includes a component that varies systematically with the 11-yr variation of UV irradiance. Only a small systematic variation is visible in the LF component, which represents a simple linear response that drifts with *F*_{s}. However, the 11-yr variation manifests itself prominently in the HF component, which corresponds to interannual variability. The systematic variation at high frequency represents a more complex, nonlinear response to the variation of UV irradiance.

Involving periods shorter than 5 yr, interannual variability includes the QBO and BO. Collectively, those components account for an 11-yr modulation in the frequency of interannual variability. It modifies the phase of interannual variability during winter when planetary waves couple the polar and equatorial stratosphere. The polar-night vortex is then sensitive to equatorial wind, which itself varies systematically with UV irradiance.

The frequency modulation of interannual variability surfaces when wintertime structure is referenced against any of several field properties. It is visible even when a property is referenced against itself. Monte Carlo simulations indicate that the systematic variation of frequency is highly significant. It intensifies upward through the roof of NCEP analyses, where the variation of ozone heating between solar min and solar max becomes substantial.

A signature of the 11-yr variation also surfaces as an amplitude modulation of interannual variability. It is magnified in each of the four decades near solar min, when polar temperature and wintertime-mean *u*_{EQ} change almost biennially. However, the signature of amplitude modulation is comparatively weak. It becomes substantial only at the highest levels of the NCEP analyses. A similar conclusion applies to the LF component of the circulation, which simply drifts with UV irradiance.

The systematic variation is prominent in the wintertime tendency of temperature, which is coupled directly to the residual mean circulation. In fact, the anomalous wintertime tendency operating coherently with *F*_{s} has the same basic structure as that operating coherently with anomalous forcing of the residual circulation. The resemblance of these anomalous structures suggests that the systematic modulation of interannual variability enters through changes of the residual circulation.

Accompanying the systematic variation of zonal-mean structure is an amplification and decay of wavenumber 1 at high latitude. It represents a poleward advance and retreat of the critical region where planetary waves experience strong absorption that forces residual motion. The critical region is advanced poleward during easterly winters near solar min, but during westerly winters near solar max.

Exactly how these changes are introduced remains uncertain. However, the upward intensification of anomalous structure suggests that they are somehow imprinted at higher levels where the variation of UV absorption becomes substantial. A likely vehicle is wind at low latitude. Influencing the critical line and thus planetary wave absorption, equatorial wind can change significantly with only a minor change of temperature.

Observed variations of wave structure and implied residual motion, in fact, parallel the systematic variation of equatorial wind. The behavior of wintertime-mean *u*_{EQ} suggests a reversal in anomalous downwelling between solar min and solar max. Establishing this connection, however, will require observations above the altitude range of meteorological analyses, altitudes that also influence polar temperature in the lower stratosphere. Although other factors cannot be ruled out, the behavior implied by wintertime-mean *u*_{EQ} at and below 10 mb is broadly consistent with the observed reversal of anomalous temperature.

The signature of anomalous downwelling over the Arctic is visible as low as 100 mb. Implied are systematic changes in the transfer of stratospheric air to the troposphere. Compensating those changes at lower latitudes is a signature of anomalous upwelling. Visible even at the tropical tropopause, the signature of anomalous upwelling implies systematic changes in the return of tropospheric air to the stratosphere. Through these transfers, a systematic variation in the stratosphere's residual circulation can influence the tropospheric circulation.

## Acknowledgments

The authors are grateful for constructive remarks provided during review. This work was supported by ASF Grant ATM.0127671.

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^{1}

Minor deviations with lag converge to zero as the ensemble size is increased.