Designed to measure high vertical resolution atmospheric radiance profiles, HIRDLS is a 21-channel limb-scanning filter radiometer on NASA's Aura satellite.

Shortly after launch in 2004 an optical blockage, believed to be a loosened flap of the Kapton^® material lining the foreoptics section of the instrument, was found to obscure around 80 % of the viewing aperture. Consequently, major corrective work has been required to produce useful atmospheric data. Measurements of temperature, cloud, and a wide range of chemical species have now been successfully retrieved and made available for scientific analysis .

One particularly productive area of research has been the detection and analysis of GWs e.g.. This would have been possible with the original horizontal and vertical scanning mode of the instrument, but the closer along-track profile spacing made possible by the lack of horizontal scanning capability has allowed measurements to be taken at a higher horizontal resolution than originally planned, facilitating such research. Measurements are made in vertical profiles: around 5500 profiles are obtained per day, spaced approximately 70–105 km apart depending on the scan direction (see Sect. ) and scanning pattern used. Due to the optical blockage, measurements are taken at a significant horizontal angle, ∼ 47∘, to the track of the satellite, as a result of which observations cannot be made south of 62∘ S and are not spatially co-located with other instruments on the Aura satellite.

V007 of the HIRDLS data set provides vertical temperature profiles from the tropopause to ∼ 80 km in altitude as a function of pressure, allowing us to produce useful gravity wave analyses at these higher altitudes. Measurements have a precision ∼ 0.5 K throughout the lower stratosphere, increasing with height to ∼ 1 K at the stratopause and 3 K or more above this, depending on latitude and season . Vertical resolution is ∼ 1 km throughout the stratosphere, smoothly declining to ∼ 2 km above this.

Data are available from late January 2005 until March 2008, when a failure of the optical chopper terminated data collection. A variety of scanning modes were used until June 2006, after which the scanning mode remained constant until the end of the mission. Consequently, we examine here data from the calendar year 2007; this provides a complete year of data at a consistent horizontal resolution, but without biasing the results by including an additional fractional year.

In parts of this paper, we also use data derived from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument on NASA's TIMED satellite to assess the methodology. A 10-channel limb-sounding infrared radiometer, SABER, was intended primarily to measure and characterise the mesosphere and lower thermosphere on a global scale, but also scans down into the stratosphere, providing ∼ 2200 vertical profiles per day with a vertical resolution of approximately 2 km and an along-track profile spacing alternating between ∼ 200 and ∼ 550 km depending on scan direction (in an equivalent manner to HIRDLS, discussed in Sect.  – see Fig. 1 of , for a diagram illustrating the comparative scanning pattern of both instruments). Kinetic temperature profiles cover the altitude range from ∼ 15 to ∼ 120 km .

SABER's scanning routine incorporates the TIMED spacecraft's yaw cycle, with the coverage region shifting north and south every 60 days to cover the poles alternately. Accordingly, while the coverage of the instrument in the tropics and at midlatitudes remains constant throughout the year, high northerly and southerly latitudes are only covered for 60 of every 120 days, in a 60-day on, 60-day off cycle, with coverage in the “off” hemisphere extending to 54∘ and in the “on” hemisphere to 87∘. We use SABER version 1.07 temperature data for 2002–2012, with a precision of ∼ 0.8 K ; the longer period is possible due to the consistent profile-to-profile scanning pattern of the instrument since launch, but provides a smaller total number of resolved wavelike features due to the coarser resolution of the data set.

In Figs.  and , we compare our results to a data set computed using the same HIRDLS V007 data, but locating only the single largest-co-varying-amplitude peak at each altitude level of each profile. This reduces our data set to one almost identical to , with minor differences due to (i) applying the noise-comparison method of , (ii) the use of a Fourier transform rather than a Stockwell transform to remove planetary-scale waves, and (iii) the data extending up to 80 km altitude. We refer to this data set as “HIRDLS-pk” and to the primary data set as “HIRDLS-all” where necessary to avoid ambiguity.

Figures and also compare our data to equivalent results computed using SABER. This analysis is methodologically equivalent to the main HIRDLS analysis method, with three differences: (i) due to the smaller numbers of usable profiles, we compute planetary waves based on a rolling 3-day mean of global temperature rather than individual days, (ii) data are interpolated to a 2 km vertical resolution rather than 1 km, and (iii) after , we remove profile pairs separated by more than 300 km. We refer to this data set as “SABER”.

For HIRDLS, the lower limit to measurable vertical wavelength λz in the stratosphere is ∼ 2 km e.g.. This is imposed by the 1 km vertical resolution of the data, resulting from the radiometer channel detectors, low radiometric noise and the choice of vertical sampling , which result in narrow averaging kernels . This corresponds to an upper limit on vertical wavenumber kz=5×10-4 cycles per metre (cpm). Due to the increased vertical width of the averaging kernels at higher altitudes , this increases to λz∼ 3.5 km (decreases to 2.9×10-4 cpm) at mesospheric altitudes. We impose a lower limit of kz=6.25×10-5 cpm (upper limit of λz = 16 km) in our analysis; the underlying reasons for this relate to particular features of our results, and are consequently discussed in Sect. 5.1.

The 70–105 km separation between profiles in principle imposes a maximum resolvable kh of 4.8–7.1 × 10-6 cpm (minimum resolvable λh of 140–215 km) depending on the scanning pattern used (see Sect.  for further details). However, limb-sensing techniques have very broad horizontal weighting functions, which imply a significant horizontal averaging. For HIRDLS, this is around 200 km in the line-of-sight (LOS) direction and 10 km in the direction perpendicular to this. Hence, whilst the instrument is in principle capable of detecting short waves propagating in a horizontal direction perpendicular to the LOS, waves propagating along the LOS shorter than ∼ 200 km will not be detected regardless of the actual profile spacing for these profiles . Adjacent profile weighting functions do not overlap.

In practice, waves with large kh will be much more challenging to detect. and Sect. 2 of discuss this for the CRISTA instrument, with broad applicability for all limb-sounding instruments including HIRDLS, and predict that a drop in sensitivity at large kh will be observed. This decline in measured amplitude is strongly related to the kz of the signal in question: at the largest kh, waves with smaller kz will tend to be detected with somewhat smaller amplitudes than an otherwise identical wave with larger kz.

The minimum resolvable horizontal wavenumber is somewhat harder to determine theoretically. For waves aligned perfectly in the zonal direction, where we filter out signals based on planetary waves of mode 7 or below, the minimum kh will correspond to that of a mode-7 planetary wave, i.e. 1.7×10-7 cpm at the Equator and higher at higher latitudes. However, in practice, the satellite scan track will be aligned in the zonal direction only at the poles and in a meridional or near-meridional direction for most of its orbit, and wavelengths longer than this will not be filtered out in these directions. We impose a cutoff of kh = 1×10-7 cpm (λh = 10 000 km) on our analysis; however, such a wave would imply Δϕi,i+1∼(2π/100) radians, which is likely to be well below any practically resolvable phase difference between two profiles, and accordingly very small horizontal wavenumbers in our results should be treated with extreme caution both because of this and because they are likely to have a large relative angle of propagation (see Sect.  below), producing a measured value much smaller than the true horizontal wavenumber of the wave.

Some variation in the maximum resolvable kh exists due to variations in the instrument scan pattern over the mission. To avoid this we use only data from 2007, when the scanning pattern remained consistent: specifically, from June 2006 onwards, HIRDLS obtained 27 pairs of vertical up and down scans of ∼ 31 s duration each, followed by a 1–2 s space view before the next 27 scan pairs .

The high velocity of a low Earth-orbiting satellite such as Aura means that, while the scanning mirror physically rotates through the whole profile, a significant geographical distance will be traversed by the satellite. Figure 1 of illustrates this effect, as does our Fig. b.

We can make an estimate of the effect of this upon our measurements. Aura completes 14.6 orbits a day and HIRDLS takes around 15.5 s to perform a vertical scan. Accordingly, in the time taken to perform a complete vertical scan, the HIRDLS measurement track will have advanced by ΔX=2π×R×14.624×60×60×15.5m, where R is the radius of a small circle around the Earth offset by 47∘ from the great circle around the poles, R∼6.4×106cos⁡47∘ m = 4.3×106 m, giving a distance travelled during each scan of ∼ 72 km.

A full vertical scan runs from the surface to a height of ∼ 121 km, and hence the horizontal distance along-track between individual height levels (i.e. after the 1 km vertical interpolation) is approximately 0.6 km. Accordingly, between the 15 and 80 km levels, the tangent point of a measurement will differ horizontally by ∼ 40 km, producing a difference in Δri,i+1 of as much as ΔX=55 % in some profile pairs at high altitudes relative to the geolocation height at 30 km. This is a larger separation than the instrument weighting functions in the narrower direction, and is hence significant. We compensate for it in our analysis by scaling the profile separation distance Δri,i+1 appropriately for each height level before calculating kh and Mi,i+1, but this means that the along-track horizontal resolution limit varies with height due to the scan direction of the profiles. Figure d, discussed in greater detail below, illustrates this effect.

This scanning effect will in principle affect vertical wavelength measurements, since the vector of the instrument scan lies at ∼(90∘-arctan1/0.6) = 31∘ to the vertical. However, this is compensated for in the retrieval, which splines the measured radiances onto a regular vertical grid.

The high velocity of the satellite allows us to consider observed waves as having been measured effectively instantaneously .

Our measurements represent only the component of the signal lying along the satellite's travel vector. Due to the low probability of the horizontal wave vector lying along this direction, our measurements will tend to underestimate the true value of kh, especially when there is a large angle between the true propagation direction and the measurement direction. If waves tend to propagate zonally rather than meridionally, this will particularly affect measurements when the satellite is travelling in a mostly meridional direction, i.e. near the Equator, and have the smallest impact when the satellite is travelling more zonally near the turnaround latitudes (∼ 62.5∘ S and ∼ 80∘ N). This effect is seen strongly in our results, and is discussed where appropriate.

Our S-Transform analysis method inherently introduces further errors into the analysis. A range of sensitivity studies using perfect wave packets were carried out by , and can be summarised as follows. These limitations apply generally to S-Transform data.

Provided the signal is above the noise level of the data, the error on the measured temperature perturbation does not depend directly on the magnitude of the “true” temperature perturbation.

To summarise, in addition to any uncertainties due to the actual measurements, we expect our analysis to systematically underestimate temperature perturbations, potentially by a very large proportion, in all conditions, and to generally underestimate vertical wavenumber in any conditions where we do not detect one or more full cycles of the same wave.

An important limitation is the ambiguity of phase cycle in our estimates of Δϕ; that is to say, we cannot know purely from our measurements whether the measured phase difference of a ζ between two adjacent profiles represents Δϕtrue, Δϕtrue+2π, Δϕtrue+4π, etc. This is referred to as aliasing , and will cause us to underestimate kh for large-wavenumber features in our data, perhaps very significantly. The effect of this on our results will be to redistribute these aliased waves across the measured wavenumber range. suggest that a large proportion of the additional smaller-scale waves detected by our method may be aliased in this way.

If we assume that such aliased waves have a random measured phase difference Δϕi,i+1, then this will distribute them evenly across the measured kh space (Eq. ). In principle, a correction factor may be applied to account for this aliasing e.g.; however, such corrections make inherent assumptions about the spectral shape of the original wave distribution, and accordingly we do not use them here.

The derivation of Eq. () assumes that the waves under consideration can be described by the midfrequency approximation. This has been shown by to account for around a 10 % difference between real and calculated values of Mi,i+1 for the CRISTA instrument.

Figure shows the global distribution of the number of observed waves, as a function of the base-10 logarithm of vertical wavenumber kz. Corresponding vertical wavelengths are provided on the top axis of the figure as a guide.

We first consider Fig. a. This shows the distribution at six height levels. At all heights, we see a broadly similar distribution, with larger numbers of observed waves at smaller wavenumbers, and a steady drop with increasing wavenumber. In particular, the number of observed waves drops by a factor of ∼ 6 between kz=10-4.2 and 10-3.5 cpm.

Heights above 55 km (orange and brown lines) are truncated at a vertical wavenumber of 10-3.6 cpm (4 km vertical wavelength) due to the reduced vertical resolution at these altitudes; the remaining lines continue to 10-3.3 cpm (2 km vertical wavelength). This truncation is introduced because, although vertical features smaller than this are “detected” by the analysis, they are clearly spurious due to the nature of the retrieved product, the resolution of which drops by a factor of ∼ 2 in this region.

Fewer small-wavenumber waves are observed at the 20, 60 and 70 km altitude levels; this is due to the proximity of the vertical ends of the data set at 80 and 15 km, which significantly reduces the possibility of properly observing a long vertical wave here. We also observe four subpeaks, centred on wavenumbers 10-3.34, 10-3.47, 10-3.64, and 10-3.95 cpm (the last only weakly visible, very broad, and shifting with height value given is for ∼ 50 km altitude).

Figure shows equivalent results for horizontal wavenumber kh. We observe a distribution which rises as a function of horizontal wavenumber. A flattening of the distribution is observed at intermediate wavenumbers, kh∼10-6.1–kh∼10-5.6 cpm; this appears to be due to the bias in observed horizontal wavenumber at equatorial latitudes due to the meridional path of the satellite (Sect. ), and is discussed further in Sect. . Aside from this flattening, the data otherwise rise consistently until a peak is reached at kh=10-5.35 cpm.

Above kh=10-5.35 cpm, a discontinuity is observed, with the absolute peak followed by a sudden drop and then by a secondary peak at kh=10-5.25 cpm. This arises due to the instrument scanning pattern (Sect. ), and is explained by Fig. , discussed below.

A general trend is seen of the distribution shifting towards higher kh with height; this will be discussed below.

Figure b and c show equivalent results for SABER and the HIRDLS-pk method. The HIRDLS-pk results show a distribution falling off at horizontal wavenumbers greater than ∼10-6.7 cpm at 20 km altitude, with the turning point in kh increasing with altitude; this suggests that the additional waves contributed by the method of may include significantly more short horizontal waves. Note, however, that it is difficult to ascertain the full effects of noise on our measurements. While the method is designed to mitigate against the inclusion of instrumental noise via the co-varying amplitude methodology and the noise-floor comparison, suggest that random fluctuations would peak at around 4 × the horizontal sampling distance, and increase with altitudes. Since we see these effects in all three data sets (HIRDLS, HIRDLS-pk and SABER), they may contribute to our distributions.

The SABER results, meanwhile, show a distribution with a form very similar to that of the primary HIRDLS results. This suggests that the anomalous blockage-induced peaks do not have a preferential apparent horizontal wavenumber, but are instead distributed across the whole observed wavenumber range.

Figure a shows the distribution of observed waves as a function of both kh (horizontal axis) and kz (vertical axis), at the 32 km altitude level. This level is chosen as it is approximately the lowest height level at which no detected wave signals could be edge-truncated (tropopause plus 16 km). It should be noted that this global-mean distribution is averaged over a vast range of geophysical regimes, and accordingly is meaningful as a reference only.

We see that the observed numerical distribution is dominated by waves with small horizontal and vertical wavenumbers, i.e. with long horizontal and vertical wavelengths, with the number of observed waves in bins in this region (bottom left to bottom centre) typically 2 or more orders of magnitude above the numbers in the least-observed region at top left. This will at least partially relate to observational effects, rather than to the geophysical distribution of such waves; a wave with a longer wavelength in either direction is likely to have a larger physical extent, and is thus more likely to be observed in a measurement profile randomly located in the vicinity of the wave. Additionally, such waves are likely to induce larger temperature perturbations (Fig. b) and thus are more likely to be above instrument noise levels. Furthermore, as shown by Fig. 4 of , the temperature perturbations of such waves will tend to be more easily detectable due to high instrument sensitivity. Duplication effects are compounded in the horizontal direction: as discussed above, it is highly non-trivial to distinguish between the same horizontal feature in adjacent profiles, and consequently long horizontal waves may well appear in several profiles, particularly when the satellite is travelling in a similar direction to the wave vector. Future refinements of the analysis method will investigate this further, using methods based upon the Fourier uncertainty principle.

The smallest number of observed waves lies in the top left of the plot, i.e. large vertical wavenumber and small horizontal wavenumber. As discussed above, the paucity of observations here in the vertical domain may well be due to random-sampling considerations; however, bias in observed horizontal waves would tend to increase rather than decrease the number of observed waves here, and detectability of their temperature perturbations should be reasonable (although the amplitude of such waves may be low). As a result, the small number of waves observed in this region is likely to be a real effect, suggesting that the atmosphere may support comparatively few short-vertical-long-horizontal-wavelength “pancake” waves.

We see the anomalous, presumably blockage-induced, spikes observed in Fig.  most strongly at top right; these do, however, extend across most of the horizontal range to at least some degree. Beneath these peaks, we continue to see a much higher number of waves than in the top left or bottom right, suggesting that even without the peaks this is a significant contributing region to the overall distribution. Finally, at bottom right, we see a region of few waves; this is consistent with the strongly reduced detectability at this combination of wavelengths. Detectability here should be by far the worst of any part of the 2-D spectrum; thus, the larger number of signals observed here than at, for example, the top left, may suggest that the atmosphere can support many short-horizontal-long-vertical-wavelength waves.

An apparent discrepancy is seen between our results and the HIRDLS-derived momentum flux wavenumber distributions of . There, a clear peak was seen in tropical waves at kh=10-5.75 cpm and kz=10-3.90 cpm. However, their method selected only the two largest signals in each rolling 10 km window, and accordingly will not have included the shorter waves that make up a large part of our distributions, hence producing results with a very different final form, more similar to our HIRDLS-pk analyses. As shown in Figs. c and c, this data set peaks at much longer vertical and horizontal wavelengths than HIRDLS-all, explaining the majority of this discrepancy. Additional differences remain in the precise location of the peak, which lies (at the 30 km level, i.e. the closest analysed height level to their analysis) at kh∼10-6.4 cpm and kz≤10-4.2 cpm in our HIRDLS-pk distribution. The difference in kh peak location arises due to the strong dependence of observed MF on wavenumber (Eq. ). Part of the difference in kz peak location is explained by the slightly larger-kz waves observed near the Equator (e.g. Fig. i and j) relative to other latitudes, but this cannot account for all of the differences, which may instead arise due to the very different analysis methods used.

Subsequent panels of this figure (Fig. b–k) are shown as percentage differences from this 32 km global distribution, with colours representing the percentage difference in the proportion of total observed waves at a given (kh,kz). This is a slightly complicated normalisation, but is chosen due to the small differences in visual appearance between un-differenced distributions; it should be noted that the distribution appears broadly identical at all heights when shown un-differenced

Compare e.g. Fig. g–k with Fig. α–ϵ, which show the same data, with this normalisation in the former case and that used in Fig. a in the latter.

Figures b–f show the difference between the observed distribution at 32 km and that at five other height levels. Grey-shaded regions in individual panels indicate areas of the spectra which may experience edge truncation at that level.

We see two key trends, with increasing height leading to (i) larger horizontal and (ii) smaller vertical wavenumbers. The first such difference is clearly visible in Fig.  for all three data sets. The latter is harder to see in the absolute observed values seen in Fig. , and only becomes apparent when the data are normalised at each level individually; as discussed above, at least part of this change with height may be associated with noise effects. The change in vertical wavelength is consistent with previous observations dating back decades (e.g. , and references therein).

Figures g–k show the differences from the global mean in Fig. a for the same analysis performed on specific 30∘ latitude bands centred on the latitude indicated in the panel, all at the same 32 km altitude level. Note that the northerly limit to observations is at 80∘ N, and thus that panel g only represents the range 60–80∘ N.

We see very clear differences, with a bias towards smaller horizontal wavenumbers near the Equator (between Fig. i and j) and towards larger horizontal wavenumbers at higher latitudes. As discussed in Sect. , this is at least partially due to the polar orbit of the instrument, which leads to the satellite travelling near-meridionally near the Equator, combined with Coriolis parameter effects which allow a broader range of wave-intrinsic frequencies near the Equator . If we assume a zonal bias to the true wave field, observations here will tend to significantly over-measure the distance between phase fronts due to the geometry of the scan, and consequently significantly under-estimate kh. There is some difference in the kz direction, with smaller kz at higher latitudes and larger kz near the Equator, but this effect is smaller than the kh effect; this is consistent with e.g. , and .

Aside from this, minimal differences are observed between these distributions. This is largely due to the kh effect drowning out such variation visually. To compensate for this, all following analyses will be normalised for latitude by comparing only within a given latitude band or a given region.

Figures and show the observed kh–kz distributions for each of our geographic regions for global summer (JJA NH (Northern Hemisphere), DJF SH (Southern Hemisphere)) and winter (DJF NH, JJA SH) respectively. The panels are normalised in a similar way to Fig. b–k above, but with the differences being not from the annual mean global mean, but from the annual mean zonal mean at that latitude (shown identically in Figs. α–ϵ and α–ϵ). This allows us to focus on variations other than the instrument- and Coriolis-parameter-induced variation in horizontal wavelength with latitude seen in Fig. g–k, which would otherwise dominate all panels. Analyses were also carried out for spring and autumn, but have been omitted for brevity as the variations they showed were much smaller than for summer and winter.

We see the largest relative variations at low latitudes and in the bottom right of each panel, i.e. low kz and high kh. Since wind-based spectral filtering in this region is primarily driven by the QBO, with a scale much longer than the year examined here, this may represent variation in the source mechanisms in this region, which is highly convectively active. Alternatively, it may indeed partially be due to QBO-related filtering, specifically the Doppler shifting of waves in and out of the observational filter of HIRDLS by the partial phase change of the QBO winds over the 6 months between Figs.  and . Examination of a longer period of time would help to elucidate this, but even the full 3 years of HIRDLS data are unlikely to provide a sufficiently long record to decouple this effect completely. Source changes are likely to be the dominant of the two factors due to the strong seasonality of convective activity in this region e.g..

Smaller differences are seen at higher latitudes, and do not display such strong seasonal variation. In particular, at the highest latitudes, we often see an enhancement relative to the annual mean at many wavelengths in both summer and winter, i.e. large numbers in these seasons and low numbers in autumn and spring (not shown). Due to this behaviour, these regions are discussed below, where whole-year time series are shown.

As we saw above, major differences between regions of the wavenumber distribution tended to manifest as peaks in the corners of each panel. Hence, it may be useful to subdivide our analysis by wavelength and to study separately the time evolution of these individual components of the distribution. Figure a accordingly divides the overall observed wave distribution into four distinct subtypes, or species, defined by wavenumber: short-vertical long-horizontal (“Sl”, top left), short-vertical short-horizontal (“Ss”, top right), long-vertical long-horizontal (“Ll”, bottom left), and long-vertical short-horizontal (“Ls”, bottom right).

Figure b shows the mean temperature anomaly associated with each kh–kz combination, indicating that the largest temperature perturbations are associated with species Ll. From this, one might initially conclude that waves of species Ll were the most important, due to their large amplitude – in particular, this implies a large potential energy per wave, (1/2)(g/N)2(T^/T‾)2. However, a vitally important geophysical quantity is the MF transported by the waves – in particular, this is one of the key parameters used in weather and climate modelling. In the mid-frequency approximation, this can be characterised by Eq. () above. There are three key variable terms in this which we can derive directly from HIRDLS data: kh, kz and T^/T‾, where kz and kh combine in the ratio kh/kz. For the waves we can observe with HIRDLS, this ratio can vary over nearly 3 orders of magnitude, as shown in Fig. c. As a consequence of this, the observed momentum flux per wave, Fig. d, is almost entirely dominated by Ls waves, particularly those at the very largest kh and smallest kz, which as shown by Figs.  and represent the bulk of the variability in our observations once k–h variations due to orbital geometry and/or the variation of the Coriolis parameter with latitude are removed. Our results suggest, therefore, that variations in the number of observed Ls waves appear critically important to the variability of the global MF distribution in a much more fundamental way than the other three species.

As shown by Fig. , the global numerical distribution of observed waves is dominated by waves of species Ll and Ss. This numerical dominance of species Ll may be due in part to their larger mean temperature perturbations (Fig. b) and consequent easier detection in temperature data; however, this clearly cannot be the whole reason, due to the relatively small mean temperature perturbations for species Ss.

Figure examines the time variation over the year 2007 for each of our four wave species, as a time series of the total number of observed waves per profile. The data exhibit significant day-to-day variability, and have been smoothed by 2 weeks to aid interpretation.

In general, the most-observed species is type Ss, with around 1.0–1.3 waves per profile (wpp) in most regions and at most times, whilst the least-observed species is type Sl, with typically ∼ 0.4 wpp. The former type includes a significant contribution from the anomalous subpeaks, which may contribute to the large number of observed signals, while the latter is difficult to detect due to limb-sounding sensitivity considerations , which may explain the comparatively low number. However, this limitation applies more strongly to waves of type Ls, and thus cannot fully explain the difference. The number of observed waves of the two long-vertical species, Ls and Ll, in general lie between these values, with both Ls and Ll varying between around 0.4 and 1.4 wpp over the course of the year.

Figure , meanwhile, shows the same data, normalised such that the annual mean value for each species equals 100. This emphasises the variability of each species with respect to time, and shows that, while in some regions and at some times all four species can vary together, at others different species can vary independently of each other, often with some apparent compensation between different species as one rises in observed frequency to take the place of another. The latter effect will be discussed further in Sect. .

Deviations from the mean are observed at Arctic latitudes in winter and spring, specifically between days 1 and 100 in Fig. c–f. The relatively low numbers during this time are consistent with filtering of waves by the polar vortex, and the variations coincide with vortex influences by large-scale planetary waves.

The next largest variations are those of type Ls, particularly at tropical and subtropical latitudes, i.e. Fig. m–x, γ–δ. This has important consequences: as discussed above, this species is by far the dominant carrier of large momentum fluxes, and the large temporal variations of these species thus contribute significantly to the temporal variability of MF. In particular, we see large peaks in the observed number of Ls waves during monsoon periods in monsoon regions, i.e. Fig. n, p, q, r, γ in NH summer and Fig. t, u, v, x, δ in SH summer . This suggests that the large momentum fluxes previously observed to be associated with the monsoon are dominantly carried by Ls waves; the correlation between our Ls wave time series and outgoing longwave radiation (OLR) over the particularly intense monsoon regions (panels q and r), for example, is ∼ -0.7. This is consistent with previous work, e.g. and . The absolute number of observed Ls waves is not especially low compared to other regions at the same latitude during the other parts of the year (e.g. Fig. m, o in NH summer), suggesting that the relatively low value during the other parts of the year are a baseline rather than a reduction in Ls waves due to other processes.

Interestingly, comparatively little variation is observed in the relative distribution of all four species around the well-known Andes/Antarctic Peninsula hotspot, in Fig. z–aa. This region has previously been observed to dominate the global MF distribution, with values in JJA typically an order of magnitude or more larger than the next largest peak at any other location or time. We see a reduction here in the number of observed waves in all four species during the period of enhanced MF, especially in Fig. aa; this reduction continues downstream of the hotspot through Fig. ab–ad and round to Fig. y, leading to a very strong such reduction in the zonal mean (Fig. ϵ). Figure  emphasises that this is an actual reduction rather than a normalisation artifact. This suggests that the increase in MF observed during this period is not due to increased dominance of any one species, but instead due to a significant increase in MF per wave carried by all four types, dominating over a reduced absolute number of waves. A possible alternative explanation could be that large-amplitude peaks in the T′kz distribution “drown out” smaller peaks which would be visible in other seasons, leading to an apparent reduction in the total number observed; however, the drop appears to apply to all four species, and this effect would therefore have to be extremely large to explain the overall reduction.

It would be tempting to conclude that the variation of the species we observe is due primarily to source mechanisms operating at tropospheric and near-surface altitudes. However, while this may be the case in many places and times, we cannot generally assume this.

A particular example of a process which will lead to a wave being generated with one species but being observed as another is illustrated in Fig. , adapted from . Here, we illustrate how critical-layer wind filtering and the consequent refraction of waves just below such a critical layer of gravity waves would lead to a wave observed at a low altitude with type L (i.e. long-vertical-wavelength) would be observed at a height nearer a critical level with type S (i.e. short-vertical-wavelength).

The wave initially propagates along a given group velocity vector vg1, with wavefronts as indicated on the diagram. An observation taken here, indicated by the dashed oval, will measure a relatively long vertical wavelength due to the large difference in geometric height between the wavefronts, with some compression of the vertical wavefronts due to the approaching critical level.

At some later time, the wave has propagated some distance and approaches a critical layer zc. As the wave approaches this layer, the vertical component of the group velocity vg2 tends towards zero, causing the wavefronts to realign. Were the observation (dashed oval) to be taken here instead of at the first location, a much shorter vertical wavelength would be measured.

As a result, the HIRDLS observation will show it to instead be of type S in the second instance and type L in the first, despite having the same source. Furthermore, an observation slightly higher in altitude will not observe the wave at all, consequently reducing the number of observed waves. Analogous processes could also operate in the horizontal direction (not illustrated), or could operate to take a wave packet completely out of our observational filter, causing it to disappear from the results despite remaining present in the physical atmosphere.

Such processes could serve to explain at least some cases where we see a reduction of waves in one type and a compensating increase in waves of another type; in such situations, it may be the case that the generation mechanism has remained constant throughout, but wind layers have altered the wave properties before the point of observation. Many other similar and dissimilar mechanisms may operate, and may operate multiple times on the same wave as it travels through the atmosphere; we only include one example to illustrate this likelihood. While our observed variations in the relative number of each wave may be indicative of source terms, there are many confounding geophysical mechanisms even assuming a perfect instrument.