## 1. Introduction

Large-scale interannual variability in the tropical stratosphere is dominated by the quasi-biennial oscillation (QBO), consisting of alternately descending westerly and easterly zonal wind regimes. The QBO period is roughly 28 months and the wind regimes descend at a rate of ≈1 km month^{−1}, although substantial cycle-to-cycle variability is present. The QBO is forced by a spectrum of waves of widely varying spatial and temporal scales, from planetary-scale Kelvin and Rossby–gravity waves to small-scale gravity waves (Baldwin et al. 2001). Although the basic wave–mean-flow interaction mechanism that forces the QBO is well established (Lindzen and Holton 1968; Holton and Lindzen 1972), determining the details of this wave spectrum remains a challenge. This presents difficulties when attempting to accurately represent the QBO from first principles in stratosphere-resolving atmospheric general circulation models (GCMs). Such models often do not spontaneously generate a QBO, and the lack of observational constraints makes it difficult to identify the most culpable model deficiencies.

The main difficulty in providing observational constraints lies in the difficulty of observing tropical stratospheric waves. Because different observational techniques are sensitive to different frequencies and spatial scales of waves (Alexander et al. 2010), no single observational dataset provides a comprehensive picture of the wave spectrum that forces the QBO. Compounding the problem is the expectation that small-scale waves, which are the most difficult to observe, make a dominant contribution to the forcing (Dunkerton 1997; Kawatani et al. 2010a). In GCMs run at spatial resolutions typical of climate models, the effects of a significant portion of the small-scale wave spectrum must be parameterized, and many of the parameters used in these schemes are poorly constrained by observations. Although the situation is improving (Geller et al. 2013), presently modelers are afforded substantial freedom when tuning their GCMs to obtain realistic QBOs, which naturally raises the question as to whether realistic QBOs are obtained for realistic reasons.

Waves that force the QBO must propagate from their source regions to the tropical stratosphere, where they dissipate. Hence the characteristics of a modeled QBO can be expected to be sensitive to the characteristics of the modeled wave sources (e.g., tropical deep convection), factors affecting wave propagation (e.g., the climatological winds at altitudes below the QBO), and the relevant dissipative mechanisms (e.g., radiative damping, diffusion of momentum). Extratropical processes may also influence the QBO via the tropical upwelling of the Brewer–Dobson circulation and the equatorward propagation of waves generated at higher latitudes. The multitude of processes involved arguably makes the QBO a sensitive test of model fidelity (Baldwin et al. 2001), yet also suggests that the combination of factors leading to a realistic QBO may not be unique. This raises the strong possibility that tuning a model to exhibit a realistic QBO can involve trade-offs between compensating model errors. Hence it is important to understand the sensitivity of the QBO to model formulation.

The vertical resolution of a GCM can strongly influence its representation of the QBO. This was first shown by Takahashi (1996), who obtained a QBO of realistic amplitude and a 1.5-yr period using a GCM with 0.5-km vertical resolution, T21 horizontal resolution, weak horizontal diffusion, and tropical convection parameterized by a moist convective adjustment scheme. Subsequent work showed that increased horizontal resolution alleviated the need for weak horizontal diffusion (Takahashi 1999) and that the very high levels of resolved-wave activity generated by the moist convective adjustment scheme could be compensated by parameterized nonorographic gravity wave drag (Scaife et al. 2000; Giorgetta et al. 2002). Yet high vertical resolution, typically finer than 1 km in the lower stratosphere, appears to remain a common requirement for GCM simulations of the QBO (Takahashi 1996, 1999; Horinouchi and Yoden 1998; Hamilton et al. 1999, 2001; Giorgetta et al. 2002, 2006; Scinocca et al. 2008; Orr et al. 2010; Evan et al. 2012; Krismer and Giorgetta 2014; Richter et al. 2014; Rind et al. 2014). That this result is found in a variety of models suggests that it is robust and, hence, may be due to a simple physical mechanism. However, there are counterexamples. Hamilton and Yuan (1992) found no sensitivity of tropical zonal-mean wind oscillations to a threefold increase in vertical resolution (as fine as 0.7 km). Boville and Randel (1992) obtained no tropical zonal-mean wind oscillations despite the fact that progressively increasing vertical resolution (2.8, 1.4, and 0.7 km) caused larger momentum deposition by Kelvin and mixed Rossby–gravity waves in the lower tropical stratosphere.

The importance of vertical resolution is consistent with the expectation that large vertical shear of the QBO winds can Doppler shift a vertically propagating wave to small vertical scale, reducing its vertical group velocity and increasing its radiative damping rate. Nevertheless, it is likely that a substantial fraction of the necessary wave forcing is completely unresolved by GCMs run at horizontal resolutions often used in climate models (e.g., spectral truncation of T63, or grid spacings of ≈1°–3°. Observations indicate that large-scale waves cannot account for all of the forcing required to maintain the QBO (Dunkerton 1997; Sato and Dunkerton 1997), and high-resolution GCM experiments with no parameterized gravity wave forcing suggest that much of the wave activity forcing the QBO occurs at small horizontal scales (Kawatani et al. 2010a,b, who used a T213 spectral truncation). Forcing a QBO at coarser horizontal resolution using only resolved waves appears to require unrealistically large amounts of upward-propagating resolved-wave activity, as can be generated by the moist convective adjustment scheme (Takahashi 1999; Horinouchi et al. 2003). Scaife et al. (2000) showed that a GCM with modest horizontal resolution (2.5° × 3.75°) and parameterized nonorographic gravity wave drag could generate a QBO. Moreover, they showed that either the Warner–McIntyre gravity wave parameterization scheme (Warner and McIntyre 1999) or the Hines scheme (Hines 1997a,b) both gave reasonable results, provided they were suitably tuned. (As noted above, the lack of observational constraints makes such tuning justifiable.) Giorgetta et al. (2002, 2006) similarly used the Hines scheme to obtain a reasonably realistic QBO in a GCM run at T42 horizontal resolution. It is worth noting that in these studies, tuning a gravity wave scheme allowed a QBO to be obtained with few other model changes required (e.g., modifying the parameterization of deep convection), thus reducing the likelihood of undesirable impacts of the tuning on other aspects of the model climate (e.g., extratropical variability). This is useful because one of the main reasons to represent the QBO in a GCM is so that its interactions with other aspects of the climate can be modeled accurately, such as in model integrations lasting many decades allowing analysis of natural variability on long time scales, or using large ensembles of initialized experiments as in seasonal forecasting.

Yet given that parameterized waves can compensate for a lack of resolved-wave forcing, it is unclear why high vertical resolution should be required. To the best of our knowledge, GCMs generally require vertical resolution in the lowermost tropical stratosphere of ≈1 km or finer in order to exhibit a QBO.^{1} This suggests that there are limitations on the degree of compensation by gravity wave drag that is possible. If this is true, then it is of interest to determine why, since the basic theory of the QBO does not restrict the particular types of waves (e.g., large scale vs small scale) that can contribute to the QBO forcing (Lindzen and Holton 1968; Holton and Lindzen 1972; Plumb 1977). While the characteristics of the wave spectrum are expected to affect the amplitude and vertical structure of the resulting oscillation (e.g., Campbell and Shepherd 2005a,b), and there is observational evidence that a wide range of wave scales is involved (Baldwin et al. 2001), it is not clear that there is a basic physical reason why a QBO could not be forced entirely by parameterized waves.

In this study we attempt to determine why fine vertical resolution appears to be required to simulate the QBO in a GCM—or to put it another way, why parameterized nonorographic gravity wave drag appears unable to drive a QBO when the vertical resolution is coarse. A description of the GCM and its nonorographic gravity wave drag parameterization is given in section 2. A benchmark QBO simulation is briefly described and compared to reanalysis data in section 3. Section 4 examines the sensitivity to vertical resolution of the QBO and its forcing by resolved waves. In section 5, an idealized one-dimensional QBO model is used to interpret the GCM results, accounting for the properties of the gravity parameterization that is used in the GCM. Section 6 summarizes the results and briefly discusses implications.

## 2. Model description

The GCM used in this study is the Canadian Middle Atmosphere Model (CMAM), which solves the hydrostatic primitive equations in spherical coordinates (Beagley et al. 1997; Scinocca et al. 2008). The model is horizontally spectral and vertically finite difference. Most of the model runs use the T47 spectral truncation, although some T63 runs are used to test the robustness of the results to a change in horizontal resolution. (Expressed as one-half the shortest represented zonal wavelength at the equator, these resolutions are 428 and 319 km, respectively.) A variety of vertical grids are employed, the details of which are described in section 4. The vertical discretization is defined by a hybrid vertical coordinate *η* that behaves as a terrain-following coordinate at low altitudes and becomes essentially equivalent to pressure levels in the stratosphere; see Beagley et al. (1997) for further details. For convenience we express values of the vertical resolution, which are actually defined by the choice of *η* levels, as the difference Δ*z* between vertical levels *z* defined by *z* = *H* log(η) where *H* = 7 km. Since *η* takes values between 1 and 0, *z* is essentially equivalent to log-pressure altitude in the stratosphere.

Radiative transfer from the surface to the midstratosphere is parameterized using the scheme of Morcrette (1991), and altitudes from the midstratosphere to the model lid at ≈0.01 hPa (η ≈ 10^{−5}, *z* ≈ 80 km) use a scheme suitable for the middle atmosphere (Fomichev et al. 1993, 2004). The transition between the two schemes occurs by linear interpolation over the altitude range from 40 to 5 hPa, or *z* ≈ 23–37 km, as described by Fomichev and Blanchet (1995). Vertical diffusion is parameterized by a fixed coefficient (0.1 m^{2} s^{−1} unless stated otherwise) with an enhancement being applied when the Richardson number (Ri) becomes small, although this small-Ri correction is generally unimportant in the stratosphere. Horizontal diffusion follows the Leith form as described in Boer et al. (1984), which behaves similarly to a high-order hyperdiffusion. Sea surface temperatures are imposed as a smoothly varying climatology and no interannual forcings are applied; all interannual variability in the GCM is internally generated. Deep convection is parameterized by the Zhang and McFarlane (1995) scheme, with large-scale precipitation acting to stabilize moist profiles that do not initiate deep convection.

The nonorographic gravity wave drag (GWD) parameterization scheme used is that of Scinocca (2003, hereafter S03), which is a spectral gravity wave parameterization in the framework of Warner and McIntyre (1996). Full details of the GWD scheme are given in S03; we summarize only the salient points here. As in other GWD schemes, a spectral density of gravity wave pseudomomentum flux (“momentum flux” for brevity) is specified at a launch level *H* = 7 km). By choosing

It is conventional in GCMs to assume the launch spectrum to be horizontally homogeneous—that is, identical at all horizontal grid points. We relax this assumption by introducing a simple variation in latitude, shown in Fig. 1. The GWD scheme is called twice at each time step in a model run, with the total launch flux being modulated by the tropical function in one call and the extratropical function in the other. This allows us to modify the tropical GWD parameters, directly affecting the QBO, without introducing extratropical changes forced directly by extratropical GWD (e.g., changes in the Brewer–Dobson circulation). The results of the two GWD scheme calls are summed to give the total GWD forcing at each model grid point. The latitudinal shape of the tropical function is derived from an annually averaged precipitation climatology obtained from a CMAM run, justified by the expectation that large amounts of small-scale gravity wave activity are generated by tropical deep convection.

The momentum flux spectrum is discretized on a *c* the phase speed with respect to the solid Earth and *U* the background flow in the azimuth. Two types of gravity wave dissipation are employed: critical-level filtering, which annihilates a spectral element if it reaches a critical level (*m*^{−3} shape seen in observed gravity wave spectra (Smith et al. 1987). It should be noted that these two mechanisms are not entirely distinct, since the dispersion relation *N* is the buoyancy frequency, implies that *m* approaches infinity as a wave approaches its critical level. In principle, sufficiently high vertical resolution could allow a spectral element of momentum flux to be almost completely absorbed during the approach to the critical level. But regardless of the vertical resolution, explicit critical-level filtering is required to ensure that unphysical propagation of waves through their critical levels does not occur. If background conditions (i.e., the resolved winds and temperatures at a GCM horizontal grid point) are such that no dissipation occurs, then the waves propagate conservatively without affecting the resolved flow.

## 3. A realistic QBO simulation

The most basic characteristics of a QBO-like oscillation are that it is downward propagating and is not synchronized with the seasonal cycle (i.e., its period is not a fixed multiple of 6 months). It is also desirable for the oscillation to have a realistic period (i.e., an average period of 2–3 yr), but the basic QBO mechanism implies that the period depends on the total wave forcing, which in the GCM is due to both resolved and parameterized waves. The GWD total launch momentum flux is only weakly constrained by observations and can be tuned to give the oscillation a realistic period, which has been previously done in versions of CMAM with fine vertical resolution (Scinocca et al. 2008; Anstey et al. 2010) as well as in other models (e.g., Scaife et al. 2000; Giorgetta et al. 2006; Richter et al. 2014). In this section we briefly describe the characteristics of a CMAM simulation using fine vertical resolution Δ*z* = 0.5 km in the lower stratosphere and total GWD flux that has been adjusted to obtain a realistic QBO period.

Figure 2 shows the time evolution of equatorial (2°S–2°N mean) zonal-mean zonal wind

A time-mean westerly bias at the lowest QBO altitudes—that is, a weak downward penetration of the QBO-E phase—is present to varying degrees in all CMAM runs we have performed (i.e., in all QBO-resolving configurations of the model). This westerly bias was larger in a previous model version and was substantially reduced in the run shown in Fig. 2 by shifting the tropical GWD launch level slightly upward to ≈90 hPa (

## 4. Vertical resolution

### a. Free-running experiments

Figure 3 shows *z*. Vertical profiles of the grid spacing Δ*z* for these runs are shown in Fig. 4a, and no model parameters differ between these runs except for their vertical grids. Figure 3a shows that at the coarse resolution of Δ*z* = 1.5 km, oscillations in *z* to 1.25 km (Fig. 3b) yields a longer period and more variation between cycles but retains a strong tendency to phase lock with the annual cycle, and Δ*z* = 1.0 km is similar but with slightly increased downward penetration of the westerlies (Fig. 3c). This tendency continues at Δ*z* = 0.75 km, showing further increases in period and low-level amplitude (Fig. 3d). Decreasing Δ*z* still further, to 0.5 km, results in the oscillation shown in Fig. 2a. For even smaller Δ*z* the oscillation remains qualitatively similar, although the period and amplitude continue to increase (e.g., as shown for Δ*z* = 0.31 km in Fig. 6a, and to be discussed below). The Δ*z* = 0.5-, 1.0-, and 1.5-km experiments will be referred to as model runs A, B, and C, respectively.

This tendency of increasing vertical resolution to break the seasonal synchronization and lengthen the period occurs in all CMAM configurations that we have tested. The key question addressed in this study is to explain why changes in vertical resolution have this effect. Although changes in GWD parameters strongly affect some characteristics of the oscillation, such as its amplitude and period, they appear to have no effect on this behavior; for all GWD settings that we have tested, *z* is coarser than ≈1 km (details of these GWD sensitivity experiments will reported in a subsequent study). This does not exclude the possibility that the GWD can be configured to induce a QBO at coarse vertical resolution, and in future work we intend to explore further modifications to the S03 scheme to try and circumvent this limitation. However, numerous experiments conducted so far have tested the effects of adjustments to most of the standard S03 scheme parameters, and these experiments have consistently produced *z* is coarser than ≈1 km. Hence, we focus here on how Δ*z* affects the resolved waves and mean flow.

One possibility is that vertical resolution in the troposphere affects the generation and propagation of waves that eventually reach the lower tropical stratosphere. The QBO is strongly dependent on the characteristics of the waves that drive it, and hence changes in tropospheric wave sources caused by Δ*z*—for example, changes in the behavior of the parameterized deep convection—could affect the QBO. This effect cannot explain the results of Fig. 3, since the vertical grids for these experiments (Fig. 4a) differ only above the tropical tropopause, but its potential importance for the QBO appears not to have been explicitly considered in previous studies. To distinguish between effects of tropospheric and stratospheric Δ*z*, the experiments having Δ*z* = 0.98 km and Δ*z* = 1.55 km shown in Fig. 4b were performed for comparison with runs B and C.^{2} They differ from B and C in that Δ*z* coarsens to its stratospheric value somewhere in the midtroposphere, rather than having Δ*z* = 0.5 km everywhere below the tropical tropopause. It was found that *z* = 0.98-km experiment strongly resemble those in B, and likewise for Δ*z* = 1.55 km and C (not shown). A similar comparison was also performed for an alternate set of runs using altered deep convective settings (as described briefly in section 3) and yielded the same result. A further sensitivity test was performed using the vertical grid labeled Δ*z* = 0.5 km in Fig. 4b, in which there is a drastic contrast between coarse upper-tropospheric Δ*z* and fine-stratospheric Δ*z*; the physical motivation for this choice is that high-stratospheric values of *N* will refract waves crossing the tropopause to smaller vertical wavelength. Using this vertical grid in run A produced essentially no change in the QBO. These comparisons suggest that mid- to upper-tropospheric Δ*z* is not the main determinant of whether a *z* ≈ 0.5–1.5 km and the tested deep convection parameters. This is consistent with the expectation that lower stratospheric Δ*z* is most important for the QBO, which is physically reasonable given that the vertical wavelengths of waves propagating upward through the tropopause should decrease as the waves encounter the increased background stratification of the stratosphere.

Since the effects of tropospheric Δ*z* appear to be minor, we focus hereafter on the effects of stratospheric Δ*z*. It is useful to briefly review the reasons why Δ*z* can be expected to affect the propagation and dissipation of waves in the tropical stratosphere. The QBO is driven by upward-propagating tropical waves that dissipate some or all of their momentum flux in the tropical stratosphere (Lindzen and Holton 1968; Holton and Lindzen 1972; Plumb 1977), and there is a broad spectrum of such waves covering a wide range of horizontal phase speeds (Baldwin et al. 2001; Horinouchi et al. 2003). Waves having zonal phase speeds within the range of

To estimate the vertical resolution that might be required to represent the waves that are relevant to the QBO, consider the dispersion relation for Kelvin waves, *c* being the ground-based zonal phase speed, with *c* > 0 eastward)—and *m* is the vertical wavenumber, which is related to the vertical wavelength *c* = 25 m s^{−1} (e.g., Wallace and Kousky 1968) propagating upward from the tropical tropopause region into a QBO westerly phase in the lower stratosphere will experience a background wind change of ^{−1}, hence a Doppler shift in ^{−1}. For a typical tropical stratospheric buoyancy of *N* = 0.022 s^{−1} (Grise et al. 2010), this corresponds to *z* > 1 km, and even the nonshifted wave (*z* ≈ 2–3-km grid. Small

From the foregoing considerations, mean-flow forcing by resolved waves in the lower tropical stratosphere would be expected to increase as Δ*z* decreases. Figure 5 shows vertical profiles of all zonal-mean-flow forcing terms for experiments A, B, and C, composited by 32-hPa westerly phase (QBO-W) onsets. The zonal momentum budget is diagnosed using the transformed Eulerian mean primitive equations (e.g., Andrews et al. 1987) in which the forcing by resolved waves is the divergence of the Eliassen–Palm (EP) flux, ∇ ⋅ **F**. As expected, Fig. 5 shows ∇ ⋅ **F** increasing with the vertical resolution (i.e., with decreasing Δ*z*). Similar composites for 32-hPa easterly phase (QBO-E) conditions indicate very little change in ∇ ⋅ **F**, showing that in this model it is mainly eastward waves that are affected by Δ*z*.

What mechanisms cause ∇ ⋅ **F** in the lowermost tropical stratosphere to increase with resolution? One of the possibilities noted above is the dependence of radiative damping rate on *z* = 0.31 km (see Fig. 4b for the vertical profile of Δ*z*), shown in Fig. 6a. The higher vertical resolution is desirable to maximize the possible effect of scale-dependent radiative damping rates. The middle atmosphere radiation scheme uses coarsened vertical resolution for computational efficiency, so as a sensitivity test we shift the transition between this scheme and the Morcrette scheme, which uses the model Δ*z*, to occur near the stratopause (see section 2 for further details). This is expected to increase the radiative damping of resolved waves, and Fig. 6b shows that the result is an increase in QBO period. To confirm that this change is due to scale-dependent radiative damping, Fig. 6c shows the same experiment as Fig. 6b but with a vertical smoothing applied to the temperature input to the Morcrette scheme in the *z* ≈ 16–31-km region. The smoothing strongly damps vertical wavelengths finer than *z* ≈ 20–25 km) than the QBO of Fig. 6c (not shown). Note that increased period and downward vertical penetration is also caused by increasing vertical resolution (Fig. 3).

### b. Nudged experiments

The results of Fig. 6 suggest that although scale-dependent radiative damping affects the wave driving of the QBO, it is not the crucial reason why fine Δ*z* is required. (If it were the crucial factor, then vertically smoothing the temperature input to the longwave radiative scheme should have significantly degraded the QBO, but Fig. 6c shows that it did not.) The key question is to address how ∇ ⋅ **F** responds to changes in Δ*z*. Experiments A, B, and C are insufficient to answer this question because both the mean flow and ∇ ⋅ **F** differ between these experiments, and hence the response of the waves to Δ*z* and to the mean-flow changes cannot be separated. To isolate the response of the waves to Δ*z* we use model relaxation experiments in which the zonal-mean state is constrained but the vertical resolution is varied. The grids for runs A, B, and C are used (Δ*z* = 0.5, 1.0, and 1.5 km, respectively), for which the relaxed runs are denoted Ar, Br, and Cr, and the global zonal-mean flow is nudged toward that of run A. Since run A exhibits a reasonably realistic QBO, the relaxation allows waves in the coarse-resolution relaxed runs (Br, Cr) to propagate through strong mean-flow vertical shears that do not occur in the corresponding free-running experiments (B, C). The relaxation is accomplished by strongly nudging (with a 12-h relaxation time scale) the zonal-mean spectral amplitudes of vorticity, divergence, and temperature at all model levels toward the time-evolving state from the free-running experiment A, which is vertically interpolated to the B and C grids for use in Br and Cr. (Hence we expect Ar and A to give essentially similar results; Ar was performed as a consistency check on the method.) Note that this method has been used in other studies with CMAM to examine tropospheric responses to constrained stratospheric flow conditions (Simpson et al. 2011; Hitchcock and Simpson 2014). Here we use the same method to separate the effects of mean-flow changes and model resolution changes in the tropical stratosphere.

Since no nudging is applied to the zonally asymmetric components of the flow, differences in ∇ ⋅ **F** between runs Ar, Br, and Cr indicate the response of resolved-wave dissipation to changes in Δ*z*. Figure 7 shows ∇ ⋅ **F** superimposed on *z* ≈ 24 km). The response to changes in vertical resolution of eastward ∇ ⋅ **F** progressively increases as the grid spacing changes from coarse to fine, following the descent of the zero-wind line. At coarse resolution (Δ*z* = 1.5 km; Fig. 7a), eastward ∇ ⋅ **F** occurs at lower altitudes within the shear zone (i.e., ∇ ⋅ **F** > 0 occurs farther away from the eastward **F** increases at intermediate resolution (Δ*z* = 1.0 km; Fig. 7b). At the finest resolution (Δ*z* = 0.5 km; Fig. 7c), eastward ∇ ⋅ **F** is strongest and reaches the farthest into the shear zone (i.e., closest to the eastward **F**. Eastward forcing shifts upward to the region where

Constructing the equivalent of Fig. 7 for the descending QBO-E phase (not shown) indicates that a similar effect on westward ∇ ⋅ **F** occurs, but its magnitude is considerably reduced in comparison to the response of eastward ∇ ⋅ **F** to changed Δ*z*. In this model the primary effect of Δ*z* is to allow eastward-propagating resolved waves to force the mean flow at smaller

The response of eastward ∇ ⋅ **F** to Δ*z* is clear in Fig. 7, but its contribution to the overall QBO forcing is modest. As shown by the vertical profiles of momentum budget terms for the free-running experiments (Fig. 5), in run A the GWD contribution is at least twice as large as ∇ ⋅ **F**. Output of the GWD tendencies from experiments Ar, Br, and Cr confirmed that the GWD shows essentially no change due to Δ*z*; that is, the GWD tendency is essentially determined by the zonal-mean state shear, and offline calculations with the GWD scheme were used to confirm this (not shown). The vertical advection term in Fig. 5 changes strongly from run C to A owing to intensification of both the shear and vertical velocity, which are related by thermal wind balance [e.g., Baldwin et al. (2001), their section 2.2]; in the relaxed experiments no appreciable differences in vertical velocity are seen, as expected since the wind shear is essentially prescribed. It is changes in eastward ∇ ⋅ **F** induced by changes in vertical resolution that cause the

### c. Further experiments related to spatial resolution

We tested the effects of other resolution-related parameters on the QBO. Regarding the vertical resolution, an obvious question is whether the chosen vertical diffusion affects the oscillation. The standard value used in the model is 0.1 m^{2} s^{−1} (section 2). Modifying run A to use larger values of 0.3 and 0.5 m^{2} s^{−1} resulted in more smeared-out shear zones of ^{2} s^{−1} yielded a slightly shorter QBO period and slightly reduced amplitude, consistent with the arguments given above to explain the response to weakened radiative damping (Fig. 6; section 4a). The lack of a large increase in the vertical shear of

The sensitivity of the QBO to horizontal resolution, horizontal diffusion, and model time step was also tested. At T47 resolution, decreasing the model time step from 450 to 300 s had no effect on the QBO. Increasing the horizontal resolution to T63 (which also required reducing the time step from 450 to 300 s) did not change the response to vertical-resolution changes seen in runs A, B, and C. Increased horizontal resolution gave increased mean-flow forcing by resolved waves, necessitating a modest reduction in the GWD strength in order to obtain a realistic QBO period, but after this adjustment the T63 and T47 QBOs looked similar. The response to changed horizontal diffusion was more dramatic, but qualitatively similar. Modest adjustment of the Leith diffusion coefficients (which control dissipation of vorticity, divergence, and temperature) produced large changes in ∇ ⋅ **F**, which is of potential concern given that these parameters are poorly constrained by observations. Nevertheless, reducing the horizontal diffusion coefficients by a factor of 4 in runs B and C did not qualitatively change

The effect of changes in horizontal diffusion and resolution on the horizontal structure of the QBO was also diagnosed. All QBOs obtained in CMAM have some tendency for the meridional width of the oscillation at lower altitudes (*z* ≈ 20–25 km) to be narrower than that at higher altitudes, while the QBO in ERA-Interim has roughly the same meridional extent at all altitudes (not shown). Following Giorgetta et al. (2006), we tested whether excessive horizontal diffusion might cause this behavior by reducing the magnitude of the horizontal diffusion applied to only the zonal-mean component of the stratospheric circulation. This did not alter the meridional width of the QBO at lower altitudes, and the origin of the attenuation with decreasing altitude remains unexplained. Possibly it may be related to the meridional extent of resolved-wave sources: although Haynes (1998) points out that the meridional width of the QBO may be determined by the width of the characteristic tropical response to forcing, rather than by the width of the forcing itself, narrowness of the forcing may still affect the width if it is narrower than the width of the region over which a tropical response occurs. Further consideration of this behavior is left for future work. It is potentially important given that the effect of the QBO on other regions of the atmosphere may be sensitive to the meridional width of the QBO (O’Sullivan and Young 1992).

### d. Convergence

It is natural to ask whether convergence has been achieved at Δ*z* = 0.5 km. Does the simulated QBO change when vertical resolution is increased further? Figure 8 compares vertical profiles of the composited *z* = 0.5 (run A), 0.25, and 0.14 km, respectively. (Vertical profiles of Δ*z* for the latter two runs are shown in Fig. 4b, and all model parameters other than the vertical grids are the same as run A.) It is evident from Fig. 8 that the amplitude and downward vertical penetration of the QBO-W phase increases with increasing vertical resolution. The QBO period also increases as Δ*z* decreases (not shown), which is reminiscent of the result for finescale radiative damping shown in Fig. 6 and consistent with the trend already seen in Fig. 3 for coarser Δ*z* values. Note that if the total wave forcing is roughly unchanged, because of tropospheric wave generation being unchanged, then larger amplitude is consistent with longer period owing to the total change in momentum between opposite QBO phases being larger.

It is apparent that decreases in Δ*z* beyond 0.5 km yield diminishing returns, suggesting that while run A (Δ*z* = 0.5 km) has not completely converged, it may be close. Whether convergence should be expected depends on the spectrum of phase speeds that force the QBO, since this determines *c* ≈ 20 m s^{−1} should be reasonably well resolved at Δ*z* = 0.5 km when the background flow is 10–15 m s^{−1}, and Horinouchi et al. (2003) showed that vertical EP-flux at ≈65 hPa in several models has substantial power at *c* ^{−1}. It is also plausible that waves with very small *c* will tend to be filtered out by the tropospheric flow before reaching the QBO region. However, the validity of these arguments clearly depends on the details of tropospheric wave sources and wave filtering, which are likely to be model dependent. Also, Fig. 8 shows that the downward penetration of the QBO increases slightly with the vertical resolution. If the QBO amplitude near the tropopause is important—for example, for possible QBO influence on deep convection (Collimore et al. 2003; Liess and Geller 2012)—then further increases in vertical resolution may be important. It is clear from Fig. 8b that the GCM strongly underestimates the downward penetration of the QBO-E phase in comparison to ERA-Interim. Underestimated QBO-E amplitude also occurs at 50 hPa, which is the altitude of QBO winds that has been found to most strongly correlate with Northern Hemisphere winter stratospheric polar vortex strength in many studies [for a summary, see Anstey and Shepherd (2014)]. Hence, the polar vortex response to the QBO could be affected by this bias in QBO-E amplitude.

## 5. Discussion: Comparison with an idealized QBO model

The results of section 4 indicate that increased mean-flow forcing by resolved waves appears to be required for the

The model is 1D, representing the QBO in the vertical direction only, so that the model state *u*(*z*, *t*) represents the vertical profile of *u*(*z*, *t*) is simply the sum of the forcings from each contributing wave component, it is straightforward to incorporate other types of wave forcing into this model, such as gravity waves undergoing saturation as formulated in the S03 GWD scheme. We use the 1D model to compare the responses of *u*(*z*, *t*) to these two types of wave forcing, which for brevity will be referred to as the “damped” and “saturated” cases.^{3} It is assumed that the combined effects of these two types of forcing in the 1D model provides an adequate conceptual model of the combined effects of resolved and parameterized wave forcing in the GCM.

*u*(

*z*,

*t*) iswhere

*H*,

*n*th wave, and

*υ*is the vertical eddy diffusivity. For the damped case, the vertical structure of

*n*th wave having zonal phase speed

*N*is the buoyancy frequency, and

*α*is the radiative damping rate. All waves, for both damped and saturated cases, are assumed to have upward group velocity, so that the forcing is imposed by specifying

*a*= 1.5; this expression is equivalent to the

*m*

^{−3}saturation bound.

^{4}

*z*and is nonzero everywhere (up to the critical level). For the saturated case, conservative propagation will occur at altitudes where

*z*to offset decreases in

*ρ*, which can occur if

*u*/∂

*z*. In contrast, as shown by P77, it follows from (2) and (3) that the forcing in the damped case,

*u*but not on ∂

*u*/∂

*z*, which prevents downward influence in the flow. By numerically integrating (1) for the case of two waves with oppositely signed

*c*(but otherwise identical properties, as in P77) and an SAO imposed at the upper levels, we have verified that seasonal synchronization is obtained when the two waves are saturated but not when they are damped.

Since downward influence can occur in the saturation forcing case, interaction of gravity waves with the SAO can produce an oscillation that propagates downward from the SAO region, as also occurred for the case of gravity wave critical-level filtering shown in Lindzen and Holton (1968). Hence in a model incorporating both types of wave forcing—both damped and saturated—seasonal synchronization should be expected when the saturation forcing dominates.^{5} This is consistent with the results of section 4, which showed that resolved-wave forcing decreased as Δ*z* coarsened (Fig. 7) and that GWD dominates the mean-flow forcing when Δ*z* is coarse (Fig. 5). In making this interpretation we are assuming that damping as expressed by (2) and (3) is an adequate model of the forcing by resolved waves.

There is a caveat, however, that downward influence need not be dominant just because it is possible. (Vertical diffusion allows downward influence, but its effect is negligible unless the flow curvature is very large, as in the “switching” region at the lowest altitudes of the 1D model.) Campbell and Shepherd (2005a) showed that it is possible to drive QBO-like oscillations in the 1D model using either the Lindzen (1981, hereafter L81) or Alexander and Dunkerton (1999, hereafter AD99) gravity wave parameterizations without requiring an SAO. While these are different parameterizations than the one used in our GCM, schemes of the Warner and McIntyre (1996) type such as S03 share with L81 and AD99 the characteristic feature that gravity waves need not encounter critical levels in order to deposit momentum in the mean flow. The L81 scheme is similar to S03 in that it assumes saturation acts to maintain waves at the threshold of stability, thereby allowing continuous profiles of forcing over finite altitude ranges and multiple wave-breaking levels.^{6} The fact that a 1D model using the L81 or AD99 scheme can exhibit a QBO appears at odds with the inability of the GCM to exhibit a QBO when GWD forcing by the S03 scheme is dominant, as it is in the coarse-

The key difference between L81 and S03, for the purpose of this comparison, is the assumption in S03 that the GWD spectrum is saturated at launch. This assumption is useful because it provides an observational constraint on the shape of the launch spectrum (i.e., that it varies as *m*^{−3} for ^{7} Numerical integration of the 1D model, for the aforementioned two-wave case, shows that if the S03 scheme is modified so that the launch spectrum for *D* < 1 (i.e., *D* = 1 would correspond to saturation at launch) then a QBO-like oscillation can occur without an SAO being required to initiate the descent of shear zones. But imposing an SAO can still act to synchronize the oscillation so that its period is an integer multiple of 6 months, and this is the case relevant to interpreting the GCM behavior since the GCM spontaneously exhibits an SAO. Further investigation of this behavior, in either the 1D model or the GCM, is left for future work.

The QBO in the GCM, and in the real atmosphere, is forced by waves having a wide range of scales and zonal phase speeds. It is therefore questionable to what degree the preceding conclusions based on an idealized QBO model, forced by two waves that obey one or the other of two idealized damping mechanisms, are applicable to the GCM or to reality. We have conducted numerous GCM experiments, analogous to runs B and C of section 4 (i.e., using Δ*z* ≥ 1 km), in which modifications were made to most of the standard parameters of the S03 scheme. Further details of these experiments will be reported in a subsequent study, but for the purposes of this study they yielded one essential null result: in the coarse-Δ*z* experiments, we have failed to induce any oscillations in *z* version of CMAM would need to be driven almost entirely by GWD). However, this null result from our coarse-Δ*z* GCM experiments, combined with the basic properties of the saturation forcing (as described in this section), strongly suggests that saturation-dominated mean-flow forcing can tend to produce seasonally synchronized mean-flow oscillations. It should be noted that this does not necessarily imply that the saturation forcing is in some way deficient or unrealistic. The real QBO is observed to partially synchronize with the seasonal cycle (Dunkerton 1990). If real gravity waves behave similarly to the parameterized gravity waves considered here—and it is worth noting that the *m*^{−3} saturation spectrum has observational and theoretical support (Warner and McIntyre 1996)—then the forcing contribution from small-scale gravity waves may help to explain the partial seasonal synchronization of the real QBO.

## 6. Conclusions

The QBO has been simulated in a GCM and its sensitivity to some aspects of model formulation has been described. The results indicate that vertical resolution better than 1 km is required in the lower tropical stratosphere for the oscillation to break synchronization with the seasonal cycle, thereby displaying quasi-biennial periodicity. None of our GCM experiments with vertical resolution coarser than 1 km have exhibited a QBO; instead, oscillations in these experiments are synchronized with the seasonal cycle. It was argued that this may result from the saturation dissipation mechanism assumed in the gravity wave parameterization scheme, which allows downward influence from the SAO. A modest increase in resolved-wave forcing circumvents this model limitation. However, parameterized wave forcing is also essential for the model to exhibit a QBO, making a dominant contribution to the zonal momentum budget and effectively amplifying the forcing by resolved waves.

A vertical resolution of Δ*z* = 0.5 km appears to be reasonably close to convergence, and it was suggested (section 4d) that this result need not depend strongly on the details of the resolved tropical wave spectrum in the model. However, results do show that further increases in QBO amplitude and downward penetration occur as the resolution is increased further, and these increases may be important to capture two-way coupling with the troposphere (Collimore et al. 2003; Liess and Geller 2012).

Previous studies of GCM simulations of the QBO have shown widely varying combinations of resolved and parameterized wave forcing. In this study we have sought to better understand the link between these forcings so as to better understand why high vertical resolution appears to be required to simulate the QBO realistically. In this context, some implications of our results for further model development are worth noting. Whether the wave forcing allows downward or only upward influence has implications for determining which model biases can affect the QBO (e.g., whether SAO biases can play a role). The fact that forcing by saturated gravity waves does not strictly require an SAO for a QBO to be generated might be evidence in favor of gravity wave parameterizations that include this mechanism and, in particular, for the possibility that waves in the lowermost stratosphere are not saturated (McLandress and Scinocca 2005). A related issue is that the modeled QBO is highly sensitive to the choice of gravity wave launch level owing to the fact that wave filtering at altitudes below the QBO can strongly bias the zonal phase speed distribution of the parameterized GWD that enters the QBO region, strongly affecting the ability of GWD to force the QBO. A possibly important effect is that parameterized waves encountering strong zonal wind vertical shear near the tropopause may be unsaturated when they reach the lowermost QBO altitudes—a situation that increases the ability of these waves to force a QBO-like oscillation (as described in section 5). Although it is still unclear what is the most realistic partitioning of wave forcing between large and small scales, our results suggest that the two types of forcing have distinct properties that may be manifest in the partial seasonal synchronization of the QBO.

## Acknowledgments

We thank Slava Kharin, Fouad Majaess, and Mike Berkley for technical assistance with various aspects of CMAM. For helpful discussions we thank Thomas Birner, George Boer, Peter Hitchcock, Jiangnan Li, Norm McFarlane, Charles McLandress, Scott Osprey, and Ted Shepherd, and we thank the three anonymous reviewers for their constructive and detailed comments. JAA acknowledges support from a C-SPARC postdoctoral fellowship.

## REFERENCES

Alexander, M. J., , and T. J. Dunkerton, 1999: A spectral parameterization of mean-flow forcing due to breaking gravity waves.

,*J. Atmos. Sci.***56**, 4167–4182, doi:10.1175/1520-0469(1999)056<4167:ASPOMF>2.0.CO;2.Alexander, M. J., and et al. , 2010: Recent developments in gravity-wave effects in climate models and the global distribution of gravity-wave momentum flux from observations and models.

,*Quart. J. Roy. Meteor. Soc.***136**, 1103–1124, doi:10.1002/qj.637.Andrews, D. G., , J. R. Holton, , and C. B. Leovy, 1987:

*Middle Atmosphere Dynamics.*International Geophysics Series, Vol. 40, Academic Press, 489 pp.Anstey, J. A., , and T. G. Shepherd, 2014: High-latitude influence of the quasi-biennial oscillation.

,*Quart. J. Roy. Meteor. Soc.***140**, 1–21, doi:10.1002/qj.2132.Anstey, J. A., , T. G. Shepherd, , and J. F. Scinocca, 2010: Influence of the quasi-biennial oscillation on the extratropical winter stratosphere in an atmospheric general circulation model and in reanalysis data.

,*J. Atmos. Sci.***67**, 1402–1419, doi:10.1175/2009JAS3292.1.Baldwin, M. P., and et al. , 2001: The quasi-biennial oscillation.

,*Rev. Geophys.***39**, 179–229, doi:10.1029/1999RG000073.Beagley, S. R., , J. de Grandpré, , J. N. Koshyk, , N. A. McFarlane, , and T. G. Shepherd, 1997: Radiative-dynamical climatology of the first-generation Canadian middle atmosphere model.

,*Atmos.–Ocean***35**, 293–331, doi:10.1080/07055900.1997.9649595.Boer, G., , N. McFarlane, , R. Laprise, , J. Henderson, , and J. Blanchet, 1984: The Canadian Climate Centre spectral atmospheric general circulation model.

,*Atmos.–Ocean***22**, 397–429, doi:10.1080/07055900.1984.9649208.Boville, B., , and W. J. Randel, 1992: Equatorial waves in a stratospheric GCM: Effects of vertical resolution.

,*J. Atmos. Sci.***49**, 785–801, doi:10.1175/1520-0469(1992)049<0785:EWIASG>2.0.CO;2.Campbell, L. J., , and T. G. Shepherd, 2005a: Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part I: Gravity wave forcing.

,*J. Atmos. Sci.***62**, 4178–4195, doi:10.1175/JAS3616.1.Campbell, L. J., , and T. G. Shepherd, 2005b: Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part II: Combined effects of gravity waves and equatorial planetary waves.

,*J. Atmos. Sci.***62**, 4196–4205, doi:10.1175/JAS3617.1.Collimore, C. C., , D. W. Martin, , M. H. Hitchman, , A. Huesmann, , and D. E. Waliser, 2003: On the relationship between the QBO and tropical deep convection.

,*J. Climate***16**, 2552–2568, doi:10.1175/1520-0442(2003)016<2552:OTRBTQ>2.0.CO;2.Dee, D., and et al. , 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system.

,*Quart. J. Roy. Meteor. Soc.***137**, 553–597, doi:10.1002/qj.828.Dunkerton, T. J., 1990: Annual variation of deseasonalized mean flow acceleration in the equatorial lower stratosphere.

,*J. Meteor. Soc. Japan***68**, 499–508.Dunkerton, T. J., 1997: The role of gravity waves in the quasi-biennial oscillation.

,*J. Geophys. Res.***102**, 26 053–26 076, doi:10.1029/96JD02999.Evan, S., , M. J. Alexander, , and J. Dudhia, 2012: WRF simulations of convectively generated gravity waves in opposite QBO phases.

,*J. Geophys. Res.***117**, D12117, doi:10.1029/2011JD017302.Fels, S. B., 1982: A parameterization of scale-dependent radiative damping rates in the middle atmosphere.

,*J. Atmos. Sci.***39**, 1141–1152, doi:10.1175/1520-0469(1982)039<1141:APOSDR>2.0.CO;2.Fomichev, V. I., , and J. P. Blanchet, 1995: Development of the new CCC/GCM longwave radiation model for extension into the middle atmosphere.

,*Atmos.–Ocean***33**, 513–529, doi:10.1080/07055900.1995.9649543.Fomichev, V. I., , A. A. Kutepov, , R. A. Akmaev, , and G. M. Shved, 1993: Parameterization of the 15 μm CO2 band cooling in the middle atmosphere (15–115 km).

,*J. Atmos. Terr. Phys.***55**, 7–18, doi:10.1016/0021-9169(93)90149-S.Fomichev, V. I., , C. Fu, , J. de Grandpré, , S. R. Beagley, , V. P. Ogibalov, , and J. C. McConnell, 2004: Model thermal response to minor radiative energy sources and sinks in the middle atmosphere.

*J. Geophys. Res.*,**109**, D19107, doi:10.1029/2004JD004892.Fritts, D. C., , and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere.

*Rev. Geophys.*,**41**, 1003, doi:10.1029/2001RG000106.Geller, M. A., and et al. , 2013: A comparison between gravity wave momentum fluxes in observations and climate models.

,*J. Climate***26**, 6383–6405, doi:10.1175/JCLI-D-12-00545.1.Giorgetta, M. A., , E. Manzini, , and E. Roeckner, 2002: Forcing of the quasi-biennial oscillation from a broad spectrum of atmospheric waves.

,*Geophys. Res. Lett.***29**, doi:10.1029/2002GL014756.Giorgetta, M. A., , E. Manzini, , E. Roeckner, , M. Esch, , and L. Bengtsson, 2006: Climatology and forcing of the quasi-biennial oscillation in the MAECHEM5 model.

,*J. Climate***19**, 3882–3901, doi:10.1175/JCLI3830.1.Grise, K. M., , D. W. J. Thompson, , and T. Birner, 2010: A global survey of static stability in the stratosphere and upper troposphere.

,*J. Climate***23**, 2275–2292, doi:10.1175/2009JCLI3369.1.Hamilton, K., 1982: A note on the interaction between a thermally forced standing internal gravity wave and the mean flow, with an application to the theory of the quasi-biennial oscillation.

,*J. Atmos. Sci.***39**, 1881–1886, doi:10.1175/1520-0469(1982)039<1881:ANOTIB>2.0.CO;2.Hamilton, K., , and L. Yuan, 1992: Experiments on tropical stratospheric mean-wind variations in a spectral general circulation model.

,*J. Atmos. Sci.***49**, 2464–2483, doi:10.1175/1520-0469(1992)049<2464:EOTSMW>2.0.CO;2.Hamilton, K., , R. J. Wilson, , and R. S. Hemler, 1999: Middle atmosphere simulated with high vertical and horizontal resolution versions of a GCM: Improvements in the cold pole bias and generation of a QBO-like oscillation in the tropics.

,*J. Atmos. Sci.***56**, 3829–3846, doi:10.1175/1520-0469(1999)056<3829:MASWHV>2.0.CO;2.Hamilton, K., , R. J. Wilson, , and R. S. Hemler, 2001: Spontaneous stratospheric QBO-like oscillations simulated by the GFDL SKYHI general circulation model.

,*J. Atmos. Sci.***58**, 3271–3292, doi:10.1175/1520-0469(2001)058<3271:SSQLOS>2.0.CO;2.Haynes, P. H., 1998: The latitudinal structure of the quasi-biennial oscillation.

,*Quart. J. Roy. Meteor. Soc.***124**, 2645–2670, doi:10.1002/qj.49712455206.Hines, C. O., 1997a: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 1: Basic formulation.

,*J. Atmos. Sol.-Terr. Phys.***59**, 371–386, doi:10.1016/S1364-6826(96)00079-X.Hines, C. O., 1997b: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 2: Broad and quasi monochromatic spectra, and implementation.

,*J. Atmos. Sol.-Terr. Phys.***59**, 387–400, doi:10.1016/S1364-6826(96)00080-6.Hitchcock, P., , and I. Simpson, 2014: The downward influence of stratospheric sudden warmings.

,*J. Atmos. Sci.***71**, 3856–3876, doi:10.1175/JAS-D-14-0012.1.Holton, J. R., , and R. S. Lindzen, 1972: An updated theory for the quasi-biennial cycle of the tropical stratosphere.

,*J. Atmos. Sci.***29**, 1076–1080, doi:10.1175/1520-0469(1972)029<1076:AUTFTQ>2.0.CO;2.Horinouchi, T., , and S. Yoden, 1998: Wave–mean flow interaction associated with a QBO-like oscillation simulated in a simplified GCM.

,*J. Atmos. Sci.***55**, 502–526, doi:10.1175/1520-0469(1998)055<0502:WMFIAW>2.0.CO;2.Horinouchi, T., and et al. , 2003: Tropical cumulus convection and upward-propagating waves in middle-atmospheric GCMs.

,*J. Atmos. Sci.***60**, 2765–2782, doi:10.1175/1520-0469(2003)060<2765:TCCAUW>2.0.CO;2.Kawatani, Y., , K. Sato, , T. J. Dunkerton, , S. Watanabe, , S. Miyahara, , and M. Takahashi, 2010a: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part I: Zonal mean wave forcing.

,*J. Atmos. Sci.***67**, 963–980, doi:10.1175/2009JAS3222.1.Kawatani, Y., , K. Sato, , T. J. Dunkerton, , S. Watanabe, , S. Miyahara, , and M. Takahashi, 2010b: The roles of equatorial trapped waves and internal inertia–gravity waves in driving the quasi-biennial oscillation. Part II: Three-dimensional distribution of wave forcing.

,*J. Atmos. Sci.***67**, 981–997, doi:10.1175/2009JAS3223.1.Kim, Y.-H., , and H.-Y. Chun, 2015: Contributions of equatorial wave modes and parameterized gravity waves to the tropical QBO in HadGEM2.

,*J. Geophys. Res. Atmos.***120**, 1065–1090, doi:10.1002/2014JD022174.Krismer, T., , and M. Giorgetta, 2014: Wave forcing of the quasi-biennial oscillation in the Max Planck Institute Earth System Model.

,*J. Atmos. Sci.***71**, 1985–2006, doi:10.1175/JAS-D-13-0310.1.Lawrence, B. N., 2001: A gravity-wave induced quasi-biennial oscillation in a three-dimensional mechanistic model.

,*Quart. J. Roy. Meteor. Soc.***127**, 2005–2021, doi:10.1002/qj.49712757608.Liess, S., , and M. A. Geller, 2012: On the relationship between QBO and distribution of tropical deep convection.

,*J. Geophys. Res.***117**, D03108, doi:10.1029/2011JD016317.Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown.

,*J. Geophys. Res.***86**, 9707–9714, doi:10.1029/JC086iC10p09707.Lindzen, R. S., , and J. R. Holton, 1968: A theory of the quasi-biennial oscillation.

,*J. Atmos. Sci.***25**, 1095–1107, doi:10.1175/1520-0469(1968)025<1095:ATOTQB>2.0.CO;2.Maruyama, T., 1991: Annual and QBO-synchronized variations of lower stratospheric equatorial wave activity over Singapore during 1961-1989.

,*J. Meteor. Soc. Japan***69**, 219–231.McLandress, C., , and J. F. Scinocca, 2005: The GCM response to current parameterizations of nonorographic gravity wave drag.

,*J. Atmos. Sci.***62**, 2394–2413, doi:10.1175/JAS3483.1.Morcrette, J.-J., 1991: Radiation and cloud radiative properties in the European Centre for Medium Range Weather Forecasts forecasting system.

*J. Geophys. Res.*,**96**, 9121–9132, doi:10.1029/89JD01597.Orr, A., , P. Bechtold, , J. Scinocca, , M. Ern, , and M. Janiskova, 2010: Improved middle atmosphere climate and forecasts in the ECMWF model through a nonorographic gravity wave drag parameterization.

,*J. Climate***23**, 5905–5926, doi:10.1175/2010JCLI3490.1.O’Sullivan, D., , and R. E. Young, 1992: Modelling the quasi-biennial oscillation’s effect on the winter stratospheric circulation.

,*J. Atmos. Sci.***49**, 2437–2448, doi:10.1175/1520-0469(1992)049<2437:MTQBOE>2.0.CO;2.Plumb, R. A., 1977: The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation.

,*J. Atmos. Sci.***34**, 1847–1858, doi:10.1175/1520-0469(1977)034<1847:TIOTIW>2.0.CO;2.Richter, J. H., , A. Solomon, , and J. T. Bacmeister, 2014: On the simulation of the quasi-biennial oscillation in the Community Atmosphere Model, version 5.

,*J. Geophys. Res. Atmos.***119**, 3045–3062, doi:10.1002/2013JD021122.Rind, D., , J. Jonas, , N. Balachandran, , G. Schmidt, , and J. Lean, 2014: The QBO in two GISS global climate models: 1. Generation of the QBO.

,*J. Geophys. Res. Atmos.***119**, 8798–8824, doi:10.1002/2014JD021678.Saravanan, R., 1990: A multiwave model of the quasi-biennial oscillation.

,*J. Atmos. Sci.***47**, 2465–2474, doi:10.1175/1520-0469(1990)047<2465:AMMOTQ>2.0.CO;2.Sato, K., , and T. J. Dunkerton, 1997: Estimates of momentum flux associated with equatorial Kelvin and gravity waves.

*J. Geophys. Res.*,**102**, 26 247–26 261, doi:10.1029/96JD02514.Scaife, A. A., , N. Butchart, , C. D. Warner, , D. Stainforth, , W. Norton, , and J. Austin, 2000: Realistic quasi-biennial oscillations in a simulation of the global climate.

,*Geophys. Res. Lett.***27**, 3481–3484, doi:10.1029/2000GL011625.Scinocca, J. F., 2002: The effect of back-reflection in the parameterization of non-orographic gravity-wave drag.

,*J. Meteor. Soc. Japan***80**, 939–962, doi:10.2151/jmsj.80.939.Scinocca, J. F., 2003: An accurate spectral nonorographic gravity wave drag parameterization for general circulation models.

,*J. Atmos. Sci.***60**, 667–682, doi:10.1175/1520-0469(2003)060<0667:AASNGW>2.0.CO;2.Scinocca, J. F., , and N. A. McFarlane, 2004: The variability of modeled tropical precipitation.

,*J. Atmos. Sci.***61**, 1993–2015, doi:10.1175/1520-0469(2004)061<1993:TVOMTP>2.0.CO;2.Scinocca, J. F., , N. A. McFarlane, , M. Lazare, , J. Li, , and D. Plummer, 2008: Technical note: The CCCma third generation AGCM and its extension into the middle atmosphere.

,*Atmos. Chem. Phys.***8**, 7055–7074, doi:10.5194/acp-8-7055-2008.Simpson, I. R., , P. Hitchcock, , T. G. Shepherd, , and J. F. Scinocca, 2011: Stratospheric variability and tropospheric annular-mode timescales.

,*Geophys. Res. Lett.***38**, L20806, doi:10.1029/2011GL049304.Smith, S. A., , D. C. Fritts, , and T. E. Vanzandt, 1987: Evidence for a saturated spectrum of atmospheric gravity waves.

,*J. Atmos. Sci.***44**, 1404–1410, doi:10.1175/1520-0469(1987)044<1404:EFASSO>2.0.CO;2.Takahashi, M., 1996: Simulation of the stratospheric quasi-biennial oscillation using a general circulation model.

,*Geophys. Res. Lett.***23**, 661–664, doi:10.1029/95GL03413.Takahashi, M., 1999: Simulation of the quasi-biennial oscillation in a general circulation model.

,*Geophys. Res. Lett.***26**, 1307–1310, doi:10.1029/1999GL900188.Wallace, J. M., , and V. E. Kousky, 1968: Observational evidence of Kelvin waves in the tropical stratosphere.

,*J. Atmos. Sci.***25**, 900–907, doi:10.1175/1520-0469(1968)025<0900:OEOKWI>2.0.CO;2.Warner, C. D., , and M. E. McIntyre, 1996: On the propagation and dissipation of gravity wave spectra through a realistic middle atmosphere.

,*J. Atmos. Sci.***53**, 3213–3235, doi:10.1175/1520-0469(1996)053<3213:OTPADO>2.0.CO;2.Warner, C. D., , and M. E. McIntyre, 1999: Toward an ultra-simple spectral gravity wave parameterization for general circulation models.

,*Earth Planets Space***51**, 475–484, doi:10.1186/BF03353209.Xue, X.-H., , H.-L. Liu, , and X.-K. Dou, 2012: Parameterization of the inertial gravity waves and generation of the quasi-biennial oscillation.

,*J. Geophys. Res.***117**, D06103, doi:10.1029/2011JD016778.Zhang, G., , and N. A. McFarlane, 1995: Sensitivity of climate simulations to the parameterization of cumulus convection in the Canadian Climate Centre general circulation model.

,*Atmos.–Ocean***33**, 407–446, doi:10.1080/07055900.1995.9649539.

^{1}

The coarsest QBO-resolving vertical grid of which we are aware is for the Met Office (UKMO) model in Horinouchi et al. (2003), which is 1.3 km in the lower stratosphere (see their Table 2). A more recent UKMO model, HadGEM2, has a vertical grid spacing in the lower stratosphere of 1.2 km (Kim and Chun 2015). Xue et al. (2012) obtained a QBO driven mainly by parameterized waves in a version of WACCM with vertical grid spacing described as being “1.1–1.4 km in the lower stratosphere.” While these models have Δ*z* that is not too far from 1 km, Lawrence (2001) obtained a QBO-like oscillation in a 3D mechanistic model with Δ*z* = 2 km using the Hines GWD parameterization. This model is considerably more idealized than typical GCMs, but the comparison is nevertheless interesting. Lawrence (2001) noted an apparent downward influence from the SAO region on the QBO; we consider similar behavior in relation to GWD in section 5.

^{2}

The reason why Δ*z* for these runs does not exactly match that of runs B and C (Δ*z* = 0.98 versus 1.0 km and Δ*z* = 1.55 versus 1.5 km) is that a slight adjustment of the level spacing was required so that all runs could use the same GWD launch level.

^{3}

As noted in section 2, critical-level filtering is also implied by saturation since the saturation bound on wave momentum flux is proportional to *m*^{−3} and *α* being proportional to

^{4}

To obtain (4), all constants in Eq. (26) of S03 have been incorporated into *N* has been assumed constant. The exponent in (4) is *p* is the exponent from the *p* is uncertain, but is believed to lie in the range

^{5}

Although seasonal synchronization could also occur for other reasons, such as time-varying upwelling of the Brewer–Dobson circulation (Dunkerton 1990) or seasonal variations in tropospheric wave sources (Maruyama 1991).

^{6}

It should be noted that Eq. (3.2) of Campbell and Shepherd (2005a) for the L81 scheme is equivalent to (5) with

^{7}

Large vertical shears might still cause saturation near the launch level, but such shears would have to be extremely large to overcome the effect of background density changes if the average saturation altitude is more than a scale height above the launch level.