Uncertainties in Kelvin Waves in ECMWF Analyses and Forecasts: Insights From Aeolus Observing System Experiments

The European Space Agency Earth Explorer mission Aeolus with the first spaceborne Doppler Wind Lidar onboard provides global coverage of wind profiles twice per day. This paper discusses the impact of assimilating Aeolus winds on the quality of tropical analyses and forecasts using the observing system experiments of the European Centre for Medium‐Range Weather (ECMWF). Presented examples show that Aeolus wind profiles bring changes to the Kelvin wave structure in the layers with a significant vertical shear during the easterly phase of the quasi‐biennial oscillation in the period May to September 2020. Comparing Kelvin waves in analyses and forecasts with and without Aeolus winds, it is argued that improved ECMWF forecasts in the tropical tropopause layer are due to vertically propagating Kelvin waves.

• Impact of assimilating Aeolus winds in the ECMWF system from May to September 2020 is coupled to the easterly QBO phase • Aeolus assimilation modifies the representation of vertically propagating Kelvin waves in the tropical UTLS • Forecast improvements in May 2020 could be associated with the alteration in the upward-propagating Kelvin waves

Supporting Information:
Supporting Information may be found in the online version of this article.

Citation:
Žagar, N., Rennie, M., & Isaksen, L. (2021).  (ESA, 1999(ESA, , 2008Reitebuch, 2012), onboard the ESA Earth Explorer mission Aeolus, on the quality of tropical analyses and forecasts in the European Centre for Medium Range Weather Forecasts (ECMWF) system. The Aeolus mission (Stoffelen et al., 2005) has been under development since the late 1990s leading to the successful launch of the Aeolus satellite in August 2018. At ECMWF, Aeolus observations have been used in operations since January 09, 2020. Prior to its operational use, a major effort was invested to validate the new measurements and to understand the origin of biases. After applying corrections for the bias, the random error of Aeolus observations in clear air is still typically about twice that of radiosondes or aircraft wind measurements, because the effective Aeolus laser signal was a factor 2-3 lower than expected prelaunch. Nevertheless, evaluation of the Aeolus forecast impact in the ECMWF operational system and observing system experiments show Aeolus' positive impact on short-range forecasts in the tropical UTLS region, with improvements in lower-stratosphere temperature forecasts extending to the medium range (Rennie et al., 2021). This was only the case after the discovery of a strong link between onboard telescope temperatures and systematic wind speed errors and its correction (Rennie & Isaksen, 2020;Rennie et al., 2021). Aeolus not only led to forecast improvements at several global NWP centers, but was also shown capable of observing gravity waves (Banyard et al., 2021) and providing useful aerosol observations (Baars et al., 2021).
Here, we present the first evidence that reported improvements in the simulations of tropical circulation are associated with a representation of the large-scale equatorial waves in UTLS, in particular the Kelvin wave. First, we show that Aeolus wind profiles improve the fit of short-term tropical forecasts to observations for other observations types, in spite of their random errors on average being significantly greater than error estimates for short-term tropical forecasts using the ensembles. We focus on analysis improvements in vertically propagating equatorial waves in the UTLS region and specifically on the Kelvin wave. By filtering the Kelvin waves from the ECMWF analyses with and without Aeolus winds, we demonstrate that Aeolus brings changes to the vertical wave structure in the layers with the strongest wind shear that is observed by Aeolus.
The Kelvin wave (KW) is the slowest eastward-propagating wave solution of the linearized primitive equations and therefore the first-order ingredient of the circulation response to tropospheric heating perturbations (e.g., Salby & Garcia, 1987), and is easily detectable in different types of observations (e.g., Alexander & Ortland, 2010;Kim & Alexander, 2013;Wheeler & Kiladis, 1999). In the stratosphere, where the KW was first discovered as a 15-day wave (Wallace & Kousky, 1968), it affects zonal mean quasi-periodic flows such as the quasi-biennial oscillation (QBO) (Baldwin et al., 2001), and it is widely considered to play a role in the dynamics of the Madden-Julian Oscillation (MJO) (Zhang, 2005). Although the KW is predominantly a planetary-scale wave (zonal wavenumber 1), it makes a significant contribution to tropical analyses uncertainties and forecast errors. The role of KWs in tropical weather predictability has been addressed by several studies in the past (Žagar, Buizza, et al., 2015;Žagar et al., 2007. For example, Žagar et al. (2007) found that the ECMWF forecast errors within the 20°N-20°S belt project on KWs significantly more in the easterly QBO phase than in the westerly phase. Žagar et al. (2016) showed that although the upper-troposphere tropical forecast errors grow more rapidly in the equatorial Rossby (balanced) modes, the analysis increments at the same levels are larger in unbalanced (non-Rossby or inertia-gravity) modes, including the KWs, suggesting shortcomings in the analysis of unbalanced tropical circulation. Furthermore, Žagar, Buizza, et al. (2015) showed that missing variance associated with KWs explains a large part of underdispersiveness of the ECMWF ensemble prediction system in the medium range in the tropics. It is unclear how well KW dynamics is represented in global climate models, as they still poorly simulate the QBO and MJO, and their connections (e.g., Kim et al., 2020).
Here, we use the ECMWF observing system experiments (OSEs) to investigate the impact of assimilating Aeolus winds on the Kelvin waves in the UTLS. This is carried out by applying a wavenumber decomposition of OSEs analyses and forecasts using classical linear wave theory, as presented in Section 2. We show the evidence of a positive impact of Aeolus winds on the tropical forecasts, followed by presentation of changes in KW analyses due to the assimilation of Aeolus winds. Section 4 contains the discussion and outlook.

Observing System Experiment With Aeolus Winds
The ALADIN instrument onboard Aeolus measures profiles of the horizontal line of sight (HLOS) winds from the backscattering of molecules (Rayleigh winds) and of aerosols (Mie winds). For each channel, the profile is classified as clear or cloudy depending on the presence of the cloud particulates (Tan et al., 2008). Only Rayleigh-clear and Mie-cloudy profiles are used here. Individual measurements with horizontal scale of about 3 km are accumulated to produce profiles representing up to 86-km scale for Rayleigh-clear, and about 12 km for Mie-cloudy profiles. In the vertical direction, the atmosphere is divided into 24 layers, socalled range-bins, that have thickness of 250-2,000 m, and the HLOS wind is assigned to the half-way up the range bin. The Rayleigh-clear winds with large estimated observation error are rejected based on the limit of 12 m/s (8.5 m/s) above (below) 200 hPa. After applying a bias correction, the random error of Aeolus observations in clear air is found to vary between 4 and 7 m/s for Rayleigh-clear winds and 2.8-3.6 m/s for Mie-cloudy winds (Rennie et al., 2021). In the presented OSEs, the Rayleigh-clear winds below 850 hPa were discarded.
The OSE was performed for the period from mid-April to September 2020 using the operational ECMWF system with 137 levels up to 1 Pa. The model version was CY47R1.1 with 12-hr continuous 4D-Var (Lean et al., 2021) and an outer loop with a resolution TcO399 which corresponds to about 30-km grid distance. The magnitude of the background errors at model levels in the UTLS, that is estimated from the spread of the ensemble of 4D-Var analyses, varies between 1 and 3 m/s (not shown). The experiment which included Aeolus winds on top of all other observations is denoted "Aeolus." The reference experiment with all observations except Aeolus will be referred to as "NoAeolus." The largest analysis increments in the tropical zonal wind occur in the layer 200-100 hPa. For example, in the analysis cycle in mid-May discussed in Section 3, the mean magnitude of zonal wind increments at Aeolus geolocations in the NoAeolus cycle is 1.28 m/s compared to 1.41 m/s in the case with Aeolus winds included in the assimilation (not shown). Rennie et al. (2021) presented the NWP impact assessment for the whole OSE period. Here, we look at the Kelvin wave analyses in relation to the background flow.

Kelvin Wave Filtering
The KW is filtered from Aeolus and NoAeolus analyses using the MODES software (Žagar, Kasahara, et al., 2015) which implements linear wave decomposition in the terrain-following σ coordinate system following Kasahara and Puri (1981). MODES performs a multivariate projection of the horizontal winds and pseudo-geopotential field, which is defined by temperature and surface pressure, on balanced and unbalanced eigensolutions of the linearized primitive equations. The framework is well suited for the KW which is a normal mode of the global atmosphere. For details of the linear wave theory and decomposition, the reader is referred to Andrews et al. (1987) and Kasahara (2020), and references therein. The equatorial wave filtering of operational ECMWF forecasts at https://modes.cen.uni-hamburg.de reveals the KW signals regularly propagating eastward and upward to the stratosphere, as illustrated in Supporting Information S1. The strongest tropospheric KW signal is found over the Indian ocean and western Pacific as reported in earlier studies of KWs in reanalysis data and operational ECMWF analyses (Blaauw & Žagar, 2018;Suzuki et al., 2010;Wheeler et al., 2000). Figure 1 shows the differences between the KW zonal winds in Aeolus and NoAeolus analyses during the period May to September 2020. It shows that the differences are larger in the eastern than in the western hemisphere. The largest differences are up to ±4 m/s which is 2-3 times greater than the average magnitude of the zonal wind analysis increments in the UTLS. Similar to the KW dynamics (Blaauw & Žagar, 2018;Flannaghan & Fueglistaler, 2012;Suzuki et al., 2010), there are periods of 1-3 weeks duration when the Aeolus-NoAeolus differences in the TTL layer are pronounced (levels at 83 and 108 hPa). In particular, mid-May 2020 is characterized by the largest difference at 83 hPa and we look at this case in more details in the next section.

Impact on Forecasts
Evidence of the impact of Aeolus winds on forecasts is presented in Figure 2 for the tropical belt 20°S-20°N at 100 hPa level. The difference in root-mean-square errors (rmse) in Aeolus and NoAeolus experiments compared to their respective analyses is normalized by the Aeolus experiment rmse for three months: May, June, and August 2020. A relative improvement in the Aeolus experiment compared to NoAeolus is shown  as negative values of the normalized rmse. The period from May to August was characterized by weakening easterly shear lines in the lower stratosphere associated with the easterly phase of the QBO. Figure 2 shows that statistically significant forecast improvement by Aeolus winds diminished from 5-6 days in May to 3-4 days in June and 2-3 days in August. We speculate that this effect is associated with Aeolus' effects on vertically propagating equatorial waves during the easterly QBO phase and seasonal changes in the balanced upper tropospheric easterlies than can explain the longitudinal structure of the KW zonal wind (e.g., Baldwin & Gray, 2005;Suzuki et al., 2010).
In the troposphere, the balanced winds in May were westerly except over the Indian ocean, while easterlies dominated during July to September period throughout the tropics. The upper troposphere easterlies, which are on average found over the western Pacific, act as an escape window for KWs throughout the year and largely explains their longitudinal structure (Flannaghan & Fueglistaler, 2013). Links to the movies with the Kelvin waves and balanced winds in the operational ECMWF analyses in the upper troposphere and stratosphere are provided in Supporting Information S1 and can also be seen on the MODES webpage.
In the next section, we use examples of KW profiles in May 2020 to discuss the importance of the QBO phase and the background winds for the analyses. The rmse of observation minus background for the period in May is provided in Supporting Information S1. An eastward-slanted, vertically propagating KW structure over the Indian ocean (60°-90°E) in Figure 3 in the vicinity of the strongest balanced easterlies is similar to the typical KW structure in observations and earlier ECMWF analyses (e.g., Alexander & Ortland, 2010;Blaauw & Žagar, 2018;Flannaghan & Fueglistaler, 2013;Suzuki & Shiotani, 2008). Similarly, a weaker KW signal below 100 hPa over the western Pacific is a known property of KWs in regions with westerly balanced flow. An equivalent of Figure 3 for the NoAeolus experiment is provided in Supporting Information S1. Other figures in Supporting Information S1 for the same three days show that KWs represent most of the unbalanced (or non-Rossby) signal. The total zonal flow appears far less smooth than the balanced winds because small-scale, divergent structures project on the inertia-gravity modes.

Kelvin Wave Winds Versus Balanced Winds Near the Equator
Figure 3 also shows that between May 13 and May 16, 2020, balanced easterlies over the Indian ocean strengthened, along with strengthening westerlies around the dateline. An even stronger enhancement of both horizontal and vertical shear can be seen in the total wind (Figure in Supporting Information S1) since the KW easterlies (westerlies) occur in the regions of balanced easterly (westerly) flow. Such strong wind shear is typical for midlatitude fronts and simulated Aeolus winds were demonstrated capable of improving frontal features (Šavli et al., 2018). On synoptic scales in midlatitudes, the thermal wind balance can be applied to derive the vertical wind shear from the horizontal temperature gradient, with the temperature field obtained from high-accuracy measurements of radiances. In the tropics, the weak-temperature gradient theory for the slow, large-scale motions relies on the smallness of the temperature gradient (Sobel et al., 2001), thereby excluding the KW. However, wherever the Kelvin and Rossby waves are present together near the equator, their zonal winds will sum up (or subtract) whereas their temperature perturbations will subtract (or sum up), since their mass-wind couplings have opposite signs. Using the horizontal structure of mass-field observations to derive the Kelvin and Rossby wave signals near the equator therefore requires quantification of their respective forecast-error variances. This is a challenging task for data assimilation, even in the perfect-model 4D-Var (Žagar et al., 2008). Direct wind observations, as argued since early days of the Aeolus project, are crucial to improve accuracy of tropical analyses.

Effects of Aeolus Winds on Kelvin Waves
There is little difference between the outputs of Aeolus experiments in Figure 3 and an equivalent figure for NoAeolus experiment, which is shown in Supporting Information S1. Differences are up to the level used for contouring, 3 m/s, which is not a small value for an experiment in which the only difference is the assimilation of Aeolus winds. Although Aeolus winds are characterized by a significant random error for the Rayleigh-clear profiles, the Mie-cloudy winds in tropical cirrus layer (top of convection) have errors 2-3 m/s, similar to that of radiosondes. To understand tropical dynamical processes affected by Aeolus assimilation, we need to look at differences between analyses or at analysis increments. The latter shows the effect of observations in a single assimilation cycle whereas the former includes the effect of observations assimilated also in the earlier cycles. The memory of observations in the tropics should be longer than in the extratropics (Fisher et al., 2005), and observations that affect the tropical mean state can be expected to have a memory of at least 10 days as it was seen for the impact of AMSU-A radiances on tropical wind forecast skill (Bormann et al., 2019). Figure 3, shows that differences between KWs in the two experiments occur at the locations of significant zonal wind shear. For example, the assimilation of Aeolus winds on May 13 enhanced easterlies near 100 hPa over Indian ocean. The vertical shape of analysis differences in Figure 4 exceeding 3 m/s at several locations suggests significant changes in the vertically propagating waves in UTLS. The vertical wave structure at two locations, 66°E and 90°E, and three consecutive days is presented in Figure 5. It shows the downward propagation of the KW phase from the lower stratosphere across the TTL. The vertical phase speed at 66°E earlier in the period has a greater amplitude than at 90°E. The most important feature is the modification of the KW zonal wind shear between 100 and 50 hPa where the amplitudes and the vertical propagation are the largest. The portion of the total zonal circulation associated with KWs exceeds the balanced zonal winds in the region of upper troposphere easterlies, but it is significantly smaller in the upper troposphere westerlies ( Figure 3). This raises the question of the amplitudes of KW analysis increments with respect to analysis increments in balanced winds. If KW analysis increments are small, this may suggest that KWs are well represented in the short-range forecasts (background fields) in both experiments. However, evaluation of tropical analysis and forecast uncertainties do not support such an argument (Podglajen et al., 2014;Žagar, 2017). Current background errors estimated from the ensembles are not much lower than the ensemble estimates from 2015 discussed in Žagar (2017) and previously discussed growth of forecast errors associated with KWs is thus likely unchanged. Differences between the zonal wind in NoAeolus and Aeolus analyses on May 19, 2020 partitioned between balanced and unbalanced parts show that the two parts have similar amplitudes in regions and layers of strong shear, where westerlies shift to easterlies or vice versa. In other regions such as the upper troposphere westerlies over the Pacific with a weaker KW signal, differences between Aeolus and NoAeolus are almost entirely in balanced modes (Figure in Supporting Information S1).

Figure 4, in comparison with
Similar analysis as in Figure 5 for later dates in July and August shows that differences between Aeolus and NoAeolus are largest in the upper troposphere and TTL. Two examples in Supporting Information S1 show KWs profiles in mid-July and at the end of August, when the differences in Figure 1 are pronounced. In these cases, as well as in the other cases in August, the KW beneath the tropopause is propagating downward. This may partly explain a smaller effect of Aeolus winds on forecasts in August compared to May in Figure 2, which is seen also at 50 hPa level (not shown). Another possible reason is that KWs in the easterly balanced flow are better represented by the model in the NoAeolus experiment. There is scope for follow-on investigations using the sensitivity studies and looking at the analysis increments. A particularly interesting question is the extent to which the Aeolus winds are correcting model deficiencies in the TTL and higher up, that are discussed in Polichtchouk et al. (2021).

Discussion and Outlook
Aeolus HLOS winds in the tropics provide predominantly zonal wind information and their positive impact on tropical analyses has been foreseen (e.g., Tan & Andersson, 2004;Tan et al., 2007). However, during the two decades since the Aeolus project started, NWP experienced large advancements with improved data assimilation and more observations used, and the current global forecast models run at resolutions around 10 km, which is much better resolution than when Aeolus was first conceived. Yet, practical predictability in extratropics remains under 10 days (Haiden et al., 2018). Reduction of tropical analysis uncertainties and a more accurate representation of tropics-extratropics interactions are argued as one way to improve medium-range and extended-range forecasting (Žagar & Szunyogh, 2020).
Our evaluation of the tropical forecasts at 100 hPa as well as the background fit to other observations shows an overall positive impact of Aeolus winds in the tropics, in spite of large random errors in the Rayleigh-clear winds. In particular, we show that the impact on forecast quality lasted longer in May than later in summer, when the easterly QBO shear lines were weak and the easterly balanced winds were present in the whole tropical belt. The coupling between the phase of QBO and the assimilated Aeolus winds is likely through the KWs that were shown to explain a significantly larger forecast-error variance in the easterly QBO phase compared to the westerly phase (Žagar et al., 2007). While the QBO phase and KWs are not explicitly accounted for in the background-error term for data assimilation in the ECM-WF system, the background-error variances are derived using the 4D-Var ensemble of data assimilations thereby accounting for the flow-dependent amplitudes of short-range forecast errors in temperature and winds. It remains to investigate how Aeolus winds affect analysis increments in the upper troposphere and TTL and the dispersiveness of the ensemble prediction system. Results presented here suggest that at least a part of reported forecast improvements in the tropical tropopause layer and the lower stratosphere is likely due to corrections to the large-scale, vertically propagating Kelvin waves in layers with a strong zonal wind shear.

Data Availability Statement
The source code of the ECMWF IFS model is not available for public use as it is intellectual property of the ECMWF and its member states. Global files with outputs of the modal decomposition of ECMWF OSEs discussed in the paper are available at https://zenodo.org/record/5207392#.YRvjby2w1Bw (doi:10.5281/ zenodo.5207392). We are indebted to the many experts who over years worked on the Aeolus mission until its launch on August 22, 2018, and to many who have been operating the satellite and contributing to the validation and processing of Aeolus measurements at ESA and within the Aeolus Data, Innovation, and Science Cluster (DISC) team. We thank Frank Sielmann and Wenlin Xiao of UHH for their help with processing outputs from MODES, and to three anonymous reviewers for their constructive comments. The MODES software can be requested through the MODES webpage http://modes.cen.uni-hamburg. de. Open access funding enabled and organized by Projekt DEAL.