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 Expressions for momentum and heat fluxes using density as the vertical coordinate are derived. These are applied in the evaluation of fluxes using data from super-pressure balloons drifting on constant density surfaces in the Antarctic lower stratosphere during the VORCORE campaign (September 2005 to February 2006). We focus on the core months of October and November. Vertical fluxes of zonal and meridional momentum are calculated using wind, pressure and height data and the vertical flux of sensible heat is calculated using temperature and height data. Calculations were performed in three band passes covering 1–13 h. We find that the largest fluxes are in the vicinity of the Antarctic Peninsula. In October the fluxes in the low period band pass (1–5 h) account for the main part of the total flux of zonal momentum, consistent with topographically forced waves. During November the vertical fluxes of zonal momentum are found mainly in longer period band passes, consistent with weaker winds. The peak campaign-averaged flux of zonal momentum in the vicinity of the Antarctic Peninsula is ∼−30 mPa. These values are ∼60% larger over the peninsula than those inferred by other authors. The flux of zonal momentum provides a zonal body force of ∼5 m s−1 day−1 assuming a saturated spectrum. We infer downward sensible heat fluxes of ∼3 W m−2. The corresponding cooling rates assuming a saturated spectrum are ∼0.6 K day−1, a significant fraction of the net radiative imbalance in the springtime Antarctic lower stratosphere.
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 Over the years several experiments to advance our understanding of the stratospheric circulation have been performed with super-pressure balloons (SPBs) drifting on constant density surfaces [e.g., Hertzog and Vial, 2001; Hertzog et al., 2007]. In this framework perturbation quantities ψ′ are necessarily defined with respect to averages on constant density surfaces. The relationships between wave perturbation quantities defined in this manner can be different from the relations that apply when means and deviations are defined with respect to isosurfaces of other parameters. For example, the linearized ideal gas law when means and perturbations are defined on constant density surfaces is since ρ′ ≡ 0, whereas when means and perturbations are defined on constant height surfaces for typical waves [Walterscheid et al., 1987]. Here ρ is density, p is pressure, T is temperature, overbars denote basic state quantities, and primes denote wave-caused deviations therefrom. Other relationships such as those for momentum, continuity and heat, and others derived from these, such as flux relations, can also be different.
 Our principal objectives in this study are twofold. First, we introduce gravity wave relations in constant density coordinates. Second, we use these relations to calculate gravity wave momentum fluxes and sensible heat fluxes over in the lower stratosphere over Antarctica using data from SPBs released in the southern hemisphere polar vortex during the VORCORE campaign of the STRATÉOLE program. Log-density coordinates are the natural coordinates to infer wave fluxes using data obtained on isopycnal surfaces.
Vincent et al. , Hertzog et al. , and Boccara et al.  have inferred momentum fluxes in the lower stratosphere above Antarctica from the SPBs launched during the VORCORE campaign. These authors modify gravity wave relations based on constant height coordinates. The relations derived by Vincent et al.  are no less valid than those obtained here, but as we shall show, they require more reliance on gravity wave relations (hence make more redundant use of information) and are more difficult to apply.
 This paper is organized as follows: In section 2 we derive gravity wave relations in constant density coordinates: in section 3 we derive flux relations; in section 4 we describe a wavelet-based analysis of VORCORE data for wave fluxes, in section 5 we show results of an analysis of VORCORE data; in section 6 we discuss the results; and in section 7 we summarize our main findings.
2. Equations of Motion in Constant Density Coordinates
 In this section we present the equations of motion, temperature, and state in log-density coordinates and derive flux relations from the linearized equations.
2.1. Nonlinear Equations
 We assume as is done for pressure and isentropic coordinates that the motion, including wave motion, is quasi-static. This limits the approach to waves for which N ≫ ωI, where N is the Brunt-Vaisala frequency and ωI is wave intrinsic frequency. The equations of motion are
and θ, the potential temperature is
Symbols not previously used have the following meanings: Q is the diabatic heating rate per unit mass, u is the horizontal velocity, is the vertical unit vector, z is the geometric height of constant density surfaces, g is gravity, R is the gas constant for air, κ = R/cp, and cp is the specific heat of air at constant pressure. Quantities subscripted with zero are constants. The quantity W is analogous to the Montgomery stream function M = CPT + gz for isentropic coordinates [Montgomery, 1937]. The symbol ∇ζ is the two-dimensional gradient operator on isopycnal surfaces (i.e., it is evaluated with ρ held constant). Henceforth we omit the subscript ζ. The set (1)–(4) is derived in Appendix A.
where overbars refer to time-averaged quantities following the motion of balloons on constant density surfaces and primes to wave-associated perturbations. Here is the Lagrangian time derivative following the mean motion on constant density surfaces [Broutman et al., 2004; Hertzog and Vial, 2001]. It is assumed that and that the dominant advective terms involve the advection of disturbance quantities by the horizontal basic-state wind. Henceforth we assume adiabatic motion unless otherwise stated, i.e., Q′ = 0. We assume solutions of the form where k is the horizontal wave number vector and ω is the wave frequency. With this waveform .
2.3. Relations Between Disturbance Quantities
 In the following development we ignore gradients in the basic state temperature and winds. This assumption is made in the spirit of a WKB approach wherein basic state quantities are considered locally constant.
2.3.1. Perturbation Stream Function
 A useful relation is given by
where and we have used the linearized form of W and
2.3.2. Total Vertical Velocity
 There are two contributions to fluxes through a level coordinate surface in terms of quantities evaluated in an isopycnal coordinate system. One is the flux through constant density surfaces inferred from the set (5)–(8). The other is the flux induced by the mean flow relative to wave-deformed density surfaces (in isentropic coordinates this would be the only contribution). This is explicit in the linearized expression for the total geometric vertical velocity
It is of interest to compare the two terms in parentheses. We evaluate isothermally, whence
where γ is the ratio of the specific heats. Thus the flux through constant density surfaces is greater than the flux induced by the mean flow relative to wave-deformed density surfaces by a factor of about 5/2. Equation (21) can be written more simply as
Boccara et al.  apply (20) to evaluate the momentum flux. This requires the evaluation of k in order to determine u′k. The horizontal direction of propagation is obtained with an ambiguity of 180° by rotating the horizontal velocity components determined from a wavelet analysis until the velocity component in the rotated direction is maximized. The ambiguity is removed by assuming that wave energy propagation is upward. Next, envelopes of wave packets are identified and the flight-mean momentum flux is computed by summing over the wave packet frequencies. The periods associated with wave packets are not limited to those below or near the inertial period; periods up to 24 h are included.
3.2. Eliassen-Palm Relation
 The 1st Eliassen-Palm theorem for the system (5)–(8) relating the energy flux to the momentum flux is obtained by multiplying (18) by and averaging. This gives
The term on the left side of (23) is the wave energy flux. For stationary waves
where kH = |k|. Equation (24) shows that for intrinsic frequencies above the inertial frequency stationary waves that transfer energy upwards transfer momentum in a sense opposed to the mean flow projected onto the wave number vector. In other words, waves with north-south oriented crests in eastward zonal flow propagate westward (negative) momentum upwards.
3.3. Sensible Heat Fluxes
 Fluxes of sensible heat are forced primarily by wave dissipation [Walterscheid, 1981]. The nonadiabatic form of (8) includes a term (γ − 1)Q′ on the right side. There is also a nonadiabatic term arising from (A11), but its contribution is negligible. Equation (4) gives upon multiplication by cpT′ and averaging
where (10) was used to eliminate the correlation involving p′. Assuming isothermality
We approximate the cooling coefficient due to scale-dependent diffusion as α = m2κe where m is the nondimensional vertical wave number in log-density coordinates and κe is the log-density value of the coefficient of eddy diffusion having units of frequency [Walterscheid, 1981]. If the corresponding dimensional quantities are denoted with a caret then
The dispersion relation for the log-density coordinate system obtained from (5), (7), (8) and the adiabatic form of (A11) is
For stationary waves (c = 0) and with evaluated isothermally (29) becomes
 Beginning on 5 September 2005, the VORCORE campaign released twenty-one SPBs with 10 m diameters and ten balloons with 8.5 m diameters from McMurdo, Antarctica (77.5°S, 166.4°E); they drifted near 50 hPa and 70 hPa, respectively. The mean flight duration of the twenty-seven surviving balloons (19 10-m diameter and 8 8.5 m diameter balloons) was 59 days and the longest flight duration was 109 days [see Hertzog et al., 2007].
 Each VORCORE SPB carried temperature and pressure sensors and a Global Positioning System (GPS) receiver. Balloon positions were recorded every 15 min with a horizontal position accuracy of 10 m so that wind speeds could be estimated with accuracy greater than 0.1 m s−1 [Hertzog et al., 2007]. The vertical position accuracy is less, being about 20 m. The geographical sampling of the Antarctic vortex core was very good with best sampling in the 60°S–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] as is typical during the spring final warming [Mechoso et al., 1988].
Figure 1 shows power spectra for the two variables p′ and z′ = ϕ′/g. One can question whether the vertical displacement is too noisy for the determination of fluxes. An indication of the noise level is the frequency where the onset of the noise floor occurs. At frequencies higher than the noise floor the spectra are dominated by noise. The spectra show that the onset of the noise floor for both variables is on the short period side of 1 h. Our analysis is limited to periods ≥ 1 h. Assuming a white noise spectrum in frequency this eliminates most of the noise, so the inference of fluxes should not be subject to significant errors from high noise levels in ϕ′. This is verified by the calculations shown below that show very small fluxes over the relatively featureless Antarctic interior.
4.2. Balloon Trajectories
Figure 2 shows balloon trajectories in two week intervals for October and November 2005. There is an interval of a few days for which no points are plotted between the early (blue curves) and late parts (red curves) of the months to help delineate the changes in the vortex over the month and its effect on balloon coverage. In October a well-defined wave-deformed vortex exists for the entire period. The shape and orientation of the vortex changes as the mix of planetary waves reflects zonal wave numbers 1, 2 and 3 in varying degrees [Manney et al., 1991]. The early part of November is similar, but toward the middle of the month the vortex begins to move off the pole. This process continues so that by the end of the month the vortex is highly elongated in the east-west direction and centered near 45°E above the periphery of Antarctica. Figure 3 shows the horizontal velocities associated with the balloon trajectories.
4.3. Wavelet Formulation
 The intermittent nature of inertial waves suggests that a wavelet analysis is an appropriate means to explore the temporal behavior of the zonal wind spectra. The time series of velocity, pressure and balloon height were analyzed with Morlet wavelets in order to evaluate the momentum flux over specific time intervals [Torrence and Compo, 1998].
where the subscript n denotes the wavelet component as a function of wave period. The overbars denoting basic state quantities refer to time averages along the balloon trajectories using a sliding 24 h window. The wavelet-transformed quantities in (31) include the standard scaling factors for the Morlet wavelet as suggested by Torrence and Compo [1998, equation (11)]. The brackets indicate temporal averages used to create the momentum flux maps. The averaging is performed using equal area (500 km × 500 km) bins. Temporal averaging is performed over 24 h periods in each spatial bin. Only bins with more than 100 data points are used to calculate flux maps. The wavelet analysis was performed for each balloon trajectory over several band passes, each of which is a sum over the wavelets in (31) in a given frequency range.
 We applied (19) to a calculation of the momentum flux from VORCORE data over one-month periods for the frequency range 1 to 13 h. This range substantially covers frequencies below the inertial cutoff frequency that are well resolved by the measurements. No significant differences were found by extending the period range to 24 h.
5.1. Vertical Momentum Flux for October and November 2005
 Results for the vertical flux of zonal momentum for October 2005 for three band passes (1–5 h, 5–9 h, 9–13 h, and 1–13 h) are shown in Figure 4. The most prominent feature by far is the area of large negative flux centered near and to the west of the southern part of the Antarctic Peninsula. The association with a region of steep terrain and the negative values of flux opposed to the mean zonal wind (eastward) suggest that the waves contributing to the flux are dominated by quasi-stationary mountain waves. The location to the west of the peninsula is consistent with a westward phase tilt (into the wind) of waves required for upward energy propagation. Mountain waves located over the peninsula [Plougonven et al., 2008] would radiate energy and momentum to the west. Some displacement is also due to the tessellation involved in the flux calculations. An examination of the results for the various band passes shows that the largest contribution comes from the 1–5 h band pass. Intrinsic frequencies in this range are consistent with topographically fixed waves with wavelengths ∼100 km and predominantly eastward flow at balloon altitudes of ∼20 m s−1 (Figure 3). Waves with longer intrinsic periods corresponding to longer spatial scales provide much smaller contributions.
Figure 5 shows the results for the vertical flux of meridional momentum. For the full band pass (1–13 h) the largest fluxes over the largest area, as for the zonal component, are found near the Antarctic Peninsula. The fluxes are positive indicating an upward flux of northward momentum. Positive momentum flux is consistent with topographically fixed waves in a southward flow. During the first half of October the flow is nearly zonal (see Figure 3). However, in the second half there are indications of a fairly strong anti-cyclonic circulation cell located just west of the peninsula. Southward flow over the peninsula is associated with this circulation feature.
 A region of negative (southward) meridional flux during October is observed near the coast at 70°S, between 0° and 60°E. The wind maps (Figure 3) show northward flow on the eastward and westward extremities of this region and predominantly zonal flow between. This is consistent with maxima in the southward flux being located near 10° (9–13 and 1–13 h band passes) and 60°E (5–9 h band pass).
 The association of the northward flux over the peninsula with southward flow is suggestive of stationary waves, but because of the complexity of the flow this association is ambiguous. Zonal flow over complex terrain can generate meridionally propagating waves and it may be possible that the predominantly northward flux reflects in part north-south terrain asymmetries favoring the generation of southward propagating waves [Hines, 1988]. Another possibility is that meridionally propagating waves are not entirely stationary, but originate in part in the obviously nonstationary nature of the flow over the peninsula during October.
 Results for the vertical flux of zonal momentum for November 2005 for each band pass are shown in Figure 6. For the full band pass (1–13 h) the largest negative (westward) fluxes are found along the Antarctic Peninsula and the coastal waters to the west. Results for the subintervals of the full band pass show that the main contribution comes from the 1–5 h band pass, consistent with topographic forcing. However, this contribution is not nearly as strong as for October. The flux of zonal momentum in the 5–9 h band pass is distributed along the peninsula and are also significant. The comparative strength of the fluxes in the 5–9 h band pass is consistent with a much reduced zonal flow during November as the month progresses (see Figure 3).
 Results for the vertical flux of meridional momentum for the various band passes are shown in Figure 7. Again the largest fluxes are associated with the Antarctic Peninsula. In contrast to October, the flux is negative, indicating an upward transport of southward momentum. Also in contrast with October, the main contribution comes from the 5–9 h band pass. This is consistent with a much reduced northward component of the flow at balloon altitudes, beginning in about the second week of November. The southward flux is consistent with a predominant northward flow over the main part of the peninsula (excluding the southern most part) during November.
5.2. Campaign Averaged Results
 The results reported in this section are the averages over the full campaign (7 September 2005–6 February 2006). Vertical fluxes of zonal and meridional momentum for the full 1–13 h band pass are shown in Figure 8. Figure 8 was generated with a somewhat smaller bin size than was used for the monthly Figures (400 × 400 instead of 500 × 500 km). This takes advantage of the fact that over the whole campaign we can accumulate the same or better sample per grid element with a finer gridding. The dominant region of vertical zonal momentum flux is located over and to the west of the Antarctic Peninsula, with maximum values near −30 mPa. This is significantly larger in magnitude than the peak values (∼−18 mPa) reported by Hertzog et al.  for the full VORCORE campaign. The meridional flux we find is largest near the peninsula and is positive (northward) with peak values near 8 mPa; this is also significantly larger than the full campaign results (∼4 mPa) reported by Hertzog et al. .
 Because the finer tessellation better preserves gradients, the peak campaign-averaged values are larger than either of the peak values from October and November. The campaign-averaged meridional flux shows that the large area of positive flux near the peninsula in October is only partially offset by the small region of negative flux in November.
 The momentum fluxes calculated for the nearly featureless interior of the Antarctic continent are small and uniform. The fluxes in the interior should indicate an upper limit to the noise level. Figure 9 shows a histogram of the fluxes calculated for the region of Antarctica poleward of 75°S over the full campaign (7 September 2005–6 February 2006). The region has good balloon coverage (Figure 2). The means are essentially nil and the standard deviations derived from best fit Gaussians are <1 mPa. We conclude that noise is not a significant contributor to our results.
5.3. Sensible Heat Flux for October and November 2005
Figure 10 shows the sensible heat flux for October and November for the full band pass (1–13 h). In both months the largest fluxes over Antarctica are associated with the Antarctic Peninsula. Secondary maxima (in the absolute value sense) are located eastward of the peninsula and extend northward from the coastal waters between ∼60°W and 0° over oceanic waters to ∼65°S. The flux is everywhere negative. In November significant fluxes occur over and near the Antarctic Peninsula, although somewhat reduced from the October values. The dominant heat flux in November is near 60°E and is associated with the landmass projection between ∼45°E and 60°E (Enderby Land).
 The momentum fluxes we infer are several times less than the fluxes inferred for singular waves over the Antarctic Peninsula reported by Alexander and Teitelbaum  (140 mPa northwestward) and comparable to the magnitudes reported by Ern et al.  (∼40–50 mPa). Although the VORCORE balloons floated at two different levels in the lower stratosphere (70 and 50 hPa), their vertical separation was insufficient and the number of balloons at the higher pressure level too few (10) to allow a direct calculation of the momentum flux divergence. Instead we calculate the momentum flux divergence by assuming a saturated spectrum [Fritts, 1984], according to which the wave amplitude is constant and the flux divergence is given by
For October 2005 we infer a peak westward acceleration of 5 m s−1 day−1 over the Antarctic Peninsula. The mean meridional wind required to balance this acceleration is ∼0.5 m s−1, in the southward direction. For higher altitude the acceleration would increase proportionately with inverse density.
 In the same vein the heating due to a saturated spectrum of gravity waves is
The data for October suggest a heating rate of ∼−0.6 K day−1 for the peak fluxes near the Antarctic Peninsula. This value is a significant fraction of the radiative imbalance (<1 K day−1) in the springtime Antarctic lower stratosphere [Kiehl and Solomon, 1986]. As in the preceding paragraph heating at higher altitudes would increase proportionately with the inverse basic state density.
 We have compared our momentum fluxes with those published by Hertzog et al. . Figure 4 of their paper shows the vertical zonal and meridional momentum fluxes averaged over the full VORCORE campaign. The regions where significant fluxes are found are similar. The main difference is that the peak fluxes we have inferred are ∼60% larger when we average data from the entire campaign (∼−30 mPa versus ∼18 mPa). Also, for October the peak fluxes we calculate are located farther west, perhaps as a result of the different tessellation used for the flux calculations. This may also account for some of the difference in the peak values over the Antarctic Peninsula. The Boccara et al.  grid is equivalent in area to ∼450 km × 450 km over the Antarctic Peninsula. To see whether this could account for the difference between our results on those of Boccara et al.  we experimented with 450 km × 450 km grid and obtained somewhat smaller peak fluxes (∼−26 mPa). Also, note that the peak campaign averaged values for the vertical flux of zonal momentum are located farther east (closer to the peninsula) suggesting that the coarser tessellation is responsible for some of the westward displacement for the monthly results.
 A possibly significant factor in the difference between our results is that we ignore thermal gradients in evaluating (31). Hertzog et al.  include thermal gradients in the expression for the vertical wave number insofar as they contribute to the static stability [cf. Boccara et al., 2008, equation (29); Einaudi and Hines, 1970, equations (30), (32) and (34)]. We have examined the temperatures for balloons drifting between 16 and 18 km and between 18 and 20 km for the full campaign 7 September 2005–6 February 2006. We averaged the data at the two levels over 5 day periods to remove the strong dependence of the temperature difference between the two sets of balloons on their respective positions within the vortex. During the first half of the campaign the differences between levels are a few K, but in early November there is a transition to differences ∼10 K (∼5 K/km). The reduction in the flux associated with this temperature gradient applied to the full campaign is found to be ∼20%.
 Another contribution to the difference might come from applying a relation like (20) where, as we have mentioned earlier the directionality of u′k is subject to a 180° ambiguity that is removed upon assuming upward energy propagation. This is a good assumption, but may occasionally introduce some errors when there is a wave source (such as reflection) above the balloon altitudes.
 Finally, while we believe that the height measurements are not subject to undue noise levels there is likely to be some noise contribution to our results. Hertzog et al. , on the other hand, used thresholding to remove suspect wavelet amplitudes. It is a matter of choice whether to accept a certain amount of noise induced flux or avoid possibly eliminating signal. We have adopted the latter approach.
 In summary, the differences between our results can largely be accounted for in terms of reasonable but different choices with regard to methodology.
 We have described a methodology for evaluating momentum and heat fluxes based on gravity wave relations in constant density coordinates that is suitable and natural for studies using data from super-pressure balloons drifting on such surfaces. Our approach does not require one to define wave packets prior to calculating momentum fluxes or to infer wave packet directionality. The formulation is straightforwardly amenable to wavelet methods and allows one to calculate momentum fluxes in various band passes. Also, our methodology makes direct use of height data as a dependent variable and avoids added reliance on gravity wave relations to develop formally equivalent expressions with a smaller set of dependent variables. Thus our approach makes greater use of independent information.
 We have applied the methodology to calculate the vertical fluxes of zonal and meridional momentum from VORCORE data for the months of October and November 2005, during which period 21 balloons drifted in the lower southern stratosphere. We find that the largest fluxes are localized to the vicinity of the Antarctic Peninsula, in agreement with the findings of Hertzog et al. . However, the fluxes we have obtained are ∼60% larger than those of Hertzog et al.  when averaged over the campaign from 7 September 2005 to 6 February 2006. The differences between our results can be accounted for in terms of reasonable but different choices with regard to methodology.
 In October the fluxes in the low period band pass (1–5 h) account for the main part of the total vertical flux of zonal momentum. This dominance is consistent with topographically forced waves. During November the dominant fluxes of zonal momentum are found in longer period band passes, consistent with weaker winds over the Antarctic Peninsula. We suggest that a significant part of the meridional flux for October originates in nonstationary waves.
 The momentum fluxes in the vicinity of the Antarctic Peninsula, assuming a saturated spectrum, provide a significant zonal acceleration of the lower stratosphere. Similarly, we have calculated cooling rates due to the wave-induced flux of sensible heat and find that they can be a significant fraction of the net radiative heating in the vicinity of the Antarctic Peninsula.
Appendix A:: Derivation of Relations in Constant Density Coordinates
A1. Hydrostatic Relation
 Here the approach is to write down the relation by analogy with pressure and isentropic coordinates and verify it a posteriori. Thus
A2. Heat Equation
 In terms of potential temperature
Henceforward it is understood that partial derivates with respect to time and the horizontal coordinates are with ζ held constant. Since
ratio of the gas constant to the specific heat (κ = R/cp).
log-density value of the coefficient of eddy diffusion having units of frequency.
dimensional value of the coefficient of eddy diffusion ( ).
potential temperature (θ = T(p0/p)κ).
geopotential (ϕ = gz).
any dependent variable.
wave intrinsic frequency (ωI = ω − k ⋅ u).
two-dimensional gradient operator evaluated with ρ held constant.
Notational conventions are as follows: Overbars applied to dependent variables denote basic state quantities; primes applied to dependent variables denote deviations from basic state values; and quantities subscripted with zero are constants.
 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 Aerospace Corporation and UCLA were supported by NSF grant ATM–0732222. Research at The Aerospace Corporation was also supported by NASA grant NNX08AM13G.