1. Introduction

In the extratropical mesosphere, gravity waves (GWs) have a first-order effect on the large-scale circulation. In contrast, GWs in the extratropical stratosphere generally have a second-order effect on the large-scale circulation, with the first provided by planetary waves (e.g., Alexander 2010). Nevertheless, GWs in the extratropical stratosphere are known to contribute to the driving of the residual-mean meridional circulation (e.g., Polichtchouk et al. 2018), be important in mitigating the cold pole bias in numerical weather (NWP) and climate prediction models (e.g., Palmer et al. 1986; Garcia and Boville 1994), contribute to the Southern Hemisphere springtime polar vortex breakdown (e.g., Gupta et al. 2021; Scaife et al. 2002), and decelerate the subtropical jets in the lowermost stratosphere above steep orography (van Niekerk et al. 2020). The impact of GWs on the large-scale wind and temperature distribution can also change planetary wave propagation, leading to nontrivial interactions and compensations between the planetary and GWs (e.g., Sigmond et al. 2008; Polichtchouk et al. 2018; White et al. 2021).

Given the importance of GWs for the stratospheric circulation and recognizing the role of the stratosphere in influencing the troposphere (e.g., Baldwin et al. 2001; Butler et al. 2019), it is important to accurately represent GWs in NWP models for more skillful seasonal weather prediction. Most global models lack the required horizontal and vertical resolution to resolve the full spectrum of GWs generated by flow over complex terrain, jet–front imbalances, and deep convection, and therefore, their effect on the circulation is parameterized using many simplifying assumptions (e.g., Plougonven et al. 2020). However, global simulations with O(10) km horizontal grid spacing in which some GWs are explicitly resolved (horizontally and vertically) have become possible in the last decade: Using the Goddard Earth Observing System Model, version 5 (GEOS-5), Nature run at 7 km grid spacing, Holt et al. (2017) showed that the distribution of resolved GW sources and the momentum fluxes in the Southern Hemisphere extratropical winter stratosphere were realistic in comparison to satellite observations but that the model underestimated GW amplitudes and overestimated horizontal wavelengths. Stephan et al. (2019) performed an intercomparison of resolved GWs with horizontal wavelengths of O(100–1000) km in global convection-permitting models for August at horizontal grid spacings between 2.5 and 9 km [including European Centre for Medium-Range Weather Forecasts (ECMWF) IFS; Stevens et al. 2019] and compared to several satellite observations. They found that the models reproduce the observed distribution of global GW momentum flux well, but the magnitude of the flux differed substantially across different models, with some models producing stronger (e.g., NICAM) and some weaker (e.g., ECMWF IFS) GWs in comparison to observations in the extratropical stratosphere.

Theoretical considerations and ray tracing models suggest that while GWs down to horizontal wavelengths of subkilometer can be generated in the troposphere, such very small wavelength waves tend to be evanescent at the source or are reflected at the tropopause, resulting in GWs only with horizontal wavelengths λ_h ≳ 10 km being able to propagate into the stratosphere (Preusse et al. 2008). Models with O(10) km horizontal grid spacing are, however, unlikely to accurately resolve most of the GW effects from 10 ≲ λ_h ≲ 100 km waves in the stratosphere as explicit and implicit numerical dissipation and, among other effects, filtering of the orography spectrum generally limit the effective resolution of a model to 6–10 times the grid spacing. Thus, a horizontal resolution significantly higher than O(10) km is likely required to resolve the full GW spectrum (e.g., Davies and Brown 2001), and, as a result, parameterizations will be needed to account for the missing (i.e., unresolved) GW forcing for some time to come.

Using the hydrostatic model of the ECMWF, global simulations at unprecedented horizontal resolutions from TCo1279 (or 9 km average grid spacing) to TCo7999 (or 1.4 km average grid spacing; Wedi et al. 2020) at a fixed vertical resolution (137 levels)¹ are performed. The aim of this study is to use these simulations to quantify how resolved zonal GW forcing (GWF) from waves generated by orography and jet/front systems changes with increase in the horizontal resolution in the extratropical stratosphere and thus answer the following questions relevant for NWP model development, both at O(10) and O(1) km global resolutions:

• Q1: Does the resolved GWF increase when the horizontal resolution increases from O(10) to O(1) km?

• Q2: Is the total GWF from resolved and parameterized waves at O(10) km horizontal resolution equal to the resolved GWF at O(1) km horizontal resolution?

• Q3: Does the change in resolved GWF with an increase in the horizontal resolution occur over orography or over nonorographic regions?

In addressing Q1–Q3, we also seek to quantify the respective roles of the long- and mesoscale GWs with horizontal wavelengths 100 ≤ λ_h < 1900 km versus the smaller-scale GWs with λ_h < 100 km. This scale separation is motivated by the need to (i) inform parameterization development, as it is important to know which scales need to be accounted for; and (ii) assess convergence of explicitly resolved long- and mesoscale waves to increase in the horizontal resolution.

Several recent studies using limited area models have already shed light on Q1 and Q2. Recent intercomparison of O(10) km horizontal resolution global models to the limited area models with 3 km grid spacing over both the Drake Passage region (Kruse et al. 2022) and the Middle East mountain range (van Niekerk et al. 2020) suggest that GWF by mountain waves continues to increase for horizontal resolutions higher than 10 km. In a nonhydrostatic model, van Niekerk and Vosper (2021) and Vosper et al. (2020) show that resolved GW momentum fluxes in the stratosphere continue to increase from 32 to 2 km horizontal resolution over the Rockies, and that the regionally averaged resolved and parameterized momentum flux is much larger at 2 km resolution, compared to that at even 9 km horizontal resolution. Thus, the implication is that the parameterization of GWF will still be required for grid spacings below 10 km.

However, given that the ability of a particular model to resolve GWs does not only depend on the horizontal and vertical grid spacing but also on the numerical methods used for the solver, the vertical resolution and the parameterization package (e.g., Stephan et al. 2019), it is important to quantify how the resolved GWF changes from O(10) to O(1) km horizontal resolution within the same model, while controlling the use of parameterized versus explicitly simulated deep convection. This type of controlled experiment allows for a better understanding of the scale dependence of these waves. Moreover, answering Q3 is only possible with a global model. Here we answer Q1–Q3 with ECMWF IFS for the first half of August 2018 and November 2019 (August and November, henceforth), when the simulations have not departed greatly from the observed state. The polar Southern and Northern Hemisphere stratosphere experienced a typical seasonal evolution then.² Therefore, focusing on these two periods likely gives insights into a typical GWF at this time of year.

The timing of the November 2018 simulations around the Antarctic springtime polar vortex breakdown (i.e., the “final warming”) gives us an opportunity to answer one final question:

• Q4: How much do GWs contribute to the deceleration of the springtime Antarctic polar vortex breakdown?

Using ERA5 at T639 horizontal resolution (or 31 km average grid spacing) Gupta et al. (2021) found that the combined contribution of parameterized and resolved GWF accounts for more than 75% of the required net mean flow deceleration of the springtime Antarctic polar vortex. By examining the zonal-mean zonal momentum budget, Gupta et al. (2021) showed that the Coriolis torque largely balances the planetary wave forcing, resulting in a relatively small contribution from the large-scale terms to the net vortex deceleration. In ERA5 a larger part of the GWF is, however, parameterized. Therefore, it is insightful to analyze the momentum budget during the Antarctic final warming at much higher horizontal resolution which better resolves GWs.

This study builds on Polichtchouk et al. (2022, henceforth P21), who already elucidated the impact of horizontal resolution and/or explicit versus parameterized representation of deep convection on convectively generated GWs in the tropical stratosphere in the simulations discussed here. It was found that with the explicit representation of deep convection, the total resolved GWF in the tropical stratosphere was unchanged from O(10) to O(1) km horizontal resolution, though at O(10) km resolution GWs with λ_h > 100 km were overestimated in comparison to those at O(1) km resolution. Thus, the conclusion was that some parameterization of deep convection is required at 9 and 4 km grid spacing together with the parameterization of nonorographic GWF to account for the unresolved forcing from smaller-scale GWs with λ_h < 100 km.

This paper is structured as follows. After briefly introducing the IFS simulations and the diagnostics used in section 2, we address Q1 and Q2 for the zonal-mean GWF in section 3. In section 4, we further examine the geographical GWF distribution to shed light on Q3 before proceeding to discuss the zonal-mean zonal momentum budget around Antarctic final warming in section 5 in order to address Q4. Finally, summary and conclusions are given in section 6.

2. Model setup and diagnostics

All simulations are performed with the full-complexity global semi-implicit semi-Lagrangian spectral ECMWF IFS atmosphere model (based on cycle 45r1; ECMWF 2018) and forced by the 0.05° OSTIA sea surface temperature and sea ice data (Donlon et al. 2012). The IFS is discretized in the horizontal via spherical harmonic expansion and a cubic-octahedral grid (“O” grid; Malardel et al. 2016). Three horizontal resolutions are considered: TCo7999, TCo2559, and TCo1279 which use 7999, 2559, and 1279 total wavenumbers in the spherical harmonic expansion, corresponding to an approximate horizontal grid spacing over the entire globe of 1.4, 4.5, and 9 km, respectively. Therefore, we refer to the TCo7999, TCo2559, and TCo1279 horizontal resolution simulations interchangeably as “1 km,” “4 km,” and “9 km,” respectively. The unprecedented TCo7999 horizontal resolution simulations have been performed on the Summit supercomputer, accessed through an Innovative and Novel Computational Impact on Theory and Experiment award (INCITE; Wedi et al. 2020). At higher horizontal resolution, smaller scales are included in the prescribed orography, as can be seen by examining the global horizontal wavenumber spectrum of resolved orography in Fig. 1. The figure suggests that mountain waves with horizontal wavelengths down to 40, 20, and 7 km can be faithfully generated by the resolved orography in the 9, 4, and 1 km simulations, respectively. In practice, atmospheric conditions (e.g., winds and stability) will dictate which scales will produce GWs. Moreover, explicit and implicit diffusion in the IFS dynamical core is likely to significantly damp the smallest resolved waves.

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Total horizontal wavenumber power spectrum of resolved orography (i.e., surface geopotential) in the different horizontal resolution simulations. Note that the power spectrum is multiplied by the total wavenumber n^5/3.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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The simulations use the parameterization package of the operational weather forecasts at ECMWF (ECMWF 2018), except that the deep convection parameterization (Tiedtke 1993; Bechtold et al. 2014) is switched off in 1 and 4 km simulations. We find that the parameterization of deep convection has little effect on the extratropical resolved GWs, unlike for the GWs in the tropical stratosphere where the parameterized deep convection inhibits resolved GW generation (see P21). The subgrid-scale orographic (Lott and Miller 1997) and the nonorographic (Scinocca 2003) GWF parameterizations at ECMWF should reduce with an increase in the horizontal resolution (ECMWF 2018) and have zero contribution in the 1 km simulations. In the 4 km simulations, the magnitude of the parameterized GWF is ≈40% of that in the 9 km simulations. Whether these parameterizations respond to changes in resolution as expected, is to be investigated.

The simulations are initialized at 0000 UTC 1 November 2018 and at 0000 UTC 1 August 2019 from the ECMWF operational analysis and run with a time step size of 60, 240, and 450 s at 1, 4, and 9 km resolutions, respectively. While the simulations are run for four months, here the focus is on the first 15 days of the simulation when the background flow and temperature distributions across the simulations are similar to each other and (re)analyses. Since the background flow and temperature structure influences the generation of GWs and where they can propagate, meaningful comparisons are difficult at later times. Note that we have also verified that all our conclusions hold when only the first 5 days of the simulation are analyzed, when the background flow and temperature structure is almost undistinguishable across the simulations.

3. Zonal-mean view of gravity wave forcing

The zonal-mean distribution of the zonal GWF in the stratosphere is examined first in Figs. 2 and 3, which show GWF averaged over the first 15 days of the simulations at different horizontal resolutions for November 2019 and August 2018. The top panels show the contribution to the total resolved GWF from the long- and mesoscale GWs (with 100 ≤ λ_h < 1900 km), the middle panels from the smaller-scale GWs (with λ_h < 100 km), and the bottom panels show the total GWF including the contribution from the parameterized orographic and nonorographic GWF (henceforth OGWF and NOGWF) for the 9 and 4 km simulations. The focus here is on the extratropical GWF, since the tropical GWF in these simulations has been extensively discussed in P21.

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Latitude–pressure distributions of zonal gravity wave forcing (GWF; shading; m s⁻¹ day⁻¹) in the stratosphere (shown for 1 < p < 300 hPa), averaged over the first 15 days for the simulations in November 2018. (a)–(c) Resolved GWF due to long- and mesoscale waves (“res LS”) with horizontal wavelengths 100 ≤ λ_h < 1900 km (or total wavenumbers 20 < n < 400) at (a) 1 km (or TCo7999), (b) 4 km (or TCo2559), and (c) 9 km (or TCo1279) horizontal resolutions. (d)–(f) As in (a)–(c), but for the resolved GWF due to small-scale waves (“res SS”) with horizontal wavelengths λ_h < 100 km (or total wavenumbers n >399). (g) Total GWF due to all resolved waves with horizontal wavelengths λ_h < 1900 km at 1 km horizontal resolution. (h),(i) Total GWF due to all resolved and parameterized orographic and nonorographic gravity waves at (h) 4 and (i) 9 km horizontal resolutions. Black contours show zonal wind (negative dashed; m s⁻¹), with 10 m s⁻¹ contour interval. Note the nonlinear contour interval for GWF.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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As in Fig. 2, but for simulations in August 2019.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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In November, GWF decelerates the Antarctic polar vortex in the upper stratosphere before the springtime polar vortex breakdown (Gupta et al. 2021) as well as the Arctic uppermost stratosphere as the polar vortex forms. In the lower stratosphere, GWF decelerates the flow above the subtropical jet maximum (e.g., van Niekerk et al. 2020). A GWF acceleration due to long- and mesoscale waves occurs in the core of the subtropical and eddy-driven jets (top panels in Fig. 2), suggesting GW generation in this region. One mechanism for GW generation here is spontaneous emission from baroclinic life cycles in the jet exit region near the tropopause, which tends to produce large-scale GWs (e.g., O’Sullivan and Dunkerton 1995; Plougonven and Snyder 2007). This behavior in the uppermost troposphere is also seen for August in Fig. 3 and implies that GWF in this region will not require parameterization in the O(10) km horizontal resolution models as it will be already well-resolved (and indeed Figs. 2c, 2i, 3c, and 3i are similar in this region). The evidence for this is the consistency of the long- and mesoscale GWF between the resolutions in this region.

The scale decomposition of GWF at a given horizontal resolution reveals another interesting feature in the region where the subtropical jets terminate: the long- and mesoscale GWs break at a lower altitude (as measured by their contribution to the GWF) than the smaller-scale GWs (cf. top and middle panels at a fixed horizontal resolution in Figs. 2 and 3). This is consistent with the vertical group velocity tending toward zero at a lower altitude for the long- and mesoscale waves than for the smaller-scale waves [i.e., vertical group velocity for midfrequency waves is $c_{gz}\sim {(c-U)}^{2}K/N$, where U is the horizontal background wind, c is the phase speed, N is the Brunt–Väisälä frequency, and K is the horizontal wavenumber]. For waves with close to stationary phase speeds (i.e., c ∼ 0), when the background winds tend to zero—as they do above the subtropical jet maximum—the vertical group velocity tends to zero and the waves cannot propagate. For smaller-scale waves (large K), however, this occurs higher up in the atmosphere. From the polarization relation for midfrequency waves u* is also larger for long- and mesoscale waves (small K), which means that larger scales saturate lower.

In August, GWF predominantly decelerates the polar night jet in the upper stratosphere. Note also the region of positive GWF due to long- and mesoscale waves just below the core of the polar vortex at 60°S. This is likely due to the southward propagation of GWs generated by the southern Andes north of 60°S (e.g., Hindley et al. 2020; Sato et al. 2012; see also Fig. 16b in Kruse et al. 2022), or the internal generation of GWs in the polar vortex (e.g., Polichtchouk and Scott 2020). Southward propagation can be due to the advection or refraction of GWs (Amemiya and Sato 2016; Sato et al. 2012). Advection is more likely than refraction for long- and mesoscale waves because propagated horizontal distance is larger for small vertical group velocities, characteristic of long horizontal wavelength GWs. As this appears to be a large-scale effect, these results suggest that the parameterization of lateral propagation and/or internal generation of GWs in the polar vortex might not be necessary for models at O(10) km horizontal resolution as these processes are already well-resolved.

For both November and August, the extratropical zonal-mean GWF from long- and mesoscale waves is almost unchanged when the horizontal resolution increases (cf. top panels in Figs. 2 and 3). This is in contrast with P21, who found nonconvergence of long- and mesoscale waves in the tropical stratosphere to the horizontal resolution. The convergence in this study is consistent with Stephan et al. (2019), who found that GW momentum flux from waves with λ_h ∼ O(100–1000) km is almost unchanged in the extratropical stratosphere when the horizontal resolution is doubled in a given convection-permitting model, including the IFS at 9 and 4 km grid spacing. Importantly, the convergence of long- and mesoscale GWs to horizontal resolution also implies that a comparison to current satellite observations that mostly resolve GWs with λ_h > 100 km is unlikely to be affected by an increase in the horizontal resolution from O(10) to O(1) km.

However, as is expected, increasing the horizontal resolution increases resolved smaller-scale GWF (cf. middle panels in Figs. 2 and 3). At 1 km resolution, smaller-scale GWF is almost as large as the long- and mesoscale GWF. This is also true for the resolved GW flux in the lower stratosphere (see Figs. S3 and S4 in the online supplementary information), and is consistent with the global resolved orography spectrum in Fig. 1, which suggests that the forcing by resolved orography should only differ across the horizontal resolutions for λ_h < 40 km. At 9 km, the resolved smaller-scale GWF is negligible. Even at 4 km grid spacing, smaller-scale GWF in the stratosphere is not fully resolved (cf. Figs. 2d,e and 3d,e). Therefore, in contrast with the behavior for the tropical GWF (see P21), the total resolved GWF in the extratropical stratosphere (i.e., the sum of the top and middle panels in Figs. 2 and 3) continues to increase as the horizontal resolution increases from O(10) to O(1) km. This implies that the parameterizations of GWF are still needed at 9 and 4 km grid spacings in the IFS. Similar conclusion for the IFS and other models was reached in Kruse et al. (2022) for the mountain waves in the Drake Passage region and in van Niekerk et al. (2020) for the mountain waves in the Middle East region.

To assess whether the total zonal-mean GWF from resolved and parameterized waves at O(10) km horizontal resolution is similar to the resolved GWF at O(1) km horizontal resolution, Figs. 2h, 2i, 3h, and 3i are compared to Figs. 2g and 3g. In the zonal mean, the parameterized NOGWF is significantly larger than OGWF in the IFS (Polichtchouk et al. 2018; see also Figs. S1 and S2 in the supplementary information). In the extratropical lower stratosphere—above the subtropical jet maximum in both hemispheres—total GWF for November and August at 9 and 4 km is remarkably close to the resolved GWF at 1 km, with 10%–15% underestimation (when pressure weighted, see Fig. 4) of negative GWF in the 70–50 hPa region at 9 km in comparison to 1 km. There is a systematic decrease of the total GWF with increasing grid spacing above the subtropical jet maximum, especially in the NH. This implies that models run at lower horizontal resolutions may have more significant underestimations of total GWF in this region. This finding is consistent with Vosper et al. (2020), who used a nonhydrostatic model, and motivate the need for more scale-aware parameterizations (van Niekerk and Vosper 2021). The underestimation is likely due to too little parameterized OGWF (van Niekerk et al. 2020), since this region is a critical layer for stationary waves.

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GWF (m s⁻¹ day⁻¹) for different horizontal resolutions, averaged over the first 15 days of the (a) November 2018 and (b) August 2019 simulations and over different areas, split into the contribution from resolved GWF over orographic and nonorographic regions and into parameterized GWF from orographic and nonorographic parameterizations. The resolved GWF is further split into contributions from all waves (“All”), large-scale waves with horizontal wavelengths 100 ≤ λ_h < 1900 km (“LS”), and small-scale waves with horizontal wavelengths λ_h < 100 km (“SS”). The number in the parentheses is the percentage of orographic points in each area. LSTS SH: Southern Hemisphere lower subtropical stratosphere area 20°–50°S, 40 < p < 150 hPa; LSTS NH: Northern Hemisphere lower subtropical stratosphere area 30°–55°N, 20 < p < 150 hPa; PV SH: Southern Hemisphere polar vortex area 40°–90°S, 1 < p < 40 hPa; PV NH: Northern Hemisphere polar vortex area 30°–90°N, 1 < p < 20 hPa.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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In contrast, in the Antarctic upper stratosphere in November the total negative GWF at O(10) km horizontal resolution is overestimated by ≈1.5 times at 9 km and by ≈1.25 times at 4 km, in comparison to the resolved GWF at 1 km. This is due to the large contribution of parameterized NOGWF at this altitude in the 9 and 4 km simulations. This, taken together with the underestimation of OGWF above subtropical jet maximum, signifies that a redistribution between the parameterized OGWF and NOGWF in IFS is required. A possible way to reduce the NOGWF over the Antarctic is to reduce the launch momentum flux over the poles.

The above findings are further quantified and summarized in Fig. 4, which shows area- and pressure-averaged GWF in the subtropical lower stratosphere and in the polar night jets in each hemisphere. We note that the findings based on the resolved GWF also apply to the zonal GW momentum flux in the lower stratosphere (see Fig. S5 in the supplementary information), indicating that it is the sources of the smaller-scale GWs rather than the dissipation of waves that is not accurately captured by the parameterizations.

4. Geographical distribution of gravity wave forcing

To inform partitioning of GWF parameterizations between OGWF and NOGWF—which is a long-standing topic of debate (Garcia et al. 2017)—we now examine whether the increase in the resolved GWF in the extratropical stratosphere with an increase in the horizontal resolution from O(10) to O(1) km occurs over the orographic or nonorographic regions. As the increase in the GWF is due to smaller-scale waves with λ_h < 100 km, we focus attention on the differences in the geographical distribution of these small-scale waves. Before this, however, it is insightful to examine the geographical distributions of the total GWF from all resolved waves in the 1 km simulations in the lower (at p = 80 hPa) and upper (at p = 10 hPa) stratosphere in Fig. 5. Note that a nonlinear contour interval is used so as to accentuate the nonorographic regions. Because of localized large amplitudes of mountain waves, GWF over steep orography dominates. For completeness, the geographical distributions of zonal GW momentum flux is also shown in the supplementary information in Figs. S3 and S4. Note that the resolved zonal GW momentum flux above the Antarctic Peninsula in November in the 50–70 hPa region is approximately −50 mPa at 1 km horizontal resolution (it is approximately −30 mPa at 9 km horizontal resolution). The magnitude of the flux in the 1 km simulation is close to the momentum flux estimates from in situ balloon observations, which measured the full GW spectrum in November 2010 in this region (see Fig. 3 in Jewtoukoff et al. 2015). However, a one-to-one comparison cannot be made due to a different year considered here and the differences in the momentum flux calculation.

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Latitude–longitude distributions of total resolved zonal GWF (shading; m s⁻¹ day⁻¹) for the 1 km simulations, averaged over the first 15 days for the simulation in (a),(b) November 2018 and (c),(d) August 2019 at (a),(c) 80 and (b),(d) 10 hPa. Gray contours show zonal wind (negative dashed; m s⁻¹), with (a)–(c) 8 and (d) 16 m s⁻¹ contour intervals. Note the nonlinear contour interval for GWF.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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In November in the extratropical lower stratosphere, the largest GWF occurs over orography, as is consistent with GW momentum flux distribution in previous observational and high-resolution model studies (e.g., Jewtoukoff et al. 2015; Holt et al. 2017; Stephan et al. 2019). In the Southern Hemisphere (SH) extratropics these regions are the Andes, the Central Plateau, Australia and New Zealand. In the Northern Hemisphere (NH) extratropics these regions are the Rockies, the Tibetan Plateau, and the Middle East mountain range. The mountain waves propagate from the troposphere to the stratosphere on the background westerlies, and where the subtropical jets terminate at ∼ 100 hPa at around 30°S and 30°N, these waves deposit westerly momentum, further decelerating and sharpening the transition from westerlies to easterlies aloft. In the upper stratosphere (Figs. 5b,d) the westerlies are located farther north (south) in November (August). As a result, only mountain waves generated in the high extratropics are able to propagate into the upper stratosphere and decelerate the flow there, where they encounter critical layers or saturate. Note that, while the magnitude of the westerly GWF over orographic regions is much larger than that over the nonorographic regions in the extratropical stratosphere, there are large areas of westerly GWF downstream of the large mountains and in the storm-track regions. Since these nonorographic regions occupy a large geographical extent, especially in the SH, they significantly contribute to the zonal-mean GWF in Figs. 2 and 3 and, thus, to the zonal-mean circulation. Locally, however, the GWF over steep orography is more than 20 times larger than over the nonorographic regions, and therefore, orographic GWs have a larger impact on the local momentum budget than the nonorographic ones. We note that it is possible that with the hydrostatic approximation the increase in GWF over steep orography in the stratosphere is overestimated due to the predominantly vertically upward propagation of orographically generated waves in a hydrostatic model versus the more horizontally tilted one in the nonhydrostatic model. Moreover, nonhydrostatic effects reduce flux and forcing for the smallest-scale waves in general. Therefore, a hydrostatic model might overestimate GW fluxes and forcing in the stratosphere. The nonhydrostatic effects remain to be quantified in a future study.

Figures 6a–d and 7a–d show the difference in the geographical distribution of smaller-scale GWF between 9 and 1 km resolutions for November and August, respectively. Both the lower (p = 80 hPa, left) and the upper (p = 10 hPa, right) stratosphere is shown. In the figures, the increase in GWF with an increase in the horizontal resolution occurs over both orographic and nonorographic regions, implying that parameterizations of both orographic and nonorographic GWF are important at O(10) km horizontal resolution. The geographical distribution of parameterized OGWF and NOGWF for the 9 km simulations is therefore shown in Figs. 6e–h and 7e–h. Figure 6 shows that in November, during the Antarctic final warming, the parameterized NOGWF in the upper stratosphere is much stronger than the resolved small-scale GWF at 1 km over the SH ocean (cf. Fig. 6b to Fig. 6h). This results in the total GWF in the Antarctic being 1.5 times larger at 9 km than at 1 km. On the other hand, the parameterized OGWF is much weaker than the resolved small-scale GWF at 1 km (cf. Figs. 6a,b and 7a,b to Figs. 6e,f and 7e,f). This further suggests that a redistribution of the parameterized GWF is required in the IFS.

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Latitude–longitude distributions of zonal GWF (shading; m s⁻¹ day⁻¹) from resolved small-scale GWs with λ_h < 100 km for the (a),(b) 1 and (c),(d) 9 km simulation, and parameterized (e),(f) orographic and (g),(h) nonorographic GWF for the 9 km simulation only. Averages over the first 15 days for the November 2018 simulations are shown at (left) 80 and (right) 10 hPa. Gray contours show zonal wind (negative dashed; m s⁻¹), with an 8 m s⁻¹ contour interval. Note the nonlinear contour interval for GWF. Note also the smaller contour interval than in Fig. 5.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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As in Fig. 6, but for simulations in August 2019. Note that the contour interval for the zonal wind at 10 hPa is 16 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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To summarize the above findings and to quantify how much the orographic versus nonorographic regions contribute to the zonal-mean GWF, Fig. 4 shows area-averaged GWF separated into orographic and nonorographic contributions. This is done following van Niekerk and Vosper (2021) by attributing resolved GWF over land points to orographic regions (setting GWF to zero elsewhere) and over sea points to nonorographic regions. We acknowledge that the GWF over, e.g., nonorographic regions can be due to orographically generated GWs propagating laterally and over orographic regions due to nonorographically generated GWs by, e.g., convection or jets and fronts. To minimize the possibility of convectively generated GWs over the Asian monsoon and the Gulf of Mexico being counted as orographic, only areas poleward of 20°N/S are considered. It is clear from Fig. 4 that the orographic regions in the NH stratosphere dominate the total resolved GWF, contributing over 70%. In the SH extratropics, despite there being significantly less land than in the NH, the nonorographic regions contribute a comparable or larger amount to the total GWF. This implies that the parameterization of OGWF and NOGWF require equal attention for the SH zonal-mean circulation. These findings also apply to the zonal GW momentum flux in the lower stratosphere (see Fig. S5 in the supplementary information).

5. Momentum budget during Antarctic final warming

Antarctic final warming has been previously identified as an event when GWF makes a significant contribution to the zonal-mean zonal momentum budget (Gupta et al. 2021). Moreover, a composite centered on the SH polar vortex breakdown event of polar-cap-averaged zonal-wind analysis increments in ERA5 suggests that the IFS experiences a too-weak polar vortex bias at 10 hPa at this time of year—as shown in Fig. 8—indicative of a too strong GWF. Given this, we now examine the transformed Eulerian-mean (TEM) zonal momentum budget (Andrews et al. 1987) leading up to the Antarctic final warming in November 2018:

${\overline{u}}_t={\overline{\upsilon }}^{*}[f-{(a\cos \varphi )}^{-1}{(\overline{u}\cos \varphi )}_{\varphi }]-{\overline{w}}^{*}{\overline{u}}_z+{(\rho a\cos \varphi )}^{-1}\mathbf{\nabla }\cdot \mathbf{F}+\overline{\text{GWF}}+\overline{X},$(4)

where the subscripts t, z, and ϕ denote partial derivatives with respect to time, vertical direction, and latitude (in radians), respectively; the first term on the right-hand side of (4) is the sum of the Coriolis term and the meridional advection by the residual mean meridional velocity ${\overline{\upsilon }}^{*}$; f is the Coriolis parameter; the second term is the vertical advection by the residual-mean vertical velocity ${\overline{w}}^{*}$; the overbar denotes zonal mean; a is Earth’s radius; ρ is density; F = (F^ϕ, F^z) is the Eliassen–Palm flux (EP flux):

$F^{\varphi }=\rho a\cos \varphi ({\overline{u}}_z\overline{{\upsilon }^{\prime }{\theta }^{\prime }}/{\overline{\theta }}_z-\overline{{\upsilon }^{\prime }{u}^{\prime }}),$(5)

$F^z=\rho a\cos \varphi \{[f-{(a\cos \varphi )}^{-1}{(\overline{u}\cos \varphi )}_{\varphi }]\overline{{\upsilon }^{\prime }{\theta }^{\prime }}/{\overline{\theta }}_z-\overline{w^{\prime }{u}^{\prime }}\},$(6)

where υ and w are the meridional and vertical wind, respectively; and θ is the potential temperature. Here, the EP-flux divergence denotes large-scale wave forcing by waves with total wavenumbers n ≤ 21 (or horizontal wavelengths λ_h ≥ 1900 km), encompassing planetary waves and very large-scale inertia–GWs. Therefore, here the primes denote deviations from the zonal mean of fields truncated to n = 21 and transformed to a T21 Gaussian grid. Similarly, in the calculation of ${\overline{\upsilon }}^{*}$ and ${\overline{w}}^{*}$ only large-scale waves (n ≤ 21) are considered. The $\overline{X}$ in (4) represents parameterized zonal-wind tendencies, consisting of vertical diffusion (all simulations) and parameterized GWF (4 and 9 km simulations only).

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(a) Southern Hemisphere polar-cap-averaged (60°–90°S) zonal wind at 10 hPa (m s⁻¹) for ERA5 and short 18-h forecasts started from ERA5 for years 1980–2021, composited on the final warming day (defined as the day when the zonal-mean zonal wind at 60°S and 10 hPa reverses sign). (b) Cumulative analysis increment (i.e., ERA5 minus the 18-h forecast) of the composited polar-cap-averaged zonal wind at 10 hPa (m s⁻¹). Note that year 2002 is excluded from the analysis, due to this year experiencing the only major sudden stratospheric warming. Note also that the forecast started from ERA5 systematically predicts weaker polar-cap-averaged zonal wind.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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Figure 9 compares the cumulative zonal momentum change at 50°–75°S and at 10 hPa from different terms in the momentum budget: the resolved GWF and the large-scale contribution from the sum of the Coriolis term, the planetary wave forcing and the advection by the large-scale residual circulation (in Fig. 9b), the sum of resolved and parameterized GWF (in Fig. 9c), and the separate contributions of the planetary wave forcing and the Coriolis term and the advection by the large-scale residual circulation (in Fig. 9d). GWF provides a steady decelerating tendency to the polar vortex throughout the simulations, whereas the near cancellation between the planetary wave forcing and the sum of the Coriolis term and the advection by the residual circulation leads to a net acceleration in the first 3 days of the simulations followed by cumulative deceleration thereafter.

As expected from the analysis in sections 3 and 4, while the resolved GWF is larger at 1 km than at 9 km (solid lines in Fig. 9b), the total (resolved + parameterized) GWF is larger at 9 km due to the large contribution from parameterized NOGWF (see Fig. 9c). At day seven, the resolved GWF contributes 30% to the polar vortex deceleration at 1 km grid spacing, whereas the contribution is 20% at 9 km grid spacing and 24% at 4 km grid spacing, respectively. The large contribution from parameterized NOGWF implies that the total GWF at day seven contributes 50% to the polar vortex deceleration at 9 km grid spacing and 35% at 4 km grid spacing. Therefore, at higher horizontal resolution the relative contribution of total GWF goes down. If we treat the resolved GWF at 1 km as truth, the parameterized NOGWF appears to be overestimated in the IFS in this region. It is likely to also be overestimated in ERA5 (as similar tuning is used in ERA5 and in the operational IFS), as is consistent with positive cumulative zonal wind analysis increments over SH polar cap at this time of year (see Fig. 8). Therefore, it is possible that the contribution of parameterized plus resolved GWF in Gupta et al. (2021) to the momentum budget is overestimated, though we note that here only one event is considered in comparison to the composite of many events in Gupta et al. (2021). Nevertheless, the one-third of the contribution of resolved GWF at O(1) km horizontal resolution is substantial and highlights the important role of GWF (together with planetary wave forcing) in the deceleration of the Antarctic springtime polar vortex.

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Accumulated 75°–50°- averaged zonal-mean zonal momentum budget at 10 hPa (m s⁻¹) for the November 2018 simulations, leading up to the Antarctic final warming. Different horizontal resolutions up to day 11 are shown. (a) Zonal-mean zonal wind change. ERA5 is shown for reference in green. (b) Resolved GWF (solid lines) and the sum of EP-flux divergence due to large-scale waves, the Coriolis term and the advection by the large-scale residual circulation (dotted lines). (c) Total GWF due to resolved and parameterized waves. Note that in the 1 km simulation, no parameterized GWF is present and in the 9 and 4 km simulations parameterized zonal-mean forcing is dominated by the nonorographic scheme. (d) EP-flux divergence due to large-scale waves (solid lines) and the Coriolis term plus the advection by the large-scale residual circulation (dotted lines).

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-22-0138.1

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Note that there appears to be some compensation in the large-scale terms in Fig. 9b at earlier times to account for smaller total GWF in the 1 km simulation so that the zonal-mean zonal wind is almost unchanged between the three simulations up to day 6 in Fig. 9a. Higher up in the stratosphere, where the difference in total GWF between the three simulations is even larger, the zonal-mean zonal wind is weaker in the 9 km simulation than in the 1 km simulation already within the first few days of the simulation (not shown). At later times, the larger-scale terms diverge more from each other in Fig. 9d. This is likely due to the chaotic nature of the simulations and the loss of the initial condition memory, or due to the response to the differing GWF at earlier times between the simulations.

6. Summary and conclusions

Using global ECMWF IFS simulations for November 2018 and August 2019 at unprecedented horizontal resolutions of 1, 4, and 9 km, this study elucidated the following questions relevant for NWP model development in the extratropical stratosphere:

• Q1: Does the resolved GWF increase when the horizontal resolution increases from O(10) to O(1) km? Yes, the resolved GWF almost doubles when the horizontal grid spacing is increased from 9 to 1 km. This is due to an increase in GWF from small-scale waves with λ_h < 100 km. These small-scale waves and their sources are better resolved at O(1) km. At 9 km grid spacing, which is the current resolution of the operational 10-day forecasts at ECMWF, GWs with λ_h < 100 km are not well resolved and need to be parameterized (even at 4 km grid spacing). It should be noted that our findings for the extratropical stratosphere are different to P21 for the tropical stratosphere. In the tropics, where convection is the source of GWs, combined resolved GWF from small-scale and large- and mesoscale GWs is almost unchanged as the horizontal resolution increases from O(10) to O(1) km, provided deep convection is represented explicitly.

• Q2: Is the total GWF from resolved and parameterized waves at O(10) km horizontal resolution equal to the resolved GWF at O(1) km horizontal resolution? Generally, no. The total GWF at O(10) km horizontal resolution is smaller than the resolved GWF at O(1) km horizontal resolution. In the lowermost stratosphere above the subtropical jet maximum, the underestimation is ≈10%–15% and is likely due to too little parameterized OGWF there. In the upper polar stratosphere the underestimation is ≈20%–25%. The exception is the Antarctic final warming in November when the total GWF from resolved and parameterized waves at O(10) km horizontal resolution is 1.5 times larger than the resolved GWF at O(1) km horizontal resolution. This is likely due to the overestimation of parameterized NOGWF in the SH polar upper stratosphere. These results suggest that a rebalancing of parameterized NOGWF and OGWF is required in the IFS.

• Q3: Does the change in resolved GWF with an increase in the horizontal resolution occur over orography or over nonorographic regions? Both nonorographic and orographic regions experience an increase in GWF when the horizontal resolution is increased. This suggests that both parameterizations of NOGWF and OGWF continue to be important at O(10) km horizontal resolutions.

• Q4: How much do GWs contribute to the deceleration of the springtime Antarctic polar vortex breakdown? At 1 km grid spacing, resolved GWs contribute approximately one-third with the rest provided by the combined contribution from large-scale planetary wave forcing and the Coriolis torque. At 9 and 4 km grid spacing the contribution of total GWF is nearly 50%, due to a large contribution from parameterized NOGWF. This suggests that the parameterized NOGWF at this time of year is overestimated in the IFS at O(10) km horizontal resolution.

This study highlights the need for the use of GWF parameterizations in models even beyond the current horizontal resolution of O(10) km of many operational weather forecasts. It also highlights the need to continue developing the parameterizations of orographic and nonorographic gravity waves to better match the distribution of GWF in the 1 km simulations, provided these are treated as “truth.”

This study only scratches the surface in terms of what is possible with the 1 km dataset and assessed GWs in the statistical sense, albeit only for 15-day averages. An assessment of individual GW life cycle in the 1 km simulations and a comparison to observations is another interesting topic to explore, which will be investigated in a future study. Moreover, it would be useful to establish if the convergence of the smaller-scale GWs in the stratosphere to horizontal resolution could already be achieved at a more affordable horizontal resolution of 2–3 km. Another interesting avenue of investigation is to use the 1 km simulations to tune parameterizations of OGWF and NOGWF, since their relative contributions can now be constrained globally. As already discussed in our companion paper P21, apart from the horizontal resolution, the impact of the vertical resolution and on the relaxation of the hydrostatic approximation on the resolved waves remains to be assessed.

Data availability statement.

The raw model output on the native grids amounts to few hundred TB, so it is not possible to make all data available for more than a few steps. However, postprocessed output used to make the figures will be retained and made available to those who request it. The analysis codes can be made available upon request.