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AMT | Articles | Volume 11, issue 3

Atmos. Meas. Tech., 11, 1363-1375, 2018

https://doi.org/10.5194/amt-11-1363-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/amt-11-1363-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Observing Atmosphere and Climate with Occultation Techniques...

**Research article**
08 Mar 2018

**Research article** | 08 Mar 2018

On the distortions in calculated GW parameters

^{1}LIDTUA, CIC, Facultad de Ingeniería, Universidad Austral and CONICET, Mariano Acosta 1611, Pilar, Provincia de Buenos Aires B1629ODT, Argentina^{2}IFIBA, CONICET, Ciudad Universitaria, C1428EGA, Buenos Aires, Argentina^{3}GFZ, GFZ German Research Centre for Geosciences, Sect. 1.1: GPS/Galileo Earth Observation, Telegrafenberg A17, 14473 Potsdam, Germany

^{1}LIDTUA, CIC, Facultad de Ingeniería, Universidad Austral and CONICET, Mariano Acosta 1611, Pilar, Provincia de Buenos Aires B1629ODT, Argentina^{2}IFIBA, CONICET, Ciudad Universitaria, C1428EGA, Buenos Aires, Argentina^{3}GFZ, GFZ German Research Centre for Geosciences, Sect. 1.1: GPS/Galileo Earth Observation, Telegrafenberg A17, 14473 Potsdam, Germany

Abstract

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The significant distortions introduced in the measured atmospheric gravity wavelengths by soundings other than those in vertical and horizontal directions, are discussed as a function of the elevation angle of the sounding path and the gravity wave aspect ratio. Under- or overestimation of real vertical wavelengths during the measurement process depends on the value of these two parameters. The consequences of these distortions on the calculation of the energy and the vertical flux of horizontal momentum are analyzed and discussed in the context of two experimental limb satellite setups: GPS-LEO radio occultations and TIMED/SABER ((Atmosphere using Broadband Emission Radiometry/Thermosphere–Ionosphere–Mesosphere–Energetics and Dynamics)) measurements. Possible discrepancies previously found between the momentum flux calculated from satellite temperature profiles, on site and from model simulations, may to a certain degree be attributed to these distortions. A recalculation of previous momentum flux climatologies based on these considerations seems to be a difficult goal.

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How to cite.

de la Torre, A., Alexander, P., Schmidt, T., Llamedo, P., and Hierro, R.: On the distortions in calculated GW parameters during slanted atmospheric soundings, Atmos. Meas. Tech., 11, 1363-1375, https://doi.org/10.5194/amt-11-1363-2018, 2018.

1 Introduction

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In the last few years, we have observed the ongoing development of several
techniques to sound the lower, middle and upper atmosphere (e.g. Wu and
Waters, 1996; Tsuda et al., 2000; Preusse et al., 2002; Alexander et al.,
2011; Hertzog et al., 2012; John and Kumar, 2013; Lieberman et al., 2013;
Oliver et al., 2013; Alexander, 2015; de Wit et al., 2017). The advantages
and disadvantages of each choice are clearly distinguishable among the
available rocket-, balloon- and satellite-borne instruments, as well as radar
and lidar ground-based devices. Regarding the retrieval of information on
atmospheric dynamics from satellite measurements, we know that both satellite
limb and nadir observing techniques are needed to resolve different parts of
the gravity wave (GW) spectrum (Wu et al., 2006) and that a better
understanding of GW complexities requires joint analyses of these data and
high-resolution model simulations. The global observation of the atmosphere
and the ionosphere using limb or nadir sounding paths, makes it possible to
obtain vertical profiles of refractivity, density, temperature (*T*),
pressure, water vapor content and electron density, which is a remarkable
achievement considering the available experimental
resources.

One of the main objectives pursued by current observations is the permanent improvement required for the understanding of GW sources of generation (such as flow over topography, convection, and jet imbalance), as well as their propagation, breaking and dissipation around and above the tropopause, forcing atmospheric circulation. This knowledge is needed in the sub-grid parameterizations in global models for climate and weather forecasting applications, in order to simulate the influence of orographic and non-orographic GWs and produce realistic wind and temperature (e.g. Fritts and Alexander, 2003; McLandress and Scinocca, 2005; Kawatani et al., 2009; Alexander et al., 2010; Shutts and Vosper, 2011; Geller et al., 2013). In these parameterizations, some parameters describe the global distributions of GW vertical flux of horizontal momentum (MF), as well as their wavelengths and frequencies. Until recently, the necessary parameters could not be determined through global observations, because the waves are small in scale and intermittent in occurrence. The parameterizations compute a momentum forcing term by making assumptions about the unresolved wave properties that have not been properly constrained by observations. The assumptions are formulated as a set of tuning parameters that are used to adjust the circulation and temperature structure in the upper troposphere and middle atmosphere (Alexander et al., 2010).

Among recently developed sounding devices, global positioning system (GPS)
radio occultation (RO) is a well-established technique for obtaining global
GW activity information. RO uses GPS signals received by low Earth-orbiting
(LEO) satellites for atmospheric limb sounding. *T* profiles are derived with
high vertical resolution and provide global coverage under any weather
condition, offering the possibility to carry out the global monitoring of
the vertical *T* structure and atmospheric wave parameters. Several authors
have contributed to global analyses of horizontal and vertical GW
wavelengths, specifically potential energy and MF distribution (Tsuda et al.,
2000; de la Torre et al., 2006; Wang and Alexander, 2010; Faber et al., 2013;
Schmidt et al., 2016; Alexander et al., 2015). In particular, Alexander
et al. (2008) (A08) stated that it is not possible to fully resolve GW from
RO measurements because there are different kinds of distortions. In each
occultation, the outcome depends on wave characteristics (essentially
wavelengths and amplitude), the line of sight (LOS) and the line of tangent
points (LTP), both with respect to the phase fronts to be detected. Ideal
conditions for accurate wave amplitude extraction in occultation retrievals
are given by quasi-horizontal wave phase surfaces or when the LOS and LTP are
nearly contained and out of those planes. Short horizontal scale
waves are weakened or even filtered out with high probability. Another result
from A08 is that the detected vertical wavelengths will always differ from
the original ones, but only the presence of inertio-GWs, which have nearly
horizontal constant phase surfaces, will ensure small discrepancies. They
concluded that extreme caution is needed when addressing the issues of
amplitude, wavelength and phase of gravity waves in occultation data. Some
years before A08, de la Torre and Alexander (1995) (TA95) already observed
and established analytically the discrepancies to be expected between
measured and real horizontal and vertical wavelengths during balloon
soundings, taking into account the motion of the gondola with respect to the
constant GW phase surfaces. This analysis was performed both from the
intrinsic and the ground frame of reference.

In Sect. 2, we analyze the distortion to be expected in the detection of real vertical and horizontal wavelengths from almost instantaneous soundings. These are different from vertical and horizontal, specifically for satellite measurements. In Sect. 3, the consequences of this distortion in the calculation of GW energy and MF are discussed. In Sect. 4, the implications of using two different satellite setups are considered in some detail. In Sect. 5, some conclusions are outlined for future applications and a possible careful reconsideration of some results and conclusions obtained in previous climatologies is suggested.

2 GW wavelengths distortion

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From TA95 and A08, it is clear that when an on-site or remote sensing
instrument sounds the atmosphere along a given direction, which is different
from the vertical or the horizontal plane, the measured vertical and
horizontal wavelengths are expected to considerably differ from “real” (or
“actual”) values. In the Appendix from TA95, (1) a stationary GW observed
from (2) a ground-fixed frame of reference (Fig. A1 and Eqs. A1–A5) was
specifically considered. Now, it may be accepted that both these conditions
are emulated by GPS-LEO RO (e.g. Kursinski et al., 1997), as well as by
TIMED–SABER (atmosphere using broadband emission
radiometry and thermosphere–ionosphere–mesosphere energetics and dynamics)
(Russell et al., 1999) measurements (see below Sect. 4). In relation to the
first condition, we may assume that satellite-based soundings yield *T*
profiles almost instantaneously. Following this reasoning, the vertical
“real” and “apparent” (or measured) wavelengths (*λ*_{z} and
${\mathit{\lambda}}_{z}^{\text{ap}}$, respectively) are related by the following
expression (TA95, Eqs. A3–A5):

$$\begin{array}{}\text{(1)}& {\displaystyle}{\displaystyle}{\mathit{\lambda}}_{Z}^{\text{ap}}={\displaystyle \frac{{\mathit{\lambda}}_{Z}}{\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)}},\end{array}$$

where *α* is the elevation angle defined by a straight sounding path
direction and the horizontal plane. In turn, cot (*ψ*) is the ratio
between the horizontal wavenumber vector (*k*_{H}) projected on the vertical
*α*-plane and the vertical wavenumber *k*_{Z} (Fig. 1). The ratio
(*k*_{H}∕*k*_{Z}) is also known as the GW aspect ratio. Figure 1, with two
arbitrary successive GW phase surfaces, *φ*_{1} and *φ*_{2},
cutting *α* plane defined, show a clear difference between real and
apparent vertical (and horizontal) wavelengths. This distortion, frequently
present in radiosoundings or satellite-based GW studies, is in
general non-negligible and affects the calculation of all magnitudes
requiring previous identification of wave parameters.

Here we recall that cot (*α*) is equal to the ratio between ${\mathit{\lambda}}_{H}^{\text{ap}}$ and ${\mathit{\lambda}}_{Z}^{\text{ap}}$, and this result will be used
below. A similar relation to Eq. (1) may be derived between
horizontal real and apparent wavelengths, from Eqs. (A3) to (A6) in TA95. The
resulting relation is as follows (not shown in TA95):

$$\begin{array}{}\text{(2)}& {\displaystyle}{\displaystyle}{\mathit{\lambda}}_{H}^{\text{ap}}={\displaystyle \frac{{\mathit{\lambda}}_{H}}{\text{abs}\left(\mathrm{1}+\mathrm{tan}\left(\mathit{\alpha}\right)\mathrm{tan}\left(\mathit{\psi}\right)\right)}}.\end{array}$$

We should mention that *λ*_{H} is real but may not be the true
horizontal wavelength, as information must be sampled along two different
horizontal directions in order to be able to calculate it (e.g. Faber
et al., 2013; Schmidt et al., 2016). We will now focus on the consequences
derived from the expected distortion in *k*_{Z} or in *λ*_{Z}. As is
known, in global atmospheric models the subgrid parameterization of GW energy
and MF is based on a successful identification of GW parameters, after proper
processing of *T* profiles. The effects of GW on large-scale circulation
have been treated via parameterizations in both climate and weather
forecasting applications. In these parameterizations, key parameters describe
a global distribution of MF, GW wavelengths and frequencies (e.g. Alexander
et al., 2010).

Equation (1) provides the magnitude of the expected departure in
${\mathit{\lambda}}_{Z}^{\text{ap}}$ from *λ*_{Z}, for each monochromatic GW
component, within a given wave ensemble at any atmospheric region. In order
to better understand this distortion, we will consider this equation as
parametric in *α* or *ψ*. As stated above, both
independent parameters are simple trigonometric functions of the apparent and
real (and horizontal or vertical) wavenumber components ratio, respectively. The
angle *α* only depends on the sounding path direction during the
observation process through progressive atmospheric layers, and *ψ*, on
the GW direction of propagation, ** k**∕

In Fig. 2, we define the distortion as the ratio:

$$\begin{array}{}\text{(3)}& {\displaystyle}{\displaystyle}D={\displaystyle \frac{{\mathit{\lambda}}_{Z}^{\text{ap}}}{{\mathit{\lambda}}_{Z}}}.\end{array}$$

Following Eq. (1), *D* may be equivalently represented as a function
of *α* leaving *ψ* as a parameter, or vice versa, making use of the
symmetric dependence on both of them. We first describe this function in
terms of *α* in Fig. 2a and b. For illustration, we show the variation
of *D* for increasing selected values of *ψ* between 0 and *π* rad. Note
that the underestimation of *λ*_{Z} occurs when (*D*<1) *ψ*=0.1,
0.5, 0.9, 1.3 rad and the overestimation of *λ*_{Z} occurs
when $(D>\mathrm{1})\mathit{\psi}=\mathrm{1.7}$, 2.1, 2.5 and 2.9 rad. For each *ψ*
value, a singular *α* value associated to two upper diverging branches
is seen. This is better appreciated in Fig. 2b. The horizontal dashed line
corresponds to the “non-distortion” *D*=1 case. Considerable
departures from this non-distortion limit are seen. Note that the functional
behavior of *D* is non-symmetric for *ψ* greater than and less than
*π*∕2 rad. Also, notice that all possible sounding and wave
orientations are covered by defining one of the angles between 0 and
*π*∕2 rad and the other one between 0 and *π* rad.

From the above arguments, we can conclude that for a given GW ensemble, a net
significant distortion of the measured spectra should be expected. This net
distortion will become more or less significant, depending on (i) the
composition of the ensemble and (ii) the specific measuring device. In the
next section we will illustrate this argument for the case of satellite-borne
measurements. A 3-D plot presents better the functional dependence of *D*
with *ψ* and *α* already shown in Fig. 2a and b, now separately for
under- and overestimations of *λ*_{Z}, below and above the plane *D*=1
(Fig. 3a and b respectively).

The 3-D plot shows the complete variability of *D* for *a* between 0 and
*π*∕2 and *ψ* between 0 and *π*. For any fixed *ψ* value, starting
at *α*=0, each *D* curve increases from zero, crosses the *D*=1
boundary diverging at a given *α* value, located (after/before) *π*∕2
depending on *ψ* is (less/greater) than *π*∕2 and decreases again to
zero, as *α* approaches the *π* limit. Due to the symmetric dependence
of *D* with both parameters, to avoid a possible confusion and redundancy, in
Fig. 2 it seems enough to show the *D* variability for *α* between 0 and
*π*∕2.

3 GW energy, spectra and momentum flux

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The computation of the specific potential energy per unit mass, Ep, for a GW ensemble, is given by the following equation:

$$\begin{array}{}\text{(4)}& {\displaystyle}{\displaystyle}\mathrm{Ep}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{g}{N}}\right)}^{\mathrm{2}}\stackrel{\mathrm{\u203e}}{{\left({\displaystyle \frac{\widehat{T}}{{T}_{\mathrm{0}}}}\right)}^{\mathrm{2}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\left({\displaystyle \frac{g}{N}}\right)}^{\mathrm{2}}{\displaystyle \frac{\mathrm{1}}{{z}_{\mathrm{2}}-{z}_{\mathrm{1}}}}\underset{{Z}_{\mathrm{1}}}{\overset{{Z}_{\mathrm{2}}}{\int}}{\left({\displaystyle \frac{\widehat{T}}{{T}_{\mathrm{0}}}}\right)}^{\mathrm{2}}\mathrm{d}z,\end{array}$$

where *z*_{1} and *z*_{2} are the minimum and maximum altitudes for
integration, *g* is the acceleration due to gravity, *N* is the buoyancy
frequency, $\widehat{T}$ and *T*_{0} are the perturbation amplitude and
background temperature, respectively, and the overbar indicates a space
averaging process. This average must be performed for the GW ensemble
considered, over at least one wavelength corresponding to the GW mode with
the largest amplitude in any direction (i.e. horizontal, slanted or, as
usually, vertical). Consistently, different choices of this direction
involving the same ensemble should ideally yield identical results.
Alternatively, the average may be also performed over a time interval at
a fixed point, considering a general non-stationary ensemble of GW. In this
case, the net contribution of stationary waves would be obviously
underestimated. In addition, we recall that the computation of instantaneous
Ep at fixed points is sometimes reported without the corresponding averaging
process, but we consider that this procedure lacks a clear, physical sense.

In Eq. (4) we must have previously removed the noise and long scale structures
from the *T* profiles. The remaining GWs should include amplitudes expected to
significantly contribute to Ep. The vertical interval for integration is
usually about 10 km. But, depending on *α*, *ψ*_{i} and the
azimuth of each one of the dominant modes in the GW ensemble, some waves may
not be contained for at least one complete cycle within the integration
interval. Then, the integral in Eq. (4) may not include at least one
full wavelength from all of these dominant modes. As a result, the individual
contribution of each mode to the net Ep will be under- or overestimated to
a significant extent.

To extend these considerations to a quite realistic scenario, let us consider
a particular modeled distribution of GW vertical wavelengths, selected among
the numerous theories developed and based on diverse experimental setups,
after the seminal paper by Dewan and Good (1986) (e.g. Smith et al., 1987;
Hines, 1991; Fritts and Alexander, 2003; Yiğit et al., 2017, and
references therein). It has been observed and broadly assumed that part of
a GW spectrum (the larger vertical wavenumbers) is saturated beyond a given
characteristic ${k}_{Z}^{C}$ value that decreases with increasing altitude.
Smaller wavenumbers than ${k}_{Z}^{c}$ are not expected to be saturated and
their amplitudes increase with increasing altitude. One example of the
spectral models proposed to describe energy density, *E*, assumes its
separability as the product of three functions *A*, *B* and *C*, depending
respectively on the vertical wave number, the intrinsic frequency, *ω*,
and the azimuthal direction of propagation, Φ (Fritts and VanZandt,
1993) as follows:

$$\begin{array}{ll}{\displaystyle}\mathrm{Ep}\left({k}_{Z},\mathit{\omega},\mathrm{\Phi}\right)& {\displaystyle}=A\left({k}_{Z}\right)B\left(\mathit{\omega}\right)C\left(\mathrm{\Phi}\right)\\ \text{(5)}& {\displaystyle}& {\displaystyle}={A}_{\mathrm{0}}{\displaystyle \frac{\mathrm{1}}{\frac{{k}_{Z}^{c}}{{k}_{Z}}+{\left(\frac{{k}_{Z}}{{k}_{Z}^{c}}\right)}^{\mathrm{3}}}}B\left(\mathit{\omega}\right)C\left(\mathrm{\Phi}\right).\end{array}$$

In the above form, *A*(*k*_{Z}) takes into account the requirement of
a positive slope (to get a finite vertical energy flux) at small wavenumbers
and the proposed ${k}_{Z}^{-\mathrm{3}}$ dependence at large wavenumber values. This
“universal model” has been the subject of several objections and variations
in the last three decades (see e.g. Fritts and Alexander, 2003). Note that
a given Ep distribution like Eq. (5) is obtained based on an
experimental setup (for example, the parameters may be derived after an
analysis of COSMIC GPS RO *T* data). Consistently, *k*_{Z} as well as
${k}_{Z}^{C}$ should then be considered apparent values, estimated after
a spectral analysis (e.g. Tsuda et al., 2011). For vertical (i.e. lidar)
soundings, apparent and real parameters are indistinguishable. Following this
argument, expressed as follows:

$$\begin{array}{}\text{(6)}& {\displaystyle}{\displaystyle}{{k}_{Z}^{\text{ap}}}_{(\mathrm{1},\mathrm{2})}={k}_{Z(\mathrm{1},\mathrm{2})}{\left[\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)\right]}^{-\mathrm{1}},\end{array}$$

we consider Eq. (5) with ${k}_{Z}^{\text{ap}}$ instead of *k*_{Z} and
${k}_{Z}^{C\text{, ap}}$ instead of ${k}_{Z}^{C}$ to quantitatively illustrate
the distortion in Ep and (below) in MF, derived from the misinterpretation
between real and apparent parameters. In doing so, the GW energy contained in
a given vertical wavenumber interval $\mathrm{\Delta}{k}_{z}^{\text{ap}}$ is as follows:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{Ep}}_{\mathrm{\Delta}{k}_{Z}}^{\text{ap}}={A}_{\mathrm{0}}B\left(\mathit{\omega}\right)C\left(\mathrm{\Phi}\right)\underset{{k}_{Z\mathrm{1}}^{\text{ap}}}{\overset{{k}_{Z\mathrm{2}}^{\text{ap}}}{\int}}{\displaystyle \frac{\mathrm{1}}{\frac{{k}_{Z}^{c,\text{ap}}}{{k}_{Z}^{\text{ap}}}+{\left(\frac{{k}_{Z}^{\text{ap}}}{{k}_{Z}^{c,\text{ap}}}\right)}^{\mathrm{3}}}}\mathrm{d}{k}_{Z}^{\text{ap}}\\ \text{(7)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}={A}_{\mathrm{0}}B\left(\mathit{\omega}\right)C\left(\mathrm{\Phi}\right)\left[{\displaystyle \frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{2}}^{{\text{ap}}^{\mathrm{2}}}}{{k}_{Z}^{c,{\text{ap}}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{c,{\text{ap}}^{-\mathrm{1}}}}}-{\displaystyle \frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{1}}^{{\text{ap}}^{\mathrm{2}}}}{{k}_{Z}^{c,{\text{ap}}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{c,{\text{ap}}^{-\mathrm{1}}}}}\right].\end{array}$$

Let us assume that from a given slanted
sounding, after extracting the GW perturbations with a wavelet or bandpass
filtering analysis, a clearly dominant quasi-monochromatic wave packet,
encompassed by two apparent wavenumber bounds, ${k}_{Z\mathrm{1}}^{\text{ap}}$ and
${k}_{Z\mathrm{2}}^{\text{ap}}$, is identified. We may calculate the wave energy
associated to this wave packet, directly from Eq. (6). The relative
error in Ep may be estimated after replacing apparent by real wavenumbers in
Eq. (6). To simplify the argument, we assume in Eq. (5) that
${k}_{Z\mathrm{1}}^{\text{C,ap}}$ and ${k}_{Z\mathrm{2}}^{\text{C,ap}}$ are close enough to assume
a parametric dependence with constant *α* and *ψ* values. The
relative error in Ep takes the following form:

$$\begin{array}{ll}{\displaystyle \frac{\mathrm{\Delta}\mathrm{Ep}}{\mathrm{Ep}}}& {\displaystyle}={\displaystyle \frac{\left[\begin{array}{c}\u2308\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{2}}^{{\text{ap}}^{\mathrm{2}}}}{{k}_{Z}^{c,{\text{ap}}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{c,{\text{ap}}^{-\mathrm{1}}}}-\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{1}}^{{\text{ap}}^{\mathrm{2}}}}{{k}_{Z}^{c,{\text{ap}}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{c,{\text{ap}}^{-\mathrm{1}}}}\u2309\\ -\left[\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{2}}^{\mathrm{2}}}{{k}_{Z}^{{c}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{{c}^{-\mathrm{1}}}}-\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{1}}^{\mathrm{2}}}{{k}_{Z}^{{c}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{{c}^{-\mathrm{1}}}}\right]\end{array}\right]}{\left[\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{2}}^{\mathrm{2}}}{{k}_{Z}^{{c}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{{c}^{-\mathrm{1}}}}-\frac{{\mathrm{tan}}^{-\mathrm{1}}\left(\frac{{k}_{Z\mathrm{1}}^{\mathrm{2}}}{{k}_{Z}^{{c}^{\mathrm{2}}}}\right)}{\mathrm{2}{k}_{Z}^{{c}^{-\mathrm{1}}}}\right]}}\\ \text{(8)}& {\displaystyle}& {\displaystyle}={\left[\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)\right]}^{-\mathrm{1}}.\end{array}$$

That is to say, under the above assumptions the relative error in Ep does not
depend on vertical wavenumbers or parameters other than simply *α* and
*ψ*.

The MF for internal GWs may be calculated under certain hypotheses based on
the existence of a dominant mode, characterized by *λ*_{Z} and
*λ*_{H} within a given intrinsic frequency range, applying the
following equation (for its detailed derivation and discussion refer to
Appendix A of Ern et al., 2004):

$$\begin{array}{}\text{(9)}& {\displaystyle}\mathrm{MF}={\displaystyle \frac{\mathit{\rho}}{\mathrm{2}}}{\displaystyle \frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}}{\left({\displaystyle \frac{g}{N}}\right)}^{\mathrm{2}}\stackrel{\mathrm{\u203e}}{{\left({\displaystyle \frac{\widehat{T}}{{T}_{\mathrm{0}}}}\right)}^{\mathrm{2}}}=\mathit{\rho}{\displaystyle \frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}}\mathrm{Ep},\end{array}$$

where *ρ* is the background density. Note that in this derivation, the
dominant mode with *λ*_{Z} and *λ*_{H} dominates within the
narrow wavenumber interval mentioned above in the discussion of the spectral
distribution of Ep. A first order estimation of the MF relative error may be
derived, by propagating up to the first order the relative errors in Ep and
(*λ*_{Z}∕*λ*_{H}). The relative error in MF will simply result in
the sum of those relative errors:

$$\begin{array}{ll}{\displaystyle \frac{\mathrm{\Delta}\left(\mathrm{MF}\right)}{\mathrm{MF}}}& {\displaystyle}=\left|{\displaystyle \frac{\mathrm{\Delta}\left(\frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}\right)}{\frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}}}\right|+\left|{\displaystyle \frac{\mathrm{\Delta}{E}_{p}}{{E}_{p}}}\right|\\ {\displaystyle}& {\displaystyle}=\left|{\displaystyle \frac{{\left(\frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}\right)}^{\text{ap}}-\left(\frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}\right)}{\left(\frac{{\mathit{\lambda}}_{Z}}{{\mathit{\lambda}}_{H}}\right)}}\right|+\left|{\displaystyle \frac{\mathrm{\Delta}{E}_{p}}{{E}_{p}}}\right|\\ {\displaystyle}& {\displaystyle}=\left|{\displaystyle \frac{\mathrm{tan}\left(\mathit{\alpha}\right)-\mathrm{cot}\left(\mathit{\psi}\right)}{\mathrm{cot}\left(\mathit{\psi}\right)}}\right|\\ \text{(10)}& {\displaystyle}& {\displaystyle}+{\left[\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)\right]}^{-\mathrm{1}},\end{array}$$

remembering that ${\mathit{\lambda}}_{Z}^{\text{ap}}/{\mathit{\lambda}}_{H}^{\text{ap}}=\mathrm{tan}\left(\mathit{\alpha}\right)$ and ${\mathit{\lambda}}_{Z}/{\mathit{\lambda}}_{H}=\mathrm{cot}\left(\mathit{\psi}\right)$. Note that, under the above assumptions, the MF relative error does not depend on the wavenumber bounds nor on the wavenumber width of the GW packet considered. Note that an erroneous replacement in Eq. (9) of apparent instead of real wavelengths, would lead to the conclusion that the MF would depend on the geometry of the sounding path.

To provide a measure of the distortion in MF from data retrieved during
a specific slanted case study, let us consider a GPS RO slanted sounding
close to Andes mountains analyzed in detail by Hierro et al. (2017, H17). In
that case study, from a collocation database between RO and cloud data and
from weather research and forecasting (WRF) mesoscale model simulations, real
and apparent vertical wavelengths during COSMIC RO soundings were identified.
From the model, coherent bi-dimensional GW structures with constant phase
surfaces oriented from SW to NE were noted. From the orographic
quasi-monochromatic structures detected below the cloud tops, averages of
*λ*_{Z}≈22.5 km and *λ*_{H}=20 km were
estimated, yielding the ratio ${\mathit{\lambda}}_{Z}/{\mathit{\lambda}}_{H}=\mathrm{1.12}$ with a wave
propagation angle $\mathit{\psi}={\mathrm{tan}}^{-\mathrm{1}}({\mathit{\lambda}}_{H}/{\mathit{\lambda}}_{Z})\approx \mathrm{0.73}\phantom{\rule{0.125em}{0ex}}\mathrm{rad}$. In this case study, the LOS stands at each TP almost
aligned to the GW phase surfaces observed, it is to say, at 190^{∘} from
north direction (dotted lines in Fig. 7 from H17). This particular geometry
between LOS and constant phase surfaces should allow to observe vertical
oscillations in the RO profile corresponding to short *λ*_{H}
structures, as described in A08. In Sect. 2 we mentioned that *α* may be
calculated from a rectilinear approximation of the LTP and cot (*α*) is
also equal to the ratio between ${\mathit{\lambda}}_{H}^{\text{ap}}$ and
${\mathit{\lambda}}_{Z}^{\text{ap}}$ in the region and altitude interval considered in
H17. From the average inclination of LTP, cot $\left(\mathit{\alpha}\right)={\mathit{\lambda}}_{H}^{\text{ap}}/{\mathit{\lambda}}_{Z}^{\text{ap}}\approx \mathrm{0.68}\phantom{\rule{0.125em}{0ex}}\mathrm{rad}$,
which considerably differs from the ratio between the corresponding real
wavelengths, ${\mathit{\lambda}}_{H}/{\mathit{\lambda}}_{Z}=\mathrm{0.89}$. From Eq. (9) the
proportionality of MF to the real wavelengths ratio indicates that when this
ratio is erroneously replaced by the apparent wavelengths ratio,
a significant error is in the general case, introduced.

As stated above, the estimation of the MF relative error for this particular Andes case study gives the following results:

$$\begin{array}{ll}{\displaystyle \frac{\mathrm{\Delta}\left(\mathrm{MF}\right)}{\mathrm{MF}}}& {\displaystyle}=\left|{\displaystyle \frac{\mathrm{tan}\left(\mathit{\alpha}\right)-\mathrm{cot}\left(\mathit{\psi}\right)}{\mathrm{cot}\left(\mathit{\psi}\right)}}\right|\\ {\displaystyle}& {\displaystyle}+{\left[\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)\right]}^{-\mathrm{1}}\\ \text{(11)}& {\displaystyle}& {\displaystyle}=\mathrm{0.31}+\mathrm{0.57}=\mathrm{0.88}.\end{array}$$

The error result should be observed as indicative, as the uncertainty
affecting the determination of the parameters *α* and *ψ* also affects this result.

Now we may wonder about the logically expected following point: would the
distortion previously described and clearly affecting a single case study be
able to affect the results and conclusions from any specific existing GW
global or local climatology? At first glance, given the slanted nature of
soundings upon which a given climatology is obtained and the anisotropic
nature of the dependence on *α* and *ψ*, we have no reason to assume
that the distortion expected on each sounding should be averaged out in the
climatology, notwithstanding the available density of soundings. To try to
answer this question, the option to accurately calculate each of the
distortions introduced respectively in each sounding is clearly not possible,
due to the unknown *ψ* parameter. Nevertheless, in an effort to address
this point, we resort to one of the idealized modeled distributions of GW available
in the literature (Alexander and Vincent, 2000). This is a linear model
describing one-dimensional GW propagation through a vertically varying
background atmosphere. It was used to clarify the relationship between GW
properties at stratospheric heights and the GW sources at the troposphere.
The authors aimed to test whether all of the observational results retrieved
from radiosonde profiles could be synthesized into a consistent physical
model of a spectrum of vertically propagating GW. In doing so, modeled
energy densities and MF were computed before they were compared with the
radiosonde results. The model uses the general dispersion relation for the
intrinsic and ground-based frequency, $\widehat{\mathit{\omega}}$ and *ω*
respectively, including a background zonal wind *u* and Coriolis acceleration
*f*, derived i.e. in Gill (1982) as follows:

$$\begin{array}{}\text{(12)}& {\displaystyle}{\widehat{\mathit{\omega}}}^{\mathrm{2}}={\left(\mathit{\omega}-{k}_{H}u\right)}^{\mathrm{2}}={\displaystyle \frac{{N}^{\mathrm{2}}{k}_{H}^{\mathrm{2}}+{f}^{\mathrm{2}}\left({k}_{Z}^{\mathrm{2}}+{\mathit{\mu}}^{\mathrm{2}}\right)}{{k}_{H}^{\mathrm{2}}+{k}_{Z}^{\mathrm{2}}+{\mathit{\mu}}^{\mathrm{2}}}},\end{array}$$

where *N* is the buoyancy frequency, $\mathit{\mu}=(\mathrm{2}H{)}^{-\mathrm{1}}$ and *H* is the density
scale height. The GW source is specified as a distribution of MF vs.
horizontal phase speed, $c=\mathit{\omega}/{k}_{H}$, for fixed *k*_{H} values. In this
model, the intrinsic frequency and vertical wavenumber vary with *u* and
stability, while *k*_{H} remains constant. The changes in $\widehat{\mathit{\omega}}$ with
*u*(*z*) refer to Doppler shifting and the changes in *k*_{Z} with *u*(*z*) are
referred to as refraction (see Alexander and Vincent, 2000 for details). From
the different GW sources proposed by these authors as spectra of MF vs. phase
speed located at fixed tropospheric heights, we illustratively consider
the following source function that is perfectly antisymmetric and isotropic:

$$\begin{array}{}\text{(13)}& {\displaystyle}{B}_{\mathrm{0}}\left(c\right)={B}_{m}\left({\displaystyle \frac{c-{u}_{\mathrm{0}}}{{c}_{w}}}\right)\mathrm{exp}\left(\mathrm{1}-\left|{\displaystyle \frac{c-{u}_{\mathrm{0}}}{{c}_{w}}}\right|\right),\end{array}$$

where *B*_{m} represents a spectral amplitude and *c*_{w} a source spectrum
width. Note that in the high-middle frequency approximation and when neglecting
*μ*, we may write the argument in Eq. (13) as follows:

$$\begin{array}{ll}{\displaystyle \frac{c-{u}_{\mathrm{0}}}{{c}_{w}}}& {\displaystyle}={\displaystyle \frac{{\widehat{\mathit{\omega}}}_{i}}{{k}_{H}{c}_{w}}}={\left[{\displaystyle \frac{{N}^{\mathrm{2}}{k}_{H}^{\mathrm{2}}}{{k}_{H}^{\mathrm{2}}+{k}_{Z}^{\mathrm{2}}}}\right]}^{\mathrm{0.5}}{\displaystyle \frac{\mathrm{1}}{{k}_{H}{c}_{w}}}\\ \text{(14)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{N\mathrm{|}\mathrm{cos}\left(\mathit{\psi}\right)\mathrm{|}}{{k}_{H}{c}_{w}}}.\end{array}$$

We now analyze the explicit inclusion of the previous distortion *D*
parameter in the scope of this model. As stated, we assume only GW within the
high or middle intrinsic frequency regime, neglecting *f* and *μ*. The
fitting of MF from modeled results (MF^{mod}) to measured
radiosonde data (MF^{mea}) at a fixed location and for constant
*k*_{H}, involves a comparison between MF profiles which are, in essence,
functions of real and apparent data, respectively. Then it looks reasonable
to fit modeled to measured data after applying the corresponding
transform to the modeled source spectrum. In doing so, we replace cos *ψ*
in Eq. (14) following Eq. (5):

$$\begin{array}{ll}{\displaystyle}D=& {\displaystyle \frac{{k}_{Z}}{{k}_{Z}^{\text{ap}}}}=\text{abs}\left(\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\right)\\ \text{(15)}& {\displaystyle}& {\displaystyle}=\left\{\begin{array}{ll}\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right),& \text{if}\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\mathrm{0}\\ -\mathrm{1}-\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right),& \text{if}\mathrm{1}+\mathrm{cot}\left(\mathit{\alpha}\right)\mathrm{cot}\left(\mathit{\psi}\right)\mathrm{0}\end{array}\right.,\end{array}$$

and in the first case,

$$\begin{array}{}\text{(16)}& {\displaystyle}& {\displaystyle}\mathit{\psi}={\mathrm{cot}}^{-\mathrm{1}}{\displaystyle \frac{D-\mathrm{1}}{\mathrm{cot}\mathit{\alpha}}}\text{(17)}& {\displaystyle}& {\displaystyle}\mathrm{cos}\mathit{\psi}=\mathrm{cos}{\mathrm{cot}}^{-\mathrm{1}}\left({\displaystyle \frac{D-\mathrm{1}}{\mathrm{cot}\mathit{\alpha}}}\right)={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{\left(\frac{\mathrm{cot}\mathit{\alpha}}{D-\mathrm{1}}\right)}^{\mathrm{2}}}}},\end{array}$$

after applying a trigonometric identity and for over- or under estimation of
*k*_{Z}, when *D* is different from one. Eq. (13) as a function
of *D*, for constant *B*_{m}, *N*, *k*_{H}, *c*_{w} and viewing path *α*
is as follows (i.e. *α* is expectedly constant during any radiosounding with
uniform and constant background wind):

$$\begin{array}{ll}{\displaystyle}{B}_{\mathrm{0}}\left(D\right)=& {\displaystyle}{B}_{m}\left({\displaystyle \frac{N}{{k}_{H}{c}_{w}}}{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{\left(\frac{\mathrm{cot}\mathit{\alpha}}{D-\mathrm{1}}\right)}^{\mathrm{2}}}}}\right)\\ \text{(18)}& {\displaystyle}& {\displaystyle}\times \mathrm{exp}\left(\mathrm{1}-{\displaystyle \frac{N}{{k}_{H}{c}_{w}}}{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{\left(\frac{\mathrm{cot}\mathit{\alpha}}{D-\mathrm{1}}\right)}^{\mathrm{2}}}}}\right).\end{array}$$

Finally, under the second case of Eq. (15), *D*−1 is to be replaced by
$-D-\mathrm{1}$. Following this reasoning, we may expect that this or any other source
function, expressed from the onset in terms of measured data that undergo
distortions due to the slanted nature of the soundings, will provide for
the optimum value of *D*≠1, the best fit to a given experimental
MF^{mea} profile. This may provide a quantitative
estimation of the distortion to be expected in a climatology at a fixed
geographic point. To resume the idea, what really matters in any quantitative
estimation of the distortion introduced by the slanting nature of atmospheric
soundings (radiosoundings, radio occultation profiles, etc) is to
consistently compare real (apparent) modeled data with real (apparent)
measured data.

4 Distortion of vertical wavelengths for specific setups

Back to toptop
To illustrate the considerations from Sects. 2 and 3, let us consider the *T*
retrievals obtained from (1) RO events detected from different LEO-GPS
satellites and from (2) SABER/TIMED measurements. A GPS-LEO RO occurs
whenever a transmitting satellite from the global navigation network at an
altitude about 20 000 km rises or sets from the standpoint of a LEO
receiving satellite at a height of about 800 km and the signal goes
across the atmospheric limb. The doppler frequency alteration produced
through refraction of the ray by the Earth's atmosphere in the trajectory
between the transmitter and the receiver is detected, and then may be
converted into slant profiles of diverse variables in the neutral atmosphere
and the ionosphere. GPS-LEO RO observations, available since 2001, have been
broadly used to study global distributions of GW energy and momentum, mainly
in the troposphere and the stratosphere (e.g. de la Torre et al., 2006;
Alexander et al., 2010; Geller et al., 2013; Schmidt et al., 2016). The RO
technique is a global limb sounding technique, sensitive under all weather
conditions to GW with small ratios of vertical to horizontal wavelengths (Wu
et al., 2006; Alexander et al., 2016). The SABER–TIMED limb measurements
provide continuous global *T* data for the latitude range
50^{∘} N–50^{∘} S, from the lower stratosphere to the lower
thermosphere and represent an unprecedented opportunity for studying in
detail the atmospheric waves, in particular GW, as well as their role in
lower and upper atmosphere coupling (e.g. Pancheva and Mukhtarov, 2011). The
TIMED satellite provides observations beginning in January 2002. It measures
CO_{2} infrared limb radiance from approximately 20 to 120 km
altitude. Kinetic temperature profiles are retrieved over these heights using
local thermodynamic equilibrium (LTE) radiative transfer in the stratosphere
and lowest part of the mesosphere (up to 60 km) and a full non-LTE
inversion in the mesosphere and lower thermosphere (i.e. Mertens et al.,
2004; Pancheva and Mukhtarov, 2011).

In Fig. 4a and b, LTPs corresponding to both setups are illustratively shown,
for the higher tropospheric and lower stratospheric regions bounded by
31–37^{∘} S and 66–72^{∘} W, close to central southern Andes
mountains, observed during January–February 2009.

Keeping in mind that the observed difference between horizontal and vertical scales in
these figures is that there is a typical distribution of the sounding path direction
(*α*) among GPS-RO occultation events and among SABER measurements. The large number of available RO as compared to SABER profiles is
evident but no significant variation with latitude was detected. The
approximation of the sounding paths by straight segments seems, at least for
our purpose here, quite reasonable. Let us now consider the global data
retrieved from both setups during January–February 2009 (RO from LEOs:
SAC-C, CHAMP, MetOp-A, and COSMIC), of which Fig. 4 only represents
a regional subset. In Fig. 5a and b the *α* distribution is shown. Here
a linear interpolation was applied to the weakly variable *α* angle in
each RO event, between the lowest and upper available LTP values. Note the
considerably narrower variability *α*-range among SABER profiles. We did
not observe remarkable differences in the general latitudinal or
geographical distribution. The possible ranges observed from both
experimental setups allow some preliminary consequences to be drawn regarding
the expected wavelength distortions. For example, for the subset in Fig. 4,
we know that very close to the Andes mountains region dominant
large-amplitude, stationary and non hydrostatic GWs are usually observed (de
la Torre et al., 1996, 2005, 2015).

Accordingly, large GW aspect ratios may be expected there (Gill, 1982). On
the other hand, at tropical latitudes, where convective GWs dominate the
scenario, or even close to polar jet regions where hydrostatic rotating or
non rotating GWs are usually found, considerably lower characteristic aspect
ratios should be dominant. In Fig. 6, we reproduce the *D*–*α* curves
selected in Fig. 2a and b, for successive *ψ* values (Δ*ψ* step =0.2), now adding in dash-dotted green and yellow squares,
the *D*–*α* ranges affected for both experimental setups. These ranges
are, respectively, [0.17–1.22] rad for GPS-RO and [0.32–0.34] rad for
SABER. For each setup, the relevant difference mainly depends on whether
*α* and *ψ* belong to the same or different $[\mathrm{0},\mathit{\pi}/\mathrm{2}]$ and $[\mathit{\pi}/\mathrm{2},\mathit{\pi}]$ intervals.

Here, we may here observe that depending on GW aspect ratio and sounding
direction, general under- and overestimations of *λ*_{Z} are both
possible throughout both experimental setups. Within a given ensemble, the
behavior of *D* is different for *ψ* lower and greater than *π*∕2. This
suggests that different modes in the ensemble may show individual distortions
less than or greater than 1. Then, some compensations contributing to Ep and
MF are expected from different modes in the ensemble, but the net distortion
should still be considerable. In Fig. 7, the *D*–*ψ* constraint imposed
to GPS-RO observations, now for constant and progressive *α* values, is
shown. Δ*α* steps of 0.02 rad and within the corresponding
bounds [0.17–1.22] rad indicated in Fig. 5, are shown. The white, light grey
and grey sectors approximately indicate the non-hydrostatic, hydrostatic
non-rotating and hydrostatic rotating GW regimes, respectively. We observe
general underestimations for *ψ* less than *π*∕2 and in the vicinity of
*π* rad. Between these sectors, under and overestimations are possible. To
illustrate the consequences on a realistic and simple scenario, let us
consider again the region situated to the east of the central Andes,
mentioned in Figs. 4a and b. Let us suppose that, consistently with
observations and numerical simulations (i.e. de la Torre et al., 2012; Jiang
et al., 2013; Fritts et al., 2016), constant and stationary GW phase surfaces
exhibit a systematic inclination with respect to the ground and a high aspect
ratio, following the almost omnipresent forcing by mean westerlies at the
mountain tops. This feature is represented in Fig. 7 by the black arrow.

This arrow spans over all possible *α* directions within the bounds
imposed by the geometry of every GPS-LEO satellites combination during each
occultation event. This assumed scenario would reveal a net underestimation
of *λ*_{Z}, regardless of the inclination of LTPs during the sounding
of the region and the considered period. In general the analysis is expected
to be more complex, given distinct LTP contributions that may under- or
overestimate *λ*_{Z}. Finally, Fig. 8 indicates the corresponding
*D*–*ψ* features for SABER measurements, similarly as in Fig. 7.

Here we observe general underestimations for *ψ*, along the 3 GW
regimes, for values less than *π*∕2 and greater than around
2.3 rad. For intermediate values, only overestimations are expected.
Note that for SABER measurements, the forbidden *D*−*ψ* region is
considerably more extended than for GPS-LEO RO measurements.

5 Discussion and conclusions

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The expected distortions observed in the measured vertical wavelengths
during any near instantaneous slanted atmospheric sounding, as may be the
case for satellite instruments, is discussed. For the particular case of
vertical or horizontal soundings, we know that no distortion is expected in
*λ*_{Z} and *λ*_{H}, respectively. The features observed are
described as a function of GW aspect ratio and the inclination of the
sounding path.

To gain a better understanding of this distortion, and making use of the
symmetric *D* dependence with *α* and *ψ*, we consider the expression
for *D* as a parametric equation in both independent variables. To illustrate
the constraints imposed to both parameters by applying different instrumental
setups and GW scenarios, we show the results conveniently in *D*–*α*
and *D*–*ψ* plots. Above and below the non-distortion limit (*D*=1),
general under and overestimations occur depending on the relative parametric
values. The main difference is produced by two possible situations: *α*
and *ψ* belonging to the same or different quadrants, taken from $[\mathrm{0},\mathit{\pi}/\mathrm{2}]$ and $[\mathit{\pi}/\mathrm{2},\mathit{\pi}]$. Given a GW ensemble and a number of measurements
within arbitrary bounds of space and time intervals, distinct wavelength under-
and overestimations should be expected.

When Ep is calculated over a GW ensemble in any individual *T* profile, an
integral must be performed over the largest wavelength along any chosen
direction. The selection of the upper and lower vertical wavelength bounds,
should include those prevailing GW amplitudes expected to mostly contribute
to Ep. Depending on *α* and the respective *ψ* values for each one of
the dominant GW modes, some dominant real wavelengths may not be fully
contained within the integration interval. The integral in Ep then will not
include at least one wavelength of every dominant mode. The Ep calculation
could be under- or overestimated up to a significant extent.

We illustrate these arguments in a realistic scenario considering a modeled distribution of GW. This is based on the usual saturation of large vertical wavenumbers and in the separability of the spectral function in the vertical wave number, the intrinsic frequency and the azimuthal direction of propagation. To calculate the wave energy associated to a given GW packet within an ensemble, we use a simple analytical result derived from the spectral model to get an idea of the distortion expected by wrongly replacing the integration limits by apparent instead of real wavenumber values. This (or any) distortion in Ep will in turn be translated to the MF, by applying a previous result obtained by Ern et al. (2004). In addition, through a multiplying factor, the MF would be then illogically dependent on the inclination angle of the sounding path.

The results are considered for two specific experimental setups: GPS-RO and
SABER measurements. For our analysis we approximate the sounding paths in
both cases by using straight segments. The relevance of this assumption was
assessed. A clearly larger number of available *T* profiles is seen from RO
events. The *α* ranges in both techniques allow the definition of forbidden
regions in *D*–*α* as well as in *D*–*ψ* diagrams, relative to the
different GW aspect ratios (the non-hydrostatic, hydrostatic non-rotating and
hydrostatic rotating regimes). Within a given GW ensemble, even expecting
some compensation when *D* is less than and greater than 1, the net
distortion effect, as well as its contribution to Ep and MF, should be
considerable. With the exception of GWs with prevailing high aspect ratio, as
for example near the Andes mountains where a net underestimation of
*λ*_{Z} should be observed, under- and overestimations are in general
expected, from both setups respectively. This occurs for *T* profiles where
*α* and *ψ* belong to the same or different quadrants $[\mathrm{0},\mathit{\pi}/\mathrm{2}]$
and $[\mathit{\pi}/\mathrm{2},\mathit{\pi}]$. For SABER measurements, the forbidden *D*–*ψ* region
is considerably more extended than the one corresponding to the GPS-RO
measurements.

In the global study of Geller et al. (2013), which compares models with
diverse parameterizations with satellite and balloon data, the faster fall
off in relation to the height of the gravity wave MF derived from satellite measurements
than in the models considered in that study was the most significant
discrepancy between measured and model fluxes. These authors concluded that
the reasons for those differences remain unknown, although various
explanations for the differences were proposed. As we know from model
simulations, the MF is not computed from Eq. (8), but from its formal
definition based on the average of the products of the three perturbed
components of the air velocity. Based on the above considerations and
regarding the dramatic distortions on vertical and horizontal wavelengths
during slanted soundings, we may infer that if MF is computed from Eq. (8),
the wavelength distortion will unavoidably be translated to the calculation
of MF. Obviously, this situation must be considered together with the
additional constraints imposed to any satellite-borne observational window,
discussed by several authors, including A08. Finally, we must admit
that the global calculation of MF from slanted *T* profiles, including all
necessary corrections, even assuming quasi-monochromatic GW packets, appears
to be a very complex task. The distortions described above are only avoided
in the calculation of MF if the atmosphere is sounded in vertical or
horizontal directions, as provided (but only locally) by lidar or radar and
balloon setups, respectively. Up to now, from the satellite data at disposal,
an attempt to quantitatively illustrate the implications and possible
misrepresentation (or distortion) of our general understanding of GW
parameters values from slanted soundings, as their global distribution and
variability, seems unrealistic. After some research to improve this simulation,
we are now working on previous GW parameter solution-schemes which were
modified for the use of close sounding-groups of RO profiles. The method is
currently being applied to calculate GW propagation direction, net MF and
real vertical and horizontal wavelength for some case studies. The
unavoidable constraint imposed to extend preliminary results to a future GW
climatological useful description is strictly conditioned by the still
largely insufficient density of satellite-based soundings.

Data availability

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Data availability.

GPS RO and TIMED/SABER data were downloaded respectively from http://cdaac-www.cosmic.ucar.edu/cdaac/products.html and http://saber.gats-inc.com/browse_data.php.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Observing Atmosphere and Climate with Occultation Techniques – Results from the OPAC-IROWG 2016 Workshop”. It is a result of the International Workshop on Occultations for Probing Atmosphere and Climate, Leibnitz, Austria, 8–14 September 2016.

Acknowledgements

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Acknowledgements.

The study has been supported by the CONICET under grants CONICET PIP
11220120100034 and ANPCYT PICT 2013-1097 and by the German Federal Ministry
of Education and Research (BMBF) under grant
01DN14001.

Edited by: Ulrich Foelsche

Reviewed by: two anonymous referees

References

Back to toptop
Alexander, M. J.: Global and seasonal variations in threedimensional gravity wave momentum flux from satellite limb-sounding temperatures, Geophys. Res. Lett., 42, 6860–6867, https://doi.org/10.1002/2015GL065234, 2015.

Alexander, M. J. and Vincent, R. A.: Gravity waves in the tropical lower stratosphere: a model study of seasonal and interannual variability, J. Geophys. Res.-Atmos., 105, 17983–17993, https://doi.org/10.1029/2000JD900197, 2000.

Alexander, M. J., Geller, M., McLandress, C., Polavarapu, S., Preusse, P., Sassi, F., Sato, K., Eckermann, S., Ern, M., Herzog, A., Kawatani, Y., Pulido, M., Shaw, T. A., Sigmond, M., Vincent, R., and Watanabe, S.: Recent developments in gavity-wave effects in climate models and the global distribution of gravity wave momentum flux from observations and models, Q. J. Roy. Meteor. Soc., 136, 1103–1124, https://doi.org/10.1002/qj.637, 2010.

Alexander, P., de la Torre, A., and Llamedo, P.: The interpretation of gravity waves signatures extracted from GPS radio occultations, J. Geophys. Res.-Atmos., 113, D16117, https://doi.org/10.1029/2007JD009390, 2008.

Alexander, P., de la Torre, A., Hierro, R., and Llamedo, P.: An improvement of the sensitivity of GPS radio occultation data to detect gravity waves through observational and modeling factors, Adv. Space Res., 57, 543–55, 2016.

Alexander, S. P., Klekociuk, A. R., and Murphy, D. J.: Rayleigh lidar observations of gravity wave activity in the winter
upper stratosphere and lower mesosphere above Davis, Antarctica (69^{∘} S, 78^{∘} E), J. Geophys. Res.-Atmos., 116,
D13109, https://doi.org/10.1029/2010JD015164, 2011.

de la Torre, A. and Alexander, P.: The interpretation of wavelengths and periods as measured from atmospheric balloons, J. Appl. Meteorol., 3, 2747–2754, 1995.

de la Torre, A. and Alexander, P.: Gravity waves above Andes detected from GPS Radio Occultation Temperature Profiles: mountain forcing?, Geophys. Res. Lett., 32, L17815, https://doi.org/10.1029/2005GL022959, 2005.

de la Torre, A., Teitelbaum, H., and Vial, F.: Stratospheric and tropospheric gravity wave measurements near the Andes Mountains, J. Atmos. Terr. Phys., 58, 521–530, 1996.

de la Torre, A., Alexander, P., Llamedo, P., Menéndez, C., Schmidt, T., and Wickert, J.: Gravity waves above Andes detected from GPS Radio Occultation Temperature Profiles. II: Jet mechanism?, Geophys. Res. Lett., 33, L24810, https://doi.org/10.1029/2006GL027343, 2006.

de la Torre, A., Alexander, P., Hierro, R., Llamedo, P., Rolla, A., Schmidt, T., and Wickert, J.: Large amplitude gravity waves above the Southern Andes, the Drake Passage and the Antarctic Peninsula, J. Geophys. Res.-Atmos., 117, D02106, https://doi.org/10.1029/2011JD016377, 2012.

de la Torre, A., Pessano, P., Hierro, R., Santos, J. R., Llamedo, P., and Alexander, P.: The influence of topography on vertical air velocity of air in relation to severe storms near the Southern Andes Mountains, Atmos. Res., 156, 91–101, https://doi.org/10.1016/j.atmosres.2014.12.020, 2015.

Dewan, E. M. and Good, R. E.: Saturation and the “universal” spectrum for vertical profiles of horizontal scalar winds in the atmosphere, J. Geophys. Res., 91, 2742–2748, 1986.

de Wit, R. J., Janches, D., Fritts, D. C., Stockwell, R. G., and Coy, L.: Unexpected climatological behavior of MLT gravity wave momentum flux in the lee of the Southern Andes hot spot, Geophys. Res. Lett., 44, 1182–1191, https://doi.org/10.1002/2016GL072311, 2017.

Ern, M., Preusse, P., Alexander, M. J., and Warner, C. D.: Absolute values of gravity wave momentum flux derived from satellite data, J. Geophys. Res.-Atmos., 109, D20103, https://doi.org/10.1029/2004JD004752, 2004.

Faber, A., Llamedo, P., Schmidt, T., de la Torre, A., and Wickert, J.: On the determination of gravity wave momentum flux from GPS radio occultation data, Atmos. Meas. Tech., 6, 3169–3180, https://doi.org/10.5194/amt-6-3169-2013, 2013.

Fritts, D. C. and Alexander, M. J.: Gravity wave dynamics and effects in the middle atmosphere, Rev. Geophys., 41, 1003, https://doi.org/10.1029/2001RG000106, 2003.

Fritts, D. C. and VanZandt, T. E.: Spectral estimates of gravity wave energy and momentum fluxes. Part I: Energy dissipation, acceleration, and constraints, J. Atmos. Sci., 50, 3685–3694, 1993.

Fritts, D. C., Smith, R. B., Taylor, M. J., Doyle, J. D., Eckermann, S. D., Dörnbrack, A., Rapp, M., Williams, B. P., Pautet, P., Bossert, K., Criddle, N. R., Reynolds, C. A., Reinecke, P. A., Uddstrom, M., Revell, M. J., Turner, R., Kaifler, B., Wagner, J. S., Mixa, T., Kruse, C. G., Nugent, A. D., Watson, C. D., Gisinger, S., Smith, S. M., Lieberman, R. S., Laughman, B., Moore, J. J., Brown, W. O., Haggerty, J. A., Rockwell, A., Stossmeister, G. J., Williams, S. F., Hernandez, G., Murphy, D. J., Klekociuk, A. R., Reid, I. M., and Ma, J.: The Deep Propagating Gravity Wave Experiment (DEEPWAVE): an airborne and ground-based exploration of gravity wave propagation and effects from their sources throughout the lower and middle atmosphere, B. Am. Meteorol. Soc., 97, 425–453, https://doi.org/10.1175/BAMS-D-14-00269.1, 2016.

Geller, M. A., Alexander, M. J., Love, P. T., Bacmeister, J., Ern, M., Hertzog, A., Manzini, E., Preusse, P., Sato, K., Scaife, A. A., and Zhou, T.: A comparison between gravity wave momentum fluxes in observations and climate models, J. Climate, 26, 6383–6405, https://doi.org/10.1175/JCLI-D-12-00545.1, 2013.

Gill, A. E.: Atmosphere and Ocean Dynamics, Academic Press, New York, 642 pp., 1982.

Hertzog, A., Alexander, M. J., and Plougonven, R.: On the intermittency of gravity wave momentum flux in the stratosphere, J. Atmos. Sci., 69, 3433–3448, https://doi.org/10.1175/JAS-D-12-09.1, 2012.

Hierro, R., Steiner, A. K., de la Torre, A., Alexander, P., Llamedo, P., and Cremades, P.: Orographic and convective gravity waves above the Alps and Andes mountains during GPS radio occultation events – a case study, Atmos. Meas. Tech. Discuss., https://doi.org/10.5194/amt-2017-245, in review, 2017.

Hines, C. O.: The saturation of gravity waves in the middle atmosphere. Part I: Critique of linear-instability theory, J. Atmos. Sci., 48, 1348–1359, 1991.

Jiang, Q., Doyle, J. D., Reinecke, A., Smith, R. B., and Eckermann, S. D.: A modeling study of stratospheric waves over the Southern Andes and Drake Passage, J. Atmos. Sci., 70, 1668–1689, https://doi.org/10.1175/JAS-D-12-0180.1, 2013.

John, S. R. and Kumar, K. K.: A discussion on the methods of extracting gravity wave perturbations from space-based measurements, Geophys. Res. Lett., 40, 2406–2410, https://doi.org/10.1002/grl.50451, 2013.

Kawatani, Y., Takahashi, M., Sato, K., Alexander, S. P., and Tsuda, T.: Global distribution of atmospheric waves in the equatorial upper troposphere and lower stratosphere: AGCM simulation of sources and propagation, J. Geophys. Res.-Atmos., 114, D01102, https://doi.org/10.1029/2008JD010374, 2009.

Kursinski, E. R., Hajj, G. A., Hardy, K. R., Schofield, J. T., and Linfield, R.: Observing Earth's atmosphere with radio occultation measurements using the Global Positioning System, J. Geophys. Res., 102, 23429–23465, 1997.

Lieberman, R. S., Akmaev, R. A., Fuller-Rowell, T. J., and Doornbos, E.: Thermospheric zonal mean winds and tides revealed by CHAMP, Geophys. Res. Lett., 40, 2439–2443, https://doi.org/10.1002/grl.50481, 2013.

McLandress, C. and Scinocca, J. F.: The GCM response to current parameterizations of nonorographic gravity wave drag, J. Atmos. Sci., 62, 2394–2413, 2005.

Mertens, C. J., Schmidlin, F. J., Goldberg, R. A., Remsberg, E. E., Pesnell, W. D., Russell III, J. M., Mlynczak, M. G., López-Puertas, M., Wintersteiner, P. P., Picard, R. H., Winick, J. R., and Gordley, L. L.: SABER observations of mesospheric temperature and comparisons with falling sphere measurements taken during the 2002 summer MaCWINE campaign, Geophys. Res. Lett., 31, J03105, https://doi.org/10.1029/2003GL018605, 2004.

Oliver, W. L., Zhang, S.-R., and Goncharenko, L. P.: Is thermospheric global cooling caused by gravity waves?, J. Geophys. Res.-Space, 118, 3898–3908, https://doi.org/10.1002/jgra.50370, 2013.

Pancheva, D. and Mukhtarov, P.: Atmospheric tides and planetary waves: recent progress based on SABER/TIMED temperature measurements (2002–2007), in: Aeronomy of the Earth's Atmosphere and Ionosphere, 19, IAGA Special Sopron Book Series 2, edited by: Abdu, M. A. and Pancheva, D., https://doi.org/10.1007/978-94-007-0326-1, Springer Dordrecht, 19–56, 2011.

Preusse, P., Dörnbrack, A., Eckermann, S. D., Riese, M., Schaeler, B., Bacmeister, J. T., Broutman, D., and Grossmann, K. U.: Space-based measurements of stratospheric mountain waves by CRISTA, 1. Sensitivity, analysis method, and a case study, J. Geophys. Res.-Atmos., 107, 8178, https://doi.org/10.1029/2001JD000699, 2002.

Russell III, J. M., Mlynczak, M. G., Gordley, L. L., Tansock, J., and Esplin, R.: An Overview of the SABER Experiment and Preliminary Calibration Results, Proceedings of the SPIE, 44th Anual Meeting, Denver, CO, 18–23 July, Vol. 3756, 277–288, 1999.

Schmidt, T., Alexander, P., and de la Torre, A.: Stratospheric gravity wave momentum flux from radio occultations, J. Geophys. Res.-Atmos., 121, 4443–4467, https://doi.org/10.1002/2015JD024135, 2016.

Shutts, G. J. and Vosper, S. B.: Stratospheric gravity waves revealed in NWP model forecasts, Q. J. Roy. Meteor. Soc., 137, 303–317, https://doi.org/10.1002/qj.763, 2011.

Smith, S. A., Fritts, D. C., and VanZandt, T. E.: Evidence for a saturated spectrum of atmospheric gravity waves, J. Atmos. Sci., 44, 1404–1410, 1987.

Tsuda, T., Nishida, M., Rocken, C., and Ware, R. H.: A global morphology of gravity wave activity in the stratosphere revealed by the GPS occultation data (GPS/MET), J. Geophys. Res., 105, 7257–7273, https://doi.org/10.1029/1999JD901005, 2000.

Tsuda, T., Lin, X., Hayashi, H., and Noersomadi: Analysis of vertical wave number spectrum of atmospheric gravity waves in the stratosphere using COSMIC GPS radio occultation data, Atmos. Meas. Tech., 4, 1627–1636, https://doi.org/10.5194/amt-4-1627-2011, 2011.

Wang, L. and Alexander, M. J.: Gobal estimates of gravity wave parameters from GPS radio occultation temperature data, J. Geophys. Res.-Atmos., 115, D21122, https://doi.org/10.1029/2010JD013860, 2010.

Wu, D. L. and Waters, J. W.: Satellite observations of atmospheric variances: a possible indication of gravity waves, Geophys. Res. Lett., 23, 3631–3634, 1996.

Wu, D. L., Preusse, P., Eckermann, S. D., Jiang, J. H., de la Torre Juarez, M., Coy, L., and Wang, D. Y.: Remote sounding of atmospheric gravity waves with satellite limb and nadir techniques, Adv. Space Res., 37, 2269–2277, 2006.

Yiğit, E., Knížováb, P., Georgieva, K., and Ward, W. E.: A review of vertical coupling in the Atmosphere–Ionosphere system: effects of waves, sudden stratospheric warmings, space weather, and of solar activity, J. Atmos. Sol-Terr. Phys., 141, 1–12, https://doi.org/10.1016/j.jastp.2016.02.011, 2017.

Short summary

This work is based on previous findings from our group of researchers, regarding the analysis of atmospheric data from slanted soundings during radiosoundings and GPS radio occulations. Several gravity wave climatologies may be found in the literature that to a certain extent are affected by considerable measurement distortions. We intend here to contribute to the interpretation of gravity waves analyses from slanted soundings in future climatologies.

This work is based on previous findings from our group of researchers, regarding the analysis of...

Atmospheric Measurement Techniques

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