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 Δz 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 m² 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 from which the parameterized waves are assumed to propagate upward. The choice of is arbitrary; most of our model runs use , ≈87.5 hPa, or = 17.1 km (using H = 7 km). By choosing at or slightly above the tropical tropopause we assume that the launch spectrum represents the gravity waves that have propagated upward across the tropopause from their source regions below. There are a range of possible gravity wave sources (Fritts and Alexander 2003), but since coupling of GWD to these sources is not accounted for in the GCM, the launch spectrum is simply assumed to be the result of their cumulative effects.

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.

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Latitudinal modulation functions that multiply the S03 GWD launch spectra, thereby defining the domains affected by the tropical and extratropical versions of the GWD scheme. The vertical axis is normalized by , the peak magnitude of the extratropical total launch flux. The total tropical launch flux can be tuned to yield a realistic QBO period simply by scaling the tropical function up or down by a constant factor, and the value shown here (with peak value ) is that used in experiment A of section 4, for which is shown in Fig. 2a. Unless noted otherwise, the shape of the tropical modulation function is identical in all model runs.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

The momentum flux spectrum is discretized on a grid, where is the intrinsic horizontal phase speed in a given azimuth (e.g., eastward) with 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 (), and nonlinear dissipation, which constrains the spectral shape at high vertical wavenumber, , to follow the 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 , where 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 Δz = 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 for the first 12 yr of the model run, and for comparison 12 yr of from ERA-Interim (Dee et al. 2011). The model QBO has similar amplitude and vertical extent to that seen in the reanalysis, although the downward penetration of the easterly (QBO-E) phase is unrealistically weak in the model. These 12-yr excerpts also suggest that the model QBO shows less intercycle variability than the real QBO, and this is borne out by analysis of the full 100-yr model time series (not shown). This reduced variability is at least partly due to the assumption of a fixed GWD launch spectrum (section 2), whereas real wave sources vary in space and time. We additionally performed CMAM experiments in which settings of the Zhang and McFarlane (1995) deep convective parameterization scheme were altered to increase the variability of upward-propagating wave activity, following the method of Scinocca and McFarlane (2004). After retuning the GWD by adjusting the total launch flux to give a realistic QBO period, these experiments showed increased intercycle variability and in one case a slightly improved downward penetration of the QBO-E phase (not shown). However, these were only minor improvements over the QBO simulation shown in Fig. 2. Since the goal here is to better understand the trade-off between resolved and parameterized wave forcing, further consideration of these model runs with altered convection is outside the scope of this study.

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Altitude–time evolution of stratospheric equatorial (2°S–2°N mean) zonal-mean zonal wind , in (a) CMAM experiment A, with Δz = 0.5 km, and (b) ERA-Interim. Representative 12-yr segments of the oscillation in each dataset are shown using 5-day means of daily data. Contour interval is 5 m s⁻¹ with westerlies red, easterlies blue, and the = 0 line in black. Log-pressure altitude is defined here and in subsequent figures by , where p is pressure (hPa), p₀ = 1000 hPa, and H = 7 km.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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 (; section 2), which effectively removed an eastward bias in GWD forcing that was caused by a region of climatological easterly flow immediately above the original launch level of ≈100 hPa. (The extratropical GWD launch level was not adjusted; it remains near 100 hPa in all experiments.) Filtering of parameterized waves by the resolved flow between the launch level and the lowest QBO altitudes has a first-order effect on the QBO structure, making the simulated QBO highly sensitive to tropical tropospheric circulation biases. The downward penetration of QBO phases is also highly sensitive to vertical resolution, as described in the next section.

4. Vertical resolution

a. Free-running experiments

Figure 3 shows for the first 6 yr of a series of CMAM runs with progressively increasing vertical resolution—that is, progressively decreasing Δz. Vertical profiles of the grid spacing Δz 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 Δz = 1.5 km, oscillations in are seasonally synchronized, with little intercycle variation. Decreasing Δz 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 Δz = 1.0 km is similar but with slightly increased downward penetration of the westerlies (Fig. 3c). This tendency continues at Δz = 0.75 km, showing further increases in period and low-level amplitude (Fig. 3d). Decreasing Δz still further, to 0.5 km, results in the oscillation shown in Fig. 2a. For even smaller Δz the oscillation remains qualitatively similar, although the period and amplitude continue to increase (e.g., as shown for Δz = 0.31 km in Fig. 6a, and to be discussed below). The Δz = 0.5-, 1.0-, and 1.5-km experiments will be referred to as model runs A, B, and C, respectively.

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As Fig. 2, but showing the evolution of for the first 6 yr of CMAM runs that use progressively increasing vertical resolution, as identified by the vertical grid spacing Δz in the tropical lower stratosphere. Δz = (a) 1.5, (b) 1.25, (c) 1.0, and (d) 0.75 km. A further increase to Δz = 0.5 km gives the oscillation shown in Fig. 2a.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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Vertical profiles of vertical grid spacing Δz used in the various CMAM experiments. Lines are labeled by the value of Δz in the 20–30-km layer. Experiments referred to by letter in the text (experiment A, etc.) are denoted by letter labels. (a) Experiments using the same vertical resolution in the tropical troposphere. (Figures 2 and 3 show for these experiments.) (b) Additional experiments, as discussed in section 4.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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, oscillations in this GCM are seasonally synchronized when Δz 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 oscillations that are seasonally synchronized when Δz is coarser than ≈1 km. Hence, we focus here on how Δz 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 Δz—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 Δz, the experiments having Δz = 0.98 km and Δz = 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 Δz coarsens to its stratospheric value somewhere in the midtroposphere, rather than having Δz = 0.5 km everywhere below the tropical tropopause. It was found that oscillations in the Δz = 0.98-km experiment strongly resemble those in B, and likewise for Δz = 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 Δz = 0.5 km in Fig. 4b, in which there is a drastic contrast between coarse upper-tropospheric Δz and fine-stratospheric Δz; 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 Δz is not the main determinant of whether a oscillation will break seasonal synchronization and develop into a QBO, at least for the range of tested resolutions Δz ≈ 0.5–1.5 km and the tested deep convection parameters. This is consistent with the expectation that lower stratospheric Δz 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 Δz appear to be minor, we focus hereafter on the effects of stratospheric Δz. It is useful to briefly review the reasons why Δz 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 values will meet critical levels and be annihilated, depositing momentum in the mean flow. Waves with phase speeds outside the range of values can also force the QBO if their vertical group velocities are slow enough that dissipative mechanisms such as radiative damping act over a long enough time to induce substantial mean-flow momentum deposition.

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, , which also applies to hydrostatic nonrotating gravity waves and inertia–gravity waves near the equator (e.g., Andrews et al. 1987; Scinocca 2002). Here is the intrinsic zonal phase speed—that is, the zonal phase speed in a reference frame moving with the background zonal flow (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 by , and thus . A Kelvin wave with 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 +15 m s⁻¹, hence a Doppler shift in from 25 to 10 m s⁻¹. For a typical tropical stratospheric buoyancy of N = 0.022 s⁻¹ (Grise et al. 2010), this corresponds to being refracted from 7.1 to 2.9 km. Evidently the Doppler-shifted wave would be poorly resolved on a vertical grid with Δz > 1 km, and even the nonshifted wave ( = 7.1 km) would not be well represented on a Δz ≈ 2–3-km grid. Small is also associated with slow vertical propagation of wave energy, since the vertical group velocity obtained from the dispersion relation, , decreases with . Hence waves of small spend more time dissipating in the lower stratosphere, while waves of large will propagate more rapidly through the QBO region and deposit momentum mainly at higher altitudes (Boville and Randel 1992). Additionally, the radiative damping rate of a temperature perturbation increases as decreases (Fels 1982). Waves of smaller should be more strongly damped, and this has been shown to affect the QBO in a simple model using parameterized planetary wave driving (Hamilton 1982).

From the foregoing considerations, mean-flow forcing by resolved waves in the lower tropical stratosphere would be expected to increase as Δz 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 Δz). 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 Δz.

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Vertical profiles of the zonal momentum budget of the equatorial (2°S–2°N mean) lower stratosphere. Data are 5-day means composited with respect to westerly phase (QBO-W) initiation at 32 hPa and then averaged over the 25-day period centered on the QBO-W initiation. (a) Zonal-mean zonal wind . (b)–(e) Tendency (m s⁻¹ day⁻¹) of due to (b) Eliassen–Palm flux divergence of resolved waves, (c) parameterized gravity wave drag, (d) vertical diffusion, and (e) vertical advection. Superimposed in each panel are the three vertical-resolution cases of Δz = 0.5, 1.0, and 1.5 km (experiments A, B, and C, respectively, as indicated in Fig. 4a). Horizontal dashed lines in (a) and (c) indicate the tropical GWD launch level.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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 . To test this, we consider an experiment with Δz = 0.31 km (see Fig. 4b for the vertical profile of Δz), 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 Δz, 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 3.5 km, and the effect is to essentially recover the original result (i.e., Fig. 6a). A more severe filter, damping 5 km, gives the same result. These experiments illustrate that resolved-wave driving of the QBO is sensitive to the choice of longwave radiative parameterization. But the change in the QBO is perhaps counterintuitive: finer-scale radiative damping (Fig. 6b), which is expected to increase the forcing, has increased the oscillation period. A possible explanation is that larger damping rates imply a slight downward shift in the forcing, which slightly increases the QBO amplitude and persistence at very low altitudes. Composites of indicate that the QBO of Fig. 6b does indeed have larger amplitude at low altitudes (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).

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Evolution of in CMAM experiments for the Δz = 0.31-km vertical grid shown in Fig. 4b. (a) As in experiment A (Δz = 0.5 km), but for the vertical-resolution change. (b) As in (a), but with the longwave radiative scheme altered to allow finescale radiative damping. (c) As in (b), but with an artificial vertical smoothing applied to the temperature input to the longwave radiation scheme. See text for details. Colors and contours as in Fig. 2.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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 Δz 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 Δz. 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 Δz and to the mean-flow changes cannot be separated. To isolate the response of the waves to Δz 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 (Δz = 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 Δz. Figure 7 shows ∇ ⋅ F superimposed on (note that is essentially identical in the three runs), lag composited by QBO-W onsets at 32 hPa (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 (Δz = 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 maximum). The intensity and upward reach of eastward ∇ ⋅ F increases at intermediate resolution (Δz = 1.0 km; Fig. 7b). At the finest resolution (Δz = 0.5 km; Fig. 7c), eastward ∇ ⋅ F is strongest and reaches the farthest into the shear zone (i.e., closest to the eastward maximum). Thus, the effect of increasing resolution is to alter the vertical structure and magnitude of ∇ ⋅ F. Eastward forcing shifts upward to the region where for eastward-propagating waves is smaller, consistent with waves of smaller being resolved by the finer vertical grid spacing.

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Resolved-wave driving, ∇ ⋅ F (expressed as acceleration, as in Fig. 5) for relaxation experiments with varying vertical resolutions of (a) Δz = 1.5 km (model run Cr), (b) Δz = 1.0 km (run Br), and (c) Δz = 0.50 km (run Ar). Contour interval is 0.05 m s⁻¹ day⁻¹ with eastward tendency red and westward blue. As in Fig. 5, data are 5-day means composited with respect to QBO-W initiation at 32 hPa (i.e., QBO-W onset corresponds to lag 0) and are smoothed with an 11-step (i.e., 55 days) running mean. Superimposed line contours show the background zonal-mean zonal wind in each experiment (which is strongly relaxed and hence virtually identical in each panel), with contour interval of 5 m s⁻¹, westerlies red and easterlies blue, and the zero contour gray.

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

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 Δz. In this model the primary effect of Δz is to allow eastward-propagating resolved waves to force the mean flow at smaller , owing to smaller being resolved. This behavior is consistent with the fact that decreases with , so that whatever dissipative mechanisms are present have increased time to act, and hence the basic effect may be independent of the precise details of the dissipative mechanism involved.

The response of eastward ∇ ⋅ F to Δz 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 Δz; 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 oscillation to break seasonal synchronization and develop quasi-biennial periodicity.

d. Convergence

It is natural to ask whether convergence has been achieved at Δz = 0.5 km. Does the simulated QBO change when vertical resolution is increased further? Figure 8 compares vertical profiles of the composited for experiments with Δz = 0.5 (run A), 0.25, and 0.14 km, respectively. (Vertical profiles of Δz 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 Δz 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 Δz 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.

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Vertical profiles of composited as in Fig. 5a for experiment A (Δz = 0.50 km) and the two cases of higher vertical resolution shown in Fig. 4b: Δz = 0.25 and 0.14 km. Tropospheric Δz (at log-pressure altitudes below z ≈ 15 km) is the same in all three experiments, and ERA-Interim is shown for comparison. Composites are with respect to (a) QBO-E onset at 32 hPa (i.e., QBO-W at altitudes below 32 hPa) and (b) QBO-W onset at 32 hPa (i.e., QBO-E below 32 hPa).

Citation: Journal of the Atmospheric Sciences 73, 4; 10.1175/JAS-D-15-0099.1

It is apparent that decreases in Δz beyond 0.5 km yield diminishing returns, suggesting that while run A (Δz = 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 and hence . For example, c ≈ 20 m s⁻¹ should be reasonably well resolved at Δz = 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 20 m s⁻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 oscillation to break synchronization with the seasonal cycle, and it was also noted (section 4a) that we have not been able to circumvent this limitation by making adjustments to the GWD parameterization scheme. To compare the two types of wave forcing in a setting that is more easily controlled than the GCM, we consider an idealized model of the QBO, following the formulation by Plumb (1977, hereafter P77). This model provided the first convincing explanation of the QBO (Lindzen and Holton 1968; Holton and Lindzen 1972) and variants of it have been widely used to understand basic properties of the QBO (e.g., Hamilton 1982; Saravanan 1990; Dunkerton 1990), including the possibility of compensation between resolved and parameterized wave forcing (Campbell and Shepherd 2005a,b). It is this last use in particular that motivates us to use the 1D model to interpret the GCM results shown in section 4.

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 that occurs in the real 3D circulation (if the simplifying assumptions of the model are valid; see P77 for further details). We assume that the damping of resolved waves in the GCM can be modeled by the WKB solution for radiatively damped waves that is conventionally used, as in Holton and Lindzen (1972) and P77. Since the total forcing 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.

The equation for the evolution of u(z, t) is

where is the background density profile having pressure scale height H, is the vertical momentum flux due to the nth wave, and υ is the vertical eddy diffusivity. For the damped case, the vertical structure of is given by

where

is the attenuation rate of the nth wave having zonal phase speed , , is the lower boundary of the model, 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 . For the saturated case, the vertical structure is given by assuming that is bounded by

following Eq. (26) of S03, and we use a = 1.5; this expression is equivalent to the m⁻³ saturation bound.⁴

What are the important differences between the damped and saturated forcing cases? For the damped case, conservative propagation—that is, —requires , which occurs only if α = 0. Since α ≠ 0 generally, the forcing varies continuously with z and is nonzero everywhere (up to the critical level). For the saturated case, conservative propagation will occur at altitudes where increases strongly enough with z to offset decreases in due to ρ, which can occur if is sufficiently large. At altitudes where conservative propagation does not occur, , and hence the saturation bound can be differentiated to give the forcing

In either case—whether conservative propagation occurs over some altitude range, or the waves are continuously saturated as they ascend—the forcing in the saturated case depends on ∂u/∂z. In contrast, as shown by P77, it follows from (2) and (3) that the forcing in the damped case, , depends on 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 Δz coarsened (Fig. 7) and that GWD dominates the mean-flow forcing when Δz 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- runs (section 4).

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 , as noted in section 2), but there is some evidence that waves should become saturated after propagating roughly 1 or 2 scale heights above the tropical tropopause (McLandress and Scinocca 2005).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 is where 0 < 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 Δz ≥ 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-Δz experiments, we have failed to induce any oscillations in that break synchronization with the seasonal cycle. This does not imply that it is impossible to do so. There are a large number of adjustable parameters in the S03 scheme (as in other GWD schemes), and we may simply have failed to discover those that are capable of driving a QBO (given that the results of section 4 indicate that a QBO in a coarse-Δz version of CMAM would need to be driven almost entirely by GWD). However, this null result from our coarse-Δz 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 Δz = 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.