## 1. Introduction

The subject of the influence of the phase of the equatorial quasi-biennial oscillation (QBO) on the extratropics has already been given considerable attention. The idea that is currently most widely accepted was first proposed by Holton and Tan (1980, 1982) and McIntyre (1982). Using 16 yr of National Meteorological Center (NMC) data, Holton and Tan noted that the winter polar geopotential height at 50 mb (21 km) was generally lower, corresponding to weaker eastward (i.e., westerly) winds, in years with an eastward equatorial QBO at 50 mb than those years with a westward (i.e., easterly) QBO.

It is thought that this QBO-related variability in the Northern Hemisphere (NH) extratropics, commonly referred to as the extratropical QBO (or Holton–Tan Oscillation), is due to the equatorial QBO shifting the latitudinal band in the Tropics and subtropics in which winds are weak and where quasi-stationary Rossby waves tend to break and dissipate. The Rossby waves propagate up into the stratosphere in the extratropics. For a westward QBO, the subtropical band of weak winds extends further poleward, so that the Rossby waves are confined closer to the polar vortex and are more influential in disturbing the vortex and causing a winter warming. However, Holton and Tan (1980, 1982) were unable to find any clear differences in the Eliassen–Palm (EP) flux between the eastward and westward phases of the QBO.

The existence of the extratropical QBO and the relevance of the proposed dynamical mechanism for it were further substantiated by observational analysis carried out by Dunkerton and Baldwin (1991) and Baldwin and Dunkerton (1991). Dunkerton et al. (1988) correlated the QBO phase and winter warmings using the World Meteorological Office definition of a major winter warming and the depth of QBO eastward winds as a QBO phase statistic, and noted that major warmings never occurred when there were deep QBO eastward winds at midwinter. Dunkerton and Baldwin (1991) used an extension (NMC 1964–88) of the dataset used by Holton and Tan (1980), and found maximum correlation by using QBO winds at 40 mb and extratropical winds at 62°N, 10 mb to measure the extent of the winter warming. The biggest correlation occurred in November–February. They found that the polar night jet was weaker in years with a westward QBO than an eastward QBO, consistent with the results of Holton and Tan (1980). They also found that the planetary wave fluxes were consistent with the suggested Holton–Tan mechanism. Baldwin and Dunkerton (1991) used new data for the period 1978–90 (including observations for the Tropics and Southern Hemisphere) to show that the extratropical QBO extended to the upper stratosphere. Most recently Naito and Yoden (2005) have used the 46-yr National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset to examine the extratropical QBO in the troposphere and stratosphere, paying particular attention to the statistical significance of the apparent signals.

Holton and Austin (1991) carried out numerical modeling studies of the influence of the QBO phase on the extratropics using a 3D mechanistic stratospheric model (similar to that used here, but not zonally truncated). Matched pairs of model simulations over a winter season were carried out under identical conditions except for being initialized with idealized QBO winds in opposing phase in the Tropics. Forcing of extratropical waves was provided by a wave-1 perturbation in the geopotential at the lower boundary (as in the 3D model used here). Three different values of wave-forcing amplitude were used. The QBO phase had little effect for weak or strong extratropical forcing. At the intermediate forcing value, stratospheric warmings occurred for both simulations, but their details depended on the QBO phase [see Holton and Austin (1991) , their Fig. 3].

O’Sullivan and Young (1992) and O’Sullivan and Dunkerton (1994) carried out matched pairs of simulations, but with the QBO being represented by a single deep eastward/westward layer, as for the “nonsheared” QBO considered later in section 4, instead of a vertically sheared wind. O’Sullivan and Young (1992) found that for intermediate values of extratropical forcing, the winter evolution was more disturbed for the simulation with westward QBO than eastward QBO. Further model simulations showed that the difference between eastward QBO and westward QBO simulations was increased by broadening the latitudinal scale of the QBO anomaly, and that the difference was decreased by raising the altitude position of the QBO by 9 km (from being centered at 29.5 km to being centered at 38.5 km). O’Sullivan and Dunkerton (1994) looked at similar model simulations, and discussed the idea of the extratropical QBO in terms of a bifurcation whose details depend on the extratropical geopotential forcing values. Naito et al. (2003) used long (10 800 day) perpetual-winter simulations in a simplified general circulation model (i.e., containing an active troposphere, but without any sophisticated physical parameterizations) to show that there is a statistically significant extratropical response to imposed steady QBO-like perturbations of the type expected from the above.

Hamilton (1998) described a general circulation model (GCM) simulated over 48 yr, including a realistic troposphere and using a relaxing scheme to force the QBO in the equatorial lower stratosphere. Using the QBO wind at 40 mb, he found that on average the NH polar winter temperature was ∼3°C warmer for years with a westward QBO than an eastward QBO, though within each phase there was significant variability.

A recent modeling study by Gray et al. (2001a) using a 3D mechanistic model and multiyear simulations found no realistic Holton–Tan oscillation when the equatorial QBO was forced over a standard (16–32 km) height range, but that a realistic Holton–Tan oscillation did occur when the QBO was forced over an extended (16–58 km) height range. Subsequent papers (Gray 2003; Gray et al. 2003) went on to clarify this influence from the upper stratosphere. The major conclusion of Gray (2003) was that the lower stratospheric QBO influences the early winter when the flow is fairly linear, but later in winter when the flow is highly nonlinear, the greatest influence may come from the region of the equatorial stratopause. Gray et al. (2003) further examined this nonlinear behavior and highlighted the possible role of traveling anticyclones (wavenumbers 3–6) in building up the Aleutian High. Gray et al. (2004) show modeling results in which the timing of stratospheric warmings is dependent on equatorial/subtropical winds.

Several authors, such as Gray et al. (2001b) and Naito and Hirota (1997) have noted that the relationship between the QBO and NH winter warmings may be affected by the phase of the 11-yr solar cycle. Here we look in detail at the influence of the equatorial QBO on the extratropics in a 3D zonally truncated mechanistic model with extratropical planetary wave forcing at the artificial lower boundary. A QBO-like oscillation is forced in the model using a relaxation scheme. Although the period of the oscillation does not necessarily match that of the observed QBO. A QBO of period exactly two years is used in order that the phase alignment between the QBO and the annual cycle repeats from year to year. The simulations are carried out over multiple years, so that the interannual nature of the extratropical response can be clearly interpreted, and where necessary examined in further details. In particular, we investigate the influence of three aspects of the equatorial dynamics on the extratropical response to the QBO. Firstly, the phase alignment between the QBO and the annual cycle is varied in two sets of model simulations, one incorporating a QBO with typical vertical shear, and the other incorporating a QBO with no vertical shear. The results are discussed in terms of the Holton–Tan mechanism. Second, a set of simulations is performed in which the height position of the QBO in the stratosphere is varied, and the results compared to those of Gray et al. (2001a). Third, the time evolution of the equatorial QBO is varied more drastically to investigate if certain times in the seasonal evolution are more important than others for the extratropical response to the Tropics.

A QBO having a period of a noninteger number of years is then used in a long multiyear simulation. This means that the phase of the QBO at any given time in the annual cycle changes from year to year, as in observations. Both the correlation between the QBO and the extratropics and the regression of the extratropics by the QBO is shown, and the effect of the amplitude of extratropical wave forcing is discussed.

The structure of the paper is as follows. Section 2 gives details of the numerical model and the forcing of the equatorial QBO wind signal. Section 3 defines a suitable measure of the amplitude of the extratropical QBO. Section 4 discusses the effect of phase alignment between equatorial QBO and annual cycle on the extratropical response. Section 5 investigates effects of varying height and time structure of the equatorial QBO. Section 6 discusses correlation and regression issues. Section 7 gives overall conclusions.

## 2. The 3D mechanistic model

The model used in this study is a mechanistic primitive equation model developed by Saravanan (1992) and used previously by Scott and Haynes (1998), with pressure as the vertical coordinate and a spectral representation in the horizontal, using a spherical domain. It solves the primitive equations using a spherical harmonic representation in the horizontal and a grid representation in the vertical. The spherical harmonic series are truncated anisotropically, including harmonics with total wavenumber up to and including 21, but only mean and wave-1 components in longitude. The severe longitudinal truncation is incorporated to allow many multiyear simulations to be carried out, which is necessary to allow careful investigation of the parameter space. This truncation is justified, to a degree, by the observation that it is the lowest zonal wavenumbers that dominate the large-scale evolution of the stratosphere that is under investigation here (e.g., Haynes and McIntyre 1987). This general approach and the same numerical model have been used by Scott and Haynes (1998, 2000, 2002) to address other issues in the seasonal and interannual variation of the stratosphere with apparently encouraging results.

The model uses pressure as the vertical coordinate, with levels chosen to be equally spaced in log pressure height. The model has height range from 10 to 70 km in log pressure coordinates. The top 10 km of the model are dominated by the use of a sponge layer to prevent spurious wave reflection. The lower boundary condition is defined by specifying the geopotential height perturbation amplitude on the lowest pressure level. A leapfrog time stepping scheme is used, with a Robert filter to damp out computational modes.

Damping terms are included in the form of Newtonian cooling, Rayleigh friction, and small-scale damping in the form of ∇^{8} hyperdiffusion. Of these, the most relevant to the investigations here is the Newtonian cooling, through which the seasonal cycle is represented by a model relaxation to a given time-dependent potential temperature distribution. This radiative equilibrium potential temperature is calculated as a sinusoidally varying superposition of two potential temperature distributions, representing summer and winter. These in turn are calculated so as to be in thermal wind balance with prescribed idealized zonal mean wind profiles for summer and winter. For details of these wind profiles and temperature distributions, see the appendix of Scott and Haynes (1998).

_{LB}, as

*λ*is longitude,

*ϕ*is latitude, and Φ

_{0}is the amplitude of the perturbation. Hence the wave forcing is only imposed in the Northern Hemisphere, with the forcing amplitude nonzero between 30°N and the Pole, with maximum at 60°N; Φ

_{LB}is kept constant in time after an initial spinup period.

*D*is the descent rate of the QBO,

*U*

_{QBO}is the QBO amplitude, Ψ defines the phase alignment between the QBO and the annual cycle,

*P*is the period of the QBO, and

_{s}*c*is a constant controlling the strength of the relaxation and

*c*= (1/50)day

^{−1},

*D*= 0.000 386 m s

^{−1}(=1 km month

^{−1}), Δ

*ϕ*

_{QBO}= 22°,

*U*

_{QBO}= 27 m s

^{−1}, and

*z*

_{1}= 10 km (∼240 mb),

*z*

_{2}= 23 km (∼37 mb),

*z*

_{3}= 40 km (∼3.3 mb). Note that the values of the constants used do not correspond exactly to the observed QBO, because the idealized QBO wind

*u*

_{QBO}is only being relaxed toward, not forced exactly. Characteristics of the prescribed QBO profile are shown in Fig. 1. Figure 1a shows the time–height structure, giving an indication of the effect of varying the parameter Ψ. Figure 1b shows the corresponding structure for the nonsheared profile used in section 4. Figure 1c shows the latitude–height structure of the idealized QBO. The period

*P*of the QBO can be varied between simulations.

_{s}## 3. Measuring the extratropical response to the QBO

Since a large number of model simulations are to be compared, it is necessary to find a measure of the extratropical response so that different simulations can be compared in a concise quantitative manner. The dominant variation in all extratropical fields is annual because of the seasonal cycle in temperature. The simulations vary in how they depart from the regular annual cycle. By inspection of variables from a range of simulations, it is seen that, for a forced QBO with a period of an integral number of years, the perturbation to the regular annual cycle is either (a) negligible, (b) a 2-yr cycle [the “internal” mode of variability found by Scott and Haynes (1998)], (c) a periodic oscillation with the same period as that of the forced equatorial QBO, or (d) interannual variability with no obvious pattern or period. For those simulations that depart from the regular annual cycle, the perturbation generally occurs in NH midwinter, in the form of varying degrees of NH polar vortex breakdown.

At any fixed latitude and height, the variation with time of the zonal mean wind, if regular, will have a period of 2 yr or the period of the relaxed QBO, or conceivably a combination of the two. Therefore, to compare the responses, it is useful to calculate the (time) Fourier components with these periods. By choosing an analysis period that is a multiple of both two and the period of the relaxed QBO, aliasing errors in the Fourier analysis are avoided. The first four years of each model simulation are neglected in order to avoid any model spinup effects.

*(*u

*t*) at a fixed latitude and height, the Fourier analysis is to be carried out for time length

*T*(yr), with

*m*model output every year. There will be

*k*=

*T m*model output used in the analysis, say

u

_{0}, . . . ,

u

_{k−1}. For the Fourier component with period

*P*(in years), the appropriate cosine and sine components are

*a*is then given by

_{P}*z*to model top

_{B}*z*) and over latitudes north of 25°N. The integral is weighted so that it is proportional to angular momentum, so that the extratropical response Fourier component

_{T}*I*of period

_{P}*P*is defined by

*a*(

_{P}*ϕ*,

*z*) is the absolute value of the Fourier component of zonal mean wind at fixed position (

*ϕ*,

*z*).

Alternatively, for simulations in which a multiple year analysis is not possible, a measure of the winter NH variability can be found by measuring the strength of the disturbance of the polar vortex. For the 3D model, this strength is well represented by using the zonal mean wind at 65°N, 49 km (∼0.91 mb) and the potential temperature at 86°N, 40 km (∼3.3 mb), both averaged from January to March: these are the positions and times that give the largest extratropical variation in the model. From here on, these are referred to as * u _{W}*,

*, respectively.*

_{W}We note that there are alternative and equally valid ways of measuring the extratropical variability. Here it suffices to note that the results presented here are qualitatively robust to changes in the measure used.

## 4. Varying the phase alignment between the QBO and the annual cycle

For a QBO having a period of an integer number of years, the phase alignment between the QBO and the annual cycle remains fixed throughout any single simulation. Here, the effect of varying this phase alignment is investigated. Butchart and Austin (1996) investigated the effect of changing the phase alignment between the equatorial QBO and the annual cycle, but only for single winter simulations instead of the multiyear simulations used here. They found that the largest extratropical response occurred for QBO phases corresponding to Ψ = (*π*/2), although QBO phases corresponding to Ψ = *π* gave responses almost as extreme. It is not straightforward to compare their results directly with those found here, since they compare single winter rather than multiannual simulations.

Figure 2 curve (a) shows the results of varying phase alignment for a QBO of period 2 yr and Φ_{0} = 285 m: the amplitude of the extratropical response of the 2-yr period, as measured by the statistic *I*_{2} defined in section 3, is plotted against the QBO phase alignment parameter Ψ, as defined in (2). The curve shows the results of 20 different model simulations, with each simulation being of length about 15 yr. Since the QBO has a 2-yr period, a phase shift of Ψ = *π* effectively just swaps around the years in which the QBO is eastward/westward, and so has no effect on the size of the extratropical response. Hence Ψ is varied between 0 and *π*. The figure shows that the phase alignment between the annual cycle and the QBO has a large effect on the amplitude of the extratropical response: *I*_{2} varies by almost a factor of 8 between the coupling that gives minimum response and that which gives maximum response. For comparison, Fig. 2 curve (b) also shows the same experiment for the case with no extratropical geopotential forcing, Φ_{0} = 0.

Comparing Fig. 2 curves (a) and (b), one sees that the extratropical response to the equatorial QBO is not negligible for Φ_{0} = 0 (i.e., no extratropical planetary wave forcing and hence zonally symmetric circulation), but that the variation in this response associated with changing Ψ is much smaller than with Φ_{0} = 285 m. The extratropical response for Φ_{0} = 0 is simply associated with the mean meridional circulation and corresponding changes in zonal mean flow associated with the equatorial QBO forcing (e.g., Plumb and Bell 1982), and is relatively large because of the form of *I _{P}*, given in (12), which is weighted toward the lowest latitudes included in the integral. This response is well predicted by zonally symmetric dynamics linearized about a resting basic state, in which there is no effect of the annual cycle. The weak dependence on Ψ is a result of the nonlinear interaction between the annual cycle in radiative zonal mean winds (and potential temperature), which results from the Newtonian relaxation, and the imposed QBO cycle.

For Φ_{0} = 285 m, the simulation giving maximum extratropical response has Ψ ∼ 0.4*π*, and QBO winds with maximum amplitude (either eastward or westward, alternating each year) at midwinter at about 22 km (∼43 mb). The simulation giving minimum response has Ψ ∼ 0.9*π*, and maximum QBO winds at midwinter at about 27 km (∼21 mb). It is of note that the phase difference between simulations giving minimum and maximum extratropical response is close to (though not exactly) (*π*/2).

If the height of the QBO maximum wind at midwinter is to be interpreted and explained by the Holton–Tan mechanism for the extratropical QBO, then westward winds are to be associated with a disturbed winter (strong winter warming), and eastward winds with an undisturbed winter. However, as discussed by O’Sullivan and Young (1992), for example, the issue is complicated by the fact that the QBO has a vertical shear in zonal mean wind, so that the winds of maximum amplitude do not occur at all heights at the same time. Since it is not clear what time during the winter evolution is most important in determining the extratropical response to the QBO (or indeed if such a time exists)—there is no reason to assume that it should be exactly midwinter—it is difficult to draw any conclusion about which winds and which heights are associated with a disturbed winter evolution and which with an undisturbed winter evolution.

*z*is now fixed so that there is no dependence of

_{F}*u*

_{vertQBO}on height. To simplify comparisons with the standard QBO we take

*z*= 23 km (and

_{F}*D*= 1 km month

^{−1}as before). The idealized nonsheared QBO is illustrated in Fig. 1b.

The variation of extratropical response *I*_{2} due to varying the phase alignment between a nonsheared QBO of period 2 yr and the annual cycle is shown in Fig. 2 curve (c), for simulations with Φ_{0} = 285 m. Not surprisingly, the nonsheared QBO gives maximum extratropical response for a different value of Ψ than the normal QBO: the maximum response occurs for the nonsheared QBO for the simulation with Ψ ∼ 0.57*π*. The equatorial zonal mean wind is eastward at midwinter for a winter that gives a relatively undisturbed winter, in broad agreement with the Holton–Tan mechanism. However, the equatorial zonal mean wind certainly has not peaked at midwinter in this case: this suggests that, if it is indeed the maximum westward and eastward winds that give the extreme responses for the strength of winter warmings, then midwinter is not quite the most influential time.

Comparing Fig. 2 curves (a) and (c) shows that the maximum possible extratropical response (with respect to varying the phase alignment) is larger for the nonsheared QBO than for the standard QBO. This is also consistent with the Holton–Tan mechanism: if it is eastward and westward winds (or more precisely the effect of those winds on planetary wave propagation) that give the extreme responses for the strength of the winter disturbance, then one would expect that using deep layers of eastward/westward winds (as in the nonsheared QBO) would give a more extreme response (and hence bigger interannual variability) than using vertically sheared winds.

These results for the standard and the nonsheared QBO can be used to give two simple methods for estimating a most important time and height in the equatorial QBO for influencing the extratropics.

First, for the simulation with nonsheared QBO giving maximum extratropical response, the QBO has maximum westward winds in the middle of November. We note that the QBO phase in the model occurring at midwinter cannot simply be deduced from Fig. 1, because this figure is for the prescribed QBO winds. The QBO winds in the model lag behind the prescribed QBO because of the use of the relaxation scheme. So if one were to assume that it is the westward QBO winds that are significant in causing the breakdown of the polar vortex, this would suggest that the middle of November is the time of maximum influence on the winter NH polar evolution.

Second, if it is assumed that this time of maximum influence is the same for the standard QBO (i.e., the QBO with vertical shear), and that for standard QBO simulations it is again the westward QBO winds that are significant for vortex breakdown, then one can find a height of maximum influence by looking at what height the QBO has maximum westward winds at this time in the annual cycle. For the simulation that gave maximum extratropical response (Ψ ∼ 0.4*π*), this implies a height of about 23 km (∼38 mb), as shown in Fig. 3. The idea of a time of maximum influence and a height of maximum influence on the evolution of the winter polar vortex is investigated in further detail in the next section.

## 5. Varying height position and time evolution of the QBO

In the simulations described previously, the equatorial QBO was forced over a standard height range (between 10 and 40 km) and evolved in a semirealistic manner over each winter. In this section the QBO structure and time evolution are altered artificially to examine whether the QBO phase at particular times or heights is more influential on the extratropical circulation than at others.

### a. Varying QBO height

*u*

_{QBO}(

*ϕ*,

*z*,

*t*) is modified. In particular, (4) becomes

*z*

_{1}<

*z*<

*z*

_{3}to give maximum control over the actual zonal wind in the Tropics. Also, the altitude dependence is taken out of the relaxation coefficient in (6), so that it becomes

The value of *z*_{2} (the center of the QBO height range) is used to set the QBO height position, and is varied between experiments. We then take *z*_{3} = *z*_{2} + 5 km, *z*_{1} = *z*_{2} − 5 km. A QBO of period 2 yr is used and Φ_{0} = 285 m for the extratropical wave forcing is used throughout. For each simulation, the strength of the 2-yr extratropical response to the equatorial forcing is measured. Note that for comparison with results of other authors, values of *z*_{2} are taken up to the top of the model. This implies a QBO that extends higher than the upper limit of 30 km or so that is often quoted. However, recent observational studies have suggested that the QBO is distinguishable to the stratopause and beyond, e.g., Baldwin et al. (2001).

The values used for the height *z*_{2} are chosen so that the QBO height range always covers the same number of vertical model grid points. Specifically, *z*_{2} is chosen to always lie halfway between two vertical grid points, so that the 10-km QBO height range always covers four vertical grid points. To avoid complications due to the shear of the QBO winds, a nonsheared QBO is used. The phase alignment between the QBO and the annual cycle is the one that gave the maximum extratropical response in section 4 for the full-height nonsheared QBO, that is, Ψ ∼ 0.57*π*.

The results are shown in Fig. 4. There is clearly a peak around 21 km (∼48 mb), suggesting that at this height the equatorial wind signal plays a more dominant role in determining the extratropical evolution than at other heights. This height of 21 km is consistent with the height of 23 km found by the simple calculation of section 4, within the limit of the model vertical grid resolution of ∼3 km.

It is also of note that above 21 km, the influence of the QBO decreases monotonically with height. This, and the fact that a QBO constrained entirely to the lower stratosphere is sufficient to give an extratropical response (as seen in Fig. 4, and also in the results of section 4) are contrary to the recent results found by Gray et al. (2001a): they found that the presence of a equatorial QBO in the lower stratosphere alone was not sufficient to give a significant extratropical response to the QBO. Rather, they found that this required an imposed QBO in both the lower and the upper stratosphere. Subsequent papers (Gray 2003; Gray et al. 2003) examined this issue in further detail, finding that the lower stratosphere influences the early winter, but that in the later winter the greatest influence may come from the equatorial stratopause. The possible role of traveling anticyclones (zonal wavenumbers 3–8) in building up the Aleutian High was highlighted. Previous modeling studies (e.g., O’Sullivan and Dunkerton 1994; Hamilton 1998) have found a significant extratropical response to an equatorial QBO limited to the lower/middle stratosphere. However, neither of these studies extended the imposed QBO above 38 km, and so did not test whether their response would have been enhanced by the inclusion of the upper stratospheric QBO. We note that the model experiments carried out in this study resolve only mean and wave-1 components in longitude, whereas those used in O’Sullivan and Dunkerton (1994), Hamilton (1998), Gray et al. (2001a), Gray (2003), and Gray et al. (2003) were fully nonlinear models. To examine whether these differing results are due to model dependency or the difference in zonally resolved wavenumbers (and hence the degree of nonlinearity), it would require carrying out further experiments with the current model, but retaining a larger number of zonal wavenumbers. Finally, we note that Gray et al. (2004) found that the timing of winter warmings was sensitive to the equatorial winds in their modeling study. This aspect of the influence of the equatorial region over the extratropics has not been investigated in this work.

### b. Varying QBO time evolution

The aim here is to find if there are times in the winter evolution that are relatively more important than others for the influence of the Tropics, specifically the equatorial QBO, over the extratropics. This is done by carrying out a series of simulations with a relaxed QBO (of standard height range as in section 4) with the prescribed QBO wind *u*_{QBO} being altered during the simulation. Since westward QBO winds are associated with a disturbed winter and eastward QBO winds with an undisturbed winter, by changing *u*_{QBO} from westward (or eastward) to zero at a certain time during the winter evolution (varying this time between simulations), the relative importance of different times over the winter to the NH winter polar vortex evolution can be investigated.

*π*. Before the prescribed QBO wind

*u*

_{QBO}is set to zero, the model is run for several years with

*u*

_{QBO}fixed as constant eastward/westward. At the time when

*u*

_{QBO}is changed to zero, the relaxation constant

*c*is also changed to a bigger value, so that the time lag between changing

*u*

_{QBO}and

*u*changing correspondingly is minimized. Specifically, the evolution of

*u*

_{QBO}in (2) becomes

*T*

_{2}is varied between simulations. For the relaxation parameters,

*c*

_{1}is chosen as before, that is,

*c*

_{1}= (50 day)

^{−1}. Then

*c*

_{2}is maximized within the limits of numerical stability without causing a numerical error:

*c*

_{2}= (1 day)

^{−1}is used here. Using this value of

*c*

_{2}, the equatorial wind changes from being strongly westward/eastward to being close to zero in less than about 0.05 yr, so that there is a significant change of equatorial wind within half a month. The investigation carried out here divides into two parts: those using simulations where the QBO wind

*u*

_{vertQBO}is westward before

*T*

_{2}, that is, where the winter evolution would be relatively undisturbed if

*u*

_{vertQBO}were not changed to zero, and those where

*u*

_{vertQBO}is eastward before

*T*

_{2}, that is, where the winter evolution would be relatively disturbed if

*u*

_{vertQBO}were not changed to zero.

*u*

_{vertQBO}is not changed to zero, that is, if, in place of (2), we use

*u*

_{vertQBO}westward, all winters would be disturbed and for

*u*

_{vertQBO}eastward, all winters would be undisturbed, with no interannual variability. This is the case for the eastward QBO: there is no interannual variability after the initial adjustment period, and all subsequent winters are relatively undisturbed, with

*(as defined at the end of section 3) of about the same amplitude (just marginally larger) as those for the undisturbed years of the simulation with nonsheared QBO when the normal time evolution is not altered. However, for the case of the westward QBO, we do get some interannual variability. Figure 5 shows*u

_{W}*when*u

_{W}*u*

_{vertQBO}is westward. The zonal mean wind is significantly smaller in every year, indicating that the winters are relatively disturbed, but there is also a biennial oscillation between moderately disturbed winters and strongly disturbed winters. This is the biennial oscillation previously noted and analyzed by Scott and Haynes (1998), but with slightly different conditions in low latitudes to the case they studied in detail.

For the simulations in which the winter evolution is changed as in (16), a measure of the strength of winter warmings is required, so that different simulations can be quantitatively compared. As discussed in section 3, the zonal mean wind at 65°N 49 km, and the zonal mean potential temperature at 86°N 40 km are used, both averaged from midwinter to 3 months after midwinter (i.e., over January, February, and March), as a measure of the strength of the winter disturbance. These quantities are denoted by * u _{W}*,

*, respectively. For the case of*

_{W}*u*

_{vertQBO}eastward, the choice of the winter in which the evolution is changed to zero (i.e., the time

*t*=

*T*

_{2}) is unimportant as all winters are identical. For

*u*

_{vertQBO}westward, the evolution is changed in one of the years that is strongly disturbed (e.g.,

*t*= 3 yr in Fig. 5), to get the most extreme possible comparison. The results of varying the time

*t*=

*T*

_{2}are shown in Fig. 6 for

*u*

_{vertQBO}eastward, and Fig. 7 for

*u*

_{vertQBO}westward. For

*u*

_{vertQBO}eastward, changing the QBO evolution to zero winds does not strongly change the evolution of the NH polar vortex: the difference between values of

*for extreme values of*u

_{W}*T*

_{2}is 7 m s

^{−1}, and for

*16°C. The most influential time (for the relatively small change that does occur) is between 2 to 3 months before midwinter. For*

_{W}*T*

_{2}between 3 and 6 months before midwinter, there is a small anomaly where the winter is slightly more disturbed than if the QBO winds had been changed to zero much earlier. There is no obvious explanation for this anomaly.

For the case of *u*_{vertQBO} westward, the effect of changing the QBO evolution to zero is much more significant: for extreme values of *T*_{2}, * u _{W}* changes by up to 75 m s

^{−1}and

*by up to 90°C, hence changing from a strongly disturbed winter to an undisturbed one. This time the transition between extreme responses of*

_{W}*,*u

_{W}*occurs more smoothly, for*

_{W}*T*

_{2}between midwinter and 5 months before midwinter.

For both Figs. 6 and 7 it is worth noting that the overall trends seen in the two measures * u _{W}* and

*are consistent, as one would hope if both statistics are to give a measure of the strength of a winter warming.*

_{W}Considering that it takes up to half a month for the zonal wind *u* to change after *T*_{2}, for the QBO eastward this gives the month around the start of November, and for the QBO westward the 5 months around the start of November, as being the times that influence the extratropical winter evolution. This is consistent with the simplistic argument given in section 4, which gave a most influential time of the middle of November. It is also consistent with the modeling studies discussed in Scott and Haynes (2002), where the early winter conditions are found to be critical for the late winter evolution.

## 6. Correlation and regression between QBO phase and NH extratropical evolution

To carry out correlation analysis for 3D model simulations, it is necessary to choose a QBO period close to that of the observed oscillation but not very close to any fraction of an integer number of years. This is so that a chosen QBO phase will vary over all possible phases of the annual cycle. Here a QBO of period 2.31 yr is used, with model simulations of length 65 yr. We focus on the case of relatively large extratropical forcing, using Φ_{0} = 250 m and Φ_{0} = 285 m.

For the extratropical zonal mean wind and polar potential temperature, we use the measures * u _{W}*,

*defined at the end of section 3. The QBO zonal wind at 22.7 km (∼39 mb) is used for the correlation, as this is the height level closest to the height of maximum influence found in section 5a. To maximize the correlation, the time period over which the QBO wind is measured is varied (because of the regular descent of the idealized QBO, this is approximately equivalent to changing the height level at which the QBO wind is taken). For the simulation with Φ*

_{W}_{0}= 285 m, Fig. 8 shows how the QBO wind at this height correlates with the extratropical wind,

*, and the potential temperature,*u

_{W}*, for various time averaging periods of three months. It can be seen that the best correlation occurs when the QBO wind is averaged around a time 15 days before midwinter, that is, about November–January. From here on,*

_{W}u

_{eq}is used to refer to this 3-month average of the equatorial wind at 22.7 km.

We note that if the correlation time were interpreted to give a most effective time for the QBO’s influence on the extratropical evolution (averaging to give mid-December) we get a time slightly later than the one that was determined by the analysis of section 5b (which gave start to mid-November). The same analysis for the Φ_{0} = 250 m simulation gives December–February as the QBO time period giving maximum correlation.

The correlation between the QBO and the extratropics is illustrated most clearly by plotting for each year of the simulation the zonal mean wind at the equator _{eq} against * u _{W}* and

*. This is shown in Fig. 9 for the simulation with Φ*

_{W}_{0}= 250 m, and in Fig. 10 for the simulation with Φ

_{0}= 285 m.

For the simulation with Φ_{0} = 250 m, the correlation is almost perfectly linear: correlation coefficient 0.988 between equatorial QBO wind and vortex edge zonal wind. However, the QBO winds do not modulate the NH evolution very strongly, that is, there is not a very big difference between the relatively disturbed and undisturbed winters. For the simulation with Φ_{0} = 285 m, the correlation is not as strong (though still with a high correlation coefficient of 0.785), but the disturbed winters are much more disturbed than for Φ_{0} = 250 m. Two aspects of this plot are notable. Firstly, the correlation is almost linear for eastward QBO winds, but the associated modulation by the QBO is relatively small. For westward winds, the correlation is not as strong, but the modulation of the polar vortex is much greater. This is consistent with the results of section 5b (see Fig. 5) and with the results of Gray et al. (2003), for example, who show little dynamical variability with an eastward equatorial wind perturbation, but enhanced variability with a westward perturbation. Secondly, the plot suggests that most of the points lie very close to a smooth curve, with the exception of about 10 points that lie slightly off the curve. We focus on the simulation with Φ_{0} = 285 m because of the stronger modulation of the NH polar vortex by the QBO and the fact that the correlation seen in Fig. 10 is not as linear as that for Φ_{0} = 250 m seen in Fig. 9.

Figures 11 and 12 show the correlation and regression of the equatorial zonal mean wind, respectively, at 22.7 km (averaged over November–January) with the zonal mean wind and zonal mean potential temperature (both averaged over January–March) everywhere in the model. The regression plots clearly show the strong modulation of the polar vortex by the QBO: the centers of strong regression at the polar vortex zonal wind jet and the cold NH pole anomaly stand out sharply. For the potential temperature, there is a second region of strong regression higher up in the NH polar mesosphere: this is consistent with the strong regression in the vortex edge zonal wind jet and thermal wind balance. The correlations found here (and also illustrated in Figs. 9 and 10) are unrealistically high when compared to observations because of the simple mechanistic nature of the model.

For the correlation plots, the regions of highest correlation are not exactly the same as those of highest regression. This highlights the limitations of relying purely on the correlation coefficient to measure the influence of the QBO on the extratropics. The correlation is indeed high at the point of largest regression (as was shown in Fig. 10), but for both the zonal wind and the potential temperature it is even higher slightly away from the centers of strong regression. The reason for this is not certain, but could be because there is smaller overall interannual variability in the regions of highest correlation. In particular, internal interannual variability may not play as big a role, so that the (albeit smaller) modulation by the QBO is very nearly the only cause of interannual variability.

Plots of the EP flux and the corresponding EP flux divergence, when the years for which winters when the QBO was strongly westward and those when the QBO was strongly eastward are separated, show that the EP flux has a stronger vertical component, and that the EP flux divergence is larger at midlatitudes, in particular at the edge of the NH polar vortex, for those years when the QBO is westward compared to those when it is eastward. Hence in the westward QBO years, the Rossby waves propagate more upward into the region of the NH polar vortex edge zonal mean jet and break there, thus having a greater effect in destroying the polar vortex zonal wind jet and causing a winter warming. This is consistent with the Holton and Tan (1980) mechanism for the cause of the extratropical QBO.

## 7. Conclusions and further work

This paper investigated details of the influence of the QBO on the extratropics, specifically on the NH winter polar vortex and the occurrence of winter warmings, using a 3D mechanistic stratospheric zonally truncated model (wave 1 plus zonal mean flow). This zonal truncation is in contrast to some other model studies that had fuller zonal representation.

The phase alignment between the QBO and the annual cycle was shown to play a crucial role in the extratropical response to the QBO. There is a factor of 8 difference in the response between phases giving minimum response and maximum response. Using a nonvertically sheared QBO gives results consistent with those of previous authors, in that the QBO westward winds are associated with a strong polar vortex and QBO eastward winds with a weak polar vortex.

Experiments varying the height and time evolution of the QBO suggest that, in the model, 21–23 km is the height playing the dominant role in the extratropical response. The 5 months around the start of November are found to be important in the QBO’s influence on the extratropical winter evolution. It would be desirable to carry out further studies with a larger number of zonal wavenumbers retained to see how further nonlinear interactions affect these results, and for comparison with the results of other modeling studies.

Correlation and regression plots for a simulation with a noninteger period QBO and strong extratropical wave forcing (Φ_{0} = 285 m) showed high correlation (up to 0.95) between the QBO winds at 22.7 km and the extratropical wind. The regression coefficient is largest in the region of the polar vortex for extratropical winds, and at the pole for extratropical temperatures. The correlation between the QBO and the extratropics was weaker for the simulation with Φ_{0} = 285 m than the simulation with Φ_{0} = 250 m, but the regression coefficient was larger.

One possible explanation of the points lying away from the curve of best fit in Fig. 10 is that they correspond to years in which the internal interannual variability found by Scott and Haynes (1998) contributes, but the nature of any interaction between this internal mode of oscillation and the Holton and Tan QBO response remains unclear.

## Acknowledgments

This work was supported by grants from the Natural Environment Research Council and the UK Meteorological Office. We could like to thank two anonymous reviewers for their useful and insightful comments.

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