## 1. Introduction

In an atmosphere at rest, the frequency spectrum of internal gravity waves is bounded above by the Brunt–Väisälä frequency and below by the inertial frequency given by the local value of the Coriolis parameter ^{−1}]. Inertial gravity waves with lower frequencies are evanescent and have very short attenuation lengths (Eckart 1960).

Atmospheric measurements in earth-fixed coordinate frames, for example, time series of stratospheric winds and electric fields, typically show a spectral peak near the inertial frequency (Thompson 1978; Sidi and Barat 1986; Hu and Holzworth 1997). In some cases, however, spectral peaks have been observed at frequencies that are about 3%–20% higher than the inertial frequency. The source of these frequency shifts is not yet fully understood; it has been suggested that Doppler and stratification effects contribute to the shifts (Mori 1990). It has also been suggested that rotational effects originating the vorticity of the background flow might contribute to higher inertial frequencies (Kunze 1985; Jones 2005; Lee and Eriksen 1997), and that these effects might be instrumental in trapping near inertial waves in vorticity minima (centers of anticyclonic motion) (Lee and Eriksen 1997). Plougonven and Zeitlin (2005) suggest a source of near inertial waves generated by the geostrophic adjustment process.

Shifts in the spectral peak have also been reported in the spectrum of horizontal velocity inferred from location time series of three superpressure balloons (SPBs) released during the Arctic Kiruna 2002 campaign (Broutman et al. 2004; Hertzog and Vial 2001). The spectral peak for two of the balloons in that campaign was slightly higher (5%–10%) than the inertial frequency (Hertzog et al. 2002). These shifts cannot be due to Doppler effects since measurements made on drifting balloons are recorded in a quasi-Lagrangian reference frame. The spectrum corresponding to a third balloon in the Arctic Kiruna 2002 campaign lacked the near-inertial spectral peak, a feature attributed to differences in the magnitude of meridional excursions, implying significant variations of the Coriolis parameter along that balloon’s trajectory.

In the present paper we examine the spectrum of zonal wind speeds derived from location time series of SPBs released in the lower stratosphere during the austral spring of 2005 by the Vorcore Antarctic campaign. One of Vorcore’s principal objectives was an improved understanding of the gravity wave field in the Antarctic polar vortex. The SPBs released by Vorcore drifted for several months at two isopycnic levels corresponding approximately to either 50 or 70 hPa. The inertial frequencies corresponding to the latitudes of the SPB locations range from 0.075 h^{−1} at 65°S to 0.083 h^{−1} at 90°S (based on ^{−1}, more than 25% higher than the largest value of the inertial frequency. We suggest that such a shift to higher frequencies in the spectral peak is consistent with the effects of the relative vorticity of the background flow on the inertial gravity wave field. The effects of the background relative vorticity cannot be ignored in the dispersion relation of the gravity wave field. For example, it has been shown that for some specified configurations of the background flow the lower bound of the inertial wave spectrum is shifted by a substantial fraction of the relative vorticity (Kunze 1985; Kunze and Boss 1998; Jones 2005). We study these effects for the background flows in which the SPBs drifted during the Vorcore campaign. Our principal finding is that the spectral peak near 0.10 h^{−1} can be interpreted as being due to inertial waves influenced by the relative vorticity of the background flow inside the Antarctic polar vortex. We show that features of the observed spectra are consistent with relative vorticity expressed as the sum of solid-body rotation and shear.

An outline of the paper is as follows. We begin with an overview of the effects of rotation on the gravity wave dispersion relation and the location of the inertial peak (effective inertial frequency). Next, we describe the datasets used, present the results of the data analysis, and compare the observed spectra with the calculated effective inertial frequency, which has contributions from both curvature and shear in the vortex. A discussion of the results and limitations of the analysis concludes the paper. A detailed analysis of how relative vorticity influences the effective inertial peak for waves propagating in a rotating and sheared background is given in the appendix.

## 2. Theory of inertial gravity waves in rotational background flow

*m*is the vertical wavenumber,

*k*is the horizontal wavenumber,

*ω*is wave frequency,

*N*

^{2}is the square of the Brunt–Väisälä frequency, and

*H*is the scale height. For

*m*

^{2}is less than zero and the wave is evanescent. The inertial-gravity wave spectrum is cut off for frequencies below

*f*.

In a background state not at rest, its effects on the gravity wave dispersion relation are considered in two steps. The first step includes translational effects on wave frequency by replacing *ω* with the intrinsic frequency *c* is the phase speed, *k* is the magnitude of the horizontal wavenumber vector **k**, and **k**. The frequency seen by a drifting balloon is approximately the intrinsic frequency (Hertzog and Vial 2001).

**z**is the vertical unit vector around the vertical axis (solid-body rotation). In a frame of reference rotating with the flow, and hence with

**u**is given bywhere

*d*

**u**/

*dt*is given bywhere

*u*,

*υ*, and

For *m ^{2}* is negative and the wave is evanescent. Thus the inertial-gravity wave spectrum measured by a balloon is cut off for intrinsic frequencies below the effective inertial frequency.

We show in the appendix that for a more general combination of shear and curvature *f*.

## 3. Data description

Beginning on 5 September 2005, the Vorcore campaign released 19 SPBs with 10-m diameters and eight balloons with 8.5-m diameters from McMurdo, Antarctica (77.5°S, 166.4°E); they drifted near 50 and 70 hPa, respectively. The mean flight duration of the 27 balloons was 59 days and the longest flight duration was 109 days (Hertzog et al. 2007).

Each SPB carried temperature and pressure sensors and a global positioning system (GPS) receiver. Balloon positions were recorded every 15 min with a position accuracy of 15 m so that wind speeds could be estimated with accuracy greater than 0.2 m s^{−1} (Hertzog et al. 2007). The geographical sampling of the Antarctic vortex core was very good with best sampling in the 60°–80°S latitude band and 60°W–120°E longitude sector since the vortex had a tendency to be centered off the pole toward South America (Hertzog et al. 2007).

Evaluation of general conditions in the polar vortex, required to understand and interpret balloon data, was done using the Modern Era Retrospective-Analyses for Research and Applications (MERRA) tool produced by National Aeronautics and Space Administration (NASA). MERRA products are produced at 3-h intervals with a spatial resolution of ½° latitude, ⅔° longitude, and 72 pressure levels to 0.01 hPa (approximately 80-km altitude). We compute the absolute vorticity using the MERRA horizontal winds to obtain vorticity maps for each 3-h interval, then interpolate to find the absolute vorticity at the time and location of each balloon measurement. An example vorticity map for 20 October 2005 derived from MERRA horizontal winds at 52 hPa is presented in the top panel of Fig. 1. The values of ^{−4} s^{−1}, a significant fraction of *f*(−1.4 × 10^{−4} s^{−1} at 70°S). An example showing the range of absolute vorticity (or effective inertial frequencies assuming pure solid-body rotation) inside the vortex is shown in the bottom panel of Fig. 1. Inside the vortex, the ^{−4} s^{−1}, approximately 25% higher than *π*). This convention is chosen to be consistent with standard practice when discussing vorticity while also allowing intuitive conversion between frequency and period for wave data.

## 4. Data analysis and results

### a. Spectral analysis

In this section we present a spectral analysis of zonal wind derived from SPB location data showing a frequency shift of the spectral peak. The SPB flights used in the analysis that follows spanned the October–November period, when, except for late November, the polar vortex remained well-defined and strong. In 2005, the vortex was very stable in September and October, moved off the pole in November, and broke up in early December (Hertzog et al. 2007).

Zonal wind velocities were derived from the SPB measurements as described in Hertzog et al. (2007). Fourier analysis of the zonal wind velocity over 2-week periods was performed individually for each balloon and then averaged. The results, presented in Fig. 2, show a distinct spectral peak at about 0.10 h^{−1}, well separated from the frequency of the semidiurnal tide (indicated by the green line), that persists from early October through November. The red hatched area shows the range of inertial frequencies for all balloons in the 2-week period; effective inertial frequencies for pure solid-body rotation *f*. The peak shifts toward the inertial frequency range in late December, which is consistent with the breakup of the vortex.

### b. Wavelet analysis

The intermittent nature of inertial waves suggests wavelet analysis as a means to explore the temporal behavior of the zonal wind spectra. The time series from each balloon was analyzed with Morlet wavelets in order to identify the spectral features as a function of time (Torrence and Compo 1998). The wavelet analysis confirmed that the dominant peak in the spectrum occurred near 0.1 h^{−1}, as indicated by the Fourier analysis discussed above. The peak wavelet power was generally found to lie between about 0.08–0.13 h^{−1}.

Figure 3 shows the results of a wavelet analysis for three sample balloon trajectories. Also shown are the values of the inertial frequency at the balloon location *f* and the local value of *f*_{eff} for the case of pure solid-body rotation). Local values of *f* (red curves), particularly in October and early November. Several instances of significant spectral peaks on the low frequency side of *f*_{eff} seen in balloon 2 during November, when a significant oscillation occurs near 20 h (the lowest plotted frequency), is probably a manifestation of the diurnal tide.

There are a number of instances in Fig. 3 for which a cutoff near *f* are presented in Fig. 4. The frequency of maximum wavelet power is determined for each balloon measurement, for example, the frequency of the peak wavelet power as a function of time for the wavelet spectra shown in Fig. 3. The results, binned in 0.05 s^{−1} intervals, are presented in Fig. 4, which shows the number of balloon measurements as a function of difference between measured peak frequency *f _{m}* and inertial frequency

*f*. Note that wave period, rather than frequency, is plotted since the wavelet algorithm used for the spectral analysis returns wavelet power as a function of period rather than frequency and is the natural way to bin the results. Statistics are presented for each 2-week period in October and November, since the wavelet analysis shown in Fig. 3 suggests that the spectral peak diverges from

The distribution of balloon measurements presented in Fig. 4 shows peaks near

## 5. Discussion

*x*axis is along the basic flow velocity vector and the

*y*axis is in the orthogonal direction consistent with a right-handed system the vorticity is written (Holton 1972)where

*x*axis and the tangent to the streamline as a function of the distance

*s*along the streamline. In terms of the local radius of curvature

*r*where

*x*direction.

In the appendix we show that for pure shear with no curvature, wave motion normal to the direction of shear gives *f*.

An example of the relative contribution from solid-body rotation and shear excess is presented for 20 October 2005 in Fig. 5. The total vorticity for this date was depicted previously in Fig. 1. The top panels of Fig. 5 show the curvature and shear vorticity,

The temporal change of the effective inertial frequency depicted in Fig. 4 suggests that the relative contributions of solid-body and shear vorticity change between the October and November observation periods. A statistical analysis of the ratio of the solid-body to shear vorticity for each two week time period is presented in Fig. 6. Solid-body and shear vorticity were calculated from MERRA data and then interpolated to the balloon locations. Figure 6 shows the percentage of balloon measurements having the indicated solid-body to shear vorticity ratio. The solid-body component clearly dominates throughout October, where nearly 14% of the balloon observations showed dominant solid-body rotation, as compared to approximately 4% of observations for which the shear component of the vorticity dominates. From this, it would be expected that the main peak in the occurrence frequency for the effective inertial frequency for October would be shifted only slightly off

By the end of November, however, the relative contributions of solid-body and shear vorticity are nearly equal, with the average value of the effective inertial frequency expected to be shifted away from

## 6. Summary and conclusions

We have applied spectral methods to analyze wind fields from SPB measurements and used vorticity fields from the MERRA analysis to identify and interpret spectral features of low-frequency inertial-gravity waves recorded by Vorcore balloons in the Antarctic stratosphere. Balloon spectra were derived using both Fourier and wavelet analyses. We have shown that the spectral peak of wind measurements made on balloons drifting with the wind is shifted to frequencies more than 25% higher than the local inertial frequency. Frequency shifts have been reported in other works but were either made in a nonintrinsic frame (Mori 1990) or showed significantly smaller shifts (Hertzog et al. 2002). The exceptionally strong Antarctic polar vortex allowed identification of the peak as that corresponding to a shift in the inertial frequency *f* (Coriolis parameter) by the relative vorticity *f*_{eff} as the inertial frequency in a coordinate frame moving with the basic flow.

We study the case in which the flow locally can be written as the superposition of solid-body rotation and simple shear. The solid-body contribution gives

A possible source for a peak at frequencies higher than *f* is waves generated by fronts and jets. These waves have frequencies near 1.4*f* (O’Sullivan and Dunkerton 1995; Plougonven and Snyder 2005, 2007). However these frequencies are significantly higher than the inertial peak we observe. We have examined the possibility that the 0.1 h^{−1} peak is representative of the semidiurnal tide Doppler shifted by balloon motion, but found this effect to be too small to account for the observed shift. Nor do we find the large-scale coherency expected for a tide. Finally, we have examined pressure variations and find minimal power in the 0.08–0.12 h^{−1} band (not shown here). This is a characteristic feature of inertial waves.

## Acknowledgments

The authors especially thank Andrew V. Tangborn, from the National Aeronautics and Space Administration, for providing the GEOS5 data. The Vorcore data were provided by the France’s Centre National de la Recherche Scientifique from their web site (http://www.lmd.polytechnique.fr/Vorcore/McMurdoE.htm). Research at The Aerospace Corporation and UCLA was supported by NSF Grant ATM 0732222. Research at The Aerospace Corporation was also supported by NASA Grant NNX08AM13G.

## APPENDIX

### Inertial Gravity Waves in Linear Shear and Rotation

In this appendix we derive expressions for the inertial frequency in a flow combining rotation and linear shear. The latter should give a reasonable representation of the shear experienced by inertial gravity waves in a slowly spatially varying background wind field. We show that depending on the flow configuration and wave directionality

#### a. Inertial waves in solid-body rotation (Cartesian formulation)

This agrees with the results obtained in section 4.

#### b. Inertial-gravity waves

*m*is the nondimensional vertical wavenumber in the log–pressure system gives the dispersion relationwhere

#### c. Inertial-gravity waves in linear shear

*l*= 0,

*k*=

*l,*and

*k*= 0.

##### 1) Propagation normal to the sheared direction

##### 2) Propagation along the sheared direction

##### 3) Propagation at 45° to the sheared direction

*x*axis (i.e.,

*m*maximizes. The real part of

*m*may be written aswhere

For reasonable background values

##### 4) Relation to solid-body rotation

Solid-body rotation gives wave propagation in flow that is simultaneously sheared along and normal to the direction of propagation. If one averages the contribution from each direction using (A30) and (A32) one obtains

##### 5) Relation to approach of Kunze (1985)

## REFERENCES

Andrews, D. G., , J. R. Holton, , and C. B. Leovy, 1987:

*Middle Atmosphere Dynamics.*Academic Press, 489 pp.Broutman, D., , R. H. J. Grimshaw, , and S. D. Eckermann, 2004: Internal waves in a Lagrangian reference frame.

,*J. Atmos. Sci.***61**, 1308–1313.Eckart, C., 1960:

*Hydrodynamics of Oceans and Atmospheres.*Pergamon Press, 290 pp.Hertzog, A., , and F. Vial, 2001: A study of the dynamics of the equatorial lower stratosphere by use of ultra-long duration balloon, 2. Gravity waves.

,*J. Geophys. Res.***106**(D19), 22 745–22 761.Hertzog, A., , F. Vial, , C. R. Mechoso, , C. Basdevant, , and P. Cocquerez, 2002: Quasi-Lagrangian measurements in the lower stratosphere reveal an energy peak associated with near-inertial waves.

,*Geophys. Res. Lett.***29**, 70–74.Hertzog, A., and Coauthors, 2007: STRATÉOLE/Vorcore—Long-duration, superpressure balloons to study the Antarctic lower stratosphere during the 2005 winter.

,*J. Atmos. Oceanic Technol.***24**, 2048–2061.Holton, J. R., 1972:

*An Introduction to Dynamic Meteorology.*Academic Press, 319 pp.Hu, H., , and R. H. Holzworth, 1997: An inertial wave-driven stratospheric horizontal electric field: New evidence.

,*J. Geophys. Res.***102**(D16), 19 717–19 727.Jones, R. M., 2005: A general dispersion relation for internal gravity waves in the atmosphere or ocean, including baroclinicity, vorticity, and rate of strain.

,*J. Geophys. Res.***110**, D22106, doi:10.1029/2004JD005654.Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear.

,*J. Phys. Oceanogr.***15**, 544–564.Kunze, E., , and E. Boss, 1998: A model for vortex-trapped internal waves.

,*J. Phys. Oceanogr.***28**, 2104–2115.Lee, C. M., , and C. C. Eriksen, 1997: Near-inertial internal wave interactions with mesoscale fronts: observations and models.

,*J. Geophys. Res.***102**, 3237–3253.Mori, Y., 1990: Evidence of inertial oscillations of the surface wind at Marcus Island.

,*J. Geophys. Res.***95**(D8), 11 777–11 783.O’Sullivan, D., , and T. J. Dunkerton, 1995: Generation of inertia–gravity waves in a simulated life cycle of baroclinic instability.

,*J. Atmos. Sci.***52**, 3695–3716.Plougonven, R., , and C. Snyder, 2005: Gravity waves excited by jets: Propagation versus generation.

,*Geophys. Res. Lett.***32**, L18892, doi:10.1029/2005GL023730.Plougonven, R., , and V. Zeitlin, 2005: Lagrangian approach to the geostrophic adjustment of frontal anomalies in a stratified fluid.

,*Geophys. Astrophys. Fluid Dyn.***99**, 101–135, doi:10.1080/03091920512331328080.Plougonven, R., , and C. Snyder, 2007: Inertia–gravity waves spontaneously generated by jets and fronts. Part I: Different baroclinic life cycles.

,*J. Atmos. Sci.***64**, 2502–2520.Salmon, R., 1998:

*Lectures on Geophysical Fluid Dynamics*. Oxford University Press, 378 pp.Sidi, C., , and J. Barat, 1986: Observational evidence of an inertial wind structure in the stratosphere.

,*J. Geophys. Res.***91**(D1), 1209–1217.Thompson, R. O. R. Y., 1978: Observation of inertial waves in the stratosphere.

,*Quart. J. Roy. Meteor. Soc.***104**, 691–698.Torrence, C. T., , and G. P. Compo, 1998: A practical guide to wavelet analysis.

,*Bull. Amer. Meteor. Soc.***71**, 61–78.