Gravity waves are important drivers of dynamic processes in particular in the middle atmosphere. To analyse atmospheric data for gravity wave signals, it is essential to separate gravity wave perturbations from atmospheric variability due to other dynamic processes. Common methods to separate small-scale gravity wave signals from a large-scale background are separation methods depending on filters in either the horizontal or vertical wavelength domain. However, gravity waves are not the only process that could lead to small-scale perturbations in the atmosphere. Recently, concerns have been raised that vertical wavelength filtering can lead to misinterpretation of other wave-like perturbations, such as inertial instability effects, as gravity wave perturbations.

In this paper we assess the ability of different spectral background removal approaches to separate gravity waves and inertial instabilities using artificial inertial instability perturbations, global model data and satellite observations. We investigate a horizontal background removal (which applies a zonal wavenumber filter with additional smoothing of the spectral components in meridional and vertical direction), a sophisticated filter based on 2D time–longitude spectral analysis (see

Critical thresholds for the vertical wavelength and zonal wavenumber are analysed. Vertical filtering has to cut deep into the gravity wave spectrum in order to remove inertial instability remnants from the perturbations (down to 6 km cutoff wavelength). Horizontal filtering, however, removes inertial instability remnants in global model data at wavenumbers far lower than the typical gravity wave scales for the case we investigated. Specifically, a cutoff zonal wavenumber of 6 in the stratosphere is sufficient to eliminate inertial instability structures. Furthermore, we show that for infrared limb-sounding satellite profiles it is possible as well to effectively separate perturbations of inertial instabilities from those of gravity waves using a cutoff zonal wavenumber of 6. We generalize the findings of our case study by examining a 1-year time series of SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) data.

In the middle atmosphere, various wave and wave-like processes shape the global structures of temperature and winds. This includes planetary waves (Rossby waves;

Recently,

Inertial instabilities in a rotating stratified fluid, like the earth's atmosphere, arise from an imbalance between the pressure gradient and the centrifugal forces when the absolute angular momentum decreases with the radius

Gravity waves are small-scale oscillations in the atmosphere that are balanced by buoyancy as restoring force. They are generated mostly in the troposphere by flow over orography, imbalances along jet streams and frontal systems, and convection, e.g. over storm systems

In the last decades, satellite measurements began to allow for insights into global distributions of gravity wave activity

Historically, different background removal techniques have been applied for different measurement methods because of their specific characteristics

In this study, we focus on the data with vertical profile information. We investigate the application of both vertical and horizontal gravity wave background removal on different temperature data sets that incorporate inertial instability signals, and we evaluate to what extent the background removals are able to remove these instability signals from the gravity wave perturbations. In particular, we evaluate what cutoff length scales are optimal as a compromise between preserving the full gravity wave spectrum and removing more remnant structures due to inertial instabilities. For a first approach to the topic, we start by analysing artificial inertial instability perturbations. The structure was modelled according to the conditions found for a strong inertial instability event in December 2015, as previously discussed in

This paper is structured as follows: Sect.

For different steps of our analysis, we utilize the characteristics of satellite observations, realistic atmospheric reanalysis data and an idealized data set that only contains inertial instability perturbations. Satellite measurements give unique information by providing global sampling on real observations. The reanalysis data set has the advantage of covering a large range of spatial and spectral scales without being restricted to a specific irregular sampling. The artificial perturbation model gives us the opportunity to see how the background removal techniques are influencing an isolated inertial-instability-like signal. Any residual fluctuations after applying the background removal then indicate remnant inertial instability signals that should not be attributed to other processes. In realistic model data sets or measurement results, possible remnants signals might belong to the gravity wave activity we eventually want to infer. This section describes the various data sets.

The Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument is operated on board the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite, which was launched in December 2001 and is still operational to this date. The limb-sounding instrument records profiles of the atmospheric radiation by scanning its viewing direction vertically and covers a large altitude range from the tropopause to well above 100 km. For this study, we are mainly interested in the stratosphere and hence consider altitudes

The TIMED satellite covers almost the whole globe with

According to

ERA5 is the newest reanalysis product based on the 4D-Var data assimilation of ECMWF's Integrated Forecast System (IFS). In the simulation, the grid consists of 137 hybrid model levels with a top at 0.01 hPa. In the stratosphere, the hybrid levels are less than 1 km apart. The IFS is a spectral system. In version cy41r2, the model has a horizontal resolution of

For our analysis, we use a global interpolated field regularly sampled to

We construct a simple model of artificial inertial instability temperature perturbations (

Following the findings from

The symmetric structure is centred around the latitude

In the vertical, we apply a sinusoid with a given vertical wavenumber

For the zonal dimension, we use a step function with smoothed transition to zero in order to restrict the structure to about a third of a latitude circle, since midlatitude inertial instability signals are often constrained in a so-called “channel”

Cross sections of temperature perturbations of the artificial inertial instability, we defined in Sect.

In order to constrain the free parameters of the three shape functions, we refer to the case of 3 December 2015. Based on a 1-year survey of GPS-RO temperature anomalies and a climatological analysis of ERA-Interim data by

Parameters adapted for this study in the artificial inertial instability perturbation model defined in Eqs. (

In the analysis, we have to distinguish between “regularly sampled” such as outputs from an atmospheric model and “sparsely sampled” such as observations by satellite instruments, though both kinds of data sets are global. This section describes the various methods for background removal applicable to these different data sets.

As a first approach to a background removal, we apply a Butterworth filter defined by a window function

with the cutoff wavelength

The Butterworth filter was used by

Where we refer to vertical filtering by default in this paper, we use

For horizontal filtering in regularly sampled, global snapshots, we apply zonal FFT in the longitude direction to provide a zonal wavenumber spectrum for each altitude and latitude. The background is defined as the low-pass band up to a cutoff wavenumber,

Zonal spectral filtering is an intuitive solution for a background removal in regularly sampled, global data sets such as general circulation model outputs of free-running models as well as assimilation products like reanalyses. The temperature and wind fields along the latitude circle are naturally periodic, and with a regular (ideally high-resolution) sampling, an FFT analysis is directly applicable.

The background removal from

SABER provides a global coverage over the course of 1 d. Applying a similar background removal, as for a global model data set, is a reasonable approach. However, data at a given latitude from the ascending or descending orbits are collected at the same local time rather than at the same universal time considering periods as short as a few days. This is the case because the orbit of the TIMED satellite is only slowly precessing. The period of 1 d, which SABER requires to collect the data for one global coverage, is close to the shortest period of planetary waves such as the quasi 2 d waves in the mesosphere (e.g.

The observation geometry of a LEO satellite limits the range of zonal wavenumbers and frequencies in which space-time spectra can be uniquely obtained

The use of time–longitude spectra also prevents influences of day-to-day variations in the atmospheric background due to short-period travelling planetary waves

In order to determine the residual temperature fluctuations, the background is reconstructed for the precise location and time of each observation. First, spectra are interpolated in altitude and latitude. The background for the observation longitude and time is then evaluated as the superposition of the single, global-scale waves with significant spectral amplitudes using their spectral amplitudes and phases. In addition, the mean background temperature defined as the 31 d average plus a linear trend is taken into account. Eventually, the method defines the gravity wave temperature perturbation at a particular location and altitude by subtracting the average local background temperature (averaged from the mostly two background temperature estimates) from the overlapping time windows. In addition, in a second step the most relevant tidal modes in stratosphere and mesosphere are explicitly removed

We apply this method as a horizontal background removal to SABER profiles in the rest of this study.

Linear gravity wave theory is based on the assumption that the total state of any atmospheric variable, e.g. temperature

The background state is the superposition of a mean temperature profile and the large-scale influences like planetary wave and synoptic inertial instability conditions at each instance in space

The variance var

In Eq. (

In addition, the zonal mean gravity wave temperature perturbation

In the previous subsections, we have introduced two different general methods to separate temperature perturbations from a background that we want to apply in this study – horizontal and vertical spectral filtering. Resulting gravity wave temperature perturbations,

In the case of horizontal filtering, in particular zonal wavenumber filtering,

In the ideal case, however,

The advantage of using

For evaluating gravity waves in real or realistic data, it would suffice to check if the spurious inertial instability remnants in the estimated gravity wave perturbations are smaller than the targeted gravity wave signals. However, the quantification of the gravity wave signal is the purpose of the analysis itself and, hence by definition, not known a priori. It therefore cannot be used as a reference for successful background removal.

For the removal of the background from real observations, it is necessary to define a success criterion. In our approach, the background with respect to gravity waves contains global-scale variations caused by, for example, planetary waves and signals from inertial instabilities. We consider a background removal as successful if the background remnants are smaller than the errors introduced by the measurement method and the retrieval. The error budget of an instrument is usually described by precision and accuracy, where accuracy describes “systematic” errors which are present in a similar magnitude on the entire data set and precision refers to the noise-like errors that affect a measured parameter randomly. In the observation of fluctuations, like gravity waves, the uncertainty is closer linked to the precision than to the accuracy of the instrument, since errors characterized by accuracy are removed together with the background. The magnitude of both accuracy and precision usually vary with altitude. Based on this approach, we use the minimal precision value of the SABER instrument reported for the stratosphere of 0.3

A spectral background removal will usually remove global-scale variations gradually (i.e. each global-scale process is described by several spectral components), and the more wavenumbers are removed the more the background signal is removed as well. The success criterion defined above then allows us to identify threshold characteristic vertical wavelengths or zonal wavenumbers for horizontal or vertical background removals, respectively. In particular, this can be directly applied on the synthetic data. In our artificial inertial instability data set, we can test the different approaches and find the criterion for a successful background removal matched when the mean squared gravity wave temperature perturbation (cf. Eq.

Figures

Zonal temperature variance of artificial inertial instability temperature perturbations after vertical filtering (fifth-order Butterworth) with a cutoff wavelength of

Zonal temperature variance of artificial inertial instability temperature perturbations after horizontal filtering (zonal mean with additional smoothing in meridional and vertical components) with cutoff zonal wavenumber

We chose most of the cutoff limits to match values used in previous satellite studies. Figure

The upper row of Fig.

Zonal mean gravity wave temperature perturbations inferred from ERA5 temperatures for 3 December 2015, 00:00 UTC, after

The other two panels in Fig.

We have seen already that the quality of a spectral filtering is highly dependent on the cutoff length scale used. Ideally, a transition between mesoscale and synoptic-scale fluctuations (previously referred to as “spectral gap”) can be identified, where the temperature variance will be insensitive to the cutoff length separating gravity wave activity at small scales and mesoscales from global-scale inertial instability signals. However, such a transition is not necessarily present in the stratospheric temperature variance (and very unlikely to be found in the troposphere).

Dependence of mean temperature variance in a box of 90

With its dense regular sampling, the realistic global ERA5 temperature data allow for a sensitivity study with respect to temperature variances on the cutoff length scales for a wide range of cutoffs of both vertical and horizontal filtering. Figure

Different sensitivities of the mean temperature variance for more rigorous filtering are evident. For vertical cutoff wavelengths larger than 15 km, the averaged variance of ERA5 temperatures increases slowly with a linear gradient. In the same cutoff wavelength range, the variance stays mostly constant in the artificial inertial instability perturbations data. The gradient becomes gradually steeper in both data sets between 15 and 10 km. In this range, where both data sets show similar behaviour, the averaged temperature variance is likely dominated by the inertial instability signal. The gradient for the ERA5 data remains similar also for cutoff vertical wavelengths shorter than 10 km, while for the artificial inertial instability data the decrease continues to steepen. This different behaviour indicates a switch to gravity waves as the dominant dynamics process, which is missing in the artificial inertial instability perturbation data.

The sensitivity of temperature variance to increasing cutoff zonal wavenumbers shows a range of cutoffs where both the realistic ERA5 data as well as the artificial inertial instability perturbation data behave in a comparable way. From cutoff zonal wavenumber 1 to 6, both curves of averaged temperature variances are decreasing fast, indicating the dominance of the inertial instability signal. At wavenumber 6 there is an abrupt change in the variance decrease in ERA5 data, and the slope continues to change between wavenumbers 6 and 12. At wavenumbers above cutoff zonal wavenumber 12, the gradient is about constant. In comparison, the decrease of the temperature variances for the artificial perturbation data continues up to wavenumber 12 and then directly switches to stagnation at extremely small averaged temperature variances. The fact that the averaged temperature variance for the synthetic inertial instability stays constant at wavenumbers 2 and 3, wavenumbers 5 and 6, and wavenumbers 10 and 12 in the artificial data is likely connected to the regular structure of the modelled inertial instability, which spans over approximately a third of the latitude circle.

Similar to the vertical filtering case, we interpret similarities and differences in the variances of the two data sets. The variance is likely dominated by the inertial instability influence up to wavenumber 6. For wavenumbers larger than 7, gravity waves are taking over as the most dominant process, though up to wavenumber 12 there may be remnants of the inertial instability. The slow decrease at high wavenumbers shows that eventually energy is taken out of the gravity wave scales as well.

The diagnostics shown for gravity wave climatologies from measurement or model data are usually energy quantities like gravity wave potential energy densities

Vertically filtered (fifth-order Butterworth) zonal mean squared gravity wave temperature perturbations (cf. Eq.

Horizontally filtered (with additional smoothing in meridional and vertical components) zonal temperature variance of ERA5 temperatures on 3 December 2015, 00:00 UTC, with cutoff zonal wavenumber

In the following, we concentrate on four structural features and their evolution under different cutoffs in the two background removals. First, Fig.

The isolated structures merge into one broad sheet of perturbation at altitudes a few kilometres below the stratopause. Decreasing the cutoff wavelength in the vertical filtering reduces the pancake structure at midlatitudes in a way that it is not further recognizable in the results from 6 km vertical filtering. The tropical band of the structure is more persistent while being reduced in magnitude but remaining especially strong at 35 to 40 km altitude. In addition, a more rigorous vertical filtering seems to narrow the vertical distance between the mean squared gravity wave temperature perturbation peaks. As discussed in Sect.

In comparison, the horizontal filtering results show less evidence for the typical pancake structure (both at midlatitudes and in the tropics). In the tropics, above 30 km altitude some layered structures emerge for low cutoff wavenumbers (cf. Fig.

At altitudes above 20 km, the general structure continues to decrease in magnitude, but it remains very similar in structure for wavenumbers 6 and larger. This is consistent with gravity waves at a wide range of different scales causing these features. Particularly pronounced is a second structure between 60 and 80

For very low cutoff zonal wavenumbers, the high-latitude maximum is concealed by a larger structure due to Rossby waves in middle and high latitudes. From cutoff wavenumber 6, it is not distorted by pancake-like overlaid structures anymore. Then, the overall shape of the enhanced mean squared gravity wave temperature perturbation distribution is not changed up to a cutoff wavenumber 18 and also keeps roughly the same magnitude. This indicates that the horizontal wavelengths of the waves amount to at most

Below 20km, the tropical tropopause and the tropopause inversion layer show up distinctively in the vertically filtered data, which are depicted by an elongated, strong mean squared gravity wave temperature perturbation signal spanning all the way from 30

Even a rigorous vertical filtering with a cutoff wavelength of 6 km retains the signal. This problem is similar to the misinterpretation of inertial instability perturbations due to their small vertical wavelength. In comparison, the horizontal filter is removing a large part of the tropopause signal from the perturbations already with cutoff wavenumber 0, i.e. only removing the zonal mean. This could be explained by the tropical tropopause and the tropopause inversion layer being rather stable in altitude over all longitudes. However, even at zonal wavenumber 42, the remnant structure is still larger than the gravity wave fluctuations above and below, indicating that the remnants of the tropopause still mask the gravity wave signal.

Finally, all of the results show enhanced mean squared gravity wave temperature perturbations in the troposphere below 15 km around 40

So far we have considered a case study for a particular, strong inertial instability event. In order to evaluate whether the results are representative, we investigate 1 year of SABER data.

As already addressed in Sects.

Like Fig.

Zonal mean gravity wave temperature perturbations and corresponding standard deviations of SABER data. The upper row shows the mean perturbation from

Figure

Time series of daily-mean gravity wave temperature perturbation profiles from SABER in the midlatitude box (90

Throughout the entire year, the vertically filtered data show a pronounced signal above the tropopause, i.e. between 20 and 25 km, and around the stratopause, i.e. between 40 and 50 km altitude. Furthermore, this data set exhibits remnant pancake structures persisting for several days through the Northern Hemisphere winter (December to mid February). This is consistent with favourable conditions for inertial instability particularly around the winter solstice

After horizontal filtering, some wave-like structures are remaining but smaller in magnitude and of shorter duration. These structures indicate very active gravity wave events dominating the average in the domain. Also, the horizontal background removal does not show the very strong signals associated with the tropopause and stratopause.

The analysis of satellite data for gravity waves relies on a scale separation between the targeted gravity waves and larger-scale structures. Recently, concern was raised that inertial instabilities might not be properly removed in generating gravity wave climatologies and thus cause spurious signals in the derived distributions

We approached this by considering a particularly strong inertial instability event on 3 December 2015, which was discussed in the study of

First, we compared the vertical and horizontal filtering results for the idealized inertial instability signal. Zonal mean gravity wave temperature perturbations and zonal mean variances show that both methods are in principle capable of removing an inertial-instability-like perturbation to a limit that it falls to a level below the SABER precision threshold. However, in vertical filtering we had to eliminate all fluctuations larger than 7 km in the vertical to achieve a sufficient signal reduction. This would lead also to the removal of a major part of the gravity wave spectrum. A comparable level of removal after horizontal filtering was achieved with a cutoff zonal wavenumber of 6, which does not limit the gravity wave analysis, in particular, if momentum flux is considered.

The investigation of ERA5 data shows that for the midlatitude stratosphere, gravity waves dominate over inertial instability signals for vertical cutoff wavelengths shorter than about 10 km and for horizontal cutoff wavenumbers larger than 6. After vertical background removal, remnants of the stratopause remain. In addition, the tropical stratosphere showed perturbations over all altitudes from 20 to 45 km that indicate Kelvin wave activity. Horizontal filtering, in contrast, removes both effects. Horizontal filtering is also more capable of dealing with the tropical tropopause, but remnants larger than the average gravity wave variance remain. In the subtropical troposphere and lowermost stratosphere, zonal wavenumbers larger than 18 are required to reduce the signal of Rossby waves. Further consideration will be needed in future if data sets should be evaluated for such lower altitudes, as well. A zonal wavenumber filter applying wavenumbers higher than 18 removes already part of the gravity wave spectrum.

A 1-year analysis of SABER data reveals inertial instability signals for the vertical filtering but no evidence for inertial instabilities after horizontal filtering. Hence, the case study of one inertial instability event is representative.

Considering existing satellite climatologies, such data sets which are based on time–longitude spectra including zonal wavenumber 6 are likely free of the influence of large-scale inertial instabilities. This includes, for instance, the GRACILE climatology

In conclusion, for the removal of a background containing inertial instabilities and gravity waves, horizontal spectral filtering with a zonal wavenumber of 6 or higher provides the best results. It is, therefore, recommended to use such an approach where the data allow for it. For altitudes around the tropopause, larger zonal wavenumbers are required, but zonal wavenumbers larger than 18 reduce the fraction of retrievable gravity waves in the stratosphere greatly.

The ERA-5 data used for this study are freely available from

CS generated the artificial inertial instability data set, performed the processing of background removals on the former and the ERA5 data, wrote the paper text and generated the figures. ME processed the background removal on the satellite data set. PP supervised the research and helped with the paper. ME and MR helped with discussions and revision on the paper text and the figures.

The authors declare that they have no conflict of interest.

This research has been supported by the Deutsche Forschungsgemeinschaft (DFG; Source Variability, grant nos. PR 919/4-2 and ER 474/4-2).The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Robert Sica and reviewed by two anonymous referees.