The importance of the tropical tropopause layer for equatorial Kelvin wave propagation
T. J. Flannaghan,
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK
Corresponding author: T. J. Flannaghan, Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd., Cambridge CB3 0WA, UK. (email@example.com)
 We analyze the propagation of equatorial Kelvin waves from the troposphere to the stratosphere using a new filtering technique applied to ERA-Interim data (very similar results for Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) temperatures) that allows separation of wave activity into number of waves and wave amplitude. The phase speed of Kelvin waves (order 20 m/s) is similar to the magnitude of zonal wind in the tropical tropopause layer (TTL), and correspondingly, we find that the seasonal and interannual variability of Kelvin wave propagation is dominated by the variability in the wind field and less by tropospheric convectively coupled wave activity. We show that local relations between wave activity and zonal wind are ambiguous, and only full ray tracing calculations can explain the observed patterns of wave activity. Easterlies amplify and deflect the eastward traveling waves upward. Westerlies have the opposite effect. During boreal winter, the strong dipole of zonal winds in the TTL centered at the dateline confines wave propagation into the stratosphere to a window over the Atlantic-Indian Ocean sector (30°W to 90°E), which casts a lasting “shadow” into the lower stratosphere that explains the remarkable zonal asymmetry in wave activity there. During boreal summer, the upper level monsoon circulation leads to maximum easterlies, and wave amplitude (but not number of waves) maximizes over the Indian Ocean sector (30°E to 90°E). Interannual variability in wave propagation due to El-Niño/Southern Oscillation, for example, is well explained by its modification of the zonal wind field.
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 Equatorial Kelvin waves are an important mode of short time scale variability in the tropical troposphere and stratosphere. In the troposphere, these waves are convectively coupled [Kiladis et al., 2009] and hence can also be seen in outgoing longwave radiation and albedo. In the stratosphere, Kelvin waves are free traveling waves with large temperature amplitudes [Wallace and Kousky, 1968], and correspondingly, they can be easily detected in temperature data. Kelvin wave activity (time variance in the Kelvin wave component of the data) in temperature peaks in the tropical tropopause layer, as a consequence of the rapid increase of static stability (N2) around the tropopause leading to amplification upon upward propagation, and subsequent radiative damping in the stratosphere. These large temperature fluctuations have also been linked to cirrus formation [Boehm and Verlinde, 2000; Immler et al., 2008; Fujiwara et al., 2009] and therefore have an effect on stratospheric water vapor.
 The propagation of Kelvin waves is strongly affected by variations in the background state, i.e., static stability and zonal wind. In the tropical tropopause layer (TTL), background conditions vary substantially with location and time on seasonal and interannual time scales [see, e.g., Fueglistaler et al., 2009], forced by changes in the distribution of tropospheric convection and latent heating. Figure 1 shows the ERA-Interim [Dee et al., 2011] climatological mean zonal wind and temperature anomalies from the zonal mean, over the period 1989–2010 at 113 hPa for January and July. During boreal winter, tropical convection is focused on the Western Pacific/Maritime continent region, and correspondingly a strong “Walker cell” (i.e., eastward flow in the upper troposphere over the central and eastern Pacific) result that extends well into the TTL. Furthermore, two highly symmetric anticyclones result on either side of the equator in response to the localized heating [Highwood and Hoskins, 1998]. During boreal summer, convection is shifted far north over the Indian subcontinent and Southeast Asia. The resulting upper level anticyclone is centered around 30°N, with the southern hemispheric counterpart being weaker and closer to the equator [see, e.g., Sardeshmukh and Hoskins, 1988). For Kelvin wave propagation, it is the very strong westward winds (easterlies) over the equatorial Indian Ocean sector that are most relevant. There is a similar response to the redistribution of convection in the El Niño-Southern Oscillation (ENSO), with a strengthened Walker circulation in La Niña conditions as convection is more localized over the Maritime Continent.
Alexander and Ortland  and Alexander et al.  find that the climatology of wave activity in the TTL is not closely related to wave activity in tropospheric convection (as observed in brightness temperature measurements). Suzuki and Shiotani  give a detailed climatology of Kelvin wave activity in the TTL using the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 reanalysis data set and also find that this is not closely related to wave activity in tropospheric convection, and suggest that zonal wind is modulating the waves. Suzuki et al.  present another interesting analysis of the longitudinal structure of Kelvin waves in the TTL and provide the basis for some of the filtering techniques used here (see Flannaghan and Fueglistaler  for a more detailed discussion of the relation of this work to ours). Ryu et al.  find that zonal wind plays an important role in modulating waves in the TTL and use an approximation to the ray tracing equations to argue that there is a direct link between zonal winds and Kelvin wave amplitude in the TTL. However, this approximation is unable to explain details of the climatology as it ties wave amplitude to local background conditions. We will show that this is not consistent with observations or the full ray tracing model, where the full background structure along the trajectory of the wave packet is critical for determining its amplitude.
 Here we analyze the climatology of Kelvin waves and, in particular, the zonal distribution of waves, using a novel method of wave tracking described in Flannaghan and Fueglistaler . Section 2 presents the data and methods used in this study. Section 3 shows that ERA-Interim reliably resolves Kelvin waves. Section 4 presents the climatology of Kelvin waves, and in section 5 we use a full ray tracing model to propagate waves through the observed climatological background wind and temperature structures. Finally, section 6 summarizes and discusses the results of our analysis.
2 Data and Methodology
2.1.1 CLAUS Brightness Temperature
 Convectively coupled Kelvin waves can be observed in cloud top brightness temperature. In this study, we use the Cloud Archive User Services (CLAUS) data set [Hodges et al., 2000], which has been used in previous Kelvin wave studies [e.g., Yang et al., 2003]. The CLAUS data set is a merged data set produced by taking data from the International Satellite Cloud Climatology Program (ISCCP) and gridding it on a 0.5° by 0.5° grid at 3 h intervals. We average these data on a 1° by 1° grid at 6 h intervals before using in this study.
 The propagation of Kelvin waves is analyzed based on temperature data from ECMWF ERA-Interim [Dee et al., 2011], using a real-space domain filter (see below). ERA-Interim data on a 1° by 1° grid at 6 h intervals from 1989 to 2010 are used throughout this paper. The data are interpolated onto pressure levels which lie closest to the model levels (a hybrid coordinate system with σ levels near the surface and pressure levels in the stratosphere), reducing interpolation error and giving the full vertical resolution of the model. This can make a significant difference to wave activity as waves are of similar vertical scale to the model resolution. The resolution in the TTL is approximately 1 km (levels at approximately 132 hPa, 113 hPa, 95 hPa, 80 hPa, and 67 hPa) and is therefore sufficient to resolve Kelvin waves. In this paper, temperature and wind averages over the ±10° latitude band are considered, in line with previous studies [see Flannaghan and Fueglistaler, 2012, and references therein].
 Kelvin waves project onto both temperature and zonal wind. In the stratosphere, waves are in approximate equipartition (geopotential energy and kinetic energy averaged over one cycle of the wave are equal), and therefore the structure of the temperature activity and zonal wind activity is expected to be similar. Due to convective coupling in the troposphere and variation in static stability, there are some differences between the projection of the waves in zonal wind and temperature in the upper troposphere. The differences are presented in Flannaghan and Fueglistaler  and do not greatly affect the results obtained at 113 hPa and above. In this study, we focus on Kelvin waves in temperature, partly as this is a better observed field in the upper troposphere/lower stratosphere region. This choice does not greatly affect the wave tracking results presented in this paper.
 The Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) [Anthes et al., 2008] is a collection of satellites using GPS occultation to produce high-resolution temperature profiles of the stratosphere and upper troposphere. COSMIC data have been used in previous studies of Kelvin waves [e.g., Alexander et al., 2008]. We grid COSMIC profiles using a modified version of the method of Alexander et al. . Profiles are gridded on a 20° longitude, 5° latitude grid, in both 24 h bins and 14 day bins. The 14 day bins define the background temperature and is subtracted from the former 24 h bins to produce the temperature anomaly. Alexander et al.  use a 7 day bin rather than the 14 day bin used here. We use a longer bin as we find that 7 day binned data contain a small amount of Kelvin wave activity (Kelvin waves have a typical period of 5 to 10 days; see Wheeler and Kiladis  and Kiladis et al. ).
2.2 Wheeler-Kiladis-Type Filter
 Many previous studies [e.g., Alexander et al., 2008; Ern and Preusse, 2009a; Suzuki et al., 2010; Fujiwara et al., 2012] use filters based on the approach developed by Wheeler and Kiladis . This filtering approach retains spectral modes with propagation speeds similar to observed Kelvin waves. There is some variation in the width of the range of propagation speeds retained, with studies in the stratosphere finding faster propagation speeds compared with those observed in tropospheric convection. In this study, we use a broad range similar to that used by Suzuki et al. , selecting spectral coefficients for which the equivalent depth, h, lies in the range 10m<h<250m, the period is above 0.05 cpd, and with zonal wave numbers 1–15 (henceforth referred to as the WK99-type filter).
 We use the WK99-type filter for some of the results in this paper (in particular in section 3). It is used in section 3 because it is a spectral method and therefore implicitly interpolates the coarse-gridded COSMIC data.
2.3 Wave Tracking Algorithm
 The filter applied to track Kelvin waves in temperature data is described in detail in Flannaghan and Fueglistaler  (henceforth referred to as FF12 filter). Briefly, the filter correlates a Kelvin wave template (generated using the observed Kelvin wave structure in the TTL) with the data. This filtering method allows the tracking of waves (both vertically and horizontally).
 The tracking algorithm can be used to track waves propagating through the TTL. In this paper, we only use waves which can be tracked through the TTL from 200 hPa to 60 hPa. This was shown in Flannaghan and Fueglistaler  to robustly identify Kelvin waves without biases present in tracking methods that rely on filtered wave activity alone. It was also shown that the FF12 filter and the WK99-type filter produce consistent wave activities. The FF12 filter is used in this paper to produce the climatology of Kelvin wave activity and wave statistics such as average wave amplitude.
3 WK99-Type Kelvin Wave Activity in ERA-Interim and COSMIC
 The representation of Kelvin waves in any reanalysis data set may be affected by the Kelvin wave representation in the underlying weather forecasting model. Kelvin waves could potentially be poorly captured by models due to their short vertical wavelength and complex convective coupling in the troposphere. Here we show that ERA-Interim is suitable for detailed study of Kelvin waves by comparing to COSMIC observations (for a comparison with other reanalyses and models, see Fujiwara et al. ).
 Since 2001, ERA-Interim has assimilated GPS occultation satellite results (e.g., the CHAMP, Gravity Recovery and Climate Experiment (GRACE), and COSMIC programs) which resolve Kelvin waves in the TTL and stratosphere. COSMIC (2006 to present) is the most comprehensive set of observations and has a sufficiently dense profile distribution to resolve Kelvin waves [Alexander et al., 2008]. Here we compare the Kelvin wave climatologies in ERA-Interim and COSMIC data in order to show the suitability of ERA Interim for Kelvin wave analyses. We use the WK99-type filter as the basis for the data comparison here because we have used a coarse grid when gridding the COSMIC data; the FF12 filter does not perform well on coarse grids.
 Figure 2 shows vertical profiles of Kelvin wave activity (the time variance of the Kelvin wave filtered data) in COSMIC and ERA-Interim, using the WK99-type filter given above. Figure 2a shows that during the COSMIC period, ERA-Interim and COSMIC are very similar. COSMIC is assimilated into ERA-Interim, and Poli et al.  show that COSMIC assimilation strongly affects ERA-Interim temperatures, so it is not surprising that COSMIC and ERA-Interim Kelvin waves match during this period. COSMIC wave activity is slightly less than the ERA-Interim wave activity (about 90% of ERA-Interim), which may be a consequence of the coarse gridding used when working with the COSMIC data (this puts an upper bound on zonal wave number that is lower than the bound used in the WK99-type filter). Figure 2b shows that the vertical structure in wave activity in the pre-GPS ERA-Interim data is similar to that in the COSMIC GPS data, with a maximum in wave activity at 80 hPa in DJF and between 80 hPa and 66 hPa in JJA. The annual cycle as captured by the difference between the DJF and JJA profiles is also similar up to 80 hPa (differences above 80 hPa could be due to different aliasing of the quasi-biennial oscillation (QBO) over each period and season). The similarities between pre-GPS ERA-Interim Kelvin wave activity and COSMIC Kelvin wave activity up to 80 hPa suggest that ERA-Interim models Kelvin wave propagation through the TTL well even in the absence of Kelvin wave resolving observations in this region.
 The pre-GPS ERA-Interim wave activity is lower than the COSMIC wave activity throughout the profile in both seasons (in particular below 100 hPa and during JJA in the stratosphere). This suggests that the ERA-Interim troposphere is not generating enough upward propagating Kelvin wave activity and that during the COSMIC period, assimilation of Kelvin wave resolving observations boosts Kelvin wave activity in the TTL.
 Figure 3a shows the climatological mean annual cycle of Kelvin wave temperature activity (computed using the WK99-type filter applied to the temperature perturbation as defined in section 2.1) in the COSMIC data from October 2006 to October 2011 on 113 hPa. As with Figure 2, the wave activity in ERA-Interim over the same period is very similar due to the assimilation of COSMIC into ERA Interim. Figure 3b shows climatological Kelvin wave activity from the pre-GPS satellite period of ERA-Interim and shows a qualitatively similar annual cycle in wave distribution, with most wave activity over the Indian Ocean region during JJA. This validates our use of ERA-Interim to investigate Kelvin wave propagation in the remainder of this paper.
4 Climatology of Kelvin Waves
4.1 Annual Cycle of Convectively Coupled Kelvin Waves in Tropical Convection
 Figure 4 shows the climatological mean variance of the WK99-type filtered CLAUS brightness temperature for the four seasons DJF, MAM, JJA, and SON. The WK99-type filter has been applied to each latitude separately. The figure shows some brightness temperature variance projection on Kelvin waves also at higher latitudes, which may be due to extratropical Rossby waves projecting onto the WK99-type filter. For equatorial Kelvin waves, the figure shows a strong seasonality in the locations of the brightness temperature variance. In particular, we note the very strong variance from boreal spring to fall over the Pacific and the local maximum during MAM over Central Africa. During DJF, we note an overall reduction in convectively coupled Kelvin wave activity and less pronounced local maxima.
 The asymmetry about the equator in brightness temperature variations related to Kelvin waves reflects the asymmetry in conditions favorable for convection; the Kelvin wave perturbation in the dynamical fields (such as zonal wind or temperature) is more symmetric about the equator [Straub and Kiladis, 2002]. We find the structure of the WK99-type filtered brightness temperature variance to be robust with respect to the proxy for convection (i.e., brightness temperatures, outgoing longwave radiation (OLR), and methods of aggregation such as averages, or frequency of events exceeding a threshold). However, we noted some differences in results obtained from filtering rainfall data provided by the Global Precipitation Climatology Project. For the purpose of this paper, however, all methods provide a sufficiently similar description of the seasonality of tropospheric Kelvin wave description, and discussion of the differences is deferred to a separate publication.
4.2 Annual Cycle of Kelvin Wave Activity and Wave Statistics in the TTL
 Figure 5a shows the climatology of wave activity, defined as the climatological variance of the filtered ERA-Interim temperature data (averaged ±10° and filtered using the FF12 filter) on 113 hPa. Figure 5b shows the number of waves detected by the tracking algorithm passing through each location on the 113 hPa level, and Figure 5c shows the average amplitude of these waves. All figures also show zonal wind (black contours). We see that wave activity has a strong annual cycle, with maxima at approximately 45°E over Africa and the Indian Ocean in boreal summer (JJA). Wave activity over the Western Hemisphere is also maximal during JJA with very little wave activity from October to April. Over the Indian Ocean, we see a small semiannual cycle, with enhanced wave activity during DJF as well as the large maximum during JJA. The number of waves (Figure 5b, computed using the tracking algorithm considering only waves which propagate through the TTL), is relatively constant throughout most of the year, but shows a very striking structure during DJF, with very few waves propagating over the Pacific during this period. We also see a reduction in the number of waves over the Maritime Continent in the summer monsoon circulation (JJA, 100° E) and over the equatorial easterlies (DJF, 120° E). The average wave amplitude shows a somewhat similar pattern to wave activity, with larger wave amplitudes in regions of large wave activity. Both wave activity and wave amplitude are very small near the dateline (180°) and larger over the Africa/Indian Ocean. The wave amplitude shows more distinct maxima over the Indian Ocean during both JJA and DJF, with a stronger semiannual component in this region. These regions of large wave amplitude are also shifted east and are closer to the easterly wind maxima in the region of strongest wind gradient. We also see that a large part of the annual cycle over the Pacific in wave activity is due to the reduction in the number of waves propagating during DJF. We note that wave activity is a squared quantity, whereas average amplitude is not, and as such the relative amplitude of the largest waves compared with other regions is smaller than that in wave activity.
 These results indicate an influence of zonal wind on wave propagation, but the exact relation between the two is complex. The annual cycles of wave activity and wave amplitude appear to be, in some regions, correlated with the annual cycle of wind, but in other regions the correlation is more with the annual cycle of zonal wind gradient (du/dx). These relationships are also dependent on pressure level chosen (not shown).
 These results are broadly consistent with the results of Suzuki and Shiotani  (in particular, Figure 6a in their paper), with maxima over the Indian Ocean at approximately 60°E in boreal summer and the Maritime Continent at approximately 100°E boreal winter. Figure 3 in this paper uses a very similar filtering technique to that used in Suzuki and Shiotani  (WK99-type filtering with a similar region of spectral space taken). An important difference between our results and the results of Suzuki and Shiotani  is the pattern of wave activity over the Pacific; in Suzuki and Shiotani , there is most wave activity over the Pacific from January to July and with little wave activity during September to November, whereas in this paper, Figures 3 and 5a show most wave activity over the Pacific from May to September. Suzuki and Shiotani  use zonal wind to compute wave activity whereas here we use temperature. For plane linear Kelvin waves, zonal wind amplitude is related to temperature amplitude by the static stability [Andrews et al., 1987], and this explains some of the differences between the patterns observed (not shown). Flannaghan and Fueglistaler  give more information about differences between zonal wind and temperature results and show that the wave statistics presented in Figure 5 are similar when zonal wind is used in place of temperature.
 Figure 6 shows the number of waves and their average amplitude throughout the TTL. All plots have been normalized removing the large vertical profiles in wave amplitude, which obscure the zonal structure. The vertical structure of wave activity is dominated by the variations in N2, and observations approximately follow the expected theoretical relationship of (where T′ denotes the temperature disturbance due to the wave), which can be derived from the conservation of vertical momentum flux, or alternatively by considering the ray tracing equations, which will be discussed in section 5, with N2 a function of z only. Typically, the vertical profile of vertical momentum flux decays with height as waves are dissipated by radiative cooling and other mechanisms, and so the observed profile in wave activity lies below the expected N3/ρprofile.
 The focus of this paper is on the more complex role of the zonally asymmetric zonal wind. During DJF, we see that waves propagate through the TTL most frequently in the 30°W to 90°E region. The pattern of wave propagation extends upward (giving an indication for the group velocity of the waves) and both the wave distribution and amplitude structure persist into the stratosphere. This band of high-wave activity and wave amplitude can be seen extending further into the stratosphere during periods when the QBO permits wave propagation (not shown). These results are consistent with those found in Flannaghan and Fueglistaler  for the whole data set (see Figures 11 and 12 in their paper). During JJA, we see less zonal structure in the number of waves detected, with many more waves detected over the Pacific region, and less waves detected in the monsoon circulation (approximately 60°E), when compared with DJF. However, wave amplitude shows a different pattern, with the largest waves over the Monsoon region in the upper troposphere. In the lower stratosphere, the waves with largest amplitude are over the meridian (0°), with these waves originating over the Western Hemisphere. As with Figure 5, these results are consistent with strong but non-trivial links to the background winds. During DJF, the propagation of waves over the Pacific is inhibited by the strong Walker circulation, and in all seasons but MAM, the equatorial easterlies/monsoon circulation inhibits wave propagation beyond 120°E.
4.3 Interannual Variability: ENSO and the QBO
 The analysis of the seasonality of Kelvin wave propagation has shown that variations in the background structure in the TTL have a major impact on the wave propagation. In addition to the seasonal cycle, the background structure of the TTL is subject to large interannual variability associated with ENSO, the QBO, and perhaps to a lesser extent, to interannual variability in the Indian/Southeast Asian monsoon. Here we will focus on variability associated with the QBO (section 4.3.1) and ENSO (section 4.3.2). The wind anomalies associated with the QBO are predominantly zonally symmetric and are confined to stratospheric levels down to about 100 hPa. Conversely, the ENSO variations are predominantly zonally asymmetric, and are largest in the troposphere and extend up to the upper levels of the TTL.
 The QBO is a zonally symmetric wind and temperature anomaly which greatly affects Kelvin wave propagation in the stratosphere [Baldwin et al., 2001; Holton and Lindzen, 1972] through the large changes in zonal wind. Kelvin waves are largest in the westerly shear phase of the QBO which is also forced by Kelvin wave breaking, accelerating the westerly phase [Holton and Lindzen, 1972; Ern and Preusse, 2009a; Alexander and Ortland, 2010]. The QBO has a period which varies but averages to approximately 28 months. With only nine cycles in the period 1989–2009, the climatological average may be weakly affected by the QBO. Additionally, the QBO transitions occur more often in some seasons than others [Dunkerton, 1990; Baldwin et al., 2001] at approximately 50 hPa (and below), which further raises the possibility of artifacts in the climatological mean.
 Figure 7 shows a composite produced by identifying the zero wind line of the descending westerly shear zone in each cycle (where Kelvin waves are most affected) at 48 hPa. Wave activity is also shown in shading, highlighting the relationship between the QBO and Kelvin waves (this type of relationship has been shown in several studies, e.g., Sato and Dunkerton , Ern and Preusse [2009b], and Alexander and Ortland ). Above 95 hPa, Kelvin wave amplitude is clearly affected by the QBO, and above 70 hPa, this impact is much larger than any annual or ENSO variation in Kelvin wave activity. However, below 95 hPa, the QBO signal is no longer clear in Figure 7, suggesting that below this level, the results presented in section 4 are not biased by aliasing with the QBO.
 The annual cycle results show that the Walker circulation and equatorial easterlies control the spatial structure of Kelvin wave activity in the TTL. ENSO also modulates these circulations in a similar way to the annual cycle and is therefore expected to have a similar impact on Kelvin wave propagation. ENSO, as defined by the Multivariate ENSO Index (MEI) [Wolter and Timlin, 1993, 1998], and the annual cycle are correlated, and therefore, the annual cycle projects onto composites with the ENSO index. In order to properly separate the two, we consider only DJF, which exhibits the largest ENSO impact on the Walker circulation.
 Figure 8a shows that the El Niño configuration (defined as months where the MEI is greater than 0.5), with a weakened Walker circulation (maximum is approximately 10ms−1), displays a more homogeneous Kelvin wave distribution than the average DJF conditions (compare with Figure 6a). The largest amplitude waves are found in the region from 0°E to 120°E, which is somewhat wider than for the climatological DJF.
 Figure 8b shows that the La Niña configuration (defined as months where the MEI is less than −0.5) leads to a high degree of nonuniformity in the Kelvin wave distribution, with more than 3 times the number of waves passing through (45°E, 113 hPa) than over the Pacific at 113 hPa. In this case, the upper level zonal winds of the Walker circulation (with maxima of approximately 30ms−1) are faster than the Kelvin wave phase speed (approximately 15ms−1to 20ms−1) and so act as a barrier to wave propagation. We also see the strong blocking of waves in the sector of the equatorial easterlies, which are also intensified during La Niña conditions (peak is approximately 20ms−1).
 In both cases, changes in wave propagation are similar to the changes observed in the annual cycle in response to similar modulation of the Walker circulation.
5 Ray Tracing Applied to Wave Propagation in the TTL
 In section 4, we found that the climatology of Kelvin waves appears to be robustly related to the zonal wind. In Figure 5 we see a very clear relationship between zonal wind and average wave amplitude, with the largest waves to the west of the strongest easterly winds at 113 hPa. However, this relationship does not hold on other levels and so cannot be captured by a direct relationship to local zonal wind. In order to investigate this problem, we use a simplified ray tracing model of wave propagation. Section 5.1 introduces the relevant theoretical basis, and section 5.2 gives the details of the method. Section 5.3 gives some simple examples to aid the discussion of the results concerning the climatology presented in section 5.4. Finally, the ray tracing results are compared with the observational results (presented in section 4.2) in section 5.5.
 Equatorial Kelvin waves travel through the TTL according to their dispersion relation. Assuming a Doppler-shifted response to background wind, this dispersion relation is
where k and m are the zonal and vertical wave number, respectively, u is the background zonal wind, and N is the buoyancy frequency [Andrews et al., 1987]. Both u and N are in general functions of location (in the context of equatorial Kelvin waves, primarily longitude x) and height z, and season.
 The simplest way to model the relationship between background conditions and wave amplitude is using the WKB approximation (ray tracing approximation) This approximation assumes that there is a scale separation between the wave and the background, with the background quantities changing over much longer length scales (and time scales) than the wave. This separation is not strictly true for the case here, but we show below that this approach provides a good qualitative explanation for the observed patterns of Kelvin wave activity. Ray tracing is performed by integrating the ray tracing equations
where dg/dt represents the material derivative along the ray and Ω is the dispersion relation. For Kelvin waves, this is
In this study, we assume that the background quantities N and u are constant in time. ω is then constant along a ray by (2c) and, given uniform initial conditions, is constant everywhere.
 Ray tracing gives k and m, but for wave amplitude, the next-order WKB approximation [Andrews et al.,1987; Ryu et al., 2008] has to be considered. Wave energy density E is defined as
where ρ is the density, u′ is the zonal wind disturbance, and Φ′is the geopotential disturbance (directly related to the temperature disturbance). The denotes the time average, and <·> denotes the meridional average across the tropics. Wave action density A=E/(ω−ku) is a quantity that obeys a simple conservation equation in the WKB approximation [Andrews et al., 1987], given by
In the case where the non-conservative effects are approximately zero (further discussed below), we can integrate this equation, giving
Given A, it is then possible to compute the wave energy density, E, from which we can estimate wave activity in temperature or zonal wind by assuming equipartition.
Ryu et al.  find expressions for wave activity that varies with local background wind and stability in the cases of waves propagating almost vertically and almost horizontally and ignoring change in any quantities orthogonal to the rays. In these simple limits, Ryu et al.  find that wave amplitude is directly related to zonal wind, u, and stability, N2. This would allow wave activity to be directly derived from the local background quantities. However, by direct inspection of wave activity patterns on different levels, such as those shown in Figure 5, we see that this relationship as evaluated in Ryu et al.  does not hold. There are two major assumptions that were made to derive their simple relationship: first, that the ray tracing equations can be used, and second, that the waves propagate either almost vertically or horizontally allowing the elimination of the ray convergence term in the direction perpendicular to the ray. The ray tracing assumption is weak because the background stability and zonal wind vary on small spatial and vertical scales compared to the waves in violation of the WKB approximation, as described above. Although waves do indeed propagate almost horizontally or vertically in some of the domain (shown by Ryu et al. ), they must transition to this propagation direction from the initial condition (i.e., from the tropospheric Kelvin wave) where this assumption is not true. The ray tracing equations are integrated along the path of the ray meaning that the value of each wave parameter is dependent on the entire history of the ray. Furthermore, we note that the ray convergence is integrated in (6) to get the wave action, meaning that wave action is also dependent on the entire history of the ray and its neighbors (as required for the evaluation of the divergence operator). In the following, we retain the assumption that ray tracing can be applied, but solve the full ray tracing problem. It will be shown that although the ray tracing assumption is weak, results greatly aid the interpretation of the observed patterns presented in section 4.
 We use the standard Runge-Kutta-Fehlberg adaptive-time step routine [Fehlberg, 1969] (often referred to as the RK45 algorithm) to integrate (2). We make a change of variables such that z is the integrating variable, and integrate from an initial condition, (x0,z0,k0,m0,ω), representing typical tropospheric conditions. An adaptive-time step method is required to deal with a range of ray slopes when integrating in z. Care must be taken to use sufficiently smooth background quantities uand N2, and here we use bicubic interpolation between grid points. To test the validity of the interpolation, it is important to test using analytical profiles of u and N2 with realistic length scales such as those used later in this section. A further check of the validity of the solution is made by computing ω from ray quantities and confirming that this remains constant along rays (and over the entire domain). Integrating (2a) and (2b) gives and along a ray from initial location (x0,z0). By integrating several trajectories initialized at single level z0 but at different longitudinal locations x0, we can fill the domain with rays. In this study, we initialize waves at approximately 400 hPa (chosen as this level lies below the strongest zonal winds) at 5° intervals, with typical values of k and m corresponding to a zonal wave number 3 wave propagating at 20ms−1. Under some conditions, the rays may not fill the domain, but this only happens when rays intersect. At ray intersections, we get nonunique values for k and m, and the solution breaks down. In general, these events signal a wave breaking event, and at this point, simple ray tracing theory is no longer appropriate.
 Given a domain-filling set of rays, we can interpolate ray properties back onto a grid, where we can compute the group velocity divergence term (ray divergence) of (5) using finite differences. The group velocity divergence term is then interpolated back onto the rays and is integrated along the rays to calculate wave action, A, given in (6). As we integrated the ray equations in z, it is simplest to perform the integral in z, giving
where A0 is the initial wave action at z0. We choose this to give an initial energy density of 1, and therefore, all energy density results from the ray tracing are relative to the initial energy density at 400 hPa. This is modeling a uniform initial distribution of wave activity in the troposphere.
 The rays, and associated wave numbers, are computed on equally spaced height levels (at approximately 150 m intervals), so the interpolation between the ray locations and the grid is done only in the x direction. Cubic interpolation onto a fine grid of approximately 1° resolution is used for this, although these choices do not have a significant impact on the results.
 Figure 9 shows the results of the ray tracing calculation for a background field with an idealized zonal wind anomaly:
where U=5m s−1 in Figure 9a and U=−5m s−1in Figure 9b. This wind anomaly has been chosen to resemble, and in particular to have similar horizontal and vertical scales to, the typical wind anomalies in the TTL (i.e., the Walker circulation and the monsoon circulation; compare with Figure 13). N2 is set to the tropical average profile from ERA-Interim and is responsible for the structure in rays that do not encounter the wind anomaly. The wave is initialized at 388 hPa with zonal wave number 3 and horizontal phase speed of 20ms−1. Wave energy density is highest in regions where the rays are close together, emphasizing the importance of the ray convergence term in (5). In both cases, the figures show some correspondence between energy density and ∂u/∂x, but this is more a result of the particular configuration of the background wind rather than being a general result. In the atmosphere, the zonal wind anomalies in the TTL associated with the Walker and monsoon circulations are similar in scale (but may have larger amplitude, with implications discussed below) as the perturbations used for this idealized calculation, which explains the correspondence between wave energy density and ∂u/∂x in observations. Closer inspection of the results shows that even for these idealized representations of the wind anomalies in the TTL, the wave propagation is fairly complex.
 In Figure 9a, rays encounter the positive zonal wind anomaly, increasing their horizontal group velocity and deflecting rays to the east. Waves passing through the center of the wind anomaly are affected most, whereas waves which do not pass through the region are not deflected at all. This difference creates a minimum in amplitude directly above the wind anomaly (region “A” in Figure 9a) and a sharp peak in amplitude to the east (region “B” in the figure) where rays are very close (corresponding to decreased vertical and horizontal wavelengths). Note that above the wind anomaly, where zonal wind tends to zero, the rays return to traveling in parallel. This can be shown to always occur since ω is constant due to lack of time dependence in the background conditions. This means that the ray spacing, and therefore energy density, is frozen along the rays as they propagate beyond the imposed wind anomaly into the stratosphere.
 Figure 9b shows a negative wind anomaly similar in form to that associated with the monsoon and equatorial easterlies. The solution to this wind field is similar to the previous case, but with opposite sign. As before, both a sharp peak and a broad minimum of wave energy density are produced, but in this case with the peak to the west.
 Figure 10 shows the solution with a wind anomaly of twice the magnitude (10ms−1) of those shown in Figure 9. Comparison of Figures 9 and 10 shows that stronger wind anomalies cause a much larger peak in wave activity, but that this peak occurs at a similar location, and the qualitative pattern of wave energy density is similar.
 Figure 11 shows the energy density E at 100 hPa, 150 hPa, and 250 hPa in the results shown in Figure 9a, plotted against the background wind u. This figure highlights the lack of simple relationship between zonal wind and wave energy density even in the case of a simple wind anomaly of modest amplitude. A similar result holds for the zonal gradient of the zonal wind, ∂u/∂x (not shown). This demonstrates the high degree of nonlocality in the ray tracing solution and is very different to the relationship given in Ryu et al. .
5.4 Results for Climatological Zonal Wind and Stability
5.4.1 Issues With Realistic Background Conditions
 Figure 12 shows the result of ray tracing when climatological DJF and JJA zonal winds are used. In this case, we see that rays are strongly deflected by the zonal wind and the neighboring rays intersect. When using JJA zonal winds, no ray intersections occur below 70 hPa, whereas in DJF, many ray intersections occur in the Western Hemisphere, with rays originating over the Pacific being most affected.
 At a ray intersection, k and m multiple values at the same location mean that downstream of a ray intersection, the wave numbers are undefined, and therefore the solution in this region is outside the realm of simple ray tracing models. In general, waves can propagate through regions where rays intersect (often with a change of phase), or can be reflected, but these phenomena cannot be modeled with the ray tracing equations. When approaching ray intersections, wave amplitudes can become very large due to ray convergence, and in these situations, nonlinear process such as vertical mixing/overturning may dissipate the wave or change its local structure. Observations and models are consistent with the notion that significant Kelvin wave-induced mixing occurs in the TTL [see Fujiwara et al., 2003; Flannaghan and Fueglistaler, 2011].
 Since we found that the locations of energy density maxima are not strongly affected by a scaling of the wind field (see section 5.3), we evaluate in the following the ray tracing with a scaled version of the true wind field. This gives qualitative information about the location of wave energy density maxima, but underestimates the magnitude of these maxima. We find that a reduction in zonal wind by a factor of 4 eliminates ray intersections in the solution. We therefore use this scaling throughout the remainder of this study.
 Background static stability N2 is also important for wave propagation. However, this is more zonally symmetric than zonal wind in the TTL and so deflects neighboring waves in a similar manner and, as a consequence, does not cause ray intersections and therefore does not need to be rescaled. By comparing ray tracing results computed with a zonally symmetric zonal mean N2 with results computed with the full zonally asymmetric N2, we find that the spatial structure of static stability has less effect on the rays than the zonal wind, and in general, its main impact is an intensification of the pattern of energy density and ray propagation produced by wind alone (not shown).
5.4.2 Results With Seasonal Zonal Wind and Stability
 Figure 13 shows the result of ray tracing when applied to the climatological DJF and JJA background conditions with wind amplitude reduced by a factor of 4. It is clear that qualitatively the rays in this solution behave similarly to those in the idealized background field perturbation shown in Figure 9, with each local extremum in background wind producing a thin band of high wave activity.
 Figure 14 shows the seasonal cycle of the ray tracing wave energy density and the corresponding temperature wave activity (approximated by assuming equipartition of energy between kinetic energy and geopotential energy) on 104 hPa, with a uniform wave source in the troposphere with energy density E0=1. The narrow regions of high wave energy density in Figure 13 are also clearly visible in Figure 14. The maximum wave amplitudes are in JJA over the monsoon circulation at approximately 70°E. Wave activity in JJA is larger than in DJF (due in part to higher N2 in JJA; compare with energy density), with low wave activity over the Pacific and South America in DJF. Further, a narrow band of large wave activity over the Atlantic at approximately 20°W is seen in DJF. Before comparing these results with the observational results of section 4, we test the sensitivity of the method to Newtonian cooling and N2 variations.
5.4.3 Sensitivity to Ray Tracing Parameters
 Results are sensitive to choice of initial wave number k0 and wave propagation speed c0. These choices affect the aspect ratio of the wave, with k0 controlling vertical group velocity, and c0affecting both vertical and horizontal group velocities. Increasing k0 leads to an increase in the vertical group velocity. Increasing c0leads to an increase in the vertical group velocity with respect to the horizontal group velocity (which also increases, albeit to a lesser extent than the vertical group velocity). Changing c0also affects the response to zonal wind, with less response when c0 is larger. This effect is similar to that of changing the amplitude of the zonal wind, as discussed above, and so is not discussed further here. The effect of varying the aspect ratio of the wave remains to be discussed.
 We shall vary the aspect ratio by varying k0. Figure 15 shows the effect of changing k0on both the climatology at 104 hPa and on the full vertical and longitudinal structure during DJF. Figures 15a, 15c, and 15e show that the qualitative structure of the climatology, as discussed above, is not sensitive to changes in k0. We see the bands of high wave activity shift eastward as k0 is decreased; the band at approximately 20°W when k0=3 is most sensitive to changes in k0shifting considerably, whereas the other two bands shift far less. The response to the westward zonal winds over the Indian Ocean/Maritime Continent is very similar in all three experiments. Figures 15b, 15d, and 15f show that the change in the direction of the group velocity is responsible for the eastward shift in band location. The reason that the band at approximately 20°W is most sensitive to k0is twofold. First, the band is caused by ray convergence at around 300 hPa, which is lower than the other bands, meaning that the change in ray direction has a bigger impact on band location at 104 hPa. Second, the band is deflected strongly by eastward wind at around 30°W when k0=2 (see Figure 15b), but when k0=4, the band does not pass through this wind anomaly (see Figure 15f). The band at 120°E associated with the westward winds over the Indian Ocean/Maritime Continent region is far less sensitive because it is caused by ray convergence close to 104 hPa.
 Simple radiative cooling can be included in the ray tracing calculation (using a simple Newtonian cooling term on the right-hand side of (5), as outlined by Andrews et al.  and Andrews and McIntyre ). This significantly changes the vertical profile of E, but has only a small effect on the pattern of wave propagation because it is linear in wave amplitude. The effect is only dependent on the time spent propagating through the region and therefore damps rays with smaller vertical group velocity, cgz, more than other rays. Figure 16 shows that linear damping does reduce the amplitude on the most westward band of wave activity (around 30°W on 104 hPa) more than the band associated with the equatorial easterlies. In general, the response to a westward wind maximum is less damped than the response to an eastward wind maximum. For the calculation shown in Figure 16, Newtonian cooling with a time scale of approximately 11 days is used. At the tropopause, this is somewhat larger than typically experienced by a Kelvin wave (typically a timescale of 20 days). This slightly larger value is used to emphasize the pattern of change.
5.5 Comparison With Observational Results
 Comparison of the observed wave amplitude distribution and wave activity distribution (Figures 6a and 6c) with corresponding ray tracing results (Figure 13) shows broad similarity and some notable differences. In particular, the ray tracing model predicts several thin regions of very high wave activity, corresponding to each local extremum in the zonal wind field. In the wave propagation results, these thin bands are not clearly separated but typically appear as a single broader band of high wave activity. This difference may arise from two different causes. First, the thinness of the bands of high wave energy density predicted by the ray tracing calculation contradicts the scale separation assumption between the wave and the background winds, as these bands are thinner than the typical Kelvin wave wavelength. In reality, we hypothesize that the lack of perfect scale separation would broaden these bands. Second, the data shown in Figure 6 are the average over many years, with each year having a different background wind structure. Hence, the observed wave distribution is the superposition of wave propagation in each year, which will be smoother than that of any single year. Similarly, for the model calculations, we initiated the rays with fixed aspect ratio and wave speed. These parameters affect subsequent wave propagation, modifying the direction of propagation and the sensitivity to the background wind, and therefore changing the location of these bands, as can be seen in Figure 15. In reality, variations in these parameters will therefore smooth the wave distribution.
 In addition to these general qualitative differences between the ray tracing solution and the observations, there are a few smaller discrepancies. The ray tracing predictions for the 104 hPa level (Figure 14b) appear to miss the strong response to the DJF easterlies around 100°E seen in observations (see Figure 5). However, Figure 13a shows that the ray tracing calculations also predict a strong maximum at approximately the right longitude, but slightly upward displaced. Likewise, the ray tracing predictions do not exhibit a local maximum in wave energy density near 100°W in boreal winter, visible in observations in Figure 5a. Figure 13a shows that the most westward band of high wave energy density does pass through 100°W but below the 104 hPa level in the ray tracing results. Figure 15 shows that the location of this band is sensitive to choice of k0; the location of this band is also sensitive to the level on which we initialize waves (not shown) and the band is displaced upward and westward if we initialize waves higher up. Both of these discrepancies do not therefore contradict the qualitative picture given by ray tracing.
 Beyond these differences, the ray tracing results do share many features with the observational results. The ray tracing does predict the correct location for the largest waves in the stratosphere, and the response to local extrema in the zonal wind is similar to that given in Figure 9. For example, in Figure 6, the strong pattern of average wave amplitude seen over South America and the Atlantic (90°W to 0°) with low amplitudes above and to the west of the wind maximum and high amplitudes to the east corresponds well with the wave activity predicted by the ray tracing calculations shown in Figure 9a. Likewise, the ray tracing response to the equatorial easterlies and the monsoon circulation resembles that shown in Figure 9, with the largest wave amplitudes to the west of the wind maximum and lower wave amplitudes to the east of the wind maximum.
 The seasonal cycle of temperature wave activity at 104 hPa using the ray tracing model (Figure 14b) and in observations at 113 hPa (Figure 5) also shows good qualitative agreement. In particular, the low activity over the DJF Pacific, the large activity associated with the JJA Monsoon, and the low activity east of 120°E throughout the year are captured well by the ray tracing model.
 We have shown that there is a pronounced seasonal cycle in Kelvin wave distribution, with the waves confined to 30°W to 90°E in boreal winter, but propagating over most longitudes in boreal summer. The equatorial easterlies prevalent throughout the year over the Indian Ocean/Maritime Continent sector deflect waves upward, leading to a reduction in wave activity to the east of the wind maximum. This effect is particularly pronounced during the periods of maximum easterlies (during boreal winter in relation to the upper level monsoonal anticyclones centered over the Western Pacific, and during boreal summer in relation to the Indian/Southeast Asian monsoon circulation (as shown in Figure 1). In addition, during boreal winter, the presence of westerlies in the TTL over the Pacific related to the Walker circulation leads to a further reduction in wave activity to the east of the equatorial easterlies. The results from the wave tracking algorithm employed here show maximum wave amplitude in the sector of easterlies in JJA and DJF, but with wave amplitude maxima to the west of the wind maxima. Full ray tracing calculations recover this interesting aspect of Kelvin wave propagation, as well as most of the longitude-height structure of wave activity and its seasonal evolution. The fact that the wave amplitude maximum is not co-located with the longitude sector with the temperature minima on monthly time scales may have implications for the humidity of air entering the stratosphere because large, short time scale fluctuations in temperature can lead to more dehydration than would be expected from inspection of the background temperature field.
 The climatology of Kelvin wave amplitude in the TTL is qualitatively similar to the results of a simple ray tracing model forced with climatological winds and a uniform wave distribution in the troposphere. This suggests that Kelvin wave amplitude in the TTL is primarily a function of the zonal winds in the TTL/upper troposphere (roughly 300 hPa to 80 hPa) rather than the tropospheric distribution of waves, which explains the noted lack of correspondence between tropospheric convectively coupled Kelvin waves (section 4.1) and Kelvin wave activity in the TTL (section 4.2). This result is supported both by variations seen related to the mean annual cycle and the changes due to El Niño/Southern Oscillation. Furthermore, we show that there is no local relationship to the zonal winds (argued for by Ryu et al. ) because the ray tracing model shows that ray convergence is critical in the TTL, and that as a consequence, the effect of zonal wind anomalies on wave activity is highly nonlocal.
 The ray tracing results of section 5 have fixed frequency ω independent of longitude, which follows directly from the use of a steady background state under the ray tracing approximation, and as a consequence, will produce the same number of wave events over a given period (as defined by local maxima in time as in FF12) at all longitudes. However, the tracking algorithm used earlier (FF12) shows that there is significant variation in the number of waves (Figure 5b), especially during boreal winter, where there are fewer waves over the Pacific than over the Indian Ocean. This result indicates that either there is significant source variability (in terms of number of wave events) in the troposphere or there are other processes not captured by ray tracing that play a role over the boreal winter Pacific. The OLR results in Wheeler et al.  and the brightness temperature results shown in Figure 4 suggest that there is tropospheric convectively coupled Kelvin wave activity in the Pacific and so we suspect that a reduction in the number of tropospheric waves over the tropical Pacific is not responsible for this observed reduction in number of wave events in the TTL in this region. On the other hand, over the Pacific, the zonal winds (the upper branch of the Walker circulation) are strong during boreal winter and are often of similar magnitude to the phase speed of a Kelvin wave (often in excess of the phase speed of a Kelvin wave). Under linear Kelvin wave theory, waves cannot propagate through such regions as the meridional extent of the wave becomes unbounded. This offers a reasonable explanation for the observed drop in the number of waves over the Pacific that cannot be captured by ray tracing alone, or by previous methods that cannot separate wave activity from the number of waves.
 We thank George Kiladis and two anonymous reviewers for their helpful comments and feedback. We thank ECMWF for providing the ERA-Interim data used throughout this study. COSMIC data were obtained from the COSMIC Data Analysis and Archive Center (CDAAC). Some of the results were obtained using the CLAUS archive held at the British Atmospheric Data Centre, produced using ISCCP source data distributed by the NASA Langley Data Center. TF was supported by a NERC studentship, and SF was supported in part by NERC Advanced Researcher fellowship. This work has been supported by the U.S. Department of Energy under DOE Award No. DE-SC0006841.