We apply multifractal analysis using exponents H1, C1, and α to straight and level stratospheric flight legs of the ER-2 high-altitude research aircraft in the inner vortex (defined as having wind speed <30 ms−1). The quantities so analyzed were ozone, wind speed s and temperature T, with the more gappy NOy data being analyzed by H1 alone. The results for ozone, wind, and temperature are presented as time-dependent data on the three possible planes of the exponents and are compared for the different variables. We relate values of H1 found in January observations of NOy to those found for ozone. Inner vortex mixing does not remove the small-scale polar stratospheric cloud-induced antipersistence (negative correlation between neighboring intervals for all choices of interval) in ozone by mid-March. Given that large particles were in evidence on all flights examined up to and including 7 March (although in greatly decreased numbers compared to January), this is reasonable. The value of α for ozone did, however, show an increase by mid-March, consistent with the widespread ozone loss evident from time series of histograms of ozone and methane. The histograms also demonstrate that inhomogeneity, with long tails in the probability distributions, is maintained throughout at the 15–25% level in both species. Interpretation is made in terms of polar stratospheric cloud (PSC) induced antipersistence competing with persistence induced by the large-scale insolation field, with the balance increasingly favoring the latter as time proceeds. Results are compared with inner vortex data obtained during earlier ER-2 flights in the Antarctic (1987) and in the Arctic (1989). The inner vortex over Antarctica showed significant increases in H1(O3) and α during mid to late September. The correlated increases are consistent with latitudinal excursions of the outer vortex after the cessation of PSC processing, with increased solar exposure increasing H1(O3) and a greater variety of filaments increasing α(O3). It is concluded that the results have implications for the calculation of photochemical ozone loss in the vortex as a function of time and show that the combined effects of Bolgiano-Obukhov k−11/5 vertical scaling and Kolmogorov k−5/3 horizontal scaling predict the scaling behavior of wind speed observed by the aircraft. Rates of change of scaling exponents are linked to horizontal mixing rates and are combined with rates of change of methane to estimate diabatic descent and ozone loss rates for the inner vortex.
 The empirical description of atmospheric dispersion of tracers by power laws was formulated by Richardson , motivated by the inadequacy of “diffusion” as a mechanism on scales larger than centimetric in the atmosphere. This inadequacy has been discussed previously in the context of stratospheric reaction kinetics [Austin et al., 1987]. Hurst  extended the idea of power law descriptions of natural phenomena, via a scaling exponent Hu, which bears his name. Power law scaling is the essence of what were called fractals by Mandelbrot, who related turbulence, the scaling exponent Hq (see section 2.1), intermittency and multifractality in a series of papers from the 1960s to the 1980s, which he has recently reprised [Mandelbrot, 1998]. However, there is no detailed analytical treatment of how the observed structure in the velocity components, temperature and pressure of a turbulent fluid, or of a passive scalar embedded in it, may be related a priori to the governing Navier-Stokes equations. This is even more so in the case where several such scalars are reacting chemically. Scalar turbulence has been reviewed recently [Shraiman and Siggia, 2000]. Here we replace our recently applied Hu analysis of stratospheric airborne data [Tuck and Hovde, 1999; Tuck et al., 1999] by a multifractal formalism [Mandelbrot, 1974; Schertzer and Lovejoy, 1985] which has been linked to theories of atmospheric turbulence. The central point is to contrast the scaling exponents of observed variables that may change because of physicochemical processes (such as species which condense in polar stratospheric clouds or which are chemically transformed by exposure to them), with the scaling exponents of variables that, within a range, appear to be invariant, such as wind speed and temperature. In addition to providing numerical model-independent evidence for the occurrence of the physicochemical processes, there may be the basis for an eventual quantitative framework in which to examine cause and effect. We refer readers to our earlier papers for an introduction to the application of fractal analysis to the lower stratosphere. In those papers, ER-2 observations of ozone, wind and temperature were analyzed for scaling behavior during many segments of horizontal flight between January 1987 and September 1997, using the Hurst exponent (Hu) [Tuck and Hovde, 1999; Tuck et al., 1999]. Power law scaling was demonstrated, and the values of Hu used to examine the relative rates of chemistry and fluid mechanical mixing as they affected ozone, in a numerical model-independent way.
 For ozone, wind speed, and temperature, we compute H1, C1, and α, which are multifractal indices that have been studied for some time [see, e.g., Pecknold et al., 1993; Davis et al., 1994; Seuront et al., 1999]. For NOy, which is a gappy data set due to calibration intervals, we compute only H1 (see section 2.1). In general, Hq computed from “structure functions” has an advantage over Hu in that it can be compared to the predictions of theories of atmospheric turbulence, both two-dimensional (2-D) (latitude and longitude) and three-dimensional. As will be seen below, multifractal analysis provides further definition than is possible with single scaling exponent approaches, which are themselves extensions of what can be achieved by conventional power spectrum analysis with its assumption of second moment, Gaussian statistics. We apply the analysis to straight and level flight legs of the ER-2 inside the Arctic vortex between 20000120 and 20000312, having wind speed s < 30 ms−1; this is a numerical model-independent definition of the inner vortex taken directly from ER-2 meteorological measurement system (MMS) observations [Chan et al., 1989] and is justified by Tuck . The observational quantities analyzed were ozone, NOy(A), NOy(B), wind speed and temperature, where NOy(A) is the established reactive nitrogen measurement, mounted on the rear of an aerodynamic separator for particles, that includes gas phase species and NOy on aerosols particles smaller than about 2 μm diameter. NOy(B) is a new measurement from a forward mounted inlet which observes the gas phase species and NOy on particles of all sizes [Fahey et al., 2001; Popp et al., 2001; Gao et al., 2001]. Large particles (up to 20 μm diameter) were observed, particularly during the late January/early February flights. The objective of the scaling exponent analysis was to see if insight could be obtained into the relationship between ozone loss and polar stratospheric cloud processing, directly from observational analysis and independent of numerical model calculation. We recall that scaling analysis of Airborne Antarctic Ozone Experiment (AAOE) ER-2 data taken in the Antarctic vortex in August−September 1987, in the Airborne Arctic Stratospheric Experiment (AASE) Arctic vortex January−February 1989 and in the Photochemical Ozone Loss in the Arctic Region in Summer (POLARIS) Arctic summer anticyclone April−September 1997 suggested that persistence (high values of Hu) in the ozone suggested that photochemistry was outpacing windshear-induced mixing (random values of Hu (s)) in certain situations [Tuck et al., 1999]. All the flight tracks during AAOE and AASE crossed the polar night jet stream between the exterior and interior of the vortex, with only some flights having any penetration to the inner vortex. These inner vortex segments (s < 30 ms−1) were not analyzed separately; we do so here. We note that the SAGE3 Ozone Loss and Validation Experiment (SOLVE), by basing the ER-2 at 68°N in January–March 2000, obtained much more extensive access to the inner vortex than either AAOE (based at 53°S) or AASE (based at 59°N).
2.1. Multifractal Scaling Analysis via H1, C1, and α
 Our approach augments the bifractal (H1, C1) formalism of Davis et al.  by the computation of the Lévy index, α, a key component of the universal multifractal formalism of Schertzer and Lovejoy , which also employs H1 and C1. The calculation of H1 is the same under both methodologies. The calculation of C1 differs but the two methods are theoretically equivalent. We use the former method.
 The quantity H1 is the scaling exponent obtained from the first-order structure function. In general, the qth-order structure function of a signal f(t) is defined as
where the positive real parameter r is called the lag, q is a nonnegative real number, and 〈…〉 denotes temporal (or spatial, if appropriate) averaging over t and ensemble averaging over f. In our case, each ensemble contains just one signal. For the lags, we normally start with the greatest integer less than half the signal length, and then divide by two after each iteration until we reach r = 1.
 If the points of a plot of log Sq(r;f) versus log r are fairly co-linear, then the slope ζ(q) of the unweighted linear least squares fit to the points defines a scaling exponent for f. Define Hq = ζ(q)/q. If Hq is roughly constant as q changes, then the signal is monofractal and the scaling exponent H = Hq. If Hq is not constant with q, then the signal is multifractal and the scaling exponent is given by H = Hq + K(q)/q [Seuront et al., 1999] where K(q) is a quantity computed for the intermittency calculation (described next). Since K(1) = 0, H1 is a good estimate of the scaling exponent in both the monofractal and multifractal cases.
 The range of H1 is from zero to one, with values near zero characterizing rough, nearly stationary signals and values near one characterizing smooth, nonstationary signals. For this reason H1, is frequently called the measure of nonstationarity.
 For the uncertainty in H1 we use the 95% confidence interval returned by the unweighted linear least squares fit, considering the points of the log-log plot to have no error. We experimented with weighting the points by a quantity related to the standard deviation from the mean shown in equation (1) but found that this yielded an unreasonably large uncertainty for synthetic signals that scaled perfectly. We have also found that instrument error up to about 10% on a 1 Hz data set has little effect on H1. Larger errors induce an increasingly evident scale break in the log-log plot of synthetic signals. We have not seen any consistent scale breaks for the data presented here and so have concluded that instrument error is not an issue for this particular analysis.
 The quantity C1 measures the intermittency of the signal and takes on values in the range from zero to one, with values near zero characterizing a signal with extremely low intermittency (e.g., a Brownian motion) and values near one characterizing a signal with extremely high intermittency (e.g., Dirac δ-function). (Natural atmospheric signals typically exhibit very low intermittency (∼0.03) and a change in intermittency on the order of 0.01 is considered significant.) Our computation follows the tutorial given by Davis et al. . We consider our signal f(t) to have been measured at the discrete times t = 1, 2, 3, …, T, and define
Although in practice we use the same rs as for the structure function analysis (described next), it is helpful to instead think of starting with r = 2 and then doubling r at each iteration, for then it becomes apparent that we are coarse-graining the signal. It turns out that for many natural signals, the quantity 〈ε(r;t)q〉 has a power law dependence on the scale r. We again compute an unweighted linear least squares fit to log〈ε(r;t)q〉 versus log r. The slope of the fit is denoted −K(q). A plot of K(q) versus q shows a convex function with K(0) = K(1) = 0 [Davis et al., 1996]. The quantity C1 is defined as K′(1), which we evaluate numerically by computing the slope defined by the points (0.9,K(0.9)) and (1.1K(1.1)). For the uncertainty in C1, we use the square root of the sum of the squares of the 95% confidence intervals returned by the unweighted linear least squares fits corresponding to q = 0.9 and q = 1.1. The resulting confidence interval is probably somewhat smaller than 95% and we continue to experiment with other procedures.
 The Lévy index α ranges from zero to two and measures the power law fall-off of the tail of the probability density of the signal increments [Pecknold et al., 1993]. A Brownian motion has α ≈ 2. As α tends to zero, the tail of the probability density of the signal increments becomes longer and thicker. Most natural signals have α > 1 [see, e.g., Pecknold et al., 1993, Table 1]. We use the double trace moment method [Seuront et al., 1999] to compute α by defining
where we typically let η range from −1.0 to 1.0 in steps of 0.1. For q = 1.5 we do an unweighted linear least squares fit to log〈ε(r,η,t)q〉 versus log r, denoting the slope by K(q,η) and the uncertainty (standard deviation) in the slope by σ(q,η). We then plot log K(q,η) versus log η. The natural signals we investigated have a region of the plot characterized by fairly co-linear points defining a positive slope. We do a weighted linear least squares fit to this region, using as weights K(q,η)ln 10/σ(q,η). The resulting slope is α, and for the uncertainty in α we take the 95% confidence interval returned by the weighted fit. We believe these confidence intervals are quite realistic for most of the signals considered here. For our temperature signals, however, the confidence intervals become quite large. The difficulty may lie in the step function appearance of the temperature trace, since we know for a fact that this characteristic causes the very linear appearance of the K(q) curve in the intermittency calculation. We continue to investigate the multifractal characteristics of step function signals as well as other procedures for computing the uncertainty in α.
 Ideally H1, C1, and α are computed only for signals containing no data gaps. With regard to H1, however, such a policy is probably unnecessarily restrictive in practice, since the analysis of many nongap signals leads us to conclude that the assumption of a general scaling law in similar signals containing gaps is not a bad one. That is, we assume that if we had been able to take measurements during the data gap, these measurements would scale in essentially the same way that the rest of the signal scales, as long as the data gap is quite small, which is a condition easily met by the signals studied here. As for C1 and α, we cannot at this time make the same argument. We believe the measure of intermittency C1 is especially sensitive to missing data, since a few seconds of data could make all the difference between C1 = 0 and C1 = 1. For these reasons, the analysis of NOy data presented here is based entirely on the computation of H1, although we do not expand the associated confidence intervals to account for the data gaps. We continue to investigate the effects of data gaps on the computation of all of the multifractal indices.
2.2. Application of Scaling Analysis via Structure Functions to the Front (B) and Rear (A) Inlet NOy Measurements
 On the SOLVE mission, the NOy instrument used for the first time an inertial separator (“the football”) to measure the NOy content of both vapor and large particle condensed phases via the front inlet, NOy(B), and the vapor phase(with possible enhancement by particles less than 2 μm diameter) via the rear inlet, NOy(A). This technique was originally used for the investigation of the water content of tropical cirrus clouds [Kelly et al., 1993]; the previous measurements of NOy [Fahey et al., 1989] were of the NOy(A) type. Application of the scaling analysis to the NOy(B) channel calls for elaboration, because it is the sum of the NOy content in the particle and vapor phases; the former is in particles large enough that they have significant fall speeds [Fahey et al., 2001] under gravity. Thus while the NOy(B) actually observed is a snapshot of the NOy content in the air mass sampled by the ER-2 at the altitude of flight, it is not clear on what timescale it would remain representative, or over what mass of air. When active denitrification via large particles is taking place, an air layer will have fluxes into it from above and out of it to underlying layers, making both NOy(A) and NOy(B) measured in such circumstances possibly unrepresentative of the inner vortex in a general sense. However, there is reason to believe that the phenomenon was widespread and deep in late January and early February, and we apply the scaling analysis to the time series of NOy(B), NOy(A) and their difference [NOy(B)-NOy(A)]. Because NOy is a technically more complicated and difficult measurement than those of ozone, wind and temperature, the scaling analysis is restricted to the first-order structure function H1, and although adequate is not of the same quality. It should be noted here that when a single, isolated large particle enters the NOy(B) inlet, it produces a sudden increase in the first 1 Hz bin in which it is detected, followed by a 20 s recovery of the instrument signal to the background value. It arises from the tendency of nitric acid to adhere to the interior surfaces of the instrument. If particles are separated by less than 4 km (aircraft velocity = 200 ms−1), these decay tails will become convoluted and have the potential to cause persistence in the scaling analysis. This implies that values of H1[NOy(B)] obtained in the presence of large particles should be regarded as maximal, since persistence produces scaling exponents tending to unity in the possible 0–1 range. In actual practice, in the presence of extensive regions of large particles, the scaling exponent for NOy(B) was less than that for the difference between NOy(B) and NOy(A), indicating that the effects of the decay tails were insignificant. This was probably because the decay was initially very rapid, followed by a longer decay of low amplitude.
 To summarize, H1[NOy(A)] is taken to represent the scaling of the vapor phase NOy, while H1[NOy(B)] represents the scaling of the total NOy content. The exponent H1[NOy(B)-NOy(A)] represents the scaling of the large particle content of NOy. In the absence of particles, the last of these three exponents will represent instrument noise.
 In view of the previous results [Tuck et al., 1999] of scaling analysis via Hu, which argued that persistence in ozone relative to wind speed was a sign that chemistry was outpacing mixing, we will establish the fact of inner vortex chemical ozone loss during SOLVE. First, however, we illustrate the meaning of scaling behavior for the wind speed in the polar night jet stream. Figure 1 shows the vector winds and their differences (shear) for a flight across it, on three different scales centered on the jet axis. These shear vectors (Figure 1b) show that on all three scales there is positional exchange between neighboring intervals of air as they flow. Figure 1c shows the associated ozone and tracer structure across the jet, which supports this argument. Similar results are obtained generally for flight legs inside and outside the vortex. Chemical ozone loss is established by plotting histograms of methane and ozone (normalized probability distribution) for each of the flight legs we analyze which were near potential temperature θ = 435 K, see Figure 2. The occurrence of ozone loss from the ER-2 data has been independently established by Richard et al. ; Figure 2 displays it in a form which simultaneously displays the temporal decreases of ozone (≈50% decrease) and methane (≈20% decrease) in the inner vortex between late January and mid March, along with their variability at θ = 436 ± 4 K. Given that in the early vortex and elsewhere in the extravortical stratosphere, methane and ozone-mixing ratios are negatively correlated up to at least 35 km altitude, with methane decreasing and ozone increasing with altitude, two points are evident: (1) the decrease in methane indicates inner vortex descent; (2) given (1), in the absence of chemical loss, ozone would have increased. The decrease indicates very substantial loss, a conclusion consistent with the analysis of satellite-based methane and ozone profiles in the Arctic vortex in earlier years [Müller et al., 1999]. A third point is that the widths (and non-Gaussian shape) in the histograms indicates that substantial variation, typically a range of 20%, is maintained. The distributions often have long tails, for example the ozone on 20000307 has a most probable value of 1700 ppbv with a tail stretching to 2500 ppbv. Sparling  has seen similar distributions in satellite data. On the isentropes sampled, there is no sign during the 52-day period that the inner vortex proceeded via dissipation within an isolated air mass to a uniformly mixed state.
 We may make estimates of the methane rates of change from the histograms in Figure 2. There are three pairs of flights which are separated in time but which have legs at nearly identical potential temperatures: 20000120 and 20000226 at mean θ = 431.4 K, 20000131 and 20000307 at mean θ = 438.3 K, 20000202 and 20000312 at mean θ = 442.4 K. The three derived rates of change for methane are −0.18% d−1, −0.51% d−1 and −0.33% d−1 respectively, indicating that descent is outpacing horizontal mixing. The balance between and interpretation of “descent” and “horizontal mixing” is discussed further in section 4.
3.1. Scaling Exponents for NOy
 The scaling exponents for NOy(A) (the rear inlet, which ingests vapor and only particles <2 μm, and NOy(B), and the forward inlet, which additionally ingests any large particles) are shown as a function of day of year in Figure 3. The calculations also include legs at rather higher average potential temperatures than the legs which appear in Figure 2; these additional segments are at potential temperatures up to 455 K. The low values on 20000120 and 20000202 for NOy(B) arise from spikiness caused by the concentration of NOy in the large single particles being detected (see Figure 4). As these large particles became rarer during the mission, the NOy(B) values of H1 became larger, until they equaled those for NOy(A) during the variable but PSC-free flight legs on 20000311 and 20000312. An example is shown in Figure 5 for 20000305, where there are only low incidences of large particles. The scaling in this figure is the worst example we have used; the remainder are better fits.
 Note that PSCs in the sense of large particles with high fall speeds were detected on all flights up to and including 20000307, although with decreasing frequency; 20000311 and 20000312 were the only flights free of such particles. Because of the low fraction of the total particle surface area accounted for by such particles, coupled with varying degrees of denitrification, there can be no unequivocal statement based on the NOy data alone about the prevalence of small-particle PSCs, in which high surface areas were available for processing the air and so setting up ozone loss. However, there was evidence of such PSCs from independent sources, such as the POAM satellite [Bevilacqua et al., 2002]; it observed PSCs from mid November to mid-March, with a period of absence from 20000207 to 20000227. Note that the ER-2 saw some large particles on 20000226.
 The H1 values may be plotted in alternative ways, which bring out other features in the data. The H1 values for the difference time series between NOy(B) and NOy(A) is plotted as a function of time in Figure 6. The scaling exponents of the difference time series are near zero early in the mission when numerous large particles were detected by the forward facing inlet for NOy(B), rising to about 0.1–0.2 by the end of the mission, when such particles were almost completely absent. The near zero values early are consistent with a high degree of spikiness, arising from the discrete nature of the particles. The values about 0.1–0.2 in their absence are a measure of the noise generated when the two channels are both measuring vapor phase NOy only; systematic differences between the NOy measurements and instrument noise prevent the difference time series produced by subtraction from being exactly zero. A plot of H1[NOy(A)]versus H1[NOy(B)] shows the points with no clouds lying on the 1:1 regression line, demonstrating the remarkable consistency between the two channels in the absence of PSC particles (Figure 7). The points below the line all contain such particles; the single point above the line is traceable to problems with the catalyst in the NOy(B) channel during the flight on 20000311. It should be noted that the associated error limits for the NOy instrument [Fahey et al., 1989, 1990, 2001] are larger than those for ozone, wind speed and temperature, and that calibration gaps also degrade the scaling exponents somewhat. Nevertheless, the increase in H1[NOy] from late January to mid-March is significant in Figures 3 and 6.
3.2. Multifractal Analysis of Ozone, Wind Speed, and Temperature
 We recall that persistence in a time series means that neighboring time intervals, for all choices of interval length, tend to be positively correlated. Such a characteristic means values of H1 nearer to unity than to zero. Intermittency has been defined in various ways [Mandelbrot, 1974; Schertzer and Lovejoy, 1985; Davis et al., 1997; Mandelbrot, 1998]. It is a measure of the degree to which dissipative activity in a turbulent fluid is sporadic, and is intimately connected with the long tails on probability distributions, in which infrequent but high amplitude events make significant contributions to the mean state. When the exponent C1 tends to zero, the fluid approaches homogeneity, when C1 approaches unity all the turbulent energy is tending to concentrate in single structures, characterized by Batchelor and Townsend  as blobs, rods, slabs and ribbons but generally called filaments or sheets in the recent stratospheric literature. Whereas C1 characterizes the “sparseness” of the mean value of the field, α characterizes the distribution of the remaining values, as described by Pecknold et al.  and Seuront et al. . It is the exponent of the power law fall-off of the tail of the probability distribution.
 The values of H1, C1 and α were calculated as described in Section 2.1, allowing assessment of the degree of multifractality of the data at up to the 15,000 s (3000 km) time series length, which was the maximum achieved by the aircraft for straight and level segments. A sample calculation of the three exponents is shown for ozone during the flight of 20000305, shown in Figure 8. Similar examples for wind speed and temperature are shown in Figures 9 and 10. We will display the results on sets of three pairwise plots of H1, C1 and α, with color-coding indicating time dependence. Accordingly we perform analyses of the inner vortex (s < 30 ms−1) flight segments for SOLVE, AAOE and AASE to place the data on the (H1, C1), (H1, α) and (C1, α) planes. The results are shown in Figures 11–15. At this point, we recall the characteristics of data at each of the four corners of the (H1, C1) square [Davis et al., 1997]. Time series at (0, 0) would have no intermittency (C1 = 0) and be 1/f noise (H1 = 0). At (0, 1) the time series would be Dirac δ-functions, at (1,0) it would be a continuously differentiable function and at (1,1) it would consist of Heaviside steps.
Figure 11 shows the ozone SOLVE scaling results, displayed on the (H1,C1), (H1,α) and (C1,α) planes with the flight dates color-coded. There is a significant time trend in α(O3) ranging from 1.27 in late January to 1.92 in mid-March. There is no significant time evolution in H1(O3) over the period although there is a tendency to increase apparent in Figure 11a, and with much less scatter, none in C1(O3).
Figure 12 displays H1(s), C1(s) and α(s) for SOLVE; the data are somewhat more tightly grouped than for ozone, but the central result is the absence of any significant temporal evolution in any of the three exponents over the period. It is noticeable that C1(O3) and C1(s) are both tightly grouped around the same average value, 0.03, while H1(s) and α(s) have a shorter range and lower mean values than the same datum for ozone.
 The scaling behavior of temperature is distinct from that of ozone and wind speed, see Figure 13. The grouping on the H1 axis of the H1(T)versus C1(T) plot is tighter than for ozone and wind speed, but with more variation around a mean C1(T) of 0.09, a factor of three larger. The large error bars on the value of α(T) may result from the steppiness in the temperature data arising from truncation in the archival software. There is no significant temporal evolution of any of the three exponents for temperature during SOLVE.
 Application of the 3-exponent analysis to the AAOE ozone data for the Antarctic inner vortex in August–September 1989 is shown in Figure 14. The temporal evolution is much clearer than during SOLVE. H1(O3) increases from values of 0.28 ± 0.08 on 9 September to a value of 0.70 ± 0.07 by 22 September. The accompanying increase in α(O3) is also just significant, but is less so and covers a smaller range than α(O3) during SOLVE. Note the positive correlation between H1(O3) and α(O3) for the flights up to and including 19870909.
 The results of the (H1, C1, α) analysis of the Arctic inner vortex ozone data from January–February 1989 (AASE) are shown in Figure 15. There are no significant time trends in any of the exponents, although the highest value of α(O3) does occur during the last inner vortex flight segment. Since there were no flights during March 1989, the evolution of the higher H1(O3) under higher springtime solar illumination was not observed, and since the 1989 AASE denitrification was later, sparser and weaker than the 2000 SOLVE denitrification, the PSC-induced lower values of H1(O3) are not evident in Figure 15 either. The less severe ozone loss deduced for 1989 [Proffitt et al., 1990] was heavily weighted toward flight segments in the more sunlit outer vortex, and is not expected to be evident in the less illuminated inner vortex data which yield Figure 15.
3.3. Third-Order Structure Function for Wind Speed
Kolmogorov  formulated a theory of 3-D turbulence for smaller scales which predicts 3H3 = 1 for wind speed. Accordingly, these third-order structure functions were evaluated for our chosen flight segments; the results are shown in Table 1. Generally, 0.38 < H3 < 0.50 with a mean value of 0.44; the value of 0.33 for only the shortest leg is not significant. Arbitrary selection of much shorter legs, down to 512 s (102.4 km) gave H3 ≈ 0.47, consistent with the majority of results in Table 1. We also report H1 in Table 1; the mean value is 0.49.
 We note that it is not possible to fly the ER-2 so that any vertical coordinate (geometric altitude, pressure altitude, and potential temperature) is truly constant. Investigation of the consequences of this fact for the comparison of observed scaling of wind speed with theory is the subject of a separate publication (S. J. Hovde et al., manuscript in preparation, 2002). Here we note that Lovejoy et al.  point out that Kolmogorov k−5/3 scaling applies in the horizontal, and Bolgiano-Obukhov k−11/5 scaling applies in the vertical, resulting in a fractal dimension of 23/9 and of 5/9, in the general vicinity of the mean value (weighted by segment length) of 0.49 ± 0.05 in Table 1.
 We preface the discussion with some remarks about the scaling of the data; the quality of the straight line fits in the calculation of Hu(O3) and Hu(s) may be seen in our earlier publications [Tuck and Hovde, 1999; Tuck et al., 1999] and is maintained here in the calculation of H1 (Figures 8 and 9). As explained in sections 2.2 and 3.1, the NOy data are from a technically more complicated and difficult measurement, and while they will support an adequate calculation of H1, the result cannot be expected to have the high degree of linearity exhibited by H1(O3) and H1(s). Two examples of NOy scaling are shown; Figure 4b shows more typical scaling for this measurement, while Figure 5b shows the worst. It is our conclusion, based on the error bars and the overall consistency of the analysis, that the NOy data are adequate to make the scaling arguments we have used.
 In our Hu analysis of the Arctic vortex in 1989 and of the Antarctic vortex in 1987 [Tuck et al., 1999] we concluded that persistence in Hu(O3) relative to Hu(s) was indicative of photochemical ozone loss outpacing fluid mechanical mixing and attributed this persistence to the large-scale correlations in ozone loss patterns arising from the field of actinic solar radiation, which is relatively smooth over large scales. This interpretation holds up well in the situations in which the great majority of the flight time in 1987 and 1989 was acquired, namely in the sunlit outer vortex in high wind speeds. However, the SOLVE mission had the great majority of its horizontal flight time in the less illuminated inner vortex (here defined as wind speed <30 ms−1), and we accordingly re-examine such inner vortex ozone for Antarctica in August–September 1987 and the Arctic in January–February 1989 on the same basis as the January–March 2000 Arctic data; that is, in the (H1, C1) plane as in Figure 11. The 1987 Antarctic results (Figure 14) show that the inner vortex ozone value of H1 increased from an average of 0.28 ± 0.08 on 19870909 to 0.70 ± 0.07 on 19870922. This change is clearest on the (H1, α) plot, since both these exponents show an increase, whereas the value of C1 did not. There is no clear theoretical guidance about what this means in terms of the interaction of chemistry and turbulence. We interpret the increase in α as the introduction of filaments containing ozone-mixing ratios that result in a greater variety of increments. We see the increase in H1 as evidence of increased insolation driving the photochemistry to produce higher near neighbor correlation on all scales. We note that the physical distortion of the vortex over time would produce both of these effects. There was no PSC activity at θ ≈ 430 K level from 19870914 until 19870929 [Tuck, 1989], and so the increase in H1(O3) is a reflection of sunlight producing persistence in the processed but cloud free air in the inner vortex.
 In the Arctic in January–February 1989, the PSC processing was more sporadic than in 2000, and the denitrification was much less prevalent [Fahey et al., 1990]; PSC activity ceased in very early February. The H1 values for ozone in the 1989 inner Arctic vortex are in the range 0.33 < H1 < 0.49 (Figure 15) and reflect a state in which sporadically illuminated air does not produce enough persistence to overcome the countervailing effects of locally variable PSC processing, some denitrification and mixing. In 2000, however, the presence of large particles, on the flights analyzed here, was continual in the inner vortex, being evident on all flights up to and including 20000307. Because of variable denitrification it was not possible to determine unequivocally the presence of the small particle PSCs by the NOy(A) channel using the methods employed previously [Fahey et al., 1989]. However, satellite observations do provide such evidence from 19991115 to 20000207 and from 20000227 to 20000315 [Bevilacqua et al., 2002]. The inhomogeneous distribution of NOy on all scales is reflected in some of the very low values of H1(NOy), particularly early in the mission (Figure 3); thus, although photochemical ozone loss is driven by the solar radiation field, it cannot cause the persistence seen in the AAOE mission because of the antipersistence produced by the continual denitrification and processing from PSCs which are spatially very variable. The spring 2000 H1(O3) values remain scattered even in mid-March (Figure 11), but are somewhat higher than in late January, consistent with a lessening of PSC activity and an increase in insolation; they are similar to the H1 values for gas phase NOy, suggesting that it is possible that the NOy-induced inhomogeneity in the ozone derives more from denitrification than from variability of the chemical processing in small particle, nonsettling PSCs. The scattered values of H1 for NOy and ozone demonstrate a difficulty in trying to calculate quantitative ozone loss by numerical modeling; the consistent 15–25% ranges of the probability distributions in Figure 2 from late January to mid March reinforce this conclusion. Scale invariance means that there is no obvious length scale over which to take averages for terms in the rate equation, and the change in ozone and NOy scaling exponents from January to March means that a fixed choice of scale, for example the grid size in a numerical model, could produce results of varying fidelity. Interestingly, the α values for ozone during SOLVE do show a consistent increase from late January to mid March, see Figure 11. As with the AAOE data, this is consistent with an increase in the filamentation as the vortex distorts.
 The above results suggest a way of putting limits on mixing rates in the inner vortex, and also expressing them as rates of change of scaling exponent H1(O3) and α(O3). We will use H1 rather than α because of the large increase in it during mid September, see Figure 14. Similar results can be obtained using α. The rate dH1(O3)/dt = 0.025 d−1 between 19870909 and 19870922 is associated with a photochemical ozone loss rate in excess of 2% d−1 [Jones et al., 1989], which is outpacing the mixing rate; note that this loss rate is consistent with the chemistry-only loss rate analyzed from observations for inner vortex ozone during SOLVE [Richard et al., 2001]. The mixing rate therefore has an upper limit of 2% d−1. Note that an independent, satellite-based estimate, using only the largest scale, for the 1994 Antarctic vortex yielded 1.3% d−1 [Rosenlof et al., 1997]. A lower limit to the mixing rate may be obtained by observing that the rate of change dH1[NOy(A)]/dt was 0.01 d−1 between 20000131 and 20000226, corresponding to 0.7% d−1. A similar rate may be obtained from NOy(B) between late January and early February. This range of mixing rates assumes equivalence of chemical ozone loss rates in the inner vortices in the Arctic and in the Antarctic for the upper limit (see above), and for the lower limit that in the absence of physical and chemical sinks, ozone and NOy would behave the same. Our use of the term “mixing rate” in this paragraph applies to the rate at which structures injected from the outer to the inner vortex are dissipated, balanced by the rate at which they are internally generated in the inner vortex.
 Returning to the isentropic rates of methane decrease calculated in section 3, we compare them with the limits on horizontal mixing rates derived above. Since the methane-mixing ratio on an isentrope is the resultant of diabatic cooling (the vertical component) and isentropic mixing (the horizontal component), we see that the rate of decrease of methane in the absence of mixing would be in the range 0.85% d−1 to 2.56% d−1, or 8 to 23 ppbv d−1. Since the mean dCH4/dθ from the ER-2 data in Figure 1 is 11.5 ppbv K−1, we estimate dθ/dt as being in the range 0.8 to 2.0 K d−1. We note that Hicke and Tuck  recently computed the effect of PSCs on radiative cooling rates in the Antarctic inner vortex during August and September, finding dθ/dt of 1.0 K d−1 in the lower stratosphere. Radiative cooling rates are generally calculated to be higher in the Arctic than in the Antarctic, particularly in winter and spring [Rosenlof, 1995]. Once the use of the scaling has allowed separation of the effects of isentropic exchange and radiative (diabatic) cooling, we may use the relationship dO3/dt = (dO3/dθ)(dθ/dt) and observed values of dO3/dθ to compute the rate of change of ozone in the inner vortex. The values of the ozone lapse rate corresponding to the three cases for dθ/dt computed in section 3.0 are 23, 14 and 16 ppbv K−1, resulting in ozone loss rates of 17, 20 and 33 ppbv d−1 respectively. These values average to 23 ppbv d−1, or half of the estimate of Richard et al.  whose analysis, unlike ours, excluded the effects of dynamics and mixing on the inner vortex. We may thus infer that the ozone loss in the inner vortex was being diluted by the fluid motion at about half the rate at which the chemistry was producing it.
 The inner vortex inhomogeneity is maintained by a balance between isentropic wind shear and horizontal gradients in diabatic cooling, driven by a wide scale range of motions. Thus, an individual tracer isopleth's motion and structure is determined by the interaction of radiative cooling and isentropic exchange. The 20% decrease in mean isentropic methane while it maintained 20% horizontal variability in the inner vortex is an illustration of this point. If one considers H1 as derived from the first-order structure function, it seems that our values of H1 are not consistent with a view of stratospheric turbulence limited to 2 horizontal dimensions (which would require H1 = 1). Schertzer and Lovejoy  came to similar conclusions, as has Lindborg  in examining the MOZAIC airborne data. Bacmeister et al.  noted that meteorological and chemical variables had different spectral slopes in discrete wavelet transform analysis of 1024 s segments of ER-2 data. We note also that Shraiman and Siggia  point out the absence of any compelling reason for the generating equations to provide universal exponents whose constancy could be used to characterize atmospheric motion. Generalized scale invariance [Pecknold et al., 1993] appears to be a fact, and it also generates a spectrum of scaling exponents which can be variable, as shown by the H1 and (H1, C1, α) results here for the inner polar vortex, and by earlier Hu analysis over hundreds of flight legs ranging between 90°N and 72°S [Tuck and Hovde, 1999; Tuck et al., 1999]. Methven and Hoskins  found a very similar result for potential vorticity in a high-resolution modeling study.
 It is clear that the scaling exponents of both NOy channels, and of their difference, changed during the mission; they increased between 20000120 and 20000312. The scaling of ozone showed lower exponents than on previous missions in the vortex, and showed some increase in 2000 between late January and mid-March. In all missions, including SOLVE, the scaling exponents of wind speed and temperature remained invariant within a range. Thus scaling analysis of the Arctic data in 2000 supports the argument that inhomogeneity in NOy, from the continual denitrification and processing caused by PSCs, imposes itself on the ozone to produce antipersistence as the photochemistry lowers its mixing ratio in the inner vortex. Similarity in the scaling of gas phase NOy and ozone in the later flights suggests that gas phase inhomogeneity caused by patchy denitrification rather than inhomogeneity in small particle PSCs may be more effective at inducing the variability in the ozone via the photochemistry. The effect is sufficient to more than counterbalance the opposing effect of the more smoothly varying large-scale nature of the solar radiation field. The effect of the latter could, however, be seen emerging in late September 1987 in the inner Antarctic vortex, in the form of a high degree of persistence in the ozone about two air parcel circuits of the vortex (∼8 days) after the last PSC occurrence. We estimate that inner vortex-mixing rates lie in the range 0.7 to 2.0% d−1, implying that the chemistry and the dynamics are proceeding at rates of the same order of magnitude as far as ozone is concerned. By combining these mixing rates with rates of change of methane, we estimate the diabatic cooling rate dθ/dt for the 2000 Arctic inner vortex to be in the range 0.8 to 2.0 K d−1. The ozone loss rate in the inner vortex was in the range 17 to 33 ppbv d−1, averaging half the loss rate in an independent analysis excluding the effects of dynamics [Richard et al., 2001]. The effects of dynamics and mixing were therefore to dilute the ozone loss at about half the rate at which the chemistry was producing it. As time progressed from late January to mid March 2000, the combined effect of dynamics and chemistry, through distortion of the vortex to lower latitudes, was to increase the α exponent characterizing ozone in the inner vortex. Increasing filamentation would increase α, increasing insolation would increase H1. This behavior was also evident during August and September 1987 over the Antarctic in both H1 and α. It indicates respectively an increase in neighbor-to-neighbor correlation (H1) and an increase in the variety of ozone-mixing ratio increments (α). While both ozone and wind speed had time-invariant values of C1 tightly grouped at about 0.03, temperature had a higher (0.09) and more variable set of values than this, although still with no time trend. This could be an artifact associated with the steppiness of the temperature data. We note that continual mixing during winter is in agreement with Plumb et al.  and the earlier conclusions of Murphy et al. , Tuck , and Tuck et al. .
 The inhomogeneity, evident in θ ≈ 436K PDFs that manifest about 15–25% variability about the mean in the methane tracer, has implications for the modeling of both chemistry and dynamics. Nonlinear correlation terms in the chemical kinetic equations can have significant effects on averaged rate calculations [Tuck, 1979; Edouard et al., 1996; Tan et al., 1998], which are for example dependent on [ClO]2[M] in an important rate-determining step for the polar vortex; the change in scaling exponents in ozone and NOy from late January to early March implies that a fixed grid size for numerical simulation could lead to results of varying fidelity over time. The scaling of the dynamical quantities is not consistent with large-scale 2-D turbulence or isotropic 3-D turbulence. It implies stirring of the inner vortex on a range of scales, probably from below. The spread of scaling exponents for a single variable over different flights, and the variety of scaling behaviors for different chemical and meteorological variables in a single flight, implies that neither stratospheric dynamics nor polar vortex chemistry can be characterized by single scaling exponents. In addition, generalized scale invariance [Pecknold et al., 1993] seems to be ubiquitous. While observed changes in scaling behavior of NOy and ozone over time can be attributed to known physicochemical processes, thereby providing numerical model-independent support for the reality of the latter, there is failure of conventional 2-D and 3-D theories of atmospheric turbulence to quantitatively predict the scaling behavior of the meteorological variables and by implication of a scalar tracer, a conclusion in agreement with Schertzer and Lovejoy , Lindborg , Cho and Lindborg  and Lovejoy et al. .
 We thank C. R. Webster for the methane data from the ALIAS instrument.